%------------------------------------------------------------------------------
% File     : ITP293_4 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Example 00027_000692
% 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    : 0097_VEBT_Example_00027_000692 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 11899 (4162 unt;1473 typ;   0 def)
%            Number of atoms       : 18913 (7178 equ)
%            Maximal formula atoms :   28 (   1 avg)
%            Number of connectives : 18881 (1516   ~; 296   |;2083   &)
%                                         (2130 <=>;12856  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   24 (   5 avg)
%            Maximal term depth    :   36 (   2 avg)
%            Number of FOOLs       :  567 ( 407 fml; 160 var)
%            Number of X terms     :  475 (   0  []; 437 ite;  38 let)
%            Number of types       :   20 (  19 usr)
%            Number of type conns  : 1264 (1137   >; 127   *;   0   +;   0  <<)
%            Number of predicates  :  209 ( 206 usr;   2 prp; 0-7 aty)
%            Number of functors    : 1260 (1260 usr;  38 con; 0-7 aty)
%            Number of variables   : 30676 (27890   !; 432   ?;30676   :)
%                                         (2354  !>;   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 23:43:10.256
%------------------------------------------------------------------------------
% Could-be-implicit typings (32)
tff(ty_t_VEBT__BuildupMemImp_OVEBTi,type,
    vEBT_VEBTi: $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_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_Enum_Ofinite__3,type,
    finite_3: $tType ).

tff(ty_t_Enum_Ofinite__2,type,
    finite_2: $tType ).

tff(ty_t_Enum_Ofinite__1,type,
    finite_1: $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_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_Set_Oset,type,
    set: $tType > $tType ).

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

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

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

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

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

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

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

% Explicit typings (1441)
tff(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
    comple5582772986160207858norder: 
      !>[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_Rings_Odvd,type,
    dvd: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_cl_Type__Length_Olen,type,
    type_len: 
      !>[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_Type__Length_Olen0,type,
    type_len0: 
      !>[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_Rings_Oidom__divide,type,
    idom_divide: 
      !>[A: $tType] : $o ).

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

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

tff(sy_cl_Cardinality_OCARD__1,type,
    cARD_1: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_cl_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_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_Least__significant__bit_Olsb,type,
    least_6119777620449941438nt_lsb: 
      !>[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_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_Ot1__space,type,
    topological_t1_space: 
      !>[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__top,type,
    bounded_lattice_top: 
      !>[A: $tType] : $o ).

tff(sy_cl_Limits_Otopological__group__add,type,
    topolo1633459387980952147up_add: 
      !>[A: $tType] : $o ).

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

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

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

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

tff(sy_cl_Real__Vector__Spaces_Oreal__field,type,
    real_V7773925162809079976_field: 
      !>[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_Limits_Otopological__monoid__mult,type,
    topolo1898628316856586783d_mult: 
      !>[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_Real__Vector__Spaces_Oreal__algebra__1,type,
    real_V2191834092415804123ebra_1: 
      !>[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_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_Limits_Otopological__comm__monoid__mult,type,
    topolo4987421752381908075d_mult: 
      !>[A: $tType] : $o ).

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

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

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

tff(sy_cl_Topological__Spaces_Odiscrete__topology,type,
    topolo8865339358273720382pology: 
      !>[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_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__ring__with__nat,type,
    euclid8789492081693882211th_nat: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__up____,type,
    aTP_Lamp_up: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uq____,type,
    aTP_Lamp_uq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ur____,type,
    aTP_Lamp_ur: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__yt____,type,
    aTP_Lamp_yt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yu____,type,
    aTP_Lamp_yu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

tff(sy_c_ATP_058Lamp__yx____,type,
    aTP_Lamp_yx: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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_Assertions_Opure__assn,type,
    pure_assn: $o > assn ).

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_Binomial_Obinomial,type,
    binomial: nat > fun(nat,nat) ).

tff(sy_c_Binomial_Ogbinomial,type,
    gbinomial: 
      !>[A: $tType] : ( A > fun(nat,A) ) ).

tff(sy_c_Bit__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_Oand__int__rel,type,
    bit_and_int_rel: fun(product_prod(int,int),fun(product_prod(int,int),$o)) ).

tff(sy_c_Bit__Operations_Oconcat__bit,type,
    bit_concat_bit: ( nat * int * int ) > int ).

tff(sy_c_Bit__Operations_Oor__not__num__neg,type,
    bit_or_not_num_neg: ( num * num ) > num ).

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

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit,type,
    bit_ri4674362597316999326ke_bit: 
      !>[A: $tType] : ( ( nat * A ) > A ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand,type,
    bit_se5824344872417868541ns_and: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit,type,
    bit_se4197421643247451524op_bit: 
      !>[A: $tType] : fun(nat,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit,type,
    bit_se8732182000553998342ip_bit: 
      !>[A: $tType] : ( ( nat * A ) > A ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask,type,
    bit_se2239418461657761734s_mask: 
      !>[A: $tType] : ( nat > A ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor,type,
    bit_se1065995026697491101ons_or: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit,type,
    bit_se4730199178511100633sh_bit: 
      !>[A: $tType] : fun(nat,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit,type,
    bit_se5668285175392031749et_bit: 
      !>[A: $tType] : ( ( nat * A ) > A ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit,type,
    bit_se2584673776208193580ke_bit: 
      !>[A: $tType] : ( ( nat * A ) > A ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit,type,
    bit_se2638667681897837118et_bit: 
      !>[A: $tType] : ( ( nat * A ) > A ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor,type,
    bit_se5824344971392196577ns_xor: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type,
    bit_se5641148757651400278ts_bit: 
      !>[A: $tType] : fun(A,fun(nat,$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] : ( ( word(A) * nat ) > word(A) ) ).

tff(sy_c_Bits__Integer_OBit__integer,type,
    bits_Bit_integer: ( code_integer * $o ) > code_integer ).

tff(sy_c_Bits__Integer_Obin__last__integer,type,
    bits_b8758750999018896077nteger: code_integer > $o ).

tff(sy_c_Bits__Integer_Obin__rest__integer,type,
    bits_b2549910563261871055nteger: code_integer > code_integer ).

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__Int__Integer__Conversion_Oint__of__integer__symbolic,type,
    code_I935103866777955880mbolic: code_integer > int ).

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_Odup,type,
    code_dup: code_integer > code_integer ).

tff(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
    code_int_of_integer: code_integer > int ).

tff(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
    code_integer_of_int: int > code_integer ).

tff(sy_c_Code__Numeral_Ointeger__of__nat,type,
    code_integer_of_nat: nat > code_integer ).

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

tff(sy_c_Code__Numeral_Onum__of__integer,type,
    code_num_of_integer: code_integer > num ).

tff(sy_c_Code__Target__Int_Onegative,type,
    code_Target_negative: fun(num,int) ).

tff(sy_c_Code__Target__Nat_Oint__of__nat,type,
    code_T6385005292777649522of_nat: fun(nat,int) ).

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_Complex_OArg,type,
    arg: complex > real ).

tff(sy_c_Complex_Ocis,type,
    cis: real > complex ).

tff(sy_c_Complex_Ocnj,type,
    cnj: complex > complex ).

tff(sy_c_Complex_Ocomplex_OComplex,type,
    complex2: ( real * real ) > complex ).

tff(sy_c_Complex_Ocomplex_OIm,type,
    im: complex > real ).

tff(sy_c_Complex_Ocomplex_ORe,type,
    re: complex > real ).

tff(sy_c_Complex_Ocsqrt,type,
    csqrt: complex > complex ).

tff(sy_c_Complex_Oimaginary__unit,type,
    imaginary_unit: complex ).

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

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_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_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_Finite__Set_Ocard,type,
    finite_card: 
      !>[B: $tType] : ( set(B) > nat ) ).

tff(sy_c_Finite__Set_Ofinite,type,
    finite_finite2: 
      !>[A: $tType] : ( set(A) > $o ) ).

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

tff(sy_c_Fun_Ocomp,type,
    comp: 
      !>[B: $tType,C: $tType,A: $tType] : ( ( fun(B,C) * fun(A,B) ) > fun(A,C) ) ).

tff(sy_c_Fun_Oinj__on,type,
    inj_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_Osemiring__1__class_Osemiring__char,type,
    semiri4206861660011772517g_char: 
      !>[A: $tType] : ( itself(A) > nat ) ).

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] : ( 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] : ( 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__class_Osum,type,
    groups7311177749621191930dd_sum: 
      !>[B: $tType,A: $tType] : ( 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(B,A) * set(B) ) > 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_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__list,type,
    hash_hash_code_list: 
      !>[A: $tType] : ( ( fun(A,uint32) * list(A) ) > uint32 ) ).

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

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

tff(sy_c_Least__significant__bit_Olsb__class_Olsb,type,
    least_8051144512741203767sb_lsb: 
      !>[A: $tType] : fun(A,$o) ).

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_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_Olist_OCons,type,
    cons: 
      !>[A: $tType] : ( A > fun(list(A),list(A)) ) ).

tff(sy_c_List_Olist_Oset,type,
    set2: 
      !>[A: $tType] : fun(list(A),set(A)) ).

tff(sy_c_List_Onth,type,
    nth: 
      !>[A: $tType] : ( list(A) > fun(nat,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_Ounion,type,
    union: 
      !>[A: $tType] : ( ( list(A) * list(A) ) > list(A) ) ).

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_Misc_Ofun__of__rel,type,
    fun_of_rel: 
      !>[B: $tType,A: $tType] : ( ( set(product_prod(B,A)) * B ) > A ) ).

tff(sy_c_Misc_Oslice,type,
    slice: 
      !>[A: $tType] : ( ( nat * nat * list(A) ) > list(A) ) ).

tff(sy_c_Most__significant__bit_Omsb__class_Omsb,type,
    most_s684356279273892711sb_msb: 
      !>[A: $tType] : ( A > $o ) ).

tff(sy_c_Multiset_Ois__empty,type,
    is_empty: 
      !>[A: $tType] : ( multiset(A) > $o ) ).

tff(sy_c_Nat_OSuc,type,
    suc: fun(nat,nat) ).

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] : ( A > nat ) ).

tff(sy_c_Nat__Bijection_Oset__decode,type,
    nat_set_decode: nat > set(nat) ).

tff(sy_c_Nat__Bijection_Oset__encode,type,
    nat_set_encode: 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,type,
    neg_numeral_dbl: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec,type,
    neg_numeral_dbl_dec: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc,type,
    neg_numeral_dbl_inc: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Num_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_Num_Opow,type,
    pow: ( num * num ) > num ).

tff(sy_c_Num_Opred__numeral,type,
    pred_numeral: num > nat ).

tff(sy_c_Numeral__Type_OAbs__bit1_H,type,
    numeral_Abs_bit1: 
      !>[A: $tType] : ( int > numeral_bit1(A) ) ).

tff(sy_c_Numeral__Type_Obit1_OAbs__bit1,type,
    numeral_Abs_bit12: 
      !>[A: $tType] : fun(int,numeral_bit1(A)) ).

tff(sy_c_Numeral__Type_Obit1_ORep__bit1,type,
    numeral_Rep_bit1: 
      !>[A: $tType] : fun(numeral_bit1(A),int) ).

tff(sy_c_Orderings_Obot__class_Obot,type,
    bot_bot: 
      !>[A: $tType] : A ).

tff(sy_c_Orderings_Oord__class_Oless,type,
    ord_less: 
      !>[A: $tType] : ( A > fun(A,$o) ) ).

tff(sy_c_Orderings_Oord__class_Oless__eq,type,
    ord_less_eq: 
      !>[A: $tType] : ( A > fun(A,$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] : ( ( A * A ) > A ) ).

tff(sy_c_Orderings_Oorder__class_OGreatest,type,
    order_Greatest: 
      !>[A: $tType] : ( fun(A,$o) > A ) ).

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

tff(sy_c_Orderings_Otop__class_Otop,type,
    top_top: 
      !>[A: $tType] : A ).

tff(sy_c_Power_Opower__class_Opower,type,
    power_power: 
      !>[A: $tType] : ( 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_Oprod_Ocase__prod,type,
    product_case_prod: 
      !>[A: $tType,B: $tType,C: $tType] : ( fun(A,fun(B,C)) > fun(product_prod(A,B),C) ) ).

tff(sy_c_Pure_Otype,type,
    type2: 
      !>[A: $tType] : itself(A) ).

tff(sy_c_Rat_OFrct,type,
    frct: product_prod(int,int) > rat ).

tff(sy_c_Rat_Onormalize,type,
    normalize: product_prod(int,int) > product_prod(int,int) ).

tff(sy_c_Rat_Oof__int,type,
    of_int: int > rat ).

tff(sy_c_Rat_Oquotient__of,type,
    quotient_of: rat > product_prod(int,int) ).

tff(sy_c_Real__Vector__Spaces_OReals,type,
    real_Vector_Reals: 
      !>[A: $tType] : set(A) ).

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_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_Oof__real,type,
    real_Vector_of_real: 
      !>[A: $tType] : fun(real,A) ).

tff(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR,type,
    real_V8093663219630862766scaleR: 
      !>[A: $tType] : ( real > fun(A,A) ) ).

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: $tType] : ( A > fun(A,$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: $tType] : ( fun(nat,A) > fun(A,$o) ) ).

tff(sy_c_Set_OCollect,type,
    collect: 
      !>[A: $tType] : ( fun(A,$o) > 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] : ( A > fun(set(A),set(A)) ) ).

tff(sy_c_Set_Ois__singleton,type,
    is_singleton: 
      !>[A: $tType] : ( set(A) > $o ) ).

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_Oord__class_OatLeast,type,
    set_ord_atLeast: 
      !>[A: $tType] : ( A > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
    set_or1337092689740270186AtMost: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan,type,
    set_or7035219750837199246ssThan: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OatMost,type,
    set_ord_atMost: 
      !>[A: $tType] : ( A > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThan,type,
    set_ord_greaterThan: 
      !>[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] : ( ( A * A ) > A ) ).

tff(sy_c_Signed__Division_Osigned__division__class_Osigned__modulo,type,
    signed6721504322012087516modulo: 
      !>[A: $tType] : ( ( A * A ) > A ) ).

tff(sy_c_Time__Reasoning_OTBOUND,type,
    time_TBOUND: 
      !>[A: $tType] : ( ( heap_Time_Heap(A) * nat ) > $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_Topological__Spaces_Ocontinuous,type,
    topolo3448309680560233919inuous: 
      !>[A: $tType,B: $tType] : ( ( filter(A) * fun(A,B) ) > $o ) ).

tff(sy_c_Topological__Spaces_Ocontinuous__on,type,
    topolo81223032696312382ous_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) ) > $o ) ).

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_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: fun(real,real) ).

tff(sy_c_Transcendental_Oarsinh,type,
    arsinh: 
      !>[A: $tType] : fun(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] : fun(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] : fun(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_ORep__uint32_H,type,
    rep_uint32: uint32 > word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))) ).

tff(sy_c_Uint32_OUint32,type,
    uint322: fun(code_integer,uint32) ).

tff(sy_c_Uint32_OUint32__signed,type,
    uint32_signed: code_integer > uint32 ).

tff(sy_c_Uint32_Odiv0__uint32,type,
    div0_uint32: uint32 > uint32 ).

tff(sy_c_Uint32_Ointeger__of__uint32,type,
    integer_of_uint32: fun(uint32,code_integer) ).

tff(sy_c_Uint32_Ointeger__of__uint32__signed,type,
    intege5370686899274169573signed: uint32 > code_integer ).

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_OAbs__uint32,type,
    abs_uint32: word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))) > uint32 ).

tff(sy_c_Uint32_Ouint32_ORep__uint32,type,
    rep_uint322: uint32 > word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))) ).

tff(sy_c_Uint32_Ouint32__div,type,
    uint32_div: ( uint32 * uint32 ) > uint32 ).

tff(sy_c_Uint32_Ouint32__divmod,type,
    uint32_divmod: ( uint32 * uint32 ) > product_prod(uint32,uint32) ).

tff(sy_c_Uint32_Ouint32__mod,type,
    uint32_mod: ( uint32 * 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__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__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__memberi,type,
    vEBT_vebt_memberi: ( vEBT_VEBTi * nat ) > heap_Time_Heap($o) ).

tff(sy_c_VEBT__DelImperative_Ovebt__deletei,type,
    vEBT_vebt_deletei: ( vEBT_VEBTi * nat ) > heap_Time_Heap(vEBT_VEBTi) ).

tff(sy_c_VEBT__Example__Setup_Omfold,type,
    vEBT_Example_mfold: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,heap_Time_Heap(B))) * list(A) * B ) > heap_Time_Heap(B) ) ).

tff(sy_c_VEBT__Intf__Imperative_Ovebt__assn,type,
    vEBT_Intf_vebt_assn: ( nat * set(nat) ) > fun(vEBT_VEBTi,assn) ).

tff(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
    vEBT_VEBT_bit_concat: ( nat * nat * 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,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_Wellfounded_Oaccp,type,
    accp: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * A ) > $o ) ).

tff(sy_c_Word_OWord,type,
    word2: 
      !>[A: $tType] : fun(int,word(A)) ).

tff(sy_c_Word_Ocast,type,
    cast: 
      !>[A: $tType,B: $tType] : ( word(A) > word(B) ) ).

tff(sy_c_Word_Oeven__word,type,
    even_word: 
      !>[A: $tType] : fun(word(A),$o) ).

tff(sy_c_Word_Oof__int,type,
    of_int2: 
      !>[A: $tType] : ( int > word(A) ) ).

tff(sy_c_Word_Oof__nat,type,
    of_nat: 
      !>[A: $tType] : ( nat > word(A) ) ).

tff(sy_c_Word_Orevcast,type,
    revcast: 
      !>[A: $tType,B: $tType] : fun(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] : ( word(A) > word(B) ) ).

tff(sy_c_Word_Osigned__drop__bit,type,
    signed_drop_bit: 
      !>[A: $tType] : ( ( nat * word(A) ) > word(A) ) ).

tff(sy_c_Word_Oslice,type,
    slice2: 
      !>[A: $tType,B: $tType] : ( nat > fun(word(A),word(B)) ) ).

tff(sy_c_Word_Oslice1,type,
    slice1: 
      !>[A: $tType,B: $tType] : ( nat > fun(word(A),word(B)) ) ).

tff(sy_c_Word_Othe__int,type,
    the_int: 
      !>[A: $tType] : ( word(A) > int ) ).

tff(sy_c_Word_Othe__nat,type,
    the_nat: 
      !>[A: $tType] : ( word(A) > nat ) ).

tff(sy_c_Word_Othe__signed__int,type,
    the_signed_int: 
      !>[A: $tType] : ( word(A) > int ) ).

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__pred,type,
    word_pred: 
      !>[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 > fun(word(A),word(A)) ) ).

tff(sy_c_Word_Oword__rotr,type,
    word_rotr: 
      !>[A: $tType] : ( nat > fun(word(A),word(A)) ) ).

tff(sy_c_Word_Oword__sle,type,
    word_sle: 
      !>[A: $tType] : ( ( word(A) * word(A) ) > $o ) ).

tff(sy_c_Word_Oword__sless,type,
    word_sless: 
      !>[A: $tType] : ( ( word(A) * 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_fChoice,type,
    fChoice: 
      !>[A: $tType] : ( fun(A,$o) > A ) ).

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_n,type,
    n: nat ).

tff(sy_v_s,type,
    s: set(nat) ).

tff(sy_v_t,type,
    t: vEBT_VEBTi ).

tff(sy_v_xs,type,
    xs: list(nat) ).

% Relevant facts (9150)
tff(fact_0_n__less__equal__power__2,axiom,
    ! [Na: nat] : aa(nat,$o,ord_less(nat,Na),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ).

% n_less_equal_power_2
tff(fact_1_semiring__norm_I85_J,axiom,
    ! [M: num] : bit0(M) != one2 ).

% semiring_norm(85)
tff(fact_2_semiring__norm_I83_J,axiom,
    ! [Na: num] : one2 != bit0(Na) ).

% semiring_norm(83)
tff(fact_3_numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: num,Na: num] :
          ( aa(A,$o,ord_less(A,aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Na))
        <=> aa(num,$o,ord_less(num,M),Na) ) ) ).

% numeral_less_iff
tff(fact_4_UnCI,axiom,
    ! [A: $tType,C2: A,B2: set(A),A2: set(A)] :
      ( ( ~ member(A,C2,B2)
       => member(A,C2,A2) )
     => member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),B2)) ) ).

% UnCI
tff(fact_5_Un__iff,axiom,
    ! [A: $tType,C2: A,A2: set(A),B2: set(A)] :
      ( member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),B2))
    <=> ( member(A,C2,A2)
        | member(A,C2,B2) ) ) ).

% Un_iff
tff(fact_6_sup_Oidem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),A3) = A3 ) ).

% sup.idem
tff(fact_7_sup__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),X) = X ) ).

% sup_idem
tff(fact_8_sup_Oleft__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) ) ).

% sup.left_idem
tff(fact_9_sup__left__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) ) ).

% sup_left_idem
tff(fact_10_sup_Oright__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)),B3) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) ) ).

% sup.right_idem
tff(fact_11_sup__apply,axiom,
    ! [A: $tType,B: $tType] :
      ( semilattice_sup(A)
     => ! [F2: fun(B,A),G: fun(B,A),X: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),sup_sup(fun(B,A)),F2),G),X) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,F2,X)),aa(B,A,G,X)) ) ).

% sup_apply
tff(fact_12_numeral__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M: num,Na: num] :
          ( ( aa(num,A,numeral_numeral(A),M) = aa(num,A,numeral_numeral(A),Na) )
        <=> ( M = Na ) ) ) ).

% numeral_eq_iff
tff(fact_13_semiring__norm_I87_J,axiom,
    ! [M: num,Na: num] :
      ( ( bit0(M) = bit0(Na) )
    <=> ( M = Na ) ) ).

% semiring_norm(87)
tff(fact_14_semiring__norm_I78_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,ord_less(num,bit0(M)),bit0(Na))
    <=> aa(num,$o,ord_less(num,M),Na) ) ).

% semiring_norm(78)
tff(fact_15_semiring__norm_I75_J,axiom,
    ! [M: num] : ~ aa(num,$o,ord_less(num,M),one2) ).

% semiring_norm(75)
tff(fact_16_semiring__norm_I76_J,axiom,
    ! [Na: num] : aa(num,$o,ord_less(num,one2),bit0(Na)) ).

% semiring_norm(76)
tff(fact_17_sup__set__def,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),B2) = collect(A,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),sup_sup(fun(A,$o)),aTP_Lamp_aa(set(A),fun(A,$o),A2)),aTP_Lamp_aa(set(A),fun(A,$o),B2))) ).

% sup_set_def
tff(fact_18_sup__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( semilattice_sup(B)
     => ! [F2: fun(A,B),G: fun(A,B),X2: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),sup_sup(fun(A,B)),F2),G),X2) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F2,X2)),aa(A,B,G,X2)) ) ).

% sup_fun_def
tff(fact_19_sup__left__commute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Z)) ) ).

% sup_left_commute
tff(fact_20_sup_Oleft__commute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B3: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),B3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),C2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(A,A,aa(A,fun(A,A),sup_sup(A),B3),C2)) ) ).

% sup.left_commute
tff(fact_21_sup__commute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),X) ) ).

% sup_commute
tff(fact_22_sup_Ocommute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) = aa(A,A,aa(A,fun(A,A),sup_sup(A),B3),A3) ) ).

% sup.commute
tff(fact_23_sup__assoc,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),Z) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z)) ) ).

% sup_assoc
tff(fact_24_sup_Oassoc,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)),C2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(A,A,aa(A,fun(A,A),sup_sup(A),B3),C2)) ) ).

% sup.assoc
tff(fact_25_inf__sup__aci_I5_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),X) ) ).

% inf_sup_aci(5)
tff(fact_26_inf__sup__aci_I6_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),Z) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z)) ) ).

% inf_sup_aci(6)
tff(fact_27_inf__sup__aci_I7_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Z)) ) ).

% inf_sup_aci(7)
tff(fact_28_inf__sup__aci_I8_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) ) ).

% inf_sup_aci(8)
tff(fact_29_Un__left__commute,axiom,
    ! [A: $tType,A2: set(A),B2: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B2),C3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),C3)) ).

% Un_left_commute
tff(fact_30_Un__left__absorb,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),B2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),B2) ).

% Un_left_absorb
tff(fact_31_Un__commute,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),B2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B2),A2) ).

% Un_commute
tff(fact_32_Un__absorb,axiom,
    ! [A: $tType,A2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),A2) = A2 ).

% Un_absorb
tff(fact_33_Un__assoc,axiom,
    ! [A: $tType,A2: set(A),B2: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),B2)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B2),C3)) ).

% Un_assoc
tff(fact_34_ball__Un,axiom,
    ! [A: $tType,A2: set(A),B2: set(A),P: fun(A,$o)] :
      ( ! [X3: A] :
          ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),B2))
         => aa(A,$o,P,X3) )
    <=> ( ! [X3: A] :
            ( member(A,X3,A2)
           => aa(A,$o,P,X3) )
        & ! [X3: A] :
            ( member(A,X3,B2)
           => aa(A,$o,P,X3) ) ) ) ).

% ball_Un
tff(fact_35_bex__Un,axiom,
    ! [A: $tType,A2: set(A),B2: set(A),P: fun(A,$o)] :
      ( ? [X3: A] :
          ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),B2))
          & aa(A,$o,P,X3) )
    <=> ( ? [X3: A] :
            ( member(A,X3,A2)
            & aa(A,$o,P,X3) )
        | ? [X3: A] :
            ( member(A,X3,B2)
            & aa(A,$o,P,X3) ) ) ) ).

% bex_Un
tff(fact_36_UnI2,axiom,
    ! [A: $tType,C2: A,B2: set(A),A2: set(A)] :
      ( member(A,C2,B2)
     => member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),B2)) ) ).

% UnI2
tff(fact_37_UnI1,axiom,
    ! [A: $tType,C2: A,A2: set(A),B2: set(A)] :
      ( member(A,C2,A2)
     => member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),B2)) ) ).

% UnI1
tff(fact_38_UnE,axiom,
    ! [A: $tType,C2: A,A2: set(A),B2: set(A)] :
      ( member(A,C2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),B2))
     => ( ~ member(A,C2,A2)
       => member(A,C2,B2) ) ) ).

% UnE
tff(fact_39_Collect__disj__eq,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] : collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ab(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),collect(A,P)),collect(A,Q)) ).

% Collect_disj_eq
tff(fact_40_Un__def,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),B2) = collect(A,aa(set(A),fun(A,$o),aTP_Lamp_ac(set(A),fun(set(A),fun(A,$o)),A2),B2)) ).

% Un_def
tff(fact_41_sup_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,C2),B3)
         => aa(A,$o,ord_less(A,C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)) ) ) ).

% sup.strict_coboundedI2
tff(fact_42_sup_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,C2),A3)
         => aa(A,$o,ord_less(A,C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)) ) ) ).

% sup.strict_coboundedI1
tff(fact_43_mem__Collect__eq,axiom,
    ! [A: $tType,A3: A,P: fun(A,$o)] :
      ( member(A,A3,collect(A,P))
    <=> aa(A,$o,P,A3) ) ).

% mem_Collect_eq
tff(fact_44_Collect__mem__eq,axiom,
    ! [A: $tType,A2: set(A)] : collect(A,aTP_Lamp_aa(set(A),fun(A,$o),A2)) = A2 ).

% Collect_mem_eq
tff(fact_45_Collect__cong,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ! [X4: A] :
          ( aa(A,$o,P,X4)
        <=> aa(A,$o,Q,X4) )
     => ( collect(A,P) = collect(A,Q) ) ) ).

% Collect_cong
tff(fact_46_ext,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),G: fun(A,B)] :
      ( ! [X4: A] : aa(A,B,F2,X4) = aa(A,B,G,X4)
     => ( F2 = G ) ) ).

% ext
tff(fact_47_sup_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,B3),A3)
        <=> ( ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) )
            & ( A3 != B3 ) ) ) ) ).

% sup.strict_order_iff
tff(fact_48_sup_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B3: A,C2: A,A3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),sup_sup(A),B3),C2)),A3)
         => ~ ( aa(A,$o,ord_less(A,B3),A3)
             => ~ aa(A,$o,ord_less(A,C2),A3) ) ) ) ).

% sup.strict_boundedE
tff(fact_49_sup_Oabsorb4,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) = B3 ) ) ) ).

% sup.absorb4
tff(fact_50_sup_Oabsorb3,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,B3),A3)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) = A3 ) ) ) ).

% sup.absorb3
tff(fact_51_less__supI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,X),B3)
         => aa(A,$o,ord_less(A,X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)) ) ) ).

% less_supI2
tff(fact_52_less__supI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,X),A3)
         => aa(A,$o,ord_less(A,X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)) ) ) ).

% less_supI1
tff(fact_53_enat__ord__number_I2_J,axiom,
    ! [M: num,Na: num] :
      ( aa(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,ord_less(nat,aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),Na)) ) ).

% enat_ord_number(2)
tff(fact_54_verit__eq__simplify_I8_J,axiom,
    ! [X22: num,Y2: num] :
      ( ( bit0(X22) = bit0(Y2) )
    <=> ( X22 = Y2 ) ) ).

% verit_eq_simplify(8)
tff(fact_55_verit__eq__simplify_I10_J,axiom,
    ! [X22: num] : one2 != bit0(X22) ).

% verit_eq_simplify(10)
tff(fact_56_dbl__simps_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_dbl(A,aa(num,A,numeral_numeral(A),K)) = aa(num,A,numeral_numeral(A),bit0(K)) ) ).

% dbl_simps(5)
tff(fact_57_one__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: num] :
          ( aa(A,$o,ord_less(A,one_one(A)),aa(num,A,numeral_numeral(A),Na))
        <=> aa(num,$o,ord_less(num,one2),Na) ) ) ).

% one_less_numeral_iff
tff(fact_58_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,I: num,Na: nat] :
          ( aa(A,$o,ord_less(A,aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),I)),Na))
        <=> aa(nat,$o,ord_less(nat,X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),I)),Na)) ) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_59_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: num,Na: nat,X: nat] :
          ( aa(A,$o,ord_less(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),I)),Na)),aa(nat,A,semiring_1_of_nat(A),X))
        <=> aa(nat,$o,ord_less(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),I)),Na)),X) ) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_60_sup__Un__eq,axiom,
    ! [A: $tType,R: set(A),S: set(A),X2: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),sup_sup(fun(A,$o)),aTP_Lamp_aa(set(A),fun(A,$o),R)),aTP_Lamp_aa(set(A),fun(A,$o),S)),X2)
    <=> member(A,X2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),R),S)) ) ).

% sup_Un_eq
tff(fact_61_vebt__heap__rules_I3_J,axiom,
    ! [X: nat,Na: nat,Sa: set(nat),Ti: vEBT_VEBTi] :
      ( aa(nat,$o,ord_less(nat,X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
     => hoare_hoare_triple(vEBT_VEBTi,aa(vEBT_VEBTi,assn,vEBT_Intf_vebt_assn(Na,Sa),Ti),vEBT_vebt_inserti(Ti,X),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),insert(nat,X),bot_bot(set(nat)))))) ) ).

% vebt_heap_rules(3)
tff(fact_62_power__numeral,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [K: num,L: num] : aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),K)),aa(num,nat,numeral_numeral(nat),L)) = aa(num,A,numeral_numeral(A),pow(K,L)) ) ).

% power_numeral
tff(fact_63_zero__less__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),bit0(one2))))
        <=> ( A3 != zero_zero(A) ) ) ) ).

% zero_less_power2
tff(fact_64_set__union,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] : aa(list(A),set(A),set2(A),union(A,Xsa,Ys)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xsa)),aa(list(A),set(A),set2(A),Ys)) ).

% set_union
tff(fact_65_power__one__right,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A] : aa(nat,A,power_power(A,A3),one_one(nat)) = A3 ) ).

% power_one_right
tff(fact_66_empty__iff,axiom,
    ! [A: $tType,C2: A] : ~ member(A,C2,bot_bot(set(A))) ).

% empty_iff
tff(fact_67_all__not__in__conv,axiom,
    ! [A: $tType,A2: set(A)] :
      ( ! [X3: A] : ~ member(A,X3,A2)
    <=> ( A2 = bot_bot(set(A)) ) ) ).

% all_not_in_conv
tff(fact_68_Collect__empty__eq,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ( collect(A,P) = bot_bot(set(A)) )
    <=> ! [X3: A] : ~ aa(A,$o,P,X3) ) ).

% Collect_empty_eq
tff(fact_69_empty__Collect__eq,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ( bot_bot(set(A)) = collect(A,P) )
    <=> ! [X3: A] : ~ aa(A,$o,P,X3) ) ).

% empty_Collect_eq
tff(fact_70_insertCI,axiom,
    ! [A: $tType,A3: A,B2: set(A),B3: A] :
      ( ( ~ member(A,A3,B2)
       => ( A3 = B3 ) )
     => member(A,A3,aa(set(A),set(A),insert(A,B3),B2)) ) ).

% insertCI
tff(fact_71_insert__iff,axiom,
    ! [A: $tType,A3: A,B3: A,A2: set(A)] :
      ( member(A,A3,aa(set(A),set(A),insert(A,B3),A2))
    <=> ( ( A3 = B3 )
        | member(A,A3,A2) ) ) ).

% insert_iff
tff(fact_72_insert__absorb2,axiom,
    ! [A: $tType,X: A,A2: set(A)] : aa(set(A),set(A),insert(A,X),aa(set(A),set(A),insert(A,X),A2)) = aa(set(A),set(A),insert(A,X),A2) ).

% insert_absorb2
tff(fact_73_i0__less,axiom,
    ! [Na: extended_enat] :
      ( aa(extended_enat,$o,ord_less(extended_enat,zero_zero(extended_enat)),Na)
    <=> ( Na != zero_zero(extended_enat) ) ) ).

% i0_less
tff(fact_74_power__one,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Na: nat] : aa(nat,A,power_power(A,one_one(A)),Na) = one_one(A) ) ).

% power_one
tff(fact_75_singletonI,axiom,
    ! [A: $tType,A3: A] : member(A,A3,aa(set(A),set(A),insert(A,A3),bot_bot(set(A)))) ).

% singletonI
tff(fact_76_sup__bot__left,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),bot_bot(A)),X) = X ) ).

% sup_bot_left
tff(fact_77_sup__bot__right,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),bot_bot(A)) = X ) ).

% sup_bot_right
tff(fact_78_bot__eq__sup__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [X: A,Y: A] :
          ( ( bot_bot(A) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) )
        <=> ( ( X = bot_bot(A) )
            & ( Y = bot_bot(A) ) ) ) ) ).

% bot_eq_sup_iff
tff(fact_79_sup__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = bot_bot(A) )
        <=> ( ( X = bot_bot(A) )
            & ( Y = bot_bot(A) ) ) ) ) ).

% sup_eq_bot_iff
tff(fact_80_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) = bot_bot(A) )
        <=> ( ( A3 = bot_bot(A) )
            & ( B3 = bot_bot(A) ) ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_81_sup__bot_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),bot_bot(A)),A3) = A3 ) ).

% sup_bot.left_neutral
tff(fact_82_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [A3: A,B3: A] :
          ( ( bot_bot(A) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) )
        <=> ( ( A3 = bot_bot(A) )
            & ( B3 = bot_bot(A) ) ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_83_sup__bot_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),bot_bot(A)) = A3 ) ).

% sup_bot.right_neutral
tff(fact_84_nat__zero__less__power__iff,axiom,
    ! [X: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(nat,nat,power_power(nat,X),Na))
    <=> ( aa(nat,$o,ord_less(nat,zero_zero(nat)),X)
        | ( Na = zero_zero(nat) ) ) ) ).

% nat_zero_less_power_iff
tff(fact_85_Un__empty,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),B2) = bot_bot(set(A)) )
    <=> ( ( A2 = bot_bot(set(A)) )
        & ( B2 = bot_bot(set(A)) ) ) ) ).

% Un_empty
tff(fact_86_Un__insert__left,axiom,
    ! [A: $tType,A3: A,B2: set(A),C3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),insert(A,A3),B2)),C3) = aa(set(A),set(A),insert(A,A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B2),C3)) ).

% Un_insert_left
tff(fact_87_Un__insert__right,axiom,
    ! [A: $tType,A2: set(A),A3: A,B2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),aa(set(A),set(A),insert(A,A3),B2)) = aa(set(A),set(A),insert(A,A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),B2)) ).

% Un_insert_right
tff(fact_88_singleton__conv,axiom,
    ! [A: $tType,A3: A] : collect(A,aTP_Lamp_ad(A,fun(A,$o),A3)) = aa(set(A),set(A),insert(A,A3),bot_bot(set(A))) ).

% singleton_conv
tff(fact_89_singleton__conv2,axiom,
    ! [A: $tType,A3: A] : collect(A,aa(A,fun(A,$o),fequal(A),A3)) = aa(set(A),set(A),insert(A,A3),bot_bot(set(A))) ).

% singleton_conv2
tff(fact_90_dbl__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl(A,zero_zero(A)) = zero_zero(A) ) ) ).

% dbl_simps(2)
tff(fact_91_numeral__eq__one__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Na: num] :
          ( ( aa(num,A,numeral_numeral(A),Na) = one_one(A) )
        <=> ( Na = one2 ) ) ) ).

% numeral_eq_one_iff
tff(fact_92_one__eq__numeral__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Na: num] :
          ( ( one_one(A) = aa(num,A,numeral_numeral(A),Na) )
        <=> ( one2 = Na ) ) ) ).

% one_eq_numeral_iff
tff(fact_93_power__inject__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,M: nat,Na: nat] :
          ( aa(A,$o,ord_less(A,one_one(A)),A3)
         => ( ( aa(nat,A,power_power(A,A3),M) = aa(nat,A,power_power(A,A3),Na) )
          <=> ( M = Na ) ) ) ) ).

% power_inject_exp
tff(fact_94_power__zero__numeral,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [K: num] : aa(nat,A,power_power(A,zero_zero(A)),aa(num,nat,numeral_numeral(nat),K)) = zero_zero(A) ) ).

% power_zero_numeral
tff(fact_95_power__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A3: A,Na: nat] :
          ( ( aa(nat,A,power_power(A,A3),Na) = zero_zero(A) )
        <=> ( ( A3 = zero_zero(A) )
            & aa(nat,$o,ord_less(nat,zero_zero(nat)),Na) ) ) ) ).

% power_eq_0_iff
tff(fact_96_of__nat__numeral,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Na: num] : aa(nat,A,semiring_1_of_nat(A),aa(num,nat,numeral_numeral(nat),Na)) = aa(num,A,numeral_numeral(A),Na) ) ).

% of_nat_numeral
tff(fact_97_semiring__1__class_Oof__nat__power,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat,Na: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,power_power(nat,M),Na)) = aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),M)),Na) ) ).

% semiring_1_class.of_nat_power
tff(fact_98_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [B3: nat,W: nat,X: nat] :
          ( ( aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),B3)),W) = aa(nat,A,semiring_1_of_nat(A),X) )
        <=> ( aa(nat,nat,power_power(nat,B3),W) = X ) ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_99_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [X: nat,B3: nat,W: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),X) = aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),B3)),W) )
        <=> ( X = aa(nat,nat,power_power(nat,B3),W) ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_100_power__strict__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B3: A,X: nat,Y: nat] :
          ( aa(A,$o,ord_less(A,one_one(A)),B3)
         => ( aa(A,$o,ord_less(A,aa(nat,A,power_power(A,B3),X)),aa(nat,A,power_power(A,B3),Y))
          <=> aa(nat,$o,ord_less(nat,X),Y) ) ) ) ).

% power_strict_increasing_iff
tff(fact_101_zero__eq__power2,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A3: A] :
          ( ( aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),bit0(one2))) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% zero_eq_power2
tff(fact_102_power__strict__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B3: A,M: nat,Na: nat] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),B3)
         => ( aa(A,$o,ord_less(A,B3),one_one(A))
           => ( aa(A,$o,ord_less(A,aa(nat,A,power_power(A,B3),M)),aa(nat,A,power_power(A,B3),Na))
            <=> aa(nat,$o,ord_less(nat,Na),M) ) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_103_of__nat__zero__less__power__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,Na: nat] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),X)),Na))
        <=> ( aa(nat,$o,ord_less(nat,zero_zero(nat)),X)
            | ( Na = zero_zero(nat) ) ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_104_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [X: num,Na: nat,Y: nat] :
          ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),X)),Na) = aa(nat,A,semiring_1_of_nat(A),Y) )
        <=> ( aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),X)),Na) = Y ) ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_105_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Y: nat,X: num,Na: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),Y) = aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),X)),Na) )
        <=> ( Y = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),X)),Na) ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_106_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B3: nat,W: nat,X: nat] :
          ( aa(A,$o,ord_less(A,aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),B3)),W)),aa(nat,A,semiring_1_of_nat(A),X))
        <=> aa(nat,$o,ord_less(nat,aa(nat,nat,power_power(nat,B3),W)),X) ) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_107_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,B3: nat,W: nat] :
          ( aa(A,$o,ord_less(A,aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),B3)),W))
        <=> aa(nat,$o,ord_less(nat,X),aa(nat,nat,power_power(nat,B3),W)) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_108_dbl__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl(A,one_one(A)) = aa(num,A,numeral_numeral(A),bit0(one2)) ) ) ).

% dbl_simps(3)
tff(fact_109_emptyE,axiom,
    ! [A: $tType,A3: A] : ~ member(A,A3,bot_bot(set(A))) ).

% emptyE
tff(fact_110_insertE,axiom,
    ! [A: $tType,A3: A,B3: A,A2: set(A)] :
      ( member(A,A3,aa(set(A),set(A),insert(A,B3),A2))
     => ( ( A3 != B3 )
       => member(A,A3,A2) ) ) ).

% insertE
tff(fact_111_equals0D,axiom,
    ! [A: $tType,A2: set(A),A3: A] :
      ( ( A2 = bot_bot(set(A)) )
     => ~ member(A,A3,A2) ) ).

% equals0D
tff(fact_112_equals0I,axiom,
    ! [A: $tType,A2: set(A)] :
      ( ! [Y3: A] : ~ member(A,Y3,A2)
     => ( A2 = bot_bot(set(A)) ) ) ).

% equals0I
tff(fact_113_insertI1,axiom,
    ! [A: $tType,A3: A,B2: set(A)] : member(A,A3,aa(set(A),set(A),insert(A,A3),B2)) ).

% insertI1
tff(fact_114_insertI2,axiom,
    ! [A: $tType,A3: A,B2: set(A),B3: A] :
      ( member(A,A3,B2)
     => member(A,A3,aa(set(A),set(A),insert(A,B3),B2)) ) ).

% insertI2
tff(fact_115_ex__in__conv,axiom,
    ! [A: $tType,A2: set(A)] :
      ( ? [X3: A] : member(A,X3,A2)
    <=> ( A2 != bot_bot(set(A)) ) ) ).

% ex_in_conv
tff(fact_116_Set_Oset__insert,axiom,
    ! [A: $tType,X: A,A2: set(A)] :
      ( member(A,X,A2)
     => ~ ! [B4: set(A)] :
            ( ( A2 = aa(set(A),set(A),insert(A,X),B4) )
           => member(A,X,B4) ) ) ).

% Set.set_insert
tff(fact_117_singletonD,axiom,
    ! [A: $tType,B3: A,A3: A] :
      ( member(A,B3,aa(set(A),set(A),insert(A,A3),bot_bot(set(A))))
     => ( B3 = A3 ) ) ).

% singletonD
tff(fact_118_insert__ident,axiom,
    ! [A: $tType,X: A,A2: set(A),B2: set(A)] :
      ( ~ member(A,X,A2)
     => ( ~ member(A,X,B2)
       => ( ( aa(set(A),set(A),insert(A,X),A2) = aa(set(A),set(A),insert(A,X),B2) )
        <=> ( A2 = B2 ) ) ) ) ).

% insert_ident
tff(fact_119_insert__absorb,axiom,
    ! [A: $tType,A3: A,A2: set(A)] :
      ( member(A,A3,A2)
     => ( aa(set(A),set(A),insert(A,A3),A2) = A2 ) ) ).

% insert_absorb
tff(fact_120_insert__eq__iff,axiom,
    ! [A: $tType,A3: A,A2: set(A),B3: A,B2: set(A)] :
      ( ~ member(A,A3,A2)
     => ( ~ member(A,B3,B2)
       => ( ( aa(set(A),set(A),insert(A,A3),A2) = aa(set(A),set(A),insert(A,B3),B2) )
        <=> $ite(
              A3 = B3,
              A2 = B2,
              ? [C4: set(A)] :
                ( ( A2 = aa(set(A),set(A),insert(A,B3),C4) )
                & ~ member(A,B3,C4)
                & ( B2 = aa(set(A),set(A),insert(A,A3),C4) )
                & ~ member(A,A3,C4) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_121_singleton__iff,axiom,
    ! [A: $tType,B3: A,A3: A] :
      ( member(A,B3,aa(set(A),set(A),insert(A,A3),bot_bot(set(A))))
    <=> ( B3 = A3 ) ) ).

% singleton_iff
tff(fact_122_insert__commute,axiom,
    ! [A: $tType,X: A,Y: A,A2: set(A)] : aa(set(A),set(A),insert(A,X),aa(set(A),set(A),insert(A,Y),A2)) = aa(set(A),set(A),insert(A,Y),aa(set(A),set(A),insert(A,X),A2)) ).

% insert_commute
tff(fact_123_Collect__conv__if,axiom,
    ! [A: $tType,A3: A,P: fun(A,$o)] :
      collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ae(A,fun(fun(A,$o),fun(A,$o)),A3),P)) = $ite(aa(A,$o,P,A3),aa(set(A),set(A),insert(A,A3),bot_bot(set(A))),bot_bot(set(A))) ).

% Collect_conv_if
tff(fact_124_Collect__conv__if2,axiom,
    ! [A: $tType,A3: A,P: fun(A,$o)] :
      collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_af(A,fun(fun(A,$o),fun(A,$o)),A3),P)) = $ite(aa(A,$o,P,A3),aa(set(A),set(A),insert(A,A3),bot_bot(set(A))),bot_bot(set(A))) ).

% Collect_conv_if2
tff(fact_125_doubleton__eq__iff,axiom,
    ! [A: $tType,A3: A,B3: A,C2: A,D2: A] :
      ( ( aa(set(A),set(A),insert(A,A3),aa(set(A),set(A),insert(A,B3),bot_bot(set(A)))) = aa(set(A),set(A),insert(A,C2),aa(set(A),set(A),insert(A,D2),bot_bot(set(A)))) )
    <=> ( ( ( A3 = C2 )
          & ( B3 = D2 ) )
        | ( ( A3 = D2 )
          & ( B3 = C2 ) ) ) ) ).

% doubleton_eq_iff
tff(fact_126_insert__not__empty,axiom,
    ! [A: $tType,A3: A,A2: set(A)] : aa(set(A),set(A),insert(A,A3),A2) != bot_bot(set(A)) ).

% insert_not_empty
tff(fact_127_singleton__inject,axiom,
    ! [A: $tType,A3: A,B3: A] :
      ( ( aa(set(A),set(A),insert(A,A3),bot_bot(set(A))) = aa(set(A),set(A),insert(A,B3),bot_bot(set(A))) )
     => ( A3 = B3 ) ) ).

% singleton_inject
tff(fact_128_not__psubset__empty,axiom,
    ! [A: $tType,A2: set(A)] : ~ aa(set(A),$o,ord_less(set(A),A2),bot_bot(set(A))) ).

% not_psubset_empty
tff(fact_129_mk__disjoint__insert,axiom,
    ! [A: $tType,A3: A,A2: set(A)] :
      ( member(A,A3,A2)
     => ? [B4: set(A)] :
          ( ( A2 = aa(set(A),set(A),insert(A,A3),B4) )
          & ~ member(A,A3,B4) ) ) ).

% mk_disjoint_insert
tff(fact_130_not__iless0,axiom,
    ! [Na: extended_enat] : ~ aa(extended_enat,$o,ord_less(extended_enat,Na),zero_zero(extended_enat)) ).

% not_iless0
tff(fact_131_enat__less__induct,axiom,
    ! [P: fun(extended_enat,$o),Na: extended_enat] :
      ( ! [N: extended_enat] :
          ( ! [M2: extended_enat] :
              ( aa(extended_enat,$o,ord_less(extended_enat,M2),N)
             => aa(extended_enat,$o,P,M2) )
         => aa(extended_enat,$o,P,N) )
     => aa(extended_enat,$o,P,Na) ) ).

% enat_less_induct
tff(fact_132_power__0__left,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Na: nat] :
          aa(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_133_power__0,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [A3: A] : aa(nat,A,power_power(A,A3),zero_zero(nat)) = one_one(A) ) ).

% power_0
tff(fact_134_less__numeral__extra_I1_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,ord_less(A,zero_zero(A)),one_one(A)) ) ).

% less_numeral_extra(1)
tff(fact_135_singleton__Un__iff,axiom,
    ! [A: $tType,X: A,A2: set(A),B2: set(A)] :
      ( ( aa(set(A),set(A),insert(A,X),bot_bot(set(A))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),B2) )
    <=> ( ( ( A2 = bot_bot(set(A)) )
          & ( B2 = aa(set(A),set(A),insert(A,X),bot_bot(set(A))) ) )
        | ( ( A2 = aa(set(A),set(A),insert(A,X),bot_bot(set(A))) )
          & ( B2 = bot_bot(set(A)) ) )
        | ( ( A2 = aa(set(A),set(A),insert(A,X),bot_bot(set(A))) )
          & ( B2 = aa(set(A),set(A),insert(A,X),bot_bot(set(A))) ) ) ) ) ).

% singleton_Un_iff
tff(fact_136_Un__singleton__iff,axiom,
    ! [A: $tType,A2: set(A),B2: set(A),X: A] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),B2) = aa(set(A),set(A),insert(A,X),bot_bot(set(A))) )
    <=> ( ( ( A2 = bot_bot(set(A)) )
          & ( B2 = aa(set(A),set(A),insert(A,X),bot_bot(set(A))) ) )
        | ( ( A2 = aa(set(A),set(A),insert(A,X),bot_bot(set(A))) )
          & ( B2 = bot_bot(set(A)) ) )
        | ( ( A2 = aa(set(A),set(A),insert(A,X),bot_bot(set(A))) )
          & ( B2 = aa(set(A),set(A),insert(A,X),bot_bot(set(A))) ) ) ) ) ).

% Un_singleton_iff
tff(fact_137_insert__is__Un,axiom,
    ! [A: $tType,A3: A,A2: set(A)] : aa(set(A),set(A),insert(A,A3),A2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),insert(A,A3),bot_bot(set(A)))),A2) ).

% insert_is_Un
tff(fact_138_int__ops_I3_J,axiom,
    ! [Na: num] : aa(nat,int,semiring_1_of_nat(int),aa(num,nat,numeral_numeral(nat),Na)) = aa(num,int,numeral_numeral(int),Na) ).

% int_ops(3)
tff(fact_139_nat__int__comparison_I2_J,axiom,
    ! [A3: nat,B3: nat] :
      ( aa(nat,$o,ord_less(nat,A3),B3)
    <=> aa(int,$o,ord_less(int,aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B3)) ) ).

% nat_int_comparison(2)
tff(fact_140_insert__Collect,axiom,
    ! [A: $tType,A3: A,P: fun(A,$o)] : aa(set(A),set(A),insert(A,A3),collect(A,P)) = collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ag(A,fun(fun(A,$o),fun(A,$o)),A3),P)) ).

% insert_Collect
tff(fact_141_insert__compr,axiom,
    ! [A: $tType,A3: A,B2: set(A)] : aa(set(A),set(A),insert(A,A3),B2) = collect(A,aa(set(A),fun(A,$o),aTP_Lamp_ah(A,fun(set(A),fun(A,$o)),A3),B2)) ).

% insert_compr
tff(fact_142_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A3: A,Na: nat] :
          ( ( A3 != zero_zero(A) )
         => ( aa(nat,A,power_power(A,A3),Na) != zero_zero(A) ) ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_143_Set_Oempty__def,axiom,
    ! [A: $tType] : bot_bot(set(A)) = collect(A,aTP_Lamp_ai(A,$o)) ).

% Set.empty_def
tff(fact_144_zero__power,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Na: nat] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
         => ( aa(nat,A,power_power(A,zero_zero(A)),Na) = zero_zero(A) ) ) ) ).

% zero_power
tff(fact_145_nat__less__as__int,axiom,
    ! [X2: nat,Xa: nat] :
      ( aa(nat,$o,ord_less(nat,X2),Xa)
    <=> aa(int,$o,ord_less(int,aa(nat,int,semiring_1_of_nat(int),X2)),aa(nat,int,semiring_1_of_nat(int),Xa)) ) ).

% nat_less_as_int
tff(fact_146_power__strict__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat,N2: nat,A3: A] :
          ( aa(nat,$o,ord_less(nat,Na),N2)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
           => ( aa(A,$o,ord_less(A,A3),one_one(A))
             => aa(A,$o,ord_less(A,aa(nat,A,power_power(A,A3),N2)),aa(nat,A,power_power(A,A3),Na)) ) ) ) ) ).

% power_strict_decreasing
tff(fact_147_one__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,ord_less(A,one_one(A)),A3)
         => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
           => aa(A,$o,ord_less(A,one_one(A)),aa(nat,A,power_power(A,A3),Na)) ) ) ) ).

% one_less_power
tff(fact_148_less__numeral__extra_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ aa(A,$o,ord_less(A,one_one(A)),one_one(A)) ) ).

% less_numeral_extra(4)
tff(fact_149_less__numeral__extra_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ aa(A,$o,ord_less(A,zero_zero(A)),zero_zero(A)) ) ).

% less_numeral_extra(3)
tff(fact_150_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_151_Un__empty__left,axiom,
    ! [A: $tType,B2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),bot_bot(set(A))),B2) = B2 ).

% Un_empty_left
tff(fact_152_Un__empty__right,axiom,
    ! [A: $tType,A2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),bot_bot(set(A))) = A2 ).

% Un_empty_right
tff(fact_153_zero__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => aa(A,$o,ord_less(A,zero_zero(A)),aa(nat,A,power_power(A,A3),Na)) ) ) ).

% zero_less_power
tff(fact_154_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),I)
     => ( aa(nat,$o,ord_less(nat,aa(nat,nat,power_power(nat,I),M)),aa(nat,nat,power_power(nat,I),Na))
       => aa(nat,$o,ord_less(nat,M),Na) ) ) ).

% nat_power_less_imp_less
tff(fact_155_insert__def,axiom,
    ! [A: $tType,A3: A,B2: set(A)] : aa(set(A),set(A),insert(A,A3),B2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),collect(A,aTP_Lamp_ad(A,fun(A,$o),A3))),B2) ).

% insert_def
tff(fact_156_power__less__imp__less__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,M: nat,Na: nat] :
          ( aa(A,$o,ord_less(A,one_one(A)),A3)
         => ( aa(A,$o,ord_less(A,aa(nat,A,power_power(A,A3),M)),aa(nat,A,power_power(A,A3),Na))
           => aa(nat,$o,ord_less(nat,M),Na) ) ) ) ).

% power_less_imp_less_exp
tff(fact_157_power__strict__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat,N2: nat,A3: A] :
          ( aa(nat,$o,ord_less(nat,Na),N2)
         => ( aa(A,$o,ord_less(A,one_one(A)),A3)
           => aa(A,$o,ord_less(A,aa(nat,A,power_power(A,A3),Na)),aa(nat,A,power_power(A,A3),N2)) ) ) ) ).

% power_strict_increasing
tff(fact_158_not__numeral__less__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: num] : ~ aa(A,$o,ord_less(A,aa(num,A,numeral_numeral(A),Na)),one_one(A)) ) ).

% not_numeral_less_one
tff(fact_159_numeral__One,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ( aa(num,A,numeral_numeral(A),one2) = one_one(A) ) ) ).

% numeral_One
tff(fact_160_zero__less__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: num] : aa(A,$o,ord_less(A,zero_zero(A)),aa(num,A,numeral_numeral(A),Na)) ) ).

% zero_less_numeral
tff(fact_161_not__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: num] : ~ aa(A,$o,ord_less(A,aa(num,A,numeral_numeral(A),Na)),zero_zero(A)) ) ).

% not_numeral_less_zero
tff(fact_162_numerals_I1_J,axiom,
    aa(num,nat,numeral_numeral(nat),one2) = one_one(nat) ).

% numerals(1)
tff(fact_163_pow_Osimps_I1_J,axiom,
    ! [X: num] : pow(X,one2) = X ).

% pow.simps(1)
tff(fact_164_one__power2,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,A,power_power(A,one_one(A)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(A) ) ) ).

% one_power2
tff(fact_165_zero__power2,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,A,power_power(A,zero_zero(A)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = zero_zero(A) ) ) ).

% zero_power2
tff(fact_166_verit__comp__simplify1_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A] : ~ aa(A,$o,ord_less(A,A3),A3) ) ).

% verit_comp_simplify1(1)
tff(fact_167_power2__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] : ~ aa(A,$o,ord_less(A,aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),zero_zero(A)) ) ).

% power2_less_0
tff(fact_168_of__nat__0__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: nat] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),Na))
        <=> aa(nat,$o,ord_less(nat,zero_zero(nat)),Na) ) ) ).

% of_nat_0_less_iff
tff(fact_169_of__nat__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,Na: nat] :
          ( aa(A,$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,ord_less(nat,M),Na) ) ) ).

% of_nat_less_iff
tff(fact_170_of__nat__less__two__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat] : aa(A,$o,ord_less(A,aa(nat,A,semiring_1_of_nat(A),Na)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) ) ).

% of_nat_less_two_power
tff(fact_171_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_172_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_173_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_174_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,Na: nat] :
      ( aa(real,$o,ord_less(real,aa(num,real,numeral_numeral(real),W)),aa(nat,real,semiring_1_of_nat(real),Na))
    <=> aa(nat,$o,ord_less(nat,aa(num,nat,numeral_numeral(nat),W)),Na) ) ).

% numeral_less_real_of_nat_iff
tff(fact_175_real__of__nat__less__numeral__iff,axiom,
    ! [Na: nat,W: num] :
      ( aa(real,$o,ord_less(real,aa(nat,real,semiring_1_of_nat(real),Na)),aa(num,real,numeral_numeral(real),W))
    <=> aa(nat,$o,ord_less(nat,Na),aa(num,nat,numeral_numeral(nat),W)) ) ).

% real_of_nat_less_numeral_iff
tff(fact_176_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_177_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_178_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_179_not__gr__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Na: A] :
          ( ~ aa(A,$o,ord_less(A,zero_zero(A)),Na)
        <=> ( Na = zero_zero(A) ) ) ) ).

% not_gr_zero
tff(fact_180_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_181_less__nat__zero__code,axiom,
    ! [Na: nat] : ~ aa(nat,$o,ord_less(nat,Na),zero_zero(nat)) ).

% less_nat_zero_code
tff(fact_182_neq0__conv,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero(nat) )
    <=> aa(nat,$o,ord_less(nat,zero_zero(nat)),Na) ) ).

% neq0_conv
tff(fact_183_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A3: nat] :
      ( ( A3 != zero_zero(nat) )
    <=> aa(nat,$o,ord_less(nat,zero_zero(nat)),A3) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_184_less__one,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,ord_less(nat,Na),one_one(nat))
    <=> ( Na = zero_zero(nat) ) ) ).

% less_one
tff(fact_185_psubsetD,axiom,
    ! [A: $tType,A2: set(A),B2: set(A),C2: A] :
      ( aa(set(A),$o,ord_less(set(A),A2),B2)
     => ( member(A,C2,A2)
       => member(A,C2,B2) ) ) ).

% psubsetD
tff(fact_186_less__set__def,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] :
      ( aa(set(A),$o,ord_less(set(A),A2),B2)
    <=> aa(fun(A,$o),$o,ord_less(fun(A,$o),aTP_Lamp_aa(set(A),fun(A,$o),A2)),aTP_Lamp_aa(set(A),fun(A,$o),B2)) ) ).

% less_set_def
tff(fact_187_psubset__trans,axiom,
    ! [A: $tType,A2: set(A),B2: set(A),C3: set(A)] :
      ( aa(set(A),$o,ord_less(set(A),A2),B2)
     => ( aa(set(A),$o,ord_less(set(A),B2),C3)
       => aa(set(A),$o,ord_less(set(A),A2),C3) ) ) ).

% psubset_trans
tff(fact_188_real__arch__pow,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),X)
     => ? [N: nat] : aa(real,$o,ord_less(real,Y),aa(nat,real,power_power(real,X),N)) ) ).

% real_arch_pow
tff(fact_189_real__arch__pow__inv,axiom,
    ! [Y: real,X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
     => ( aa(real,$o,ord_less(real,X),one_one(real))
       => ? [N: nat] : aa(real,$o,ord_less(real,aa(nat,real,power_power(real,X),N)),Y) ) ) ).

% real_arch_pow_inv
tff(fact_190_bot__enat__def,axiom,
    bot_bot(extended_enat) = zero_zero(extended_enat) ).

% bot_enat_def
tff(fact_191_bot__nat__def,axiom,
    bot_bot(nat) = zero_zero(nat) ).

% bot_nat_def
tff(fact_192_int__if,axiom,
    ! [P: $o,A3: nat,B3: nat] :
      aa(nat,int,semiring_1_of_nat(int),
        $ite((P),A3,B3)) = $ite((P),aa(nat,int,semiring_1_of_nat(int),A3),aa(nat,int,semiring_1_of_nat(int),B3)) ).

% int_if
tff(fact_193_zero__one__enat__neq_I1_J,axiom,
    zero_zero(extended_enat) != one_one(extended_enat) ).

% zero_one_enat_neq(1)
tff(fact_194_nat__int__comparison_I1_J,axiom,
    ! [A3: nat,B3: nat] :
      ( ( A3 = B3 )
    <=> ( aa(nat,int,semiring_1_of_nat(int),A3) = aa(nat,int,semiring_1_of_nat(int),B3) ) ) ).

% nat_int_comparison(1)
tff(fact_195_int__ops_I1_J,axiom,
    aa(nat,int,semiring_1_of_nat(int),zero_zero(nat)) = zero_zero(int) ).

% int_ops(1)
tff(fact_196_int__ops_I2_J,axiom,
    aa(nat,int,semiring_1_of_nat(int),one_one(nat)) = one_one(int) ).

% int_ops(2)
tff(fact_197_bot__empty__eq,axiom,
    ! [A: $tType,X2: A] :
      ( aa(A,$o,bot_bot(fun(A,$o)),X2)
    <=> member(A,X2,bot_bot(set(A))) ) ).

% bot_empty_eq
tff(fact_198_bot__set__def,axiom,
    ! [A: $tType] : bot_bot(set(A)) = collect(A,bot_bot(fun(A,$o))) ).

% bot_set_def
tff(fact_199_zero__reorient,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [X: A] :
          ( ( zero_zero(A) = X )
        <=> ( X = zero_zero(A) ) ) ) ).

% zero_reorient
tff(fact_200_measure__induct__rule,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F2: fun(A,B),P: fun(A,$o),A3: A] :
          ( ! [X4: A] :
              ( ! [Y4: A] :
                  ( aa(B,$o,ord_less(B,aa(A,B,F2,Y4)),aa(A,B,F2,X4))
                 => aa(A,$o,P,Y4) )
             => aa(A,$o,P,X4) )
         => aa(A,$o,P,A3) ) ) ).

% measure_induct_rule
tff(fact_201_measure__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F2: fun(A,B),P: fun(A,$o),A3: A] :
          ( ! [X4: A] :
              ( ! [Y4: A] :
                  ( aa(B,$o,ord_less(B,aa(A,B,F2,Y4)),aa(A,B,F2,X4))
                 => aa(A,$o,P,Y4) )
             => aa(A,$o,P,X4) )
         => aa(A,$o,P,A3) ) ) ).

% measure_induct
tff(fact_202_one__reorient,axiom,
    ! [A: $tType] :
      ( one(A)
     => ! [X: A] :
          ( ( one_one(A) = X )
        <=> ( X = one_one(A) ) ) ) ).

% one_reorient
tff(fact_203_infinite__descent__measure,axiom,
    ! [A: $tType,P: fun(A,$o),V: fun(A,nat),X: A] :
      ( ! [X4: A] :
          ( ~ aa(A,$o,P,X4)
         => ? [Y4: A] :
              ( aa(nat,$o,ord_less(nat,aa(A,nat,V,Y4)),aa(A,nat,V,X4))
              & ~ aa(A,$o,P,Y4) ) )
     => aa(A,$o,P,X) ) ).

% infinite_descent_measure
tff(fact_204_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ aa(nat,$o,ord_less(nat,X),Y)
       => aa(nat,$o,ord_less(nat,Y),X) ) ) ).

% linorder_neqE_nat
tff(fact_205_infinite__descent,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( ! [N: nat] :
          ( ~ aa(nat,$o,P,N)
         => ? [M2: nat] :
              ( aa(nat,$o,ord_less(nat,M2),N)
              & ~ aa(nat,$o,P,M2) ) )
     => aa(nat,$o,P,Na) ) ).

% infinite_descent
tff(fact_206_nat__less__induct,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( ! [N: nat] :
          ( ! [M2: nat] :
              ( aa(nat,$o,ord_less(nat,M2),N)
             => aa(nat,$o,P,M2) )
         => aa(nat,$o,P,N) )
     => aa(nat,$o,P,Na) ) ).

% nat_less_induct
tff(fact_207_less__irrefl__nat,axiom,
    ! [Na: nat] : ~ aa(nat,$o,ord_less(nat,Na),Na) ).

% less_irrefl_nat
tff(fact_208_less__not__refl3,axiom,
    ! [Sa: nat,Ta: nat] :
      ( aa(nat,$o,ord_less(nat,Sa),Ta)
     => ( Sa != Ta ) ) ).

% less_not_refl3
tff(fact_209_less__not__refl2,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,Na),M)
     => ( M != Na ) ) ).

% less_not_refl2
tff(fact_210_less__not__refl,axiom,
    ! [Na: nat] : ~ aa(nat,$o,ord_less(nat,Na),Na) ).

% less_not_refl
tff(fact_211_nat__neq__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( M != Na )
    <=> ( aa(nat,$o,ord_less(nat,M),Na)
        | aa(nat,$o,ord_less(nat,Na),M) ) ) ).

% nat_neq_iff
tff(fact_212_infinite__descent0__measure,axiom,
    ! [A: $tType,V: fun(A,nat),P: fun(A,$o),X: A] :
      ( ! [X4: A] :
          ( ( aa(A,nat,V,X4) = zero_zero(nat) )
         => aa(A,$o,P,X4) )
     => ( ! [X4: A] :
            ( aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(A,nat,V,X4))
           => ( ~ aa(A,$o,P,X4)
             => ? [Y4: A] :
                  ( aa(nat,$o,ord_less(nat,aa(A,nat,V,Y4)),aa(A,nat,V,X4))
                  & ~ aa(A,$o,P,Y4) ) ) )
       => aa(A,$o,P,X) ) ) ).

% infinite_descent0_measure
tff(fact_213_infinite__descent0,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ! [N: nat] :
            ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
           => ( ~ aa(nat,$o,P,N)
             => ? [M2: nat] :
                  ( aa(nat,$o,ord_less(nat,M2),N)
                  & ~ aa(nat,$o,P,M2) ) ) )
       => aa(nat,$o,P,Na) ) ) ).

% infinite_descent0
tff(fact_214_gr__implies__not0,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,M),Na)
     => ( Na != zero_zero(nat) ) ) ).

% gr_implies_not0
tff(fact_215_less__zeroE,axiom,
    ! [Na: nat] : ~ aa(nat,$o,ord_less(nat,Na),zero_zero(nat)) ).

% less_zeroE
tff(fact_216_not__less0,axiom,
    ! [Na: nat] : ~ aa(nat,$o,ord_less(nat,Na),zero_zero(nat)) ).

% not_less0
tff(fact_217_not__gr0,axiom,
    ! [Na: nat] :
      ( ~ aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
    <=> ( Na = zero_zero(nat) ) ) ).

% not_gr0
tff(fact_218_gr0I,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero(nat) )
     => aa(nat,$o,ord_less(nat,zero_zero(nat)),Na) ) ).

% gr0I
tff(fact_219_bot__nat__0_Oextremum__strict,axiom,
    ! [A3: nat] : ~ aa(nat,$o,ord_less(nat,A3),zero_zero(nat)) ).

% bot_nat_0.extremum_strict
tff(fact_220_field__lbound__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D1: A,D22: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),D1)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),D22)
           => ? [E: A] :
                ( aa(A,$o,ord_less(A,zero_zero(A)),E)
                & aa(A,$o,ord_less(A,E),D1)
                & aa(A,$o,ord_less(A,E),D22) ) ) ) ) ).

% field_lbound_gt_zero
tff(fact_221_zero__less__iff__neq__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Na: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Na)
        <=> ( Na != zero_zero(A) ) ) ) ).

% zero_less_iff_neq_zero
tff(fact_222_gr__implies__not__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [M: A,Na: A] :
          ( aa(A,$o,ord_less(A,M),Na)
         => ( Na != zero_zero(A) ) ) ) ).

% gr_implies_not_zero
tff(fact_223_not__less__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Na: A] : ~ aa(A,$o,ord_less(A,Na),zero_zero(A)) ) ).

% not_less_zero
tff(fact_224_gr__zeroI,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Na: A] :
          ( ( Na != zero_zero(A) )
         => aa(A,$o,ord_less(A,zero_zero(A)),Na) ) ) ).

% gr_zeroI
tff(fact_225_of__nat__less__0__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat] : ~ aa(A,$o,ord_less(A,aa(nat,A,semiring_1_of_nat(A),M)),zero_zero(A)) ) ).

% of_nat_less_0_iff
tff(fact_226_of__nat__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,Na: nat] :
          ( aa(A,$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,ord_less(nat,M),Na) ) ) ).

% of_nat_less_imp_less
tff(fact_227_less__imp__of__nat__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,ord_less(nat,M),Na)
         => aa(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_228_int__eq__iff__numeral,axiom,
    ! [M: nat,V2: num] :
      ( ( aa(nat,int,semiring_1_of_nat(int),M) = aa(num,int,numeral_numeral(int),V2) )
    <=> ( M = aa(num,nat,numeral_numeral(nat),V2) ) ) ).

% int_eq_iff_numeral
tff(fact_229_pos2,axiom,
    aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% pos2
tff(fact_230_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),K)
     => ? [N: nat] :
          ( aa(nat,$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_231_pos__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),K)
     => ~ ! [N: nat] :
            ( ( K = aa(nat,int,semiring_1_of_nat(int),N) )
           => ~ aa(nat,$o,ord_less(nat,zero_zero(nat)),N) ) ) ).

% pos_int_cases
tff(fact_232_word__unat__power,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] : aa(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)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ) ).

% word_unat_power
tff(fact_233_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,ord_less(nat,zero_zero(nat)),K) ) ) ).

% of_nat_gt_0
tff(fact_234_sup1CI,axiom,
    ! [A: $tType,B2: fun(A,$o),X: A,A2: fun(A,$o)] :
      ( ( ~ aa(A,$o,B2,X)
       => aa(A,$o,A2,X) )
     => aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),sup_sup(fun(A,$o)),A2),B2),X) ) ).

% sup1CI
tff(fact_235_realpow__pos__nth__unique,axiom,
    ! [Na: nat,A3: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
       => ? [X4: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),X4)
            & ( aa(nat,real,power_power(real,X4),Na) = A3 )
            & ! [Y4: real] :
                ( ( aa(real,$o,ord_less(real,zero_zero(real)),Y4)
                  & ( aa(nat,real,power_power(real,Y4),Na) = A3 ) )
               => ( Y4 = X4 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_236_realpow__pos__nth,axiom,
    ! [Na: nat,A3: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
       => ? [R2: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),R2)
            & ( aa(nat,real,power_power(real,R2),Na) = A3 ) ) ) ) ).

% realpow_pos_nth
tff(fact_237_p2__eq__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( ( aa(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_238_not__one__less__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ aa(A,$o,ord_less(A,one_one(A)),zero_zero(A)) ) ).

% not_one_less_zero
tff(fact_239_word__gt__0__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: num] :
          ( aa(word(A),$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_240_word__less__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),X),one_one(word(A)))
        <=> ( X = zero_zero(word(A)) ) ) ) ).

% word_less_1
tff(fact_241_linorder__neqE__linordered__idom,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( ( X != Y )
         => ( ~ aa(A,$o,ord_less(A,X),Y)
           => aa(A,$o,ord_less(A,Y),X) ) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_242_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_243_boolean__algebra__cancel_Osup1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,K: A,A3: A,B3: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),sup_sup(A),K),A3) )
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B3) = aa(A,A,aa(A,fun(A,A),sup_sup(A),K),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)) ) ) ) ).

% boolean_algebra_cancel.sup1
tff(fact_244_boolean__algebra__cancel_Osup2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,K: A,B3: A,A3: A] :
          ( ( B2 = aa(A,A,aa(A,fun(A,A),sup_sup(A),K),B3) )
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),K),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)) ) ) ) ).

% boolean_algebra_cancel.sup2
tff(fact_245_less__int__code_I1_J,axiom,
    ~ aa(int,$o,ord_less(int,zero_zero(int)),zero_zero(int)) ).

% less_int_code(1)
tff(fact_246_int__int__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,int,semiring_1_of_nat(int),M) = aa(nat,int,semiring_1_of_nat(int),Na) )
    <=> ( M = Na ) ) ).

% int_int_eq
tff(fact_247_sup1E,axiom,
    ! [A: $tType,A2: fun(A,$o),B2: fun(A,$o),X: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),sup_sup(fun(A,$o)),A2),B2),X)
     => ( ~ aa(A,$o,A2,X)
       => aa(A,$o,B2,X) ) ) ).

% sup1E
tff(fact_248_sup1I1,axiom,
    ! [A: $tType,A2: fun(A,$o),X: A,B2: fun(A,$o)] :
      ( aa(A,$o,A2,X)
     => aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),sup_sup(fun(A,$o)),A2),B2),X) ) ).

% sup1I1
tff(fact_249_sup1I2,axiom,
    ! [A: $tType,B2: fun(A,$o),X: A,A2: fun(A,$o)] :
      ( aa(A,$o,B2,X)
     => aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),sup_sup(fun(A,$o)),A2),B2),X) ) ).

% sup1I2
tff(fact_250_zero__neq__one,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( zero_zero(A) != one_one(A) ) ) ).

% zero_neq_one
tff(fact_251_boolean__algebra_Odisj__zero__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),bot_bot(A)) = X ) ).

% boolean_algebra.disj_zero_right
tff(fact_252_zero__less__one,axiom,
    ! [A: $tType] :
      ( zero_less_one(A)
     => aa(A,$o,ord_less(A,zero_zero(A)),one_one(A)) ) ).

% zero_less_one
tff(fact_253_bot__apply,axiom,
    ! [B: $tType,A: $tType] :
      ( bot(A)
     => ! [X: B] : aa(B,A,bot_bot(fun(B,A)),X) = bot_bot(A) ) ).

% bot_apply
tff(fact_254_false__rule,axiom,
    ! [A: $tType,C2: heap_Time_Heap(A),Q: fun(A,assn)] : hoare_hoare_triple(A,bot_bot(assn),C2,Q) ).

% false_rule
tff(fact_255_the__elem__eq,axiom,
    ! [A: $tType,X: A] : the_elem(A,aa(set(A),set(A),insert(A,X),bot_bot(set(A)))) = X ).

% the_elem_eq
tff(fact_256_Collect__empty__eq__bot,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ( collect(A,P) = bot_bot(set(A)) )
    <=> ( P = bot_bot(fun(A,$o)) ) ) ).

% Collect_empty_eq_bot
tff(fact_257_vebt__buildupi__rule__basic,axiom,
    ! [Na: nat] :
      ( aa(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_258_bot__fun__def,axiom,
    ! [A: $tType,B: $tType] :
      ( bot(B)
     => ! [X2: A] : aa(A,B,bot_bot(fun(A,B)),X2) = bot_bot(B) ) ).

% bot_fun_def
tff(fact_259_is__singletonI,axiom,
    ! [A: $tType,X: A] : is_singleton(A,aa(set(A),set(A),insert(A,X),bot_bot(set(A)))) ).

% is_singletonI
tff(fact_260_forall__finite_I1_J,axiom,
    ! [P: fun(nat,$o),I2: nat] :
      ( aa(nat,$o,ord_less(nat,I2),zero_zero(nat))
     => aa(nat,$o,P,I2) ) ).

% forall_finite(1)
tff(fact_261_reals__Archimedean2,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [N: nat] : aa(A,$o,ord_less(A,X),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% reals_Archimedean2
tff(fact_262_bot_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] : ~ aa(A,$o,ord_less(A,A3),bot_bot(A)) ) ).

% bot.extremum_strict
tff(fact_263_bot_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] :
          ( ( A3 != bot_bot(A) )
        <=> aa(A,$o,ord_less(A,bot_bot(A)),A3) ) ) ).

% bot.not_eq_extremum
tff(fact_264_word__not__simps_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : ~ aa(word(A),$o,ord_less(word(A),X),zero_zero(word(A))) ) ).

% word_not_simps(1)
tff(fact_265_word__coorder_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : ~ aa(word(A),$o,ord_less(word(A),A3),zero_zero(word(A))) ) ).

% word_coorder.extremum_strict
tff(fact_266_word__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),Y)
        <=> ( zero_zero(word(A)) != Y ) ) ) ).

% word_gt_0
tff(fact_267_word__neq__0__conv,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( ( W != zero_zero(word(A)) )
        <=> aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),W) ) ) ).

% word_neq_0_conv
tff(fact_268_word__gt__a__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),Na: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),A3),Na)
         => aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),Na) ) ) ).

% word_gt_a_gt_0
tff(fact_269_word__greater__zero__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),A3)
        <=> ( A3 != zero_zero(word(A)) ) ) ) ).

% word_greater_zero_iff
tff(fact_270_is__singleton__the__elem,axiom,
    ! [A: $tType,A2: set(A)] :
      ( is_singleton(A,A2)
    <=> ( A2 = aa(set(A),set(A),insert(A,the_elem(A,A2)),bot_bot(set(A))) ) ) ).

% is_singleton_the_elem
tff(fact_271_is__singletonI_H,axiom,
    ! [A: $tType,A2: set(A)] :
      ( ( A2 != bot_bot(set(A)) )
     => ( ! [X4: A,Y3: A] :
            ( member(A,X4,A2)
           => ( member(A,Y3,A2)
             => ( X4 = Y3 ) ) )
       => is_singleton(A,A2) ) ) ).

% is_singletonI'
tff(fact_272_order__less__imp__not__less,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less(A,X),Y)
         => ~ aa(A,$o,ord_less(A,Y),X) ) ) ).

% order_less_imp_not_less
tff(fact_273_order__less__imp__not__eq2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less(A,X),Y)
         => ( Y != X ) ) ) ).

% order_less_imp_not_eq2
tff(fact_274_order__less__imp__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less(A,X),Y)
         => ( X != Y ) ) ) ).

% order_less_imp_not_eq
tff(fact_275_linorder__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less(A,X),Y)
          | ( X = Y )
          | aa(A,$o,ord_less(A,Y),X) ) ) ).

% linorder_less_linear
tff(fact_276_order__less__imp__triv,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,P: $o] :
          ( aa(A,$o,ord_less(A,X),Y)
         => ( aa(A,$o,ord_less(A,Y),X)
           => (P) ) ) ) ).

% order_less_imp_triv
tff(fact_277_order__less__not__sym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less(A,X),Y)
         => ~ aa(A,$o,ord_less(A,Y),X) ) ) ).

% order_less_not_sym
tff(fact_278_order__less__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,B3: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(B,$o,ord_less(B,aa(A,B,F2,B3)),C2)
           => ( ! [X4: A,Y3: A] :
                  ( aa(A,$o,ord_less(A,X4),Y3)
                 => aa(B,$o,ord_less(B,aa(A,B,F2,X4)),aa(A,B,F2,Y3)) )
             => aa(B,$o,ord_less(B,aa(A,B,F2,A3)),C2) ) ) ) ) ).

% order_less_subst2
tff(fact_279_order__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,F2: fun(B,A),B3: B,C2: B] :
          ( aa(A,$o,ord_less(A,A3),aa(B,A,F2,B3))
         => ( aa(B,$o,ord_less(B,B3),C2)
           => ( ! [X4: B,Y3: B] :
                  ( aa(B,$o,ord_less(B,X4),Y3)
                 => aa(A,$o,ord_less(A,aa(B,A,F2,X4)),aa(B,A,F2,Y3)) )
             => aa(A,$o,ord_less(A,A3),aa(B,A,F2,C2)) ) ) ) ) ).

% order_less_subst1
tff(fact_280_order__less__irrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A] : ~ aa(A,$o,ord_less(A,X),X) ) ).

% order_less_irrefl
tff(fact_281_ord__less__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A3: A,B3: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( ( aa(A,B,F2,B3) = C2 )
           => ( ! [X4: A,Y3: A] :
                  ( aa(A,$o,ord_less(A,X4),Y3)
                 => aa(B,$o,ord_less(B,aa(A,B,F2,X4)),aa(A,B,F2,Y3)) )
             => aa(B,$o,ord_less(B,aa(A,B,F2,A3)),C2) ) ) ) ) ).

% ord_less_eq_subst
tff(fact_282_ord__eq__less__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A3: A,F2: fun(B,A),B3: B,C2: B] :
          ( ( A3 = aa(B,A,F2,B3) )
         => ( aa(B,$o,ord_less(B,B3),C2)
           => ( ! [X4: B,Y3: B] :
                  ( aa(B,$o,ord_less(B,X4),Y3)
                 => aa(A,$o,ord_less(A,aa(B,A,F2,X4)),aa(B,A,F2,Y3)) )
             => aa(A,$o,ord_less(A,A3),aa(B,A,F2,C2)) ) ) ) ) ).

% ord_eq_less_subst
tff(fact_283_order__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,Z: A] :
          ( aa(A,$o,ord_less(A,X),Y)
         => ( aa(A,$o,ord_less(A,Y),Z)
           => aa(A,$o,ord_less(A,X),Z) ) ) ) ).

% order_less_trans
tff(fact_284_order__less__asym_H,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ~ aa(A,$o,ord_less(A,B3),A3) ) ) ).

% order_less_asym'
tff(fact_285_linorder__neq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ( X != Y )
        <=> ( aa(A,$o,ord_less(A,X),Y)
            | aa(A,$o,ord_less(A,Y),X) ) ) ) ).

% linorder_neq_iff
tff(fact_286_order__less__asym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less(A,X),Y)
         => ~ aa(A,$o,ord_less(A,Y),X) ) ) ).

% order_less_asym
tff(fact_287_linorder__neqE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ( X != Y )
         => ( ~ aa(A,$o,ord_less(A,X),Y)
           => aa(A,$o,ord_less(A,Y),X) ) ) ) ).

% linorder_neqE
tff(fact_288_dual__order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,B3),A3)
         => ( A3 != B3 ) ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_289_order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( A3 != B3 ) ) ) ).

% order.strict_implies_not_eq
tff(fact_290_dual__order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,ord_less(A,B3),A3)
         => ( aa(A,$o,ord_less(A,C2),B3)
           => aa(A,$o,ord_less(A,C2),A3) ) ) ) ).

% dual_order.strict_trans
tff(fact_291_not__less__iff__gr__or__eq,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ aa(A,$o,ord_less(A,X),Y)
        <=> ( aa(A,$o,ord_less(A,Y),X)
            | ( X = Y ) ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_292_order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(A,$o,ord_less(A,B3),C2)
           => aa(A,$o,ord_less(A,A3),C2) ) ) ) ).

% order.strict_trans
tff(fact_293_linorder__less__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,fun(A,$o)),A3: A,B3: A] :
          ( ! [A4: A,B5: A] :
              ( aa(A,$o,ord_less(A,A4),B5)
             => aa(A,$o,aa(A,fun(A,$o),P,A4),B5) )
         => ( ! [A4: A] : aa(A,$o,aa(A,fun(A,$o),P,A4),A4)
           => ( ! [A4: A,B5: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),P,B5),A4)
                 => aa(A,$o,aa(A,fun(A,$o),P,A4),B5) )
             => aa(A,$o,aa(A,fun(A,$o),P,A3),B3) ) ) ) ) ).

% linorder_less_wlog
tff(fact_294_exists__least__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o)] :
          ( ? [X_1: A] : aa(A,$o,P,X_1)
        <=> ? [N3: A] :
              ( aa(A,$o,P,N3)
              & ! [M3: A] :
                  ( aa(A,$o,ord_less(A,M3),N3)
                 => ~ aa(A,$o,P,M3) ) ) ) ) ).

% exists_least_iff
tff(fact_295_dual__order_Oirrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A] : ~ aa(A,$o,ord_less(A,A3),A3) ) ).

% dual_order.irrefl
tff(fact_296_dual__order_Oasym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,B3),A3)
         => ~ aa(A,$o,ord_less(A,A3),B3) ) ) ).

% dual_order.asym
tff(fact_297_linorder__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ aa(A,$o,ord_less(A,X),Y)
         => ( ( X != Y )
           => aa(A,$o,ord_less(A,Y),X) ) ) ) ).

% linorder_cases
tff(fact_298_antisym__conv3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,X: A] :
          ( ~ aa(A,$o,ord_less(A,Y),X)
         => ( ~ aa(A,$o,ord_less(A,X),Y)
          <=> ( X = Y ) ) ) ) ).

% antisym_conv3
tff(fact_299_less__induct,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),A3: A] :
          ( ! [X4: A] :
              ( ! [Y4: A] :
                  ( aa(A,$o,ord_less(A,Y4),X4)
                 => aa(A,$o,P,Y4) )
             => aa(A,$o,P,X4) )
         => aa(A,$o,P,A3) ) ) ).

% less_induct
tff(fact_300_ord__less__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( ( B3 = C2 )
           => aa(A,$o,ord_less(A,A3),C2) ) ) ) ).

% ord_less_eq_trans
tff(fact_301_ord__eq__less__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B3: A,C2: A] :
          ( ( A3 = B3 )
         => ( aa(A,$o,ord_less(A,B3),C2)
           => aa(A,$o,ord_less(A,A3),C2) ) ) ) ).

% ord_eq_less_trans
tff(fact_302_order_Oasym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ~ aa(A,$o,ord_less(A,B3),A3) ) ) ).

% order.asym
tff(fact_303_less__imp__neq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less(A,X),Y)
         => ( X != Y ) ) ) ).

% less_imp_neq
tff(fact_304_dense,axiom,
    ! [A: $tType] :
      ( dense_order(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less(A,X),Y)
         => ? [Z2: A] :
              ( aa(A,$o,ord_less(A,X),Z2)
              & aa(A,$o,ord_less(A,Z2),Y) ) ) ) ).

% dense
tff(fact_305_gt__ex,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [X: A] :
        ? [X_12: A] : aa(A,$o,ord_less(A,X),X_12) ) ).

% gt_ex
tff(fact_306_lt__ex,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [X: A] :
        ? [Y3: A] : aa(A,$o,ord_less(A,Y3),X) ) ).

% lt_ex
tff(fact_307_split__rule,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),R: fun(A,assn),Q: assn] :
      ( hoare_hoare_triple(A,P,C2,R)
     => ( hoare_hoare_triple(A,Q,C2,R)
       => hoare_hoare_triple(A,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),P),Q),C2,R) ) ) ).

% split_rule
tff(fact_308_if__rule,axiom,
    ! [A: $tType,B3: $o,P: assn,F2: heap_Time_Heap(A),Q: fun(A,assn),G: heap_Time_Heap(A)] :
      ( ( (B3)
       => hoare_hoare_triple(A,P,F2,Q) )
     => ( ( ~ (B3)
         => hoare_hoare_triple(A,P,G,Q) )
       => hoare_hoare_triple(A,P,
            $ite((B3),F2,G),
            Q) ) ) ).

% if_rule
tff(fact_309_is__singleton__def,axiom,
    ! [A: $tType,A2: set(A)] :
      ( is_singleton(A,A2)
    <=> ? [X3: A] : A2 = aa(set(A),set(A),insert(A,X3),bot_bot(set(A))) ) ).

% is_singleton_def
tff(fact_310_is__singletonE,axiom,
    ! [A: $tType,A2: set(A)] :
      ( is_singleton(A,A2)
     => ~ ! [X4: A] : A2 != aa(set(A),set(A),insert(A,X4),bot_bot(set(A))) ) ).

% is_singletonE
tff(fact_311_assn__basic__inequalities_I3_J,axiom,
    bot_bot(assn) != one_one(assn) ).

% assn_basic_inequalities(3)
tff(fact_312_uint__2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),aa(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),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) = aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na) ) ) ) ).

% uint_2p
tff(fact_313_vebt__heap__rules_I1_J,axiom,
    ! [Na: nat] : hoare_hoare_triple(vEBT_VEBTi,pure_assn(aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)),vEBT_vebt_buildupi(Na),vEBT_Intf_vebt_assn(Na,bot_bot(set(nat)))) ).

% vebt_heap_rules(1)
tff(fact_314_dbl__simps_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl(A,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% dbl_simps(4)
tff(fact_315_log2__of__power__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,M),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
       => aa(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_316_shiftl__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : bit_Sh4282982442137083160shiftl(A,one_one(A),Na) = aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na) ) ).

% shiftl_1
tff(fact_317_power2__eq__iff__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),X)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Y)
           => ( ( aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
            <=> ( X = Y ) ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_318_power2__less__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),zero_zero(A))
        <=> ( A3 = zero_zero(A) ) ) ) ).

% power2_less_eq_zero_iff
tff(fact_319_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_320_power2__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less(A,aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Y)
           => aa(A,$o,ord_less(A,X),Y) ) ) ) ).

% power2_less_imp_less
tff(fact_321_sum__power2__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,plus_plus(A,aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = zero_zero(A) )
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_322_order__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A] : aa(A,$o,ord_less_eq(A,X),X) ) ).

% order_refl
tff(fact_323_dual__order_Orefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A] : aa(A,$o,ord_less_eq(A,A3),A3) ) ).

% dual_order.refl
tff(fact_324_semiring__norm_I6_J,axiom,
    ! [M: num,Na: num] : aa(num,num,plus_plus(num,bit0(M)),bit0(Na)) = bit0(aa(num,num,plus_plus(num,M),Na)) ).

% semiring_norm(6)
tff(fact_325_semiring__norm_I71_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,ord_less_eq(num,bit0(M)),bit0(Na))
    <=> aa(num,$o,ord_less_eq(num,M),Na) ) ).

% semiring_norm(71)
tff(fact_326_semiring__norm_I68_J,axiom,
    ! [Na: num] : aa(num,$o,ord_less_eq(num,one2),Na) ).

% semiring_norm(68)
tff(fact_327_add__left__cancel,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( ( aa(A,A,plus_plus(A,A3),B3) = aa(A,A,plus_plus(A,A3),C2) )
        <=> ( B3 = C2 ) ) ) ).

% add_left_cancel
tff(fact_328_add__right__cancel,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [B3: A,A3: A,C2: A] :
          ( ( aa(A,A,plus_plus(A,B3),A3) = aa(A,A,plus_plus(A,C2),A3) )
        <=> ( B3 = C2 ) ) ) ).

% add_right_cancel
tff(fact_329_add__is__0,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,plus_plus(nat,M),Na) = zero_zero(nat) )
    <=> ( ( M = zero_zero(nat) )
        & ( Na = zero_zero(nat) ) ) ) ).

% add_is_0
tff(fact_330_Nat_Oadd__0__right,axiom,
    ! [M: nat] : aa(nat,nat,plus_plus(nat,M),zero_zero(nat)) = M ).

% Nat.add_0_right
tff(fact_331_bot__nat__0_Oextremum,axiom,
    ! [A3: nat] : aa(nat,$o,ord_less_eq(nat,zero_zero(nat)),A3) ).

% bot_nat_0.extremum
tff(fact_332_le0,axiom,
    ! [Na: nat] : aa(nat,$o,ord_less_eq(nat,zero_zero(nat)),Na) ).

% le0
tff(fact_333_add_Oinverse__inverse,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,uminus_uminus(A),aa(A,A,uminus_uminus(A),A3)) = A3 ) ).

% add.inverse_inverse
tff(fact_334_neg__equal__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,uminus_uminus(A),A3) = aa(A,A,uminus_uminus(A),B3) )
        <=> ( A3 = B3 ) ) ) ).

% neg_equal_iff_equal
tff(fact_335_uminus__apply,axiom,
    ! [A: $tType,B: $tType] :
      ( uminus(A)
     => ! [A2: fun(B,A),X: B] : aa(B,A,aa(fun(B,A),fun(B,A),uminus_uminus(fun(B,A)),A2),X) = aa(A,A,uminus_uminus(A),aa(B,A,A2,X)) ) ).

% uminus_apply
tff(fact_336_verit__minus__simplify_I4_J,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [B3: A] : aa(A,A,uminus_uminus(A),aa(A,A,uminus_uminus(A),B3)) = B3 ) ).

% verit_minus_simplify(4)
tff(fact_337_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,uminus_uminus(A),aa(A,A,uminus_uminus(A),X)) = X ) ).

% boolean_algebra_class.boolean_algebra.double_compl
tff(fact_338_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,uminus_uminus(A),X) = aa(A,A,uminus_uminus(A),Y) )
        <=> ( X = Y ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
tff(fact_339_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,plus_plus(nat,K),M)),aa(nat,nat,plus_plus(nat,K),Na))
    <=> aa(nat,$o,ord_less(nat,M),Na) ) ).

% nat_add_left_cancel_less
tff(fact_340_subset__empty,axiom,
    ! [A: $tType,A2: set(A)] :
      ( aa(set(A),$o,ord_less_eq(set(A),A2),bot_bot(set(A)))
    <=> ( A2 = bot_bot(set(A)) ) ) ).

% subset_empty
tff(fact_341_empty__subsetI,axiom,
    ! [A: $tType,A2: set(A)] : aa(set(A),$o,ord_less_eq(set(A),bot_bot(set(A))),A2) ).

% empty_subsetI
tff(fact_342_real__add__minus__iff,axiom,
    ! [X: real,A3: real] :
      ( ( aa(real,real,plus_plus(real,X),aa(real,real,uminus_uminus(real),A3)) = zero_zero(real) )
    <=> ( X = A3 ) ) ).

% real_add_minus_iff
tff(fact_343_insert__subset,axiom,
    ! [A: $tType,X: A,A2: set(A),B2: set(A)] :
      ( aa(set(A),$o,ord_less_eq(set(A),aa(set(A),set(A),insert(A,X),A2)),B2)
    <=> ( member(A,X,B2)
        & aa(set(A),$o,ord_less_eq(set(A),A2),B2) ) ) ).

% insert_subset
tff(fact_344_Un__subset__iff,axiom,
    ! [A: $tType,A2: set(A),B2: set(A),C3: set(A)] :
      ( aa(set(A),$o,ord_less_eq(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),B2)),C3)
    <=> ( aa(set(A),$o,ord_less_eq(set(A),A2),C3)
        & aa(set(A),$o,ord_less_eq(set(A),B2),C3) ) ) ).

% Un_subset_iff
tff(fact_345_negative__zle,axiom,
    ! [Na: nat,M: nat] : aa(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_346_max__word__less__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),W)
        <=> ( W = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% max_word_less_eq_iff
tff(fact_347_word__minus__one__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),X)
        <=> ( X = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% word_minus_one_le
tff(fact_348_max__word__max,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A)] : aa(word(A),$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_349_word__n1__ge,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A)] : aa(word(A),$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_350_word__order_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),A3)
        <=> ( A3 = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% word_order.extremum_unique
tff(fact_351_word__order_Oextremum,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : aa(word(A),$o,ord_less_eq(word(A),A3),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) ) ).

% word_order.extremum
tff(fact_352_word__coorder_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),A3),zero_zero(word(A)))
        <=> ( A3 = zero_zero(word(A)) ) ) ) ).

% word_coorder.extremum_unique
tff(fact_353_word__le__0__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),X),zero_zero(word(A)))
        <=> ( X = zero_zero(word(A)) ) ) ) ).

% word_le_0_iff
tff(fact_354_psubsetI,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] :
      ( aa(set(A),$o,ord_less_eq(set(A),A2),B2)
     => ( ( A2 != B2 )
       => aa(set(A),$o,ord_less(set(A),A2),B2) ) ) ).

% psubsetI
tff(fact_355_le__zero__eq,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Na: A] :
          ( aa(A,$o,ord_less_eq(A,Na),zero_zero(A))
        <=> ( Na = zero_zero(A) ) ) ) ).

% le_zero_eq
tff(fact_356_numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: num,Na: num] :
          ( aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Na))
        <=> aa(num,$o,ord_less_eq(num,M),Na) ) ) ).

% numeral_le_iff
tff(fact_357_double__eq__0__iff,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( ( aa(A,A,plus_plus(A,A3),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% double_eq_0_iff
tff(fact_358_add__0,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [A3: A] : aa(A,A,plus_plus(A,zero_zero(A)),A3) = A3 ) ).

% add_0
tff(fact_359_zero__eq__add__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [X: A,Y: A] :
          ( ( zero_zero(A) = aa(A,A,plus_plus(A,X),Y) )
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_360_add__eq__0__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,plus_plus(A,X),Y) = zero_zero(A) )
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_361_add__cancel__right__right,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = aa(A,A,plus_plus(A,A3),B3) )
        <=> ( B3 = zero_zero(A) ) ) ) ).

% add_cancel_right_right
tff(fact_362_add__cancel__right__left,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = aa(A,A,plus_plus(A,B3),A3) )
        <=> ( B3 = zero_zero(A) ) ) ) ).

% add_cancel_right_left
tff(fact_363_add__cancel__left__right,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,plus_plus(A,A3),B3) = A3 )
        <=> ( B3 = zero_zero(A) ) ) ) ).

% add_cancel_left_right
tff(fact_364_add__cancel__left__left,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [B3: A,A3: A] :
          ( ( aa(A,A,plus_plus(A,B3),A3) = A3 )
        <=> ( B3 = zero_zero(A) ) ) ) ).

% add_cancel_left_left
tff(fact_365_double__zero__sym,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( ( zero_zero(A) = aa(A,A,plus_plus(A,A3),A3) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% double_zero_sym
tff(fact_366_add_Oright__neutral,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [A3: A] : aa(A,A,plus_plus(A,A3),zero_zero(A)) = A3 ) ).

% add.right_neutral
tff(fact_367_add__le__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,plus_plus(A,C2),A3)),aa(A,A,plus_plus(A,C2),B3))
        <=> aa(A,$o,ord_less_eq(A,A3),B3) ) ) ).

% add_le_cancel_left
tff(fact_368_add__le__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,plus_plus(A,A3),C2)),aa(A,A,plus_plus(A,B3),C2))
        <=> aa(A,$o,ord_less_eq(A,A3),B3) ) ) ).

% add_le_cancel_right
tff(fact_369_add__less__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,plus_plus(A,C2),A3)),aa(A,A,plus_plus(A,C2),B3))
        <=> aa(A,$o,ord_less(A,A3),B3) ) ) ).

% add_less_cancel_left
tff(fact_370_add__less__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,plus_plus(A,A3),C2)),aa(A,A,plus_plus(A,B3),C2))
        <=> aa(A,$o,ord_less(A,A3),B3) ) ) ).

% add_less_cancel_right
tff(fact_371_add__numeral__left,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [V2: num,W: num,Z: A] : aa(A,A,plus_plus(A,aa(num,A,numeral_numeral(A),V2)),aa(A,A,plus_plus(A,aa(num,A,numeral_numeral(A),W)),Z)) = aa(A,A,plus_plus(A,aa(num,A,numeral_numeral(A),aa(num,num,plus_plus(num,V2),W))),Z) ) ).

% add_numeral_left
tff(fact_372_numeral__plus__numeral,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [M: num,Na: num] : aa(A,A,plus_plus(A,aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Na)) = aa(num,A,numeral_numeral(A),aa(num,num,plus_plus(num,M),Na)) ) ).

% numeral_plus_numeral
tff(fact_373_semiring__norm_I2_J,axiom,
    aa(num,num,plus_plus(num,one2),one2) = bit0(one2) ).

% semiring_norm(2)
tff(fact_374_semiring__norm_I69_J,axiom,
    ! [M: num] : ~ aa(num,$o,ord_less_eq(num,bit0(M)),one2) ).

% semiring_norm(69)
tff(fact_375_neg__equal__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( ( aa(A,A,uminus_uminus(A),A3) = A3 )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% neg_equal_zero
tff(fact_376_equal__neg__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( ( A3 = aa(A,A,uminus_uminus(A),A3) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% equal_neg_zero
tff(fact_377_neg__equal__0__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] :
          ( ( aa(A,A,uminus_uminus(A),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% neg_equal_0_iff_equal
tff(fact_378_neg__0__equal__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] :
          ( ( zero_zero(A) = aa(A,A,uminus_uminus(A),A3) )
        <=> ( zero_zero(A) = A3 ) ) ) ).

% neg_0_equal_iff_equal
tff(fact_379_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_380_neg__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),B3)),aa(A,A,uminus_uminus(A),A3))
        <=> aa(A,$o,ord_less_eq(A,A3),B3) ) ) ).

% neg_le_iff_le
tff(fact_381_compl__le__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y))
        <=> aa(A,$o,ord_less_eq(A,Y),X) ) ) ).

% compl_le_compl_iff
tff(fact_382_compl__less__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y))
        <=> aa(A,$o,ord_less(A,Y),X) ) ) ).

% compl_less_compl_iff
tff(fact_383_neg__less__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),B3)),aa(A,A,uminus_uminus(A),A3))
        <=> aa(A,$o,ord_less(A,A3),B3) ) ) ).

% neg_less_iff_less
tff(fact_384_neg__numeral__eq__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [M: num,Na: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na)) )
        <=> ( M = Na ) ) ) ).

% neg_numeral_eq_iff
tff(fact_385_add__minus__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] : aa(A,A,plus_plus(A,A3),aa(A,A,plus_plus(A,aa(A,A,uminus_uminus(A),A3)),B3)) = B3 ) ).

% add_minus_cancel
tff(fact_386_minus__add__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] : aa(A,A,plus_plus(A,aa(A,A,uminus_uminus(A),A3)),aa(A,A,plus_plus(A,A3),B3)) = B3 ) ).

% minus_add_cancel
tff(fact_387_minus__add__distrib,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,B3: A] : aa(A,A,uminus_uminus(A),aa(A,A,plus_plus(A,A3),B3)) = aa(A,A,plus_plus(A,aa(A,A,uminus_uminus(A),A3)),aa(A,A,uminus_uminus(A),B3)) ) ).

% minus_add_distrib
tff(fact_388_of__nat__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,Na: nat] :
          ( aa(A,$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,ord_less_eq(nat,M),Na) ) ) ).

% of_nat_le_iff
tff(fact_389_le__sup__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A,Z: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),Z)
        <=> ( aa(A,$o,ord_less_eq(A,X),Z)
            & aa(A,$o,ord_less_eq(A,Y),Z) ) ) ) ).

% le_sup_iff
tff(fact_390_sup_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B3: A,C2: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),sup_sup(A),B3),C2)),A3)
        <=> ( aa(A,$o,ord_less_eq(A,B3),A3)
            & aa(A,$o,ord_less_eq(A,C2),A3) ) ) ) ).

% sup.bounded_iff
tff(fact_391_add__gr__0,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(nat,nat,plus_plus(nat,M),Na))
    <=> ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
        | aa(nat,$o,ord_less(nat,zero_zero(nat)),Na) ) ) ).

% add_gr_0
tff(fact_392_of__nat__add,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat,Na: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,plus_plus(nat,M),Na)) = aa(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_393_subset__Compl__singleton,axiom,
    ! [A: $tType,A2: set(A),B3: A] :
      ( aa(set(A),$o,ord_less_eq(set(A),A2),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),insert(A,B3),bot_bot(set(A)))))
    <=> ~ member(A,B3,A2) ) ).

% subset_Compl_singleton
tff(fact_394_singleton__insert__inj__eq,axiom,
    ! [A: $tType,B3: A,A3: A,A2: set(A)] :
      ( ( aa(set(A),set(A),insert(A,B3),bot_bot(set(A))) = aa(set(A),set(A),insert(A,A3),A2) )
    <=> ( ( A3 = B3 )
        & aa(set(A),$o,ord_less_eq(set(A),A2),aa(set(A),set(A),insert(A,B3),bot_bot(set(A)))) ) ) ).

% singleton_insert_inj_eq
tff(fact_395_singleton__insert__inj__eq_H,axiom,
    ! [A: $tType,A3: A,A2: set(A),B3: A] :
      ( ( aa(set(A),set(A),insert(A,A3),A2) = aa(set(A),set(A),insert(A,B3),bot_bot(set(A))) )
    <=> ( ( A3 = B3 )
        & aa(set(A),$o,ord_less_eq(set(A),A2),aa(set(A),set(A),insert(A,B3),bot_bot(set(A)))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_396_Word_Oof__nat__unat,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & semiring_1(A) )
     => ! [W: word(B)] : aa(nat,A,semiring_1_of_nat(A),aa(word(B),nat,semiring_1_unsigned(B,nat),W)) = aa(word(B),A,semiring_1_unsigned(B,A),W) ) ).

% Word.of_nat_unat
tff(fact_397_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_398_zle__add1__eq__le,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,ord_less(int,W),aa(int,int,plus_plus(int,Z),one_one(int)))
    <=> aa(int,$o,ord_less_eq(int,W),Z) ) ).

% zle_add1_eq_le
tff(fact_399_uint__nonnegative,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(word(A),int,semiring_1_unsigned(A,int),W)) ) ).

% uint_nonnegative
tff(fact_400_uint__le__0__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( aa(int,$o,ord_less_eq(int,aa(word(A),int,semiring_1_unsigned(A,int),X)),zero_zero(int))
        <=> ( aa(word(A),int,semiring_1_unsigned(A,int),X) = zero_zero(int) ) ) ) ).

% uint_le_0_iff
tff(fact_401_uint__ge__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(word(A),int,semiring_1_unsigned(A,int),X)) ) ).

% uint_ge_0
tff(fact_402_word__le__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(num,word(A),numeral_numeral(word(A)),B3))
        <=> aa(int,$o,ord_less_eq(int,aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),A3))),aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),B3))) ) ) ).

% word_le_no
tff(fact_403_shiftl__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : bit_Sh4282982442137083160shiftl(A,zero_zero(A),Na) = zero_zero(A) ) ).

% shiftl_0
tff(fact_404_pure__true,axiom,
    pure_assn($true) = one_one(assn) ).

% pure_true
tff(fact_405_pure__assn__eq__emp__iff,axiom,
    ! [P: $o] :
      ( ( pure_assn((P)) = one_one(assn) )
    <=> (P) ) ).

% pure_assn_eq_emp_iff
tff(fact_406_shiftl__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : bit_Sh4282982442137083160shiftl(A,A3,zero_zero(nat)) = A3 ) ).

% shiftl_of_0
tff(fact_407_pure__false,axiom,
    pure_assn($false) = bot_bot(assn) ).

% pure_false
tff(fact_408_pure__assn__eq__false__iff,axiom,
    ! [P: $o] :
      ( ( pure_assn((P)) = bot_bot(assn) )
    <=> ~ (P) ) ).

% pure_assn_eq_false_iff
tff(fact_409_enat__ord__number_I1_J,axiom,
    ! [M: num,Na: num] :
      ( aa(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,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),Na)) ) ).

% enat_ord_number(1)
tff(fact_410_add__le__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,plus_plus(A,B3),A3)),B3)
        <=> aa(A,$o,ord_less_eq(A,A3),zero_zero(A)) ) ) ).

% add_le_same_cancel1
tff(fact_411_add__le__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,plus_plus(A,A3),B3)),B3)
        <=> aa(A,$o,ord_less_eq(A,A3),zero_zero(A)) ) ) ).

% add_le_same_cancel2
tff(fact_412_le__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),aa(A,A,plus_plus(A,A3),B3))
        <=> aa(A,$o,ord_less_eq(A,zero_zero(A)),B3) ) ) ).

% le_add_same_cancel1
tff(fact_413_le__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),aa(A,A,plus_plus(A,B3),A3))
        <=> aa(A,$o,ord_less_eq(A,zero_zero(A)),B3) ) ) ).

% le_add_same_cancel2
tff(fact_414_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,plus_plus(A,A3),A3)),zero_zero(A))
        <=> aa(A,$o,ord_less_eq(A,A3),zero_zero(A)) ) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_415_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,plus_plus(A,A3),A3))
        <=> aa(A,$o,ord_less_eq(A,zero_zero(A)),A3) ) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_416_add__less__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,plus_plus(A,B3),A3)),B3)
        <=> aa(A,$o,ord_less(A,A3),zero_zero(A)) ) ) ).

% add_less_same_cancel1
tff(fact_417_add__less__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,plus_plus(A,A3),B3)),B3)
        <=> aa(A,$o,ord_less(A,A3),zero_zero(A)) ) ) ).

% add_less_same_cancel2
tff(fact_418_less__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),aa(A,A,plus_plus(A,A3),B3))
        <=> aa(A,$o,ord_less(A,zero_zero(A)),B3) ) ) ).

% less_add_same_cancel1
tff(fact_419_less__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),aa(A,A,plus_plus(A,B3),A3))
        <=> aa(A,$o,ord_less(A,zero_zero(A)),B3) ) ) ).

% less_add_same_cancel2
tff(fact_420_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,plus_plus(A,A3),A3)),zero_zero(A))
        <=> aa(A,$o,ord_less(A,A3),zero_zero(A)) ) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_421_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,plus_plus(A,A3),A3))
        <=> aa(A,$o,ord_less(A,zero_zero(A)),A3) ) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_422_neg__0__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,uminus_uminus(A),A3))
        <=> aa(A,$o,ord_less_eq(A,A3),zero_zero(A)) ) ) ).

% neg_0_le_iff_le
tff(fact_423_neg__le__0__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),A3)),zero_zero(A))
        <=> aa(A,$o,ord_less_eq(A,zero_zero(A)),A3) ) ) ).

% neg_le_0_iff_le
tff(fact_424_less__eq__neg__nonpos,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),aa(A,A,uminus_uminus(A),A3))
        <=> aa(A,$o,ord_less_eq(A,A3),zero_zero(A)) ) ) ).

% less_eq_neg_nonpos
tff(fact_425_neg__less__eq__nonneg,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),A3)),A3)
        <=> aa(A,$o,ord_less_eq(A,zero_zero(A)),A3) ) ) ).

% neg_less_eq_nonneg
tff(fact_426_less__neg__neg,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,A3),aa(A,A,uminus_uminus(A),A3))
        <=> aa(A,$o,ord_less(A,A3),zero_zero(A)) ) ) ).

% less_neg_neg
tff(fact_427_neg__less__pos,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),A3)),A3)
        <=> aa(A,$o,ord_less(A,zero_zero(A)),A3) ) ) ).

% neg_less_pos
tff(fact_428_neg__0__less__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,uminus_uminus(A),A3))
        <=> aa(A,$o,ord_less(A,A3),zero_zero(A)) ) ) ).

% neg_0_less_iff_less
tff(fact_429_neg__less__0__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),A3)),zero_zero(A))
        <=> aa(A,$o,ord_less(A,zero_zero(A)),A3) ) ) ).

% neg_less_0_iff_less
tff(fact_430_neg__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Na: num] :
          ( aa(A,$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,ord_less_eq(num,Na),M) ) ) ).

% neg_numeral_le_iff
tff(fact_431_add_Oright__inverse,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,plus_plus(A,A3),aa(A,A,uminus_uminus(A),A3)) = zero_zero(A) ) ).

% add.right_inverse
tff(fact_432_ab__left__minus,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,plus_plus(A,aa(A,A,uminus_uminus(A),A3)),A3) = zero_zero(A) ) ).

% ab_left_minus
tff(fact_433_semiring__norm_I168_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [V2: num,W: num,Y: A] : aa(A,A,plus_plus(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),aa(A,A,plus_plus(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),Y)) = aa(A,A,plus_plus(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,plus_plus(num,V2),W)))),Y) ) ).

% semiring_norm(168)
tff(fact_434_add__neg__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,Na: num] : aa(A,A,plus_plus(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))) = aa(A,A,uminus_uminus(A),aa(A,A,plus_plus(A,aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Na))) ) ).

% add_neg_numeral_simps(3)
tff(fact_435_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_436_unsigned__1,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semiring_1(A)
        & type_len(B) )
     => ( aa(word(B),A,semiring_1_unsigned(B,A),one_one(word(B))) = one_one(A) ) ) ).

% unsigned_1
tff(fact_437_numeral__le__real__of__nat__iff,axiom,
    ! [Na: num,M: nat] :
      ( aa(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,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),Na)),M) ) ).

% numeral_le_real_of_nat_iff
tff(fact_438_uint__lt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : ~ aa(int,$o,ord_less(int,aa(word(A),int,semiring_1_unsigned(A,int),X)),zero_zero(int)) ) ).

% uint_lt_0
tff(fact_439_log__one,axiom,
    ! [A3: real] : aa(real,real,log(A3),one_one(real)) = zero_zero(real) ).

% log_one
tff(fact_440_norm__pre__pure__iff__sng,axiom,
    ! [A: $tType,B3: $o,F2: heap_Time_Heap(A),Q: fun(A,assn)] :
      ( hoare_hoare_triple(A,pure_assn((B3)),F2,Q)
    <=> ( (B3)
       => hoare_hoare_triple(A,one_one(assn),F2,Q) ) ) ).

% norm_pre_pure_iff_sng
tff(fact_441_dbl__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_dbl(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K))) = aa(A,A,uminus_uminus(A),neg_numeral_dbl(A,aa(num,A,numeral_numeral(A),K))) ) ).

% dbl_simps(1)
tff(fact_442_dbl__inc__simps_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl_inc(A,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% dbl_inc_simps(4)
tff(fact_443_numeral__le__one__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: num] :
          ( aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),Na)),one_one(A))
        <=> aa(num,$o,ord_less_eq(num,Na),one2) ) ) ).

% numeral_le_one_iff
tff(fact_444_numeral__plus__one,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Na: num] : aa(A,A,plus_plus(A,aa(num,A,numeral_numeral(A),Na)),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,plus_plus(num,Na),one2)) ) ).

% numeral_plus_one
tff(fact_445_one__plus__numeral,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Na: num] : aa(A,A,plus_plus(A,one_one(A)),aa(num,A,numeral_numeral(A),Na)) = aa(num,A,numeral_numeral(A),aa(num,num,plus_plus(num,one2),Na)) ) ).

% one_plus_numeral
tff(fact_446_power__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B3: A,X: nat,Y: nat] :
          ( aa(A,$o,ord_less(A,one_one(A)),B3)
         => ( aa(A,$o,ord_less_eq(A,aa(nat,A,power_power(A,B3),X)),aa(nat,A,power_power(A,B3),Y))
          <=> aa(nat,$o,ord_less_eq(nat,X),Y) ) ) ) ).

% power_increasing_iff
tff(fact_447_add__neg__numeral__special_I8_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(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_448_add__neg__numeral__special_I7_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(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_449_neg__one__eq__numeral__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Na: num] :
          ( ( aa(A,A,uminus_uminus(A),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na)) )
        <=> ( Na = one2 ) ) ) ).

% neg_one_eq_numeral_iff
tff(fact_450_numeral__eq__neg__one__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Na: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na)) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> ( Na = one2 ) ) ) ).

% numeral_eq_neg_one_iff
tff(fact_451_of__nat__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat] :
          ( aa(A,$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_452_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B3: nat,W: nat,X: nat] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),B3)),W)),aa(nat,A,semiring_1_of_nat(A),X))
        <=> aa(nat,$o,ord_less_eq(nat,aa(nat,nat,power_power(nat,B3),W)),X) ) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_453_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,B3: nat,W: nat] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),B3)),W))
        <=> aa(nat,$o,ord_less_eq(nat,X),aa(nat,nat,power_power(nat,B3),W)) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_454_log__le__cancel__iff,axiom,
    ! [A3: real,X: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),X)
       => ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
         => ( aa(real,$o,ord_less_eq(real,aa(real,real,log(A3),X)),aa(real,real,log(A3),Y))
          <=> aa(real,$o,ord_less_eq(real,X),Y) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_455_log__le__one__cancel__iff,axiom,
    ! [A3: real,X: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),X)
       => ( aa(real,$o,ord_less_eq(real,aa(real,real,log(A3),X)),one_one(real))
        <=> aa(real,$o,ord_less_eq(real,X),A3) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_456_one__le__log__cancel__iff,axiom,
    ! [A3: real,X: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),X)
       => ( aa(real,$o,ord_less_eq(real,one_one(real)),aa(real,real,log(A3),X))
        <=> aa(real,$o,ord_less_eq(real,A3),X) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_457_log__le__zero__cancel__iff,axiom,
    ! [A3: real,X: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),X)
       => ( aa(real,$o,ord_less_eq(real,aa(real,real,log(A3),X)),zero_zero(real))
        <=> aa(real,$o,ord_less_eq(real,X),one_one(real)) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_458_zero__le__log__cancel__iff,axiom,
    ! [A3: real,X: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),X)
       => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,log(A3),X))
        <=> aa(real,$o,ord_less_eq(real,one_one(real)),X) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_459_zero__less__log__cancel__iff,axiom,
    ! [A3: real,X: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),X)
       => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,log(A3),X))
        <=> aa(real,$o,ord_less(real,one_one(real)),X) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_460_log__less__zero__cancel__iff,axiom,
    ! [A3: real,X: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),X)
       => ( aa(real,$o,ord_less(real,aa(real,real,log(A3),X)),zero_zero(real))
        <=> aa(real,$o,ord_less(real,X),one_one(real)) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_461_one__less__log__cancel__iff,axiom,
    ! [A3: real,X: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),X)
       => ( aa(real,$o,ord_less(real,one_one(real)),aa(real,real,log(A3),X))
        <=> aa(real,$o,ord_less(real,A3),X) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_462_log__less__one__cancel__iff,axiom,
    ! [A3: real,X: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),X)
       => ( aa(real,$o,ord_less(real,aa(real,real,log(A3),X)),one_one(real))
        <=> aa(real,$o,ord_less(real,X),A3) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_463_log__less__cancel__iff,axiom,
    ! [A3: real,X: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),X)
       => ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
         => ( aa(real,$o,ord_less(real,aa(real,real,log(A3),X)),aa(real,real,log(A3),Y))
          <=> aa(real,$o,ord_less(real,X),Y) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_464_log__eq__one,axiom,
    ! [A3: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
     => ( ( A3 != one_one(real) )
       => ( aa(real,real,log(A3),A3) = one_one(real) ) ) ) ).

% log_eq_one
tff(fact_465_word__less__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( aa(word(A),$o,ord_less(word(A),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(num,word(A),numeral_numeral(word(A)),B3))
        <=> aa(int,$o,ord_less(int,aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),A3))),aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),B3))) ) ) ).

% word_less_no
tff(fact_466_neg__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Na: num] :
          ( aa(A,$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,ord_less(num,Na),M) ) ) ).

% neg_numeral_less_iff
tff(fact_467_power__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B3: A,M: nat,Na: nat] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),B3)
         => ( aa(A,$o,ord_less(A,B3),one_one(A))
           => ( aa(A,$o,ord_less_eq(A,aa(nat,A,power_power(A,B3),M)),aa(nat,A,power_power(A,B3),Na))
            <=> aa(nat,$o,ord_less_eq(nat,Na),M) ) ) ) ) ).

% power_decreasing_iff
tff(fact_468_one__add__one,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ( aa(A,A,plus_plus(A,one_one(A)),one_one(A)) = aa(num,A,numeral_numeral(A),bit0(one2)) ) ) ).

% one_add_one
tff(fact_469_not__neg__one__le__neg__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] :
          ( ~ aa(A,$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_470_neg__numeral__less__neg__one__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] :
          ( aa(A,$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_471_power__mono__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,B3: A,Na: nat] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B3)
           => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
             => ( aa(A,$o,ord_less_eq(A,aa(nat,A,power_power(A,A3),Na)),aa(nat,A,power_power(A,B3),Na))
              <=> aa(A,$o,ord_less_eq(A,A3),B3) ) ) ) ) ) ).

% power_mono_iff
tff(fact_472_power2__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A] : aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),A3)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% power2_minus
tff(fact_473_add__neg__numeral__special_I9_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,plus_plus(A,aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% add_neg_numeral_special(9)
tff(fact_474_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: num,Na: nat,X: nat] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),I)),Na)),aa(nat,A,semiring_1_of_nat(A),X))
        <=> aa(nat,$o,ord_less_eq(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),I)),Na)),X) ) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_475_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,I: num,Na: nat] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),I)),Na))
        <=> aa(nat,$o,ord_less_eq(nat,X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),I)),Na)) ) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_476_log__pow__cancel,axiom,
    ! [A3: real,B3: nat] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
     => ( ( A3 != one_one(real) )
       => ( aa(real,real,log(A3),aa(nat,real,power_power(real,A3),B3)) = aa(nat,real,semiring_1_of_nat(real),B3) ) ) ) ).

% log_pow_cancel
tff(fact_477_no__plus__overflow__neg,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),X),aa(word(A),word(A),uminus_uminus(word(A)),Y))
         => aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),plus_plus(word(A),X),Y)) ) ) ).

% no_plus_overflow_neg
tff(fact_478_overflow__plus__one__self,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),plus_plus(word(A),one_one(word(A))),P2)),P2)
        <=> ( P2 = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% overflow_plus_one_self
tff(fact_479_real__add__le__0__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,plus_plus(real,X),Y)),zero_zero(real))
    <=> aa(real,$o,ord_less_eq(real,Y),aa(real,real,uminus_uminus(real),X)) ) ).

% real_add_le_0_iff
tff(fact_480_real__0__le__add__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,plus_plus(real,X),Y))
    <=> aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),X)),Y) ) ).

% real_0_le_add_iff
tff(fact_481_plus__1__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),plus_plus(word(A),X),one_one(word(A)))),X)
        <=> ( X = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% plus_1_less
tff(fact_482_uint__add__ge0,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xa2: word(A),X: word(B)] : aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,plus_plus(int,aa(word(A),int,semiring_1_unsigned(A,int),Xa2)),aa(word(B),int,semiring_1_unsigned(B,int),X))) ) ).

% uint_add_ge0
tff(fact_483_nle__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( ~ aa(A,$o,ord_less_eq(A,A3),B3)
        <=> ( aa(A,$o,ord_less_eq(A,B3),A3)
            & ( B3 != A3 ) ) ) ) ).

% nle_le
tff(fact_484_le__cases3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Z: A] :
          ( ( aa(A,$o,ord_less_eq(A,X),Y)
           => ~ aa(A,$o,ord_less_eq(A,Y),Z) )
         => ( ( aa(A,$o,ord_less_eq(A,Y),X)
             => ~ aa(A,$o,ord_less_eq(A,X),Z) )
           => ( ( aa(A,$o,ord_less_eq(A,X),Z)
               => ~ aa(A,$o,ord_less_eq(A,Z),Y) )
             => ( ( aa(A,$o,ord_less_eq(A,Z),Y)
                 => ~ aa(A,$o,ord_less_eq(A,Y),X) )
               => ( ( aa(A,$o,ord_less_eq(A,Y),Z)
                   => ~ aa(A,$o,ord_less_eq(A,Z),X) )
                 => ~ ( aa(A,$o,ord_less_eq(A,Z),X)
                     => ~ aa(A,$o,ord_less_eq(A,X),Y) ) ) ) ) ) ) ) ).

% le_cases3
tff(fact_485_order__class_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( ( X = Y )
        <=> ( aa(A,$o,ord_less_eq(A,X),Y)
            & aa(A,$o,ord_less_eq(A,Y),X) ) ) ) ).

% order_class.order_eq_iff
tff(fact_486_ord__eq__le__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B3: A,C2: A] :
          ( ( A3 = B3 )
         => ( aa(A,$o,ord_less_eq(A,B3),C2)
           => aa(A,$o,ord_less_eq(A,A3),C2) ) ) ) ).

% ord_eq_le_trans
tff(fact_487_ord__le__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( ( B3 = C2 )
           => aa(A,$o,ord_less_eq(A,A3),C2) ) ) ) ).

% ord_le_eq_trans
tff(fact_488_order__antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,X),Y)
         => ( aa(A,$o,ord_less_eq(A,Y),X)
           => ( X = Y ) ) ) ) ).

% order_antisym
tff(fact_489_order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,B3),C2)
           => aa(A,$o,ord_less_eq(A,A3),C2) ) ) ) ).

% order.trans
tff(fact_490_order__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,Z: A] :
          ( aa(A,$o,ord_less_eq(A,X),Y)
         => ( aa(A,$o,ord_less_eq(A,Y),Z)
           => aa(A,$o,ord_less_eq(A,X),Z) ) ) ) ).

% order_trans
tff(fact_491_linorder__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,fun(A,$o)),A3: A,B3: A] :
          ( ! [A4: A,B5: A] :
              ( aa(A,$o,ord_less_eq(A,A4),B5)
             => aa(A,$o,aa(A,fun(A,$o),P,A4),B5) )
         => ( ! [A4: A,B5: A] :
                ( aa(A,$o,aa(A,fun(A,$o),P,B5),A4)
               => aa(A,$o,aa(A,fun(A,$o),P,A4),B5) )
           => aa(A,$o,aa(A,fun(A,$o),P,A3),B3) ) ) ) ).

% linorder_wlog
tff(fact_492_dual__order_Oeq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = B3 )
        <=> ( aa(A,$o,ord_less_eq(A,B3),A3)
            & aa(A,$o,ord_less_eq(A,A3),B3) ) ) ) ).

% dual_order.eq_iff
tff(fact_493_dual__order_Oantisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
         => ( aa(A,$o,ord_less_eq(A,A3),B3)
           => ( A3 = B3 ) ) ) ) ).

% dual_order.antisym
tff(fact_494_dual__order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
         => ( aa(A,$o,ord_less_eq(A,C2),B3)
           => aa(A,$o,ord_less_eq(A,C2),A3) ) ) ) ).

% dual_order.trans
tff(fact_495_antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,B3),A3)
           => ( A3 = B3 ) ) ) ) ).

% antisym
tff(fact_496_le__funD,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B),X: A] :
          ( aa(fun(A,B),$o,ord_less_eq(fun(A,B),F2),G)
         => aa(B,$o,ord_less_eq(B,aa(A,B,F2,X)),aa(A,B,G,X)) ) ) ).

% le_funD
tff(fact_497_le__funE,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B),X: A] :
          ( aa(fun(A,B),$o,ord_less_eq(fun(A,B),F2),G)
         => aa(B,$o,ord_less_eq(B,aa(A,B,F2,X)),aa(A,B,G,X)) ) ) ).

% le_funE
tff(fact_498_le__funI,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( ! [X4: A] : aa(B,$o,ord_less_eq(B,aa(A,B,F2,X4)),aa(A,B,G,X4))
         => aa(fun(A,B),$o,ord_less_eq(fun(A,B),F2),G) ) ) ).

% le_funI
tff(fact_499_le__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( aa(fun(A,B),$o,ord_less_eq(fun(A,B),F2),G)
        <=> ! [X3: A] : aa(B,$o,ord_less_eq(B,aa(A,B,F2,X3)),aa(A,B,G,X3)) ) ) ).

% le_fun_def
tff(fact_500_Orderings_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = B3 )
        <=> ( aa(A,$o,ord_less_eq(A,A3),B3)
            & aa(A,$o,ord_less_eq(A,B3),A3) ) ) ) ).

% Orderings.order_eq_iff
tff(fact_501_order__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,F2: fun(B,A),B3: B,C2: B] :
          ( aa(A,$o,ord_less_eq(A,A3),aa(B,A,F2,B3))
         => ( aa(B,$o,ord_less_eq(B,B3),C2)
           => ( ! [X4: B,Y3: B] :
                  ( aa(B,$o,ord_less_eq(B,X4),Y3)
                 => aa(A,$o,ord_less_eq(A,aa(B,A,F2,X4)),aa(B,A,F2,Y3)) )
             => aa(A,$o,ord_less_eq(A,A3),aa(B,A,F2,C2)) ) ) ) ) ).

% order_subst1
tff(fact_502_order__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,B3: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(B,$o,ord_less_eq(B,aa(A,B,F2,B3)),C2)
           => ( ! [X4: A,Y3: A] :
                  ( aa(A,$o,ord_less_eq(A,X4),Y3)
                 => aa(B,$o,ord_less_eq(B,aa(A,B,F2,X4)),aa(A,B,F2,Y3)) )
             => aa(B,$o,ord_less_eq(B,aa(A,B,F2,A3)),C2) ) ) ) ) ).

% order_subst2
tff(fact_503_order__eq__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( ( X = Y )
         => aa(A,$o,ord_less_eq(A,X),Y) ) ) ).

% order_eq_refl
tff(fact_504_linorder__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,X),Y)
          | aa(A,$o,ord_less_eq(A,Y),X) ) ) ).

% linorder_linear
tff(fact_505_ord__eq__le__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A3: A,F2: fun(B,A),B3: B,C2: B] :
          ( ( A3 = aa(B,A,F2,B3) )
         => ( aa(B,$o,ord_less_eq(B,B3),C2)
           => ( ! [X4: B,Y3: B] :
                  ( aa(B,$o,ord_less_eq(B,X4),Y3)
                 => aa(A,$o,ord_less_eq(A,aa(B,A,F2,X4)),aa(B,A,F2,Y3)) )
             => aa(A,$o,ord_less_eq(A,A3),aa(B,A,F2,C2)) ) ) ) ) ).

% ord_eq_le_subst
tff(fact_506_ord__le__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A3: A,B3: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( ( aa(A,B,F2,B3) = C2 )
           => ( ! [X4: A,Y3: A] :
                  ( aa(A,$o,ord_less_eq(A,X4),Y3)
                 => aa(B,$o,ord_less_eq(B,aa(A,B,F2,X4)),aa(A,B,F2,Y3)) )
             => aa(B,$o,ord_less_eq(B,aa(A,B,F2,A3)),C2) ) ) ) ) ).

% ord_le_eq_subst
tff(fact_507_linorder__le__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ aa(A,$o,ord_less_eq(A,X),Y)
         => aa(A,$o,ord_less_eq(A,Y),X) ) ) ).

% linorder_le_cases
tff(fact_508_order__antisym__conv,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,ord_less_eq(A,Y),X)
         => ( aa(A,$o,ord_less_eq(A,X),Y)
          <=> ( X = Y ) ) ) ) ).

% order_antisym_conv
tff(fact_509_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,plus_plus(A,I),K) = aa(A,A,plus_plus(A,J),L) ) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_510_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,ord_less_eq(A,I),J)
            & ( K = L ) )
         => aa(A,$o,ord_less_eq(A,aa(A,A,plus_plus(A,I),K)),aa(A,A,plus_plus(A,J),L)) ) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_511_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,ord_less_eq(A,K),L) )
         => aa(A,$o,ord_less_eq(A,aa(A,A,plus_plus(A,I),K)),aa(A,A,plus_plus(A,J),L)) ) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_512_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,ord_less_eq(A,I),J)
            & aa(A,$o,ord_less_eq(A,K),L) )
         => aa(A,$o,ord_less_eq(A,aa(A,A,plus_plus(A,I),K)),aa(A,A,plus_plus(A,J),L)) ) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_513_group__cancel_Oadd1,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: A,K: A,A3: A,B3: A] :
          ( ( A2 = aa(A,A,plus_plus(A,K),A3) )
         => ( aa(A,A,plus_plus(A,A2),B3) = aa(A,A,plus_plus(A,K),aa(A,A,plus_plus(A,A3),B3)) ) ) ) ).

% group_cancel.add1
tff(fact_514_group__cancel_Oadd2,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [B2: A,K: A,B3: A,A3: A] :
          ( ( B2 = aa(A,A,plus_plus(A,K),B3) )
         => ( aa(A,A,plus_plus(A,A3),B2) = aa(A,A,plus_plus(A,K),aa(A,A,plus_plus(A,A3),B3)) ) ) ) ).

% group_cancel.add2
tff(fact_515_group__cancel_Oneg1,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,K: A,A3: A] :
          ( ( A2 = aa(A,A,plus_plus(A,K),A3) )
         => ( aa(A,A,uminus_uminus(A),A2) = aa(A,A,plus_plus(A,aa(A,A,uminus_uminus(A),K)),aa(A,A,uminus_uminus(A),A3)) ) ) ) ).

% group_cancel.neg1
tff(fact_516_add_Oassoc,axiom,
    ! [A: $tType] :
      ( semigroup_add(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,plus_plus(A,aa(A,A,plus_plus(A,A3),B3)),C2) = aa(A,A,plus_plus(A,A3),aa(A,A,plus_plus(A,B3),C2)) ) ).

% add.assoc
tff(fact_517_add_Oleft__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( ( aa(A,A,plus_plus(A,A3),B3) = aa(A,A,plus_plus(A,A3),C2) )
        <=> ( B3 = C2 ) ) ) ).

% add.left_cancel
tff(fact_518_add_Oright__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [B3: A,A3: A,C2: A] :
          ( ( aa(A,A,plus_plus(A,B3),A3) = aa(A,A,plus_plus(A,C2),A3) )
        <=> ( B3 = C2 ) ) ) ).

% add.right_cancel
tff(fact_519_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [A3: A,B3: A] : aa(A,A,plus_plus(A,A3),B3) = aa(A,A,plus_plus(A,B3),A3) ) ).

% ab_semigroup_add_class.add.commute
tff(fact_520_equation__minus__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = aa(A,A,uminus_uminus(A),B3) )
        <=> ( B3 = aa(A,A,uminus_uminus(A),A3) ) ) ) ).

% equation_minus_iff
tff(fact_521_minus__equation__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,uminus_uminus(A),A3) = B3 )
        <=> ( aa(A,A,uminus_uminus(A),B3) = A3 ) ) ) ).

% minus_equation_iff
tff(fact_522_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [B3: A,A3: A,C2: A] : aa(A,A,plus_plus(A,B3),aa(A,A,plus_plus(A,A3),C2)) = aa(A,A,plus_plus(A,A3),aa(A,A,plus_plus(A,B3),C2)) ) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_523_le__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),aa(A,A,uminus_uminus(A),B3))
        <=> aa(A,$o,ord_less_eq(A,B3),aa(A,A,uminus_uminus(A),A3)) ) ) ).

% le_minus_iff
tff(fact_524_minus__le__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),A3)),B3)
        <=> aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),B3)),A3) ) ) ).

% minus_le_iff
tff(fact_525_add__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A3: A,B3: A,C2: A,D2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,C2),D2)
           => aa(A,$o,ord_less_eq(A,aa(A,A,plus_plus(A,A3),C2)),aa(A,A,plus_plus(A,B3),D2)) ) ) ) ).

% add_mono
tff(fact_526_add_Oinverse__distrib__swap,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] : aa(A,A,uminus_uminus(A),aa(A,A,plus_plus(A,A3),B3)) = aa(A,A,plus_plus(A,aa(A,A,uminus_uminus(A),B3)),aa(A,A,uminus_uminus(A),A3)) ) ).

% add.inverse_distrib_swap
tff(fact_527_le__imp__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),B3)),aa(A,A,uminus_uminus(A),A3)) ) ) ).

% le_imp_neg_le
tff(fact_528_add__left__imp__eq,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( ( aa(A,A,plus_plus(A,A3),B3) = aa(A,A,plus_plus(A,A3),C2) )
         => ( B3 = C2 ) ) ) ).

% add_left_imp_eq
tff(fact_529_add__right__imp__eq,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [B3: A,A3: A,C2: A] :
          ( ( aa(A,A,plus_plus(A,B3),A3) = aa(A,A,plus_plus(A,C2),A3) )
         => ( B3 = C2 ) ) ) ).

% add_right_imp_eq
tff(fact_530_add__left__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => aa(A,$o,ord_less_eq(A,aa(A,A,plus_plus(A,C2),A3)),aa(A,A,plus_plus(A,C2),B3)) ) ) ).

% add_left_mono
tff(fact_531_less__eqE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ~ ! [C5: A] : B3 != aa(A,A,plus_plus(A,A3),C5) ) ) ).

% less_eqE
tff(fact_532_add__right__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => aa(A,$o,ord_less_eq(A,aa(A,A,plus_plus(A,A3),C2)),aa(A,A,plus_plus(A,B3),C2)) ) ) ).

% add_right_mono
tff(fact_533_le__iff__add,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
        <=> ? [C6: A] : B3 = aa(A,A,plus_plus(A,A3),C6) ) ) ).

% le_iff_add
tff(fact_534_add__le__imp__le__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,plus_plus(A,C2),A3)),aa(A,A,plus_plus(A,C2),B3))
         => aa(A,$o,ord_less_eq(A,A3),B3) ) ) ).

% add_le_imp_le_left
tff(fact_535_add__le__imp__le__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,plus_plus(A,A3),C2)),aa(A,A,plus_plus(A,B3),C2))
         => aa(A,$o,ord_less_eq(A,A3),B3) ) ) ).

% add_le_imp_le_right
tff(fact_536_is__num__normalize_I8_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [A3: A,B3: A] : aa(A,A,uminus_uminus(A),aa(A,A,plus_plus(A,A3),B3)) = aa(A,A,plus_plus(A,aa(A,A,uminus_uminus(A),B3)),aa(A,A,uminus_uminus(A),A3)) ) ).

% is_num_normalize(8)
tff(fact_537_is__num__normalize_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,plus_plus(A,aa(A,A,plus_plus(A,A3),B3)),C2) = aa(A,A,plus_plus(A,A3),aa(A,A,plus_plus(A,B3),C2)) ) ).

% is_num_normalize(1)
tff(fact_538_fun__Compl__def,axiom,
    ! [B: $tType,A: $tType] :
      ( uminus(B)
     => ! [A2: fun(A,B),X2: A] : aa(A,B,aa(fun(A,B),fun(A,B),uminus_uminus(fun(A,B)),A2),X2) = aa(B,B,uminus_uminus(B),aa(A,B,A2,X2)) ) ).

% fun_Compl_def
tff(fact_539_verit__comp__simplify1_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A] : aa(A,$o,ord_less_eq(A,A3),A3) ) ).

% verit_comp_simplify1(2)
tff(fact_540_verit__negate__coefficient_I3_J,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = B3 )
         => ( aa(A,A,uminus_uminus(A),A3) = aa(A,A,uminus_uminus(A),B3) ) ) ) ).

% verit_negate_coefficient(3)
tff(fact_541_verit__la__disequality,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = B3 )
          | ~ aa(A,$o,ord_less_eq(A,A3),B3)
          | ~ aa(A,$o,ord_less_eq(A,B3),A3) ) ) ).

% verit_la_disequality
tff(fact_542_unsigned__word__eqI,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & semiring_char_0(A) )
     => ! [V2: word(B),W: word(B)] :
          ( ( aa(word(B),A,semiring_1_unsigned(B,A),V2) = aa(word(B),A,semiring_1_unsigned(B,A),W) )
         => ( V2 = W ) ) ) ).

% unsigned_word_eqI
tff(fact_543_word__eq__iff__unsigned,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & semiring_char_0(B) )
     => ! [V2: word(A),W: word(A)] :
          ( ( V2 = W )
        <=> ( aa(word(A),B,semiring_1_unsigned(A,B),V2) = aa(word(A),B,semiring_1_unsigned(A,B),W) ) ) ) ).

% word_eq_iff_unsigned
tff(fact_544_compl__mono,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,X),Y)
         => aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),Y)),aa(A,A,uminus_uminus(A),X)) ) ) ).

% compl_mono
tff(fact_545_compl__le__swap1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,ord_less_eq(A,Y),aa(A,A,uminus_uminus(A),X))
         => aa(A,$o,ord_less_eq(A,X),aa(A,A,uminus_uminus(A),Y)) ) ) ).

% compl_le_swap1
tff(fact_546_compl__le__swap2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),Y)),X)
         => aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),X)),Y) ) ) ).

% compl_le_swap2
tff(fact_547_word__less__eq__iff__unsigned,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & linordered_semidom(B) )
     => ! [A3: word(A),B3: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),A3),B3)
        <=> aa(B,$o,ord_less_eq(B,aa(word(A),B,semiring_1_unsigned(A,B),A3)),aa(word(A),B,semiring_1_unsigned(A,B),B3)) ) ) ).

% word_less_eq_iff_unsigned
tff(fact_548_uint__cong,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ( X = Y )
         => ( aa(word(A),int,semiring_1_unsigned(A,int),X) = aa(word(A),int,semiring_1_unsigned(A,int),Y) ) ) ) ).

% uint_cong
tff(fact_549_uint__add__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] : aa(int,$o,ord_less_eq(int,aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),plus_plus(word(A),X),Y))),aa(int,int,plus_plus(int,aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Y))) ) ).

% uint_add_le
tff(fact_550_word__le__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),A3),B3)
        <=> aa(int,$o,ord_less_eq(int,aa(word(A),int,semiring_1_unsigned(A,int),A3)),aa(word(A),int,semiring_1_unsigned(A,int),B3)) ) ) ).

% word_le_def
tff(fact_551_word__uint__eqI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          ( ( aa(word(A),int,semiring_1_unsigned(A,int),A3) = aa(word(A),int,semiring_1_unsigned(A,int),B3) )
         => ( A3 = B3 ) ) ) ).

% word_uint_eqI
tff(fact_552_uint__plus__simple,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),plus_plus(word(A),X),Y))
         => ( aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),plus_plus(word(A),X),Y)) = aa(int,int,plus_plus(int,aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ) ) ).

% uint_plus_simple
tff(fact_553_word__uint__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          ( ( A3 = B3 )
        <=> ( aa(word(A),int,semiring_1_unsigned(A,int),A3) = aa(word(A),int,semiring_1_unsigned(A,int),B3) ) ) ) ).

% word_uint_eq_iff
tff(fact_554_uint__plus__simple__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),plus_plus(word(A),X),Y))
        <=> ( aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),plus_plus(word(A),X),Y)) = aa(int,int,plus_plus(int,aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ) ) ).

% uint_plus_simple_iff
tff(fact_555_add__eq__0__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,plus_plus(A,A3),B3) = zero_zero(A) )
        <=> ( B3 = aa(A,A,uminus_uminus(A),A3) ) ) ) ).

% add_eq_0_iff
tff(fact_556_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A] : aa(A,A,plus_plus(A,aa(A,A,uminus_uminus(A),A3)),A3) = zero_zero(A) ) ).

% ab_group_add_class.ab_left_minus
tff(fact_557_add_Oinverse__unique,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,plus_plus(A,A3),B3) = zero_zero(A) )
         => ( aa(A,A,uminus_uminus(A),A3) = B3 ) ) ) ).

% add.inverse_unique
tff(fact_558_eq__neg__iff__add__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = aa(A,A,uminus_uminus(A),B3) )
        <=> ( aa(A,A,plus_plus(A,A3),B3) = zero_zero(A) ) ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_559_neg__eq__iff__add__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,uminus_uminus(A),A3) = B3 )
        <=> ( aa(A,A,plus_plus(A,A3),B3) = zero_zero(A) ) ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_560_not__numeral__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Na: num] : ~ aa(A,$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_561_neg__numeral__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Na: num] : aa(A,$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_562_add__decreasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),zero_zero(A))
         => ( aa(A,$o,ord_less_eq(A,C2),B3)
           => aa(A,$o,ord_less_eq(A,aa(A,A,plus_plus(A,A3),C2)),B3) ) ) ) ).

% add_decreasing
tff(fact_563_add__increasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less_eq(A,B3),C2)
           => aa(A,$o,ord_less_eq(A,B3),aa(A,A,plus_plus(A,A3),C2)) ) ) ) ).

% add_increasing
tff(fact_564_add__decreasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,C2),zero_zero(A))
         => ( aa(A,$o,ord_less_eq(A,A3),B3)
           => aa(A,$o,ord_less_eq(A,aa(A,A,plus_plus(A,A3),C2)),B3) ) ) ) ).

% add_decreasing2
tff(fact_565_add__increasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
         => ( aa(A,$o,ord_less_eq(A,B3),A3)
           => aa(A,$o,ord_less_eq(A,B3),aa(A,A,plus_plus(A,A3),C2)) ) ) ) ).

% add_increasing2
tff(fact_566_add__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B3)
           => aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,plus_plus(A,A3),B3)) ) ) ) ).

% add_nonneg_nonneg
tff(fact_567_add__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),zero_zero(A))
         => ( aa(A,$o,ord_less_eq(A,B3),zero_zero(A))
           => aa(A,$o,ord_less_eq(A,aa(A,A,plus_plus(A,A3),B3)),zero_zero(A)) ) ) ) ).

% add_nonpos_nonpos
tff(fact_568_add__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),X)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Y)
           => ( ( aa(A,A,plus_plus(A,X),Y) = zero_zero(A) )
            <=> ( ( X = zero_zero(A) )
                & ( Y = zero_zero(A) ) ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_569_add__nonpos__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,X),zero_zero(A))
         => ( aa(A,$o,ord_less_eq(A,Y),zero_zero(A))
           => ( ( aa(A,A,plus_plus(A,X),Y) = zero_zero(A) )
            <=> ( ( X = zero_zero(A) )
                & ( Y = zero_zero(A) ) ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_570_add__less__le__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A3: A,B3: A,C2: A,D2: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,C2),D2)
           => aa(A,$o,ord_less(A,aa(A,A,plus_plus(A,A3),C2)),aa(A,A,plus_plus(A,B3),D2)) ) ) ) ).

% add_less_le_mono
tff(fact_571_add__le__less__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A3: A,B3: A,C2: A,D2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less(A,C2),D2)
           => aa(A,$o,ord_less(A,aa(A,A,plus_plus(A,A3),C2)),aa(A,A,plus_plus(A,B3),D2)) ) ) ) ).

% add_le_less_mono
tff(fact_572_add__mono__thms__linordered__field_I3_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,ord_less(A,I),J)
            & aa(A,$o,ord_less_eq(A,K),L) )
         => aa(A,$o,ord_less(A,aa(A,A,plus_plus(A,I),K)),aa(A,A,plus_plus(A,J),L)) ) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_573_add__mono__thms__linordered__field_I4_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,ord_less_eq(A,I),J)
            & aa(A,$o,ord_less(A,K),L) )
         => aa(A,$o,ord_less(A,aa(A,A,plus_plus(A,I),K)),aa(A,A,plus_plus(A,J),L)) ) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_574_le__minus__one__simps_I4_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ aa(A,$o,ord_less_eq(A,one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% le_minus_one_simps(4)
tff(fact_575_le__minus__one__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),one_one(A))),one_one(A)) ) ).

% le_minus_one_simps(2)
tff(fact_576_subset__Compl__self__eq,axiom,
    ! [A: $tType,A2: set(A)] :
      ( aa(set(A),$o,ord_less_eq(set(A),A2),aa(set(A),set(A),uminus_uminus(set(A)),A2))
    <=> ( A2 = bot_bot(set(A)) ) ) ).

% subset_Compl_self_eq
tff(fact_577_unsigned__greater__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & unique1627219031080169319umeral(A) )
     => ! [W: word(B)] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(word(B),A,semiring_1_unsigned(B,A),W)) ) ).

% unsigned_greater_eq
tff(fact_578_Abs__fnat__hom__add,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [A3: nat,B3: nat] : aa(A,A,plus_plus(A,aa(nat,A,semiring_1_of_nat(A),A3)),aa(nat,A,semiring_1_of_nat(A),B3)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,plus_plus(nat,A3),B3)) ) ).

% Abs_fnat_hom_add
tff(fact_579_mono__nat__linear__lb,axiom,
    ! [F2: fun(nat,nat),M: nat,K: nat] :
      ( ! [M4: nat,N: nat] :
          ( aa(nat,$o,ord_less(nat,M4),N)
         => aa(nat,$o,ord_less(nat,aa(nat,nat,F2,M4)),aa(nat,nat,F2,N)) )
     => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,plus_plus(nat,aa(nat,nat,F2,M)),K)),aa(nat,nat,F2,aa(nat,nat,plus_plus(nat,M),K))) ) ).

% mono_nat_linear_lb
tff(fact_580_of__nat__mono,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [I: nat,J: nat] :
          ( aa(nat,$o,ord_less_eq(nat,I),J)
         => aa(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_581_int__ge__induct,axiom,
    ! [K: int,I: int,P: fun(int,$o)] :
      ( aa(int,$o,ord_less_eq(int,K),I)
     => ( aa(int,$o,P,K)
       => ( ! [I3: int] :
              ( aa(int,$o,ord_less_eq(int,K),I3)
             => ( aa(int,$o,P,I3)
               => aa(int,$o,P,aa(int,int,plus_plus(int,I3),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_ge_induct
tff(fact_582_word__order_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),A3)
         => ( A3 = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% word_order.extremum_uniqueI
tff(fact_583_zadd__int__left,axiom,
    ! [M: nat,Na: nat,Z: int] : aa(int,int,plus_plus(int,aa(nat,int,semiring_1_of_nat(int),M)),aa(int,int,plus_plus(int,aa(nat,int,semiring_1_of_nat(int),Na)),Z)) = aa(int,int,plus_plus(int,aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,plus_plus(nat,M),Na))),Z) ).

% zadd_int_left
tff(fact_584_int__ops_I5_J,axiom,
    ! [A3: nat,B3: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,plus_plus(nat,A3),B3)) = aa(int,int,plus_plus(int,aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B3)) ).

% int_ops(5)
tff(fact_585_int__plus,axiom,
    ! [Na: nat,M: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,plus_plus(nat,Na),M)) = aa(int,int,plus_plus(int,aa(nat,int,semiring_1_of_nat(int),Na)),aa(nat,int,semiring_1_of_nat(int),M)) ).

% int_plus
tff(fact_586_zle__iff__zadd,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,ord_less_eq(int,W),Z)
    <=> ? [N3: nat] : Z = aa(int,int,plus_plus(int,W),aa(nat,int,semiring_1_of_nat(int),N3)) ) ).

% zle_iff_zadd
tff(fact_587_neq__0__no__wrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),plus_plus(word(A),X),Y))
         => ( ( X != zero_zero(word(A)) )
           => ( aa(word(A),word(A),plus_plus(word(A),X),Y) != zero_zero(word(A)) ) ) ) ) ).

% neq_0_no_wrap
tff(fact_588_word__plus__strict__mono__right,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Z: word(A),X: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Y),Z)
         => ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),plus_plus(word(A),X),Z))
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),plus_plus(word(A),X),Y)),aa(word(A),word(A),plus_plus(word(A),X),Z)) ) ) ) ).

% word_plus_strict_mono_right
tff(fact_589_plus__le__left__cancel__nowrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y5: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),plus_plus(word(A),X),Y5))
         => ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),plus_plus(word(A),X),Y))
           => ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),plus_plus(word(A),X),Y5)),aa(word(A),word(A),plus_plus(word(A),X),Y))
            <=> aa(word(A),$o,ord_less(word(A),Y5),Y) ) ) ) ) ).

% plus_le_left_cancel_nowrap
tff(fact_590_word__le__plus,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A),C2: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),A3),aa(word(A),word(A),plus_plus(word(A),A3),B3))
         => ( aa(word(A),$o,ord_less(word(A),C2),B3)
           => aa(word(A),$o,ord_less_eq(word(A),A3),aa(word(A),word(A),plus_plus(word(A),A3),C2)) ) ) ) ).

% word_le_plus
tff(fact_591_zle__int,axiom,
    ! [M: nat,Na: nat] :
      ( aa(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,ord_less_eq(nat,M),Na) ) ).

% zle_int
tff(fact_592_nat__int__comparison_I3_J,axiom,
    ! [A3: nat,B3: nat] :
      ( aa(nat,$o,ord_less_eq(nat,A3),B3)
    <=> aa(int,$o,ord_less_eq(int,aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B3)) ) ).

% nat_int_comparison(3)
tff(fact_593_word__le__add,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),X),Y)
         => ? [N: nat] : Y = aa(word(A),word(A),plus_plus(word(A),X),aa(nat,word(A),semiring_1_of_nat(word(A)),N)) ) ) ).

% word_le_add
tff(fact_594_word__le__make__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A)] :
          ( ( Y != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
         => ( aa(word(A),$o,ord_less_eq(word(A),X),Y)
          <=> aa(word(A),$o,ord_less(word(A),X),aa(word(A),word(A),plus_plus(word(A),Y),one_one(word(A)))) ) ) ) ).

% word_le_make_less
tff(fact_595_word__Suc__leq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A),X: word(A)] :
          ( ( K != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
         => ( aa(word(A),$o,ord_less(word(A),X),aa(word(A),word(A),plus_plus(word(A),K),one_one(word(A))))
          <=> aa(word(A),$o,ord_less_eq(word(A),X),K) ) ) ) ).

% word_Suc_leq
tff(fact_596_word__Suc__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),K: word(A)] :
          ( ( X != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
         => ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),plus_plus(word(A),X),one_one(word(A)))),K)
          <=> aa(word(A),$o,ord_less(word(A),X),K) ) ) ) ).

% word_Suc_le
tff(fact_597_dbl__inc__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [X: A] : neg_numeral_dbl_inc(A,X) = aa(A,A,plus_plus(A,aa(A,A,plus_plus(A,X),X)),one_one(A)) ) ).

% dbl_inc_def
tff(fact_598_nat__leq__as__int,axiom,
    ! [X2: nat,Xa: nat] :
      ( aa(nat,$o,ord_less_eq(nat,X2),Xa)
    <=> aa(int,$o,ord_less_eq(int,aa(nat,int,semiring_1_of_nat(int),X2)),aa(nat,int,semiring_1_of_nat(int),Xa)) ) ).

% nat_leq_as_int
tff(fact_599_neg__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: num] : aa(A,$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_600_not__zero__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: num] : ~ aa(A,$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_601_le__minus__one__simps_I3_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% le_minus_one_simps(3)
tff(fact_602_le__minus__one__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),one_one(A))),zero_zero(A)) ) ).

% le_minus_one_simps(1)
tff(fact_603_add__neg__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),zero_zero(A))
         => ( aa(A,$o,ord_less_eq(A,B3),zero_zero(A))
           => aa(A,$o,ord_less(A,aa(A,A,plus_plus(A,A3),B3)),zero_zero(A)) ) ) ) ).

% add_neg_nonpos
tff(fact_604_add__nonneg__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),B3)
           => aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,plus_plus(A,A3),B3)) ) ) ) ).

% add_nonneg_pos
tff(fact_605_add__nonpos__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),zero_zero(A))
         => ( aa(A,$o,ord_less(A,B3),zero_zero(A))
           => aa(A,$o,ord_less(A,aa(A,A,plus_plus(A,A3),B3)),zero_zero(A)) ) ) ) ).

% add_nonpos_neg
tff(fact_606_add__pos__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B3)
           => aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,plus_plus(A,A3),B3)) ) ) ) ).

% add_pos_nonneg
tff(fact_607_add__strict__increasing,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less_eq(A,B3),C2)
           => aa(A,$o,ord_less(A,B3),aa(A,A,plus_plus(A,A3),C2)) ) ) ) ).

% add_strict_increasing
tff(fact_608_add__strict__increasing2,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less(A,B3),C2)
           => aa(A,$o,ord_less(A,B3),aa(A,A,plus_plus(A,A3),C2)) ) ) ) ).

% add_strict_increasing2
tff(fact_609_neg__numeral__le__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$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_610_neg__one__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$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_611_neg__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$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_612_not__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$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_613_not__one__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$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_614_power__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat,N2: nat,A3: A] :
          ( aa(nat,$o,ord_less_eq(nat,Na),N2)
         => ( aa(A,$o,ord_less_eq(A,one_one(A)),A3)
           => aa(A,$o,ord_less_eq(A,aa(nat,A,power_power(A,A3),Na)),aa(nat,A,power_power(A,A3),N2)) ) ) ) ).

% power_increasing
tff(fact_615_real__add__less__0__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,plus_plus(real,X),Y)),zero_zero(real))
    <=> aa(real,$o,ord_less(real,Y),aa(real,real,uminus_uminus(real),X)) ) ).

% real_add_less_0_iff
tff(fact_616_real__0__less__add__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,plus_plus(real,X),Y))
    <=> aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),X)),Y) ) ).

% real_0_less_add_iff
tff(fact_617_int__zle__neg,axiom,
    ! [Na: nat,M: nat] :
      ( aa(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_618_nonpos__int__cases,axiom,
    ! [K: int] :
      ( aa(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_619_negative__zle__0,axiom,
    ! [Na: nat] : aa(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_620_add1__zle__eq,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,ord_less_eq(int,aa(int,int,plus_plus(int,W),one_one(int))),Z)
    <=> aa(int,$o,ord_less(int,W),Z) ) ).

% add1_zle_eq
tff(fact_621_zless__imp__add1__zle,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,ord_less(int,W),Z)
     => aa(int,$o,ord_less_eq(int,aa(int,int,plus_plus(int,W),one_one(int))),Z) ) ).

% zless_imp_add1_zle
tff(fact_622_max__word__wrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( ( aa(word(A),word(A),plus_plus(word(A),X),one_one(word(A))) = zero_zero(word(A)) )
         => ( X = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% max_word_wrap
tff(fact_623_less__x__plus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ( X != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
         => ( aa(word(A),$o,ord_less(word(A),Y),aa(word(A),word(A),plus_plus(word(A),X),one_one(word(A))))
          <=> ( aa(word(A),$o,ord_less(word(A),Y),X)
              | ( Y = X ) ) ) ) ) ).

% less_x_plus_1
tff(fact_624_word__add__no__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),X),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))))
         => aa(word(A),$o,ord_less(word(A),X),aa(word(A),word(A),plus_plus(word(A),X),one_one(word(A)))) ) ) ).

% word_add_no_overflow
tff(fact_625_word__plus__one__nonzero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),plus_plus(word(A),X),Y))
         => ( ( Y != zero_zero(word(A)) )
           => ( aa(word(A),word(A),plus_plus(word(A),X),one_one(word(A))) != zero_zero(word(A)) ) ) ) ) ).

% word_plus_one_nonzero
tff(fact_626_inc__i,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),M: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),one_one(word(A))),I)
         => ( aa(word(A),$o,ord_less(word(A),I),M)
           => ( aa(word(A),$o,ord_less_eq(word(A),one_one(word(A))),aa(word(A),word(A),plus_plus(word(A),I),one_one(word(A))))
              & aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),plus_plus(word(A),I),one_one(word(A)))),M) ) ) ) ) ).

% inc_i
tff(fact_627_inc__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),M: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),I),M)
         => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),plus_plus(word(A),I),one_one(word(A)))),M) ) ) ).

% inc_le
tff(fact_628_word__le__plus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Na: word(A),A3: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Y),aa(word(A),word(A),plus_plus(word(A),Y),Na))
         => ( aa(word(A),$o,ord_less(word(A),A3),Na)
           => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),plus_plus(word(A),Y),A3)),aa(word(A),word(A),plus_plus(word(A),aa(word(A),word(A),plus_plus(word(A),Y),A3)),one_one(word(A)))) ) ) ) ).

% word_le_plus_1
tff(fact_629_plus__one__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Na: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),X),aa(word(A),word(A),plus_plus(word(A),Na),one_one(word(A))))
         => aa(word(A),$o,ord_less_eq(word(A),X),Na) ) ) ).

% plus_one_helper
tff(fact_630_compl__less__swap1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,ord_less(A,Y),aa(A,A,uminus_uminus(A),X))
         => aa(A,$o,ord_less(A,X),aa(A,A,uminus_uminus(A),Y)) ) ) ).

% compl_less_swap1
tff(fact_631_compl__less__swap2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),Y)),X)
         => aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),X)),Y) ) ) ).

% compl_less_swap2
tff(fact_632_verit__negate__coefficient_I2_J,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),B3)),aa(A,A,uminus_uminus(A),A3)) ) ) ).

% verit_negate_coefficient(2)
tff(fact_633_less__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),aa(A,A,uminus_uminus(A),B3))
        <=> aa(A,$o,ord_less(A,B3),aa(A,A,uminus_uminus(A),A3)) ) ) ).

% less_minus_iff
tff(fact_634_minus__less__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),A3)),B3)
        <=> aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),B3)),A3) ) ) ).

% minus_less_iff
tff(fact_635_neg__numeral__neq__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [M: num,Na: num] : aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)) != aa(num,A,numeral_numeral(A),Na) ) ).

% neg_numeral_neq_numeral
tff(fact_636_numeral__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [M: num,Na: num] : aa(num,A,numeral_numeral(A),M) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na)) ) ).

% numeral_neq_neg_numeral
tff(fact_637_verit__sum__simplify,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A] : aa(A,A,plus_plus(A,A3),zero_zero(A)) = A3 ) ).

% verit_sum_simplify
tff(fact_638_add_Ogroup__left__neutral,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,plus_plus(A,zero_zero(A)),A3) = A3 ) ).

% add.group_left_neutral
tff(fact_639_add_Ocomm__neutral,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: A] : aa(A,A,plus_plus(A,A3),zero_zero(A)) = A3 ) ).

% add.comm_neutral
tff(fact_640_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: A] : aa(A,A,plus_plus(A,zero_zero(A)),A3) = A3 ) ).

% comm_monoid_add_class.add_0
tff(fact_641_one__neq__neg__one,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ( one_one(A) != aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% one_neq_neg_one
tff(fact_642_add__mono__thms__linordered__field_I5_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,ord_less(A,I),J)
            & aa(A,$o,ord_less(A,K),L) )
         => aa(A,$o,ord_less(A,aa(A,A,plus_plus(A,I),K)),aa(A,A,plus_plus(A,J),L)) ) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_643_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,ord_less(A,K),L) )
         => aa(A,$o,ord_less(A,aa(A,A,plus_plus(A,I),K)),aa(A,A,plus_plus(A,J),L)) ) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_644_add__mono__thms__linordered__field_I1_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,ord_less(A,I),J)
            & ( K = L ) )
         => aa(A,$o,ord_less(A,aa(A,A,plus_plus(A,I),K)),aa(A,A,plus_plus(A,J),L)) ) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_645_add__strict__mono,axiom,
    ! [A: $tType] :
      ( strict9044650504122735259up_add(A)
     => ! [A3: A,B3: A,C2: A,D2: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(A,$o,ord_less(A,C2),D2)
           => aa(A,$o,ord_less(A,aa(A,A,plus_plus(A,A3),C2)),aa(A,A,plus_plus(A,B3),D2)) ) ) ) ).

% add_strict_mono
tff(fact_646_add__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => aa(A,$o,ord_less(A,aa(A,A,plus_plus(A,C2),A3)),aa(A,A,plus_plus(A,C2),B3)) ) ) ).

% add_strict_left_mono
tff(fact_647_add__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => aa(A,$o,ord_less(A,aa(A,A,plus_plus(A,A3),C2)),aa(A,A,plus_plus(A,B3),C2)) ) ) ).

% add_strict_right_mono
tff(fact_648_add__less__imp__less__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,plus_plus(A,C2),A3)),aa(A,A,plus_plus(A,C2),B3))
         => aa(A,$o,ord_less(A,A3),B3) ) ) ).

% add_less_imp_less_left
tff(fact_649_add__less__imp__less__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,plus_plus(A,A3),C2)),aa(A,A,plus_plus(A,B3),C2))
         => aa(A,$o,ord_less(A,A3),B3) ) ) ).

% add_less_imp_less_right
tff(fact_650_add__One__commute,axiom,
    ! [Na: num] : aa(num,num,plus_plus(num,one2),Na) = aa(num,num,plus_plus(num,Na),one2) ).

% add_One_commute
tff(fact_651_le__numeral__extra_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,ord_less_eq(A,zero_zero(A)),zero_zero(A)) ) ).

% le_numeral_extra(3)
tff(fact_652_zero__le,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [X: A] : aa(A,$o,ord_less_eq(A,zero_zero(A)),X) ) ).

% zero_le
tff(fact_653_verit__comp__simplify1_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B6: A,A5: A] :
          ( ~ aa(A,$o,ord_less_eq(A,B6),A5)
        <=> aa(A,$o,ord_less(A,A5),B6) ) ) ).

% verit_comp_simplify1(3)
tff(fact_654_leD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,ord_less_eq(A,Y),X)
         => ~ aa(A,$o,ord_less(A,X),Y) ) ) ).

% leD
tff(fact_655_leI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ aa(A,$o,ord_less(A,X),Y)
         => aa(A,$o,ord_less_eq(A,Y),X) ) ) ).

% leI
tff(fact_656_nless__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( ~ aa(A,$o,ord_less(A,A3),B3)
        <=> ( ~ aa(A,$o,ord_less_eq(A,A3),B3)
            | ( A3 = B3 ) ) ) ) ).

% nless_le
tff(fact_657_antisym__conv1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( ~ aa(A,$o,ord_less(A,X),Y)
         => ( aa(A,$o,ord_less_eq(A,X),Y)
          <=> ( X = Y ) ) ) ) ).

% antisym_conv1
tff(fact_658_antisym__conv2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,X),Y)
         => ( ~ aa(A,$o,ord_less(A,X),Y)
          <=> ( X = Y ) ) ) ) ).

% antisym_conv2
tff(fact_659_dense__ge,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z: A,Y: A] :
          ( ! [X4: A] :
              ( aa(A,$o,ord_less(A,Z),X4)
             => aa(A,$o,ord_less_eq(A,Y),X4) )
         => aa(A,$o,ord_less_eq(A,Y),Z) ) ) ).

% dense_ge
tff(fact_660_dense__le,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Y: A,Z: A] :
          ( ! [X4: A] :
              ( aa(A,$o,ord_less(A,X4),Y)
             => aa(A,$o,ord_less_eq(A,X4),Z) )
         => aa(A,$o,ord_less_eq(A,Y),Z) ) ) ).

% dense_le
tff(fact_661_less__le__not__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less(A,X),Y)
        <=> ( aa(A,$o,ord_less_eq(A,X),Y)
            & ~ aa(A,$o,ord_less_eq(A,Y),X) ) ) ) ).

% less_le_not_le
tff(fact_662_not__le__imp__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,X: A] :
          ( ~ aa(A,$o,ord_less_eq(A,Y),X)
         => aa(A,$o,ord_less(A,X),Y) ) ) ).

% not_le_imp_less
tff(fact_663_order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
        <=> ( aa(A,$o,ord_less(A,A3),B3)
            | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
tff(fact_664_order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
        <=> ( aa(A,$o,ord_less_eq(A,A3),B3)
            & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
tff(fact_665_order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less(A,B3),C2)
           => aa(A,$o,ord_less(A,A3),C2) ) ) ) ).

% order.strict_trans1
tff(fact_666_order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,B3),C2)
           => aa(A,$o,ord_less(A,A3),C2) ) ) ) ).

% order.strict_trans2
tff(fact_667_order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
        <=> ( aa(A,$o,ord_less_eq(A,A3),B3)
            & ~ aa(A,$o,ord_less_eq(A,B3),A3) ) ) ) ).

% order.strict_iff_not
tff(fact_668_dense__ge__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z: A,X: A,Y: A] :
          ( aa(A,$o,ord_less(A,Z),X)
         => ( ! [W2: A] :
                ( aa(A,$o,ord_less(A,Z),W2)
               => ( aa(A,$o,ord_less(A,W2),X)
                 => aa(A,$o,ord_less_eq(A,Y),W2) ) )
           => aa(A,$o,ord_less_eq(A,Y),Z) ) ) ) ).

% dense_ge_bounded
tff(fact_669_dense__le__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [X: A,Y: A,Z: A] :
          ( aa(A,$o,ord_less(A,X),Y)
         => ( ! [W2: A] :
                ( aa(A,$o,ord_less(A,X),W2)
               => ( aa(A,$o,ord_less(A,W2),Y)
                 => aa(A,$o,ord_less_eq(A,W2),Z) ) )
           => aa(A,$o,ord_less_eq(A,Y),Z) ) ) ) ).

% dense_le_bounded
tff(fact_670_dual__order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
        <=> ( aa(A,$o,ord_less(A,B3),A3)
            | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
tff(fact_671_dual__order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,B3),A3)
        <=> ( aa(A,$o,ord_less_eq(A,B3),A3)
            & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
tff(fact_672_dual__order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
         => ( aa(A,$o,ord_less(A,C2),B3)
           => aa(A,$o,ord_less(A,C2),A3) ) ) ) ).

% dual_order.strict_trans1
tff(fact_673_dual__order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,ord_less(A,B3),A3)
         => ( aa(A,$o,ord_less_eq(A,C2),B3)
           => aa(A,$o,ord_less(A,C2),A3) ) ) ) ).

% dual_order.strict_trans2
tff(fact_674_dual__order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,B3),A3)
        <=> ( aa(A,$o,ord_less_eq(A,B3),A3)
            & ~ aa(A,$o,ord_less_eq(A,A3),B3) ) ) ) ).

% dual_order.strict_iff_not
tff(fact_675_order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => aa(A,$o,ord_less_eq(A,A3),B3) ) ) ).

% order.strict_implies_order
tff(fact_676_dual__order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,B3),A3)
         => aa(A,$o,ord_less_eq(A,B3),A3) ) ) ).

% dual_order.strict_implies_order
tff(fact_677_order__le__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,X),Y)
        <=> ( aa(A,$o,ord_less(A,X),Y)
            | ( X = Y ) ) ) ) ).

% order_le_less
tff(fact_678_order__less__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less(A,X),Y)
        <=> ( aa(A,$o,ord_less_eq(A,X),Y)
            & ( X != Y ) ) ) ) ).

% order_less_le
tff(fact_679_linorder__not__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ aa(A,$o,ord_less_eq(A,X),Y)
        <=> aa(A,$o,ord_less(A,Y),X) ) ) ).

% linorder_not_le
tff(fact_680_linorder__not__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ aa(A,$o,ord_less(A,X),Y)
        <=> aa(A,$o,ord_less_eq(A,Y),X) ) ) ).

% linorder_not_less
tff(fact_681_order__less__imp__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less(A,X),Y)
         => aa(A,$o,ord_less_eq(A,X),Y) ) ) ).

% order_less_imp_le
tff(fact_682_order__le__neq__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( ( A3 != B3 )
           => aa(A,$o,ord_less(A,A3),B3) ) ) ) ).

% order_le_neq_trans
tff(fact_683_order__neq__le__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != B3 )
         => ( aa(A,$o,ord_less_eq(A,A3),B3)
           => aa(A,$o,ord_less(A,A3),B3) ) ) ) ).

% order_neq_le_trans
tff(fact_684_order__le__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,Z: A] :
          ( aa(A,$o,ord_less_eq(A,X),Y)
         => ( aa(A,$o,ord_less(A,Y),Z)
           => aa(A,$o,ord_less(A,X),Z) ) ) ) ).

% order_le_less_trans
tff(fact_685_order__less__le__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,Z: A] :
          ( aa(A,$o,ord_less(A,X),Y)
         => ( aa(A,$o,ord_less_eq(A,Y),Z)
           => aa(A,$o,ord_less(A,X),Z) ) ) ) ).

% order_less_le_trans
tff(fact_686_order__le__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,F2: fun(B,A),B3: B,C2: B] :
          ( aa(A,$o,ord_less_eq(A,A3),aa(B,A,F2,B3))
         => ( aa(B,$o,ord_less(B,B3),C2)
           => ( ! [X4: B,Y3: B] :
                  ( aa(B,$o,ord_less(B,X4),Y3)
                 => aa(A,$o,ord_less(A,aa(B,A,F2,X4)),aa(B,A,F2,Y3)) )
             => aa(A,$o,ord_less(A,A3),aa(B,A,F2,C2)) ) ) ) ) ).

% order_le_less_subst1
tff(fact_687_order__le__less__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,B3: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(B,$o,ord_less(B,aa(A,B,F2,B3)),C2)
           => ( ! [X4: A,Y3: A] :
                  ( aa(A,$o,ord_less_eq(A,X4),Y3)
                 => aa(B,$o,ord_less_eq(B,aa(A,B,F2,X4)),aa(A,B,F2,Y3)) )
             => aa(B,$o,ord_less(B,aa(A,B,F2,A3)),C2) ) ) ) ) ).

% order_le_less_subst2
tff(fact_688_order__less__le__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,F2: fun(B,A),B3: B,C2: B] :
          ( aa(A,$o,ord_less(A,A3),aa(B,A,F2,B3))
         => ( aa(B,$o,ord_less_eq(B,B3),C2)
           => ( ! [X4: B,Y3: B] :
                  ( aa(B,$o,ord_less_eq(B,X4),Y3)
                 => aa(A,$o,ord_less_eq(A,aa(B,A,F2,X4)),aa(B,A,F2,Y3)) )
             => aa(A,$o,ord_less(A,A3),aa(B,A,F2,C2)) ) ) ) ) ).

% order_less_le_subst1
tff(fact_689_order__less__le__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,B3: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(B,$o,ord_less_eq(B,aa(A,B,F2,B3)),C2)
           => ( ! [X4: A,Y3: A] :
                  ( aa(A,$o,ord_less(A,X4),Y3)
                 => aa(B,$o,ord_less(B,aa(A,B,F2,X4)),aa(A,B,F2,Y3)) )
             => aa(B,$o,ord_less(B,aa(A,B,F2,A3)),C2) ) ) ) ) ).

% order_less_le_subst2
tff(fact_690_linorder__le__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,X),Y)
          | aa(A,$o,ord_less(A,Y),X) ) ) ).

% linorder_le_less_linear
tff(fact_691_order__le__imp__less__or__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,X),Y)
         => ( aa(A,$o,ord_less(A,X),Y)
            | ( X = Y ) ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_692_le__num__One__iff,axiom,
    ! [X: num] :
      ( aa(num,$o,ord_less_eq(num,X),one2)
    <=> ( X = one2 ) ) ).

% le_num_One_iff
tff(fact_693_le__numeral__extra_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,ord_less_eq(A,one_one(A)),one_one(A)) ) ).

% le_numeral_extra(4)
tff(fact_694_bot_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),bot_bot(A))
         => ( A3 = bot_bot(A) ) ) ) ).

% bot.extremum_uniqueI
tff(fact_695_bot_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),bot_bot(A))
        <=> ( A3 = bot_bot(A) ) ) ) ).

% bot.extremum_unique
tff(fact_696_bot_Oextremum,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] : aa(A,$o,ord_less_eq(A,bot_bot(A)),A3) ) ).

% bot.extremum
tff(fact_697_plus__nat_Oadd__0,axiom,
    ! [Na: nat] : aa(nat,nat,plus_plus(nat,zero_zero(nat)),Na) = Na ).

% plus_nat.add_0
tff(fact_698_add__eq__self__zero,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,plus_plus(nat,M),Na) = M )
     => ( Na = zero_zero(nat) ) ) ).

% add_eq_self_zero
tff(fact_699_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,plus_plus(nat,I),J)),K)
     => aa(nat,$o,ord_less(nat,I),K) ) ).

% add_lessD1
tff(fact_700_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa(nat,$o,ord_less(nat,I),J)
     => ( aa(nat,$o,ord_less(nat,K),L)
       => aa(nat,$o,ord_less(nat,aa(nat,nat,plus_plus(nat,I),K)),aa(nat,nat,plus_plus(nat,J),L)) ) ) ).

% add_less_mono
tff(fact_701_not__add__less1,axiom,
    ! [I: nat,J: nat] : ~ aa(nat,$o,ord_less(nat,aa(nat,nat,plus_plus(nat,I),J)),I) ).

% not_add_less1
tff(fact_702_not__add__less2,axiom,
    ! [J: nat,I: nat] : ~ aa(nat,$o,ord_less(nat,aa(nat,nat,plus_plus(nat,J),I)),I) ).

% not_add_less2
tff(fact_703_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,I),J)
     => aa(nat,$o,ord_less(nat,aa(nat,nat,plus_plus(nat,I),K)),aa(nat,nat,plus_plus(nat,J),K)) ) ).

% add_less_mono1
tff(fact_704_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,I),J)
     => aa(nat,$o,ord_less(nat,I),aa(nat,nat,plus_plus(nat,J),M)) ) ).

% trans_less_add1
tff(fact_705_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,I),J)
     => aa(nat,$o,ord_less(nat,I),aa(nat,nat,plus_plus(nat,M),J)) ) ).

% trans_less_add2
tff(fact_706_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,K),L)
     => ( ( aa(nat,nat,plus_plus(nat,M),L) = aa(nat,nat,plus_plus(nat,K),Na) )
       => aa(nat,$o,ord_less(nat,M),Na) ) ) ).

% less_add_eq_less
tff(fact_707_uminus__int__code_I1_J,axiom,
    aa(int,int,uminus_uminus(int),zero_zero(int)) = zero_zero(int) ).

% uminus_int_code(1)
tff(fact_708_plus__int__code_I1_J,axiom,
    ! [K: int] : aa(int,int,plus_plus(int,K),zero_zero(int)) = K ).

% plus_int_code(1)
tff(fact_709_plus__int__code_I2_J,axiom,
    ! [L: int] : aa(int,int,plus_plus(int,zero_zero(int)),L) = L ).

% plus_int_code(2)
tff(fact_710_less__eq__nat_Osimps_I1_J,axiom,
    ! [Na: nat] : aa(nat,$o,ord_less_eq(nat,zero_zero(nat)),Na) ).

% less_eq_nat.simps(1)
tff(fact_711_bot__nat__0_Oextremum__unique,axiom,
    ! [A3: nat] :
      ( aa(nat,$o,ord_less_eq(nat,A3),zero_zero(nat))
    <=> ( A3 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_unique
tff(fact_712_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A3: nat] :
      ( aa(nat,$o,ord_less_eq(nat,A3),zero_zero(nat))
     => ( A3 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_713_le__0__eq,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,Na),zero_zero(nat))
    <=> ( Na = zero_zero(nat) ) ) ).

% le_0_eq
tff(fact_714_nat__less__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,M),Na)
    <=> ( aa(nat,$o,ord_less_eq(nat,M),Na)
        & ( M != Na ) ) ) ).

% nat_less_le
tff(fact_715_less__imp__le__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,M),Na)
     => aa(nat,$o,ord_less_eq(nat,M),Na) ) ).

% less_imp_le_nat
tff(fact_716_le__eq__less__or__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),Na)
    <=> ( aa(nat,$o,ord_less(nat,M),Na)
        | ( M = Na ) ) ) ).

% le_eq_less_or_eq
tff(fact_717_less__or__eq__imp__le,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,$o,ord_less(nat,M),Na)
        | ( M = Na ) )
     => aa(nat,$o,ord_less_eq(nat,M),Na) ) ).

% less_or_eq_imp_le
tff(fact_718_le__neq__implies__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),Na)
     => ( ( M != Na )
       => aa(nat,$o,ord_less(nat,M),Na) ) ) ).

% le_neq_implies_less
tff(fact_719_less__mono__imp__le__mono,axiom,
    ! [F2: fun(nat,nat),I: nat,J: nat] :
      ( ! [I3: nat,J2: nat] :
          ( aa(nat,$o,ord_less(nat,I3),J2)
         => aa(nat,$o,ord_less(nat,aa(nat,nat,F2,I3)),aa(nat,nat,F2,J2)) )
     => ( aa(nat,$o,ord_less_eq(nat,I),J)
       => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,F2,I)),aa(nat,nat,F2,J)) ) ) ).

% less_mono_imp_le_mono
tff(fact_720_real__arch__simple,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [N: nat] : aa(A,$o,ord_less_eq(A,X),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% real_arch_simple
tff(fact_721_inf__sup__ord_I4_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Y: A,X: A] : aa(A,$o,ord_less_eq(A,Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) ) ).

% inf_sup_ord(4)
tff(fact_722_inf__sup__ord_I3_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A] : aa(A,$o,ord_less_eq(A,X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) ) ).

% inf_sup_ord(3)
tff(fact_723_le__supE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B3: A,X: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)),X)
         => ~ ( aa(A,$o,ord_less_eq(A,A3),X)
             => ~ aa(A,$o,ord_less_eq(A,B3),X) ) ) ) ).

% le_supE
tff(fact_724_le__supI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,X: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),X)
         => ( aa(A,$o,ord_less_eq(A,B3),X)
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)),X) ) ) ) ).

% le_supI
tff(fact_725_sup__ge1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A] : aa(A,$o,ord_less_eq(A,X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) ) ).

% sup_ge1
tff(fact_726_sup__ge2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,X: A] : aa(A,$o,ord_less_eq(A,Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) ) ).

% sup_ge2
tff(fact_727_le__supI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,X),A3)
         => aa(A,$o,ord_less_eq(A,X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)) ) ) ).

% le_supI1
tff(fact_728_le__supI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,B3: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,X),B3)
         => aa(A,$o,ord_less_eq(A,X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)) ) ) ).

% le_supI2
tff(fact_729_sup_Omono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,A3: A,D2: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,C2),A3)
         => ( aa(A,$o,ord_less_eq(A,D2),B3)
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),sup_sup(A),C2),D2)),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)) ) ) ) ).

% sup.mono
tff(fact_730_sup__mono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,C2: A,B3: A,D2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),C2)
         => ( aa(A,$o,ord_less_eq(A,B3),D2)
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)),aa(A,A,aa(A,fun(A,A),sup_sup(A),C2),D2)) ) ) ) ).

% sup_mono
tff(fact_731_sup__least,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,X: A,Z: A] :
          ( aa(A,$o,ord_less_eq(A,Y),X)
         => ( aa(A,$o,ord_less_eq(A,Z),X)
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z)),X) ) ) ) ).

% sup_least
tff(fact_732_le__iff__sup,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,X),Y)
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = Y ) ) ) ).

% le_iff_sup
tff(fact_733_sup_OorderE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
         => ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) ) ) ) ).

% sup.orderE
tff(fact_734_sup_OorderI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) )
         => aa(A,$o,ord_less_eq(A,B3),A3) ) ) ).

% sup.orderI
tff(fact_735_sup__unique,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [F2: fun(A,fun(A,A)),X: A,Y: A] :
          ( ! [X4: A,Y3: A] : aa(A,$o,ord_less_eq(A,X4),aa(A,A,aa(A,fun(A,A),F2,X4),Y3))
         => ( ! [X4: A,Y3: A] : aa(A,$o,ord_less_eq(A,Y3),aa(A,A,aa(A,fun(A,A),F2,X4),Y3))
           => ( ! [X4: A,Y3: A,Z2: A] :
                  ( aa(A,$o,ord_less_eq(A,Y3),X4)
                 => ( aa(A,$o,ord_less_eq(A,Z2),X4)
                   => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),F2,Y3),Z2)),X4) ) )
             => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = aa(A,A,aa(A,fun(A,A),F2,X),Y) ) ) ) ) ) ).

% sup_unique
tff(fact_736_sup_Oabsorb1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) = A3 ) ) ) ).

% sup.absorb1
tff(fact_737_sup_Oabsorb2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) = B3 ) ) ) ).

% sup.absorb2
tff(fact_738_sup__absorb1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,ord_less_eq(A,Y),X)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = X ) ) ) ).

% sup_absorb1
tff(fact_739_sup__absorb2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,X),Y)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = Y ) ) ) ).

% sup_absorb2
tff(fact_740_sup_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B3: A,C2: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),sup_sup(A),B3),C2)),A3)
         => ~ ( aa(A,$o,ord_less_eq(A,B3),A3)
             => ~ aa(A,$o,ord_less_eq(A,C2),A3) ) ) ) ).

% sup.boundedE
tff(fact_741_sup_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
         => ( aa(A,$o,ord_less_eq(A,C2),A3)
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),sup_sup(A),B3),C2)),A3) ) ) ) ).

% sup.boundedI
tff(fact_742_sup_Oorder__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
        <=> ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) ) ) ) ).

% sup.order_iff
tff(fact_743_sup_Ocobounded1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B3: A] : aa(A,$o,ord_less_eq(A,A3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)) ) ).

% sup.cobounded1
tff(fact_744_sup_Ocobounded2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B3: A,A3: A] : aa(A,$o,ord_less_eq(A,B3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)) ) ).

% sup.cobounded2
tff(fact_745_sup_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) = A3 ) ) ) ).

% sup.absorb_iff1
tff(fact_746_sup_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) = B3 ) ) ) ).

% sup.absorb_iff2
tff(fact_747_sup_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,C2),A3)
         => aa(A,$o,ord_less_eq(A,C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)) ) ) ).

% sup.coboundedI1
tff(fact_748_sup_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,C2),B3)
         => aa(A,$o,ord_less_eq(A,C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)) ) ) ).

% sup.coboundedI2
tff(fact_749_insert__mono,axiom,
    ! [A: $tType,C3: set(A),D3: set(A),A3: A] :
      ( aa(set(A),$o,ord_less_eq(set(A),C3),D3)
     => aa(set(A),$o,ord_less_eq(set(A),aa(set(A),set(A),insert(A,A3),C3)),aa(set(A),set(A),insert(A,A3),D3)) ) ).

% insert_mono
tff(fact_750_subset__insert,axiom,
    ! [A: $tType,X: A,A2: set(A),B2: set(A)] :
      ( ~ member(A,X,A2)
     => ( aa(set(A),$o,ord_less_eq(set(A),A2),aa(set(A),set(A),insert(A,X),B2))
      <=> aa(set(A),$o,ord_less_eq(set(A),A2),B2) ) ) ).

% subset_insert
tff(fact_751_subset__insertI,axiom,
    ! [A: $tType,B2: set(A),A3: A] : aa(set(A),$o,ord_less_eq(set(A),B2),aa(set(A),set(A),insert(A,A3),B2)) ).

% subset_insertI
tff(fact_752_subset__insertI2,axiom,
    ! [A: $tType,A2: set(A),B2: set(A),B3: A] :
      ( aa(set(A),$o,ord_less_eq(set(A),A2),B2)
     => aa(set(A),$o,ord_less_eq(set(A),A2),aa(set(A),set(A),insert(A,B3),B2)) ) ).

% subset_insertI2
tff(fact_753_less__eq__real__def,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,X),Y)
    <=> ( aa(real,$o,ord_less(real,X),Y)
        | ( X = Y ) ) ) ).

% less_eq_real_def
tff(fact_754_less__eq__int__code_I1_J,axiom,
    aa(int,$o,ord_less_eq(int,zero_zero(int)),zero_zero(int)) ).

% less_eq_int_code(1)
tff(fact_755_subset__code_I1_J,axiom,
    ! [A: $tType,Xsa: list(A),B2: set(A)] :
      ( aa(set(A),$o,ord_less_eq(set(A),aa(list(A),set(A),set2(A),Xsa)),B2)
    <=> ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xsa))
         => member(A,X3,B2) ) ) ).

% subset_code(1)
tff(fact_756_int__cases2,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),N)) ) ).

% int_cases2
tff(fact_757_plus__le__left__cancel__wrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y5: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),plus_plus(word(A),X),Y5)),X)
         => ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),plus_plus(word(A),X),Y)),X)
           => ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),plus_plus(word(A),X),Y5)),aa(word(A),word(A),plus_plus(word(A),X),Y))
            <=> aa(word(A),$o,ord_less(word(A),Y5),Y) ) ) ) ) ).

% plus_le_left_cancel_wrap
tff(fact_758_Un__mono,axiom,
    ! [A: $tType,A2: set(A),C3: set(A),B2: set(A),D3: set(A)] :
      ( aa(set(A),$o,ord_less_eq(set(A),A2),C3)
     => ( aa(set(A),$o,ord_less_eq(set(A),B2),D3)
       => aa(set(A),$o,ord_less_eq(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),B2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C3),D3)) ) ) ).

% Un_mono
tff(fact_759_Un__least,axiom,
    ! [A: $tType,A2: set(A),C3: set(A),B2: set(A)] :
      ( aa(set(A),$o,ord_less_eq(set(A),A2),C3)
     => ( aa(set(A),$o,ord_less_eq(set(A),B2),C3)
       => aa(set(A),$o,ord_less_eq(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),B2)),C3) ) ) ).

% Un_least
tff(fact_760_Un__upper1,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] : aa(set(A),$o,ord_less_eq(set(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),B2)) ).

% Un_upper1
tff(fact_761_Un__upper2,axiom,
    ! [A: $tType,B2: set(A),A2: set(A)] : aa(set(A),$o,ord_less_eq(set(A),B2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),B2)) ).

% Un_upper2
tff(fact_762_Un__absorb1,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] :
      ( aa(set(A),$o,ord_less_eq(set(A),A2),B2)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),B2) = B2 ) ) ).

% Un_absorb1
tff(fact_763_Un__absorb2,axiom,
    ! [A: $tType,B2: set(A),A2: set(A)] :
      ( aa(set(A),$o,ord_less_eq(set(A),B2),A2)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),B2) = A2 ) ) ).

% Un_absorb2
tff(fact_764_subset__UnE,axiom,
    ! [A: $tType,C3: set(A),A2: set(A),B2: set(A)] :
      ( aa(set(A),$o,ord_less_eq(set(A),C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),B2))
     => ~ ! [A6: set(A)] :
            ( aa(set(A),$o,ord_less_eq(set(A),A6),A2)
           => ! [B7: set(A)] :
                ( aa(set(A),$o,ord_less_eq(set(A),B7),B2)
               => ( C3 != aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B7) ) ) ) ) ).

% subset_UnE
tff(fact_765_subset__Un__eq,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] :
      ( aa(set(A),$o,ord_less_eq(set(A),A2),B2)
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),B2) = B2 ) ) ).

% subset_Un_eq
tff(fact_766_word__coorder_Oextremum,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : aa(word(A),$o,ord_less_eq(word(A),zero_zero(word(A))),A3) ) ).

% word_coorder.extremum
tff(fact_767_word__coorder_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),A3),zero_zero(word(A)))
         => ( A3 = zero_zero(word(A)) ) ) ) ).

% word_coorder.extremum_uniqueI
tff(fact_768_word__zero__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A)] : aa(word(A),$o,ord_less_eq(word(A),zero_zero(word(A))),Y) ) ).

% word_zero_le
tff(fact_769_word__le__not__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: word(A),A3: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),B3),A3)
        <=> ~ aa(word(A),$o,ord_less(word(A),A3),B3) ) ) ).

% word_le_not_less
tff(fact_770_word__not__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ~ aa(word(A),$o,ord_less_eq(word(A),X),Y)
        <=> aa(word(A),$o,ord_less(word(A),Y),X) ) ) ).

% word_not_le
tff(fact_771_word__le__less__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),X),Y)
        <=> ( ( X = Y )
            | aa(word(A),$o,ord_less(word(A),X),Y) ) ) ) ).

% word_le_less_eq
tff(fact_772_subset__iff__psubset__eq,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] :
      ( aa(set(A),$o,ord_less_eq(set(A),A2),B2)
    <=> ( aa(set(A),$o,ord_less(set(A),A2),B2)
        | ( A2 = B2 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_773_subset__psubset__trans,axiom,
    ! [A: $tType,A2: set(A),B2: set(A),C3: set(A)] :
      ( aa(set(A),$o,ord_less_eq(set(A),A2),B2)
     => ( aa(set(A),$o,ord_less(set(A),B2),C3)
       => aa(set(A),$o,ord_less(set(A),A2),C3) ) ) ).

% subset_psubset_trans
tff(fact_774_subset__not__subset__eq,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] :
      ( aa(set(A),$o,ord_less(set(A),A2),B2)
    <=> ( aa(set(A),$o,ord_less_eq(set(A),A2),B2)
        & ~ aa(set(A),$o,ord_less_eq(set(A),B2),A2) ) ) ).

% subset_not_subset_eq
tff(fact_775_psubset__subset__trans,axiom,
    ! [A: $tType,A2: set(A),B2: set(A),C3: set(A)] :
      ( aa(set(A),$o,ord_less(set(A),A2),B2)
     => ( aa(set(A),$o,ord_less_eq(set(A),B2),C3)
       => aa(set(A),$o,ord_less(set(A),A2),C3) ) ) ).

% psubset_subset_trans
tff(fact_776_psubset__imp__subset,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] :
      ( aa(set(A),$o,ord_less(set(A),A2),B2)
     => aa(set(A),$o,ord_less_eq(set(A),A2),B2) ) ).

% psubset_imp_subset
tff(fact_777_psubset__eq,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] :
      ( aa(set(A),$o,ord_less(set(A),A2),B2)
    <=> ( aa(set(A),$o,ord_less_eq(set(A),A2),B2)
        & ( A2 != B2 ) ) ) ).

% psubset_eq
tff(fact_778_psubsetE,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] :
      ( aa(set(A),$o,ord_less(set(A),A2),B2)
     => ~ ( aa(set(A),$o,ord_less_eq(set(A),A2),B2)
         => aa(set(A),$o,ord_less_eq(set(A),B2),A2) ) ) ).

% psubsetE
tff(fact_779_iadd__is__0,axiom,
    ! [M: extended_enat,Na: extended_enat] :
      ( ( aa(extended_enat,extended_enat,plus_plus(extended_enat,M),Na) = zero_zero(extended_enat) )
    <=> ( ( M = zero_zero(extended_enat) )
        & ( Na = zero_zero(extended_enat) ) ) ) ).

% iadd_is_0
tff(fact_780_i0__lb,axiom,
    ! [Na: extended_enat] : aa(extended_enat,$o,ord_less_eq(extended_enat,zero_zero(extended_enat)),Na) ).

% i0_lb
tff(fact_781_ile0__eq,axiom,
    ! [Na: extended_enat] :
      ( aa(extended_enat,$o,ord_less_eq(extended_enat,Na),zero_zero(extended_enat))
    <=> ( Na = zero_zero(extended_enat) ) ) ).

% ile0_eq
tff(fact_782_power__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat,N2: nat,A3: A] :
          ( aa(nat,$o,ord_less_eq(nat,Na),N2)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
           => ( aa(A,$o,ord_less_eq(A,A3),one_one(A))
             => aa(A,$o,ord_less_eq(A,aa(nat,A,power_power(A,A3),N2)),aa(nat,A,power_power(A,A3),Na)) ) ) ) ) ).

% power_decreasing
tff(fact_783_power__le__imp__le__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,M: nat,Na: nat] :
          ( aa(A,$o,ord_less(A,one_one(A)),A3)
         => ( aa(A,$o,ord_less_eq(A,aa(nat,A,power_power(A,A3),M)),aa(nat,A,power_power(A,A3),Na))
           => aa(nat,$o,ord_less_eq(nat,M),Na) ) ) ) ).

% power_le_imp_le_exp
tff(fact_784_le__imp__0__less,axiom,
    ! [Z: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z)
     => aa(int,$o,ord_less(int,zero_zero(int)),aa(int,int,plus_plus(int,one_one(int)),Z)) ) ).

% le_imp_0_less
tff(fact_785_le__log__of__power,axiom,
    ! [B3: real,Na: nat,M: real] :
      ( aa(real,$o,ord_less_eq(real,aa(nat,real,power_power(real,B3),Na)),M)
     => ( aa(real,$o,ord_less(real,one_one(real)),B3)
       => aa(real,$o,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),Na)),aa(real,real,log(B3),M)) ) ) ).

% le_log_of_power
tff(fact_786_nat__less__real__le,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,Na),M)
    <=> aa(real,$o,ord_less_eq(real,aa(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_787_nat__le__real__less,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,ord_less_eq(nat,Na),M)
    <=> aa(real,$o,ord_less(real,aa(nat,real,semiring_1_of_nat(real),Na)),aa(real,real,plus_plus(real,aa(nat,real,semiring_1_of_nat(real),M)),one_one(real))) ) ).

% nat_le_real_less
tff(fact_788_plus__one__helper2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Na: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),X),Na)
         => ( ( aa(word(A),word(A),plus_plus(word(A),Na),one_one(word(A))) != zero_zero(word(A)) )
           => aa(word(A),$o,ord_less(word(A),X),aa(word(A),word(A),plus_plus(word(A),Na),one_one(word(A)))) ) ) ) ).

% plus_one_helper2
tff(fact_789_Collect__imp__eq,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] : collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_aj(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),collect(A,P))),collect(A,Q)) ).

% Collect_imp_eq
tff(fact_790_le__log2__of__power,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),M)
     => aa(real,$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_791_word__1__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A),X: nat] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),plus_plus(word(A),A3),one_one(word(A)))),B3)
         => ( aa(word(A),$o,ord_less(word(A),A3),aa(nat,word(A),semiring_1_of_nat(word(A)),X))
           => aa(word(A),$o,ord_less(word(A),A3),B3) ) ) ) ).

% word_1_0
tff(fact_792_dbl__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [X: A] : neg_numeral_dbl(A,X) = aa(A,A,plus_plus(A,X),X) ) ).

% dbl_def
tff(fact_793_realpow__square__minus__le,axiom,
    ! [U: real,X: real] : aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(nat,real,power_power(real,U),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% realpow_square_minus_le
tff(fact_794_less__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( aa(fun(A,B),$o,ord_less(fun(A,B),F2),G)
        <=> ( aa(fun(A,B),$o,ord_less_eq(fun(A,B),F2),G)
            & ~ aa(fun(A,B),$o,ord_less_eq(fun(A,B),G),F2) ) ) ) ).

% less_fun_def
tff(fact_795_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_796_neg__numeral__less__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Na: num] : aa(A,$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_797_not__numeral__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Na: num] : ~ aa(A,$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_798_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_799_less__minus__one__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),one_one(A))),one_one(A)) ) ).

% less_minus_one_simps(2)
tff(fact_800_less__minus__one__simps_I4_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ aa(A,$o,ord_less(A,one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% less_minus_one_simps(4)
tff(fact_801_one__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Na: num] : one_one(A) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na)) ) ).

% one_neq_neg_numeral
tff(fact_802_numeral__neq__neg__one,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Na: num] : aa(num,A,numeral_numeral(A),Na) != aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% numeral_neq_neg_one
tff(fact_803_add__less__zeroD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,plus_plus(A,X),Y)),zero_zero(A))
         => ( aa(A,$o,ord_less(A,X),zero_zero(A))
            | aa(A,$o,ord_less(A,Y),zero_zero(A)) ) ) ) ).

% add_less_zeroD
tff(fact_804_add__neg__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),zero_zero(A))
         => ( aa(A,$o,ord_less(A,B3),zero_zero(A))
           => aa(A,$o,ord_less(A,aa(A,A,plus_plus(A,A3),B3)),zero_zero(A)) ) ) ) ).

% add_neg_neg
tff(fact_805_add__pos__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),B3)
           => aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,plus_plus(A,A3),B3)) ) ) ) ).

% add_pos_pos
tff(fact_806_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ~ ! [C5: A] :
                ( ( B3 = aa(A,A,plus_plus(A,A3),C5) )
               => ( C5 = zero_zero(A) ) ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_807_pos__add__strict,axiom,
    ! [A: $tType] :
      ( strict7427464778891057005id_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less(A,B3),C2)
           => aa(A,$o,ord_less(A,B3),aa(A,A,plus_plus(A,A3),C2)) ) ) ) ).

% pos_add_strict
tff(fact_808_numeral__Bit0,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Na: num] : aa(num,A,numeral_numeral(A),bit0(Na)) = aa(A,A,plus_plus(A,aa(num,A,numeral_numeral(A),Na)),aa(num,A,numeral_numeral(A),Na)) ) ).

% numeral_Bit0
tff(fact_809_less__add__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A] : aa(A,$o,ord_less(A,A3),aa(A,A,plus_plus(A,A3),one_one(A))) ) ).

% less_add_one
tff(fact_810_add__mono1,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => aa(A,$o,ord_less(A,aa(A,A,plus_plus(A,A3),one_one(A))),aa(A,A,plus_plus(A,B3),one_one(A))) ) ) ).

% add_mono1
tff(fact_811_one__plus__numeral__commute,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [X: num] : aa(A,A,plus_plus(A,one_one(A)),aa(num,A,numeral_numeral(A),X)) = aa(A,A,plus_plus(A,aa(num,A,numeral_numeral(A),X)),one_one(A)) ) ).

% one_plus_numeral_commute
tff(fact_812_uint__0__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( ( aa(word(A),int,semiring_1_unsigned(A,int),X) = zero_zero(int) )
        <=> ( X = zero_zero(word(A)) ) ) ) ).

% uint_0_iff
tff(fact_813_uint__0__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),int,semiring_1_unsigned(A,int),zero_zero(word(A))) = zero_zero(int) ) ) ).

% uint_0_eq
tff(fact_814_not__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: num] : ~ aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),Na)),zero_zero(A)) ) ).

% not_numeral_le_zero
tff(fact_815_zero__le__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: num] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(num,A,numeral_numeral(A),Na)) ) ).

% zero_le_numeral
tff(fact_816_not__one__le__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ aa(A,$o,ord_less_eq(A,one_one(A)),zero_zero(A)) ) ).

% not_one_le_zero
tff(fact_817_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,ord_less_eq(A,zero_zero(A)),one_one(A)) ) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_818_zero__less__one__class_Ozero__le__one,axiom,
    ! [A: $tType] :
      ( zero_less_one(A)
     => aa(A,$o,ord_less_eq(A,zero_zero(A)),one_one(A)) ) ).

% zero_less_one_class.zero_le_one
tff(fact_819_uint__1__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),int,semiring_1_unsigned(A,int),one_one(word(A))) = one_one(int) ) ) ).

% uint_1_eq
tff(fact_820_one__le__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: num] : aa(A,$o,ord_less_eq(A,one_one(A)),aa(num,A,numeral_numeral(A),Na)) ) ).

% one_le_numeral
tff(fact_821_word__less__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),A3),B3)
        <=> aa(int,$o,ord_less(int,aa(word(A),int,semiring_1_unsigned(A,int),A3)),aa(word(A),int,semiring_1_unsigned(A,int),B3)) ) ) ).

% word_less_def
tff(fact_822_zero__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,power_power(A,A3),Na)) ) ) ).

% zero_le_power
tff(fact_823_power__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,B3: A,Na: nat] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
           => aa(A,$o,ord_less_eq(A,aa(nat,A,power_power(A,A3),Na)),aa(nat,A,power_power(A,B3),Na)) ) ) ) ).

% power_mono
tff(fact_824_norm__pre__pure__rule2,axiom,
    ! [A: $tType,B3: $o,F2: heap_Time_Heap(A),Q: fun(A,assn)] :
      ( ( (B3)
       => hoare_hoare_triple(A,one_one(assn),F2,Q) )
     => hoare_hoare_triple(A,pure_assn((B3)),F2,Q) ) ).

% norm_pre_pure_rule2
tff(fact_825_one__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,ord_less_eq(A,one_one(A)),A3)
         => aa(A,$o,ord_less_eq(A,one_one(A)),aa(nat,A,power_power(A,A3),Na)) ) ) ).

% one_le_power
tff(fact_826_of__nat__0__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: nat] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),Na)) ) ).

% of_nat_0_le_iff
tff(fact_827_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,ord_less(nat,I),J)
     => ? [K2: nat] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K2)
          & ( aa(nat,nat,plus_plus(nat,I),K2) = J ) ) ) ).

% less_imp_add_positive
tff(fact_828_log2__of__power__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
       => aa(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_829_ex__least__nat__le,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( aa(nat,$o,P,Na)
     => ( ~ aa(nat,$o,P,zero_zero(nat))
       => ? [K2: nat] :
            ( aa(nat,$o,ord_less_eq(nat,K2),Na)
            & ! [I2: nat] :
                ( aa(nat,$o,ord_less(nat,I2),K2)
               => ~ aa(nat,$o,P,I2) )
            & aa(nat,$o,P,K2) ) ) ) ).

% ex_least_nat_le
tff(fact_830_subset__singletonD,axiom,
    ! [A: $tType,A2: set(A),X: A] :
      ( aa(set(A),$o,ord_less_eq(set(A),A2),aa(set(A),set(A),insert(A,X),bot_bot(set(A))))
     => ( ( A2 = bot_bot(set(A)) )
        | ( A2 = aa(set(A),set(A),insert(A,X),bot_bot(set(A))) ) ) ) ).

% subset_singletonD
tff(fact_831_subset__singleton__iff,axiom,
    ! [A: $tType,X5: set(A),A3: A] :
      ( aa(set(A),$o,ord_less_eq(set(A),X5),aa(set(A),set(A),insert(A,A3),bot_bot(set(A))))
    <=> ( ( X5 = bot_bot(set(A)) )
        | ( X5 = aa(set(A),set(A),insert(A,A3),bot_bot(set(A))) ) ) ) ).

% subset_singleton_iff
tff(fact_832_odd__nonzero,axiom,
    ! [Z: int] : aa(int,int,plus_plus(int,aa(int,int,plus_plus(int,one_one(int)),Z)),Z) != zero_zero(int) ).

% odd_nonzero
tff(fact_833_zless__add1__eq,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,ord_less(int,W),aa(int,int,plus_plus(int,Z),one_one(int)))
    <=> ( aa(int,$o,ord_less(int,W),Z)
        | ( W = Z ) ) ) ).

% zless_add1_eq
tff(fact_834_int__gr__induct,axiom,
    ! [K: int,I: int,P: fun(int,$o)] :
      ( aa(int,$o,ord_less(int,K),I)
     => ( aa(int,$o,P,aa(int,int,plus_plus(int,K),one_one(int)))
       => ( ! [I3: int] :
              ( aa(int,$o,ord_less(int,K),I3)
             => ( aa(int,$o,P,I3)
               => aa(int,$o,P,aa(int,int,plus_plus(int,I3),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_gr_induct
tff(fact_835_not__int__zless__negative,axiom,
    ! [Na: nat,M: nat] : ~ aa(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_836_max__word__not__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) != zero_zero(word(A)) ) ) ).

% max_word_not_0
tff(fact_837_word__not__simps_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A)] : ~ aa(word(A),$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_838_word__order_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : ~ aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),A3) ) ).

% word_order.extremum_strict
tff(fact_839_word__order_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] :
          ( ( A3 != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
        <=> aa(word(A),$o,ord_less(word(A),A3),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) ) ) ).

% word_order.not_eq_extremum
tff(fact_840_max__word__not__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : ~ aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),X) ) ).

% max_word_not_less
tff(fact_841_nonneg__int__cases,axiom,
    ! [K: int] :
      ( aa(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_842_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( aa(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_843_word__induct2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(word(A),$o),Na: word(A)] :
          ( aa(word(A),$o,P,zero_zero(word(A)))
         => ( ! [N: word(A)] :
                ( ( aa(word(A),word(A),plus_plus(word(A),one_one(word(A))),N) != zero_zero(word(A)) )
               => ( aa(word(A),$o,P,N)
                 => aa(word(A),$o,P,aa(word(A),word(A),plus_plus(word(A),one_one(word(A))),N)) ) )
           => aa(word(A),$o,P,Na) ) ) ) ).

% word_induct2
tff(fact_844_word__induct,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(word(A),$o),M: word(A)] :
          ( aa(word(A),$o,P,zero_zero(word(A)))
         => ( ! [N: word(A)] :
                ( aa(word(A),$o,P,N)
               => aa(word(A),$o,P,aa(word(A),word(A),plus_plus(word(A),one_one(word(A))),N)) )
           => aa(word(A),$o,P,M) ) ) ) ).

% word_induct
tff(fact_845_lt1__neq0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),one_one(word(A))),X)
        <=> ( X != zero_zero(word(A)) ) ) ) ).

% lt1_neq0
tff(fact_846_unsigned__eq__0__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & semiring_char_0(A) )
     => ! [W: word(B)] :
          ( ( aa(word(B),A,semiring_1_unsigned(B,A),W) = zero_zero(A) )
        <=> ( W = zero_zero(word(B)) ) ) ) ).

% unsigned_eq_0_iff
tff(fact_847_word__less__iff__unsigned,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & linordered_semidom(B) )
     => ! [A3: word(A),B3: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),A3),B3)
        <=> aa(B,$o,ord_less(B,aa(word(A),B,semiring_1_unsigned(A,B),A3)),aa(word(A),B,semiring_1_unsigned(A,B),B3)) ) ) ).

% word_less_iff_unsigned
tff(fact_848_log__of__power__le,axiom,
    ! [M: nat,B3: real,Na: nat] :
      ( aa(real,$o,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),M)),aa(nat,real,power_power(real,B3),Na))
     => ( aa(real,$o,ord_less(real,one_one(real)),B3)
       => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
         => aa(real,$o,ord_less_eq(real,aa(real,real,log(B3),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),Na)) ) ) ) ).

% log_of_power_le
tff(fact_849_square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),one_one(A))),X)
         => ( aa(A,$o,ord_less_eq(A,X),one_one(A))
           => aa(A,$o,ord_less_eq(A,aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)) ) ) ) ).

% square_le_1
tff(fact_850_sum__power2__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,plus_plus(A,aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% sum_power2_ge_zero
tff(fact_851_sum__power2__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,plus_plus(A,aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))),zero_zero(A))
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_852_numeral__code_I2_J,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Na: num] :
          aa(num,A,numeral_numeral(A),bit0(Na)) = $let(
            m: A,
            m:= aa(num,A,numeral_numeral(A),Na),
            aa(A,A,plus_plus(A,m),m) ) ) ).

% numeral_code(2)
tff(fact_853_less__log__of__power,axiom,
    ! [B3: real,Na: nat,M: real] :
      ( aa(real,$o,ord_less(real,aa(nat,real,power_power(real,B3),Na)),M)
     => ( aa(real,$o,ord_less(real,one_one(real)),B3)
       => aa(real,$o,ord_less(real,aa(nat,real,semiring_1_of_nat(real),Na)),aa(real,real,log(B3),M)) ) ) ).

% less_log_of_power
tff(fact_854_log__of__power__eq,axiom,
    ! [M: nat,B3: real,Na: nat] :
      ( ( aa(nat,real,semiring_1_of_nat(real),M) = aa(nat,real,power_power(real,B3),Na) )
     => ( aa(real,$o,ord_less(real,one_one(real)),B3)
       => ( aa(nat,real,semiring_1_of_nat(real),Na) = aa(real,real,log(B3),aa(nat,real,semiring_1_of_nat(real),M)) ) ) ) ).

% log_of_power_eq
tff(fact_855_neg__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: num] : aa(A,$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_856_not__zero__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: num] : ~ aa(A,$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_857_ex__power__ivl1,axiom,
    ! [B3: nat,K: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),B3)
     => ( aa(nat,$o,ord_less_eq(nat,one_one(nat)),K)
       => ? [N: nat] :
            ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,power_power(nat,B3),N)),K)
            & aa(nat,$o,ord_less(nat,K),aa(nat,nat,power_power(nat,B3),aa(nat,nat,plus_plus(nat,N),one_one(nat)))) ) ) ) ).

% ex_power_ivl1
tff(fact_858_ex__power__ivl2,axiom,
    ! [B3: nat,K: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),B3)
     => ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
       => ? [N: nat] :
            ( aa(nat,$o,ord_less(nat,aa(nat,nat,power_power(nat,B3),N)),K)
            & aa(nat,$o,ord_less_eq(nat,K),aa(nat,nat,power_power(nat,B3),aa(nat,nat,plus_plus(nat,N),one_one(nat)))) ) ) ) ).

% ex_power_ivl2
tff(fact_859_less__minus__one__simps_I3_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% less_minus_one_simps(3)
tff(fact_860_less__minus__one__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),one_one(A))),zero_zero(A)) ) ).

% less_minus_one_simps(1)
tff(fact_861_neg__numeral__less__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$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_862_neg__one__less__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$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_863_not__numeral__less__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$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_864_not__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$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_865_not__neg__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$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_866_uminus__numeral__One,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),one2)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% uminus_numeral_One
tff(fact_867_zero__less__two,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,plus_plus(A,one_one(A)),one_one(A))) ) ).

% zero_less_two
tff(fact_868_power__minus__Bit0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: A,K: num] : aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),X)),aa(num,nat,numeral_numeral(nat),bit0(K))) = aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(K))) ) ).

% power_minus_Bit0
tff(fact_869_power__less__imp__less__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Na: nat,B3: A] :
          ( aa(A,$o,ord_less(A,aa(nat,A,power_power(A,A3),Na)),aa(nat,A,power_power(A,B3),Na))
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B3)
           => aa(A,$o,ord_less(A,A3),B3) ) ) ) ).

% power_less_imp_less_base
tff(fact_870_power__le__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less_eq(A,A3),one_one(A))
           => aa(A,$o,ord_less_eq(A,aa(nat,A,power_power(A,A3),Na)),one_one(A)) ) ) ) ).

% power_le_one
tff(fact_871_int__cases4,axiom,
    ! [M: int] :
      ( ! [N: nat] : M != aa(nat,int,semiring_1_of_nat(int),N)
     => ~ ! [N: nat] :
            ( aa(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_872_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( aa(int,$o,ord_less(int,aa(int,int,plus_plus(int,aa(int,int,plus_plus(int,one_one(int)),Z)),Z)),zero_zero(int))
    <=> aa(int,$o,ord_less(int,Z),zero_zero(int)) ) ).

% odd_less_0_iff
tff(fact_873_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( aa(int,$o,ord_less_eq(int,one_one(int)),Z)
    <=> aa(int,$o,ord_less(int,zero_zero(int)),Z) ) ).

% int_one_le_iff_zero_less
tff(fact_874_word__induct__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(word(A),$o),M: word(A)] :
          ( aa(word(A),$o,P,zero_zero(word(A)))
         => ( ! [N: word(A)] :
                ( aa(word(A),$o,ord_less(word(A),N),M)
               => ( aa(word(A),$o,P,N)
                 => aa(word(A),$o,P,aa(word(A),word(A),plus_plus(word(A),one_one(word(A))),N)) ) )
           => aa(word(A),$o,P,M) ) ) ) ).

% word_induct_less
tff(fact_875_word__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),X),aa(word(A),word(A),plus_plus(word(A),X),one_one(word(A))))
          | ( aa(word(A),word(A),plus_plus(word(A),X),one_one(word(A))) = zero_zero(word(A)) ) ) ) ).

% word_overflow
tff(fact_876_word__gr0__conv__Suc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),M)
         => ? [N: word(A)] : M = aa(word(A),word(A),plus_plus(word(A),N),one_one(word(A))) ) ) ).

% word_gr0_conv_Suc
tff(fact_877_less__is__non__zero__p1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),K: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),A3),K)
         => ( aa(word(A),word(A),plus_plus(word(A),A3),one_one(word(A))) != zero_zero(word(A)) ) ) ) ).

% less_is_non_zero_p1
tff(fact_878_power2__eq__iff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [X: A,Y: A] :
          ( ( aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
        <=> ( ( X = Y )
            | ( X = aa(A,A,uminus_uminus(A),Y) ) ) ) ) ).

% power2_eq_iff
tff(fact_879_power__eq__iff__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat,A3: A,B3: A] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
           => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B3)
             => ( ( aa(nat,A,power_power(A,A3),Na) = aa(nat,A,power_power(A,B3),Na) )
              <=> ( A3 = B3 ) ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_880_power__eq__imp__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Na: nat,B3: A] :
          ( ( aa(nat,A,power_power(A,A3),Na) = aa(nat,A,power_power(A,B3),Na) )
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
           => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B3)
             => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
               => ( A3 = B3 ) ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_881_log2__of__power__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ( M = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) )
     => ( 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)) ) ) ).

% log2_of_power_eq
tff(fact_882_self__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,ord_less_eq(A,one_one(A)),A3)
         => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
           => aa(A,$o,ord_less_eq(A,A3),aa(nat,A,power_power(A,A3),Na)) ) ) ) ).

% self_le_power
tff(fact_883_nat__1__add__1,axiom,
    aa(nat,nat,plus_plus(nat,one_one(nat)),one_one(nat)) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% nat_1_add_1
tff(fact_884_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero(int) )
     => ( ! [N: nat] :
            ( ( K = aa(nat,int,semiring_1_of_nat(int),N) )
           => ~ aa(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,ord_less(nat,zero_zero(nat)),N) ) ) ) ).

% int_cases3
tff(fact_885_log__of__power__less,axiom,
    ! [M: nat,B3: real,Na: nat] :
      ( aa(real,$o,ord_less(real,aa(nat,real,semiring_1_of_nat(real),M)),aa(nat,real,power_power(real,B3),Na))
     => ( aa(real,$o,ord_less(real,one_one(real)),B3)
       => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
         => aa(real,$o,ord_less(real,aa(real,real,log(B3),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),Na)) ) ) ) ).

% log_of_power_less
tff(fact_886_self__le__ge2__pow,axiom,
    ! [K: nat,M: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
     => aa(nat,$o,ord_less_eq(nat,M),aa(nat,nat,power_power(nat,K),M)) ) ).

% self_le_ge2_pow
tff(fact_887_power2__nat__le__eq__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,power_power(nat,M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,power_power(nat,Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))
    <=> aa(nat,$o,ord_less_eq(nat,M),Na) ) ).

% power2_nat_le_eq_le
tff(fact_888_power2__nat__le__imp__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,power_power(nat,M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Na)
     => aa(nat,$o,ord_less_eq(nat,M),Na) ) ).

% power2_nat_le_imp_le
tff(fact_889_power2__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [A3: A] :
          ( ( aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(A) )
        <=> ( ( A3 = one_one(A) )
            | ( A3 = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).

% power2_eq_1_iff
tff(fact_890_zero__le__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% zero_le_power2
tff(fact_891_power2__eq__imp__eq,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y: A] :
          ( ( aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),X)
           => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Y)
             => ( X = Y ) ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_892_power2__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Y)
           => aa(A,$o,ord_less_eq(A,X),Y) ) ) ) ).

% power2_le_imp_le
tff(fact_893_power__strict__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,B3: A,Na: nat] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
           => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
             => aa(A,$o,ord_less(A,aa(nat,A,power_power(A,A3),Na)),aa(nat,A,power_power(A,B3),Na)) ) ) ) ) ).

% power_strict_mono
tff(fact_894_neg__int__cases,axiom,
    ! [K: int] :
      ( aa(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,ord_less(nat,zero_zero(nat)),N) ) ) ).

% neg_int_cases
tff(fact_895_two__realpow__ge__one,axiom,
    ! [Na: nat] : aa(real,$o,ord_less_eq(real,one_one(real)),aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),bit0(one2))),Na)) ).

% two_realpow_ge_one
tff(fact_896_less__log2__of__power,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),M)
     => aa(real,$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_897_not__sum__power2__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] : ~ aa(A,$o,ord_less(A,aa(A,A,plus_plus(A,aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))),zero_zero(A)) ) ).

% not_sum_power2_lt_zero
tff(fact_898_sum__power2__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,plus_plus(A,aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
        <=> ( ( X != zero_zero(A) )
            | ( Y != zero_zero(A) ) ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_899_arcosh__1,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,arcosh(A),one_one(A)) = zero_zero(A) ) ) ).

% arcosh_1
tff(fact_900_VEBT__internal_Otwo__realpow__ge__two,axiom,
    ! [Na: nat] :
      ( aa(real,$o,ord_less_eq(real,one_one(real)),aa(nat,real,semiring_1_of_nat(real),Na))
     => aa(real,$o,ord_less_eq(real,aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),bit0(one2))),Na)) ) ).

% VEBT_internal.two_realpow_ge_two
tff(fact_901_not__exp__less__eq__0__int,axiom,
    ! [Na: nat] : ~ aa(int,$o,ord_less_eq(int,aa(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_902_nat__induct2,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( aa(nat,$o,P,one_one(nat))
       => ( ! [N: nat] :
              ( aa(nat,$o,P,N)
             => aa(nat,$o,P,aa(nat,nat,plus_plus(nat,N),aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
         => aa(nat,$o,P,Na) ) ) ) ).

% nat_induct2
tff(fact_903_exp__add__not__zero__imp__left,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Na: nat] :
          ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,plus_plus(nat,M),Na)) != zero_zero(A) )
         => ( aa(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_904_exp__add__not__zero__imp__right,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Na: nat] :
          ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,plus_plus(nat,M),Na)) != zero_zero(A) )
         => ( aa(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_905_discrete,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
        <=> aa(A,$o,ord_less_eq(A,aa(A,A,plus_plus(A,A3),one_one(A))),B3) ) ) ).

% discrete
tff(fact_906_field__le__epsilon,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( ! [E: A] :
              ( aa(A,$o,ord_less(A,zero_zero(A)),E)
             => aa(A,$o,ord_less_eq(A,X),aa(A,A,plus_plus(A,Y),E)) )
         => aa(A,$o,ord_less_eq(A,X),Y) ) ) ).

% field_le_epsilon
tff(fact_907_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_908_arsinh__0,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,arsinh(A),zero_zero(A)) = zero_zero(A) ) ) ).

% arsinh_0
tff(fact_909_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_ak(A,A),Na,zero_zero(A)) ) ).

% of_nat_code
tff(fact_910_subsetI,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] :
      ( ! [X4: A] :
          ( member(A,X4,A2)
         => member(A,X4,B2) )
     => aa(set(A),$o,ord_less_eq(set(A),A2),B2) ) ).

% subsetI
tff(fact_911_subset__antisym,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] :
      ( aa(set(A),$o,ord_less_eq(set(A),A2),B2)
     => ( aa(set(A),$o,ord_less_eq(set(A),B2),A2)
       => ( A2 = B2 ) ) ) ).

% subset_antisym
tff(fact_912_ComplI,axiom,
    ! [A: $tType,C2: A,A2: set(A)] :
      ( ~ member(A,C2,A2)
     => member(A,C2,aa(set(A),set(A),uminus_uminus(set(A)),A2)) ) ).

% ComplI
tff(fact_913_Compl__iff,axiom,
    ! [A: $tType,C2: A,A2: set(A)] :
      ( member(A,C2,aa(set(A),set(A),uminus_uminus(set(A)),A2))
    <=> ~ member(A,C2,A2) ) ).

% Compl_iff
tff(fact_914_Compl__eq__Compl__iff,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] :
      ( ( aa(set(A),set(A),uminus_uminus(set(A)),A2) = aa(set(A),set(A),uminus_uminus(set(A)),B2) )
    <=> ( A2 = B2 ) ) ).

% Compl_eq_Compl_iff
tff(fact_915_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,plus_plus(nat,K),M)),aa(nat,nat,plus_plus(nat,K),Na))
    <=> aa(nat,$o,ord_less_eq(nat,M),Na) ) ).

% nat_add_left_cancel_le
tff(fact_916_Compl__anti__mono,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] :
      ( aa(set(A),$o,ord_less_eq(set(A),A2),B2)
     => aa(set(A),$o,ord_less_eq(set(A),aa(set(A),set(A),uminus_uminus(set(A)),B2)),aa(set(A),set(A),uminus_uminus(set(A)),A2)) ) ).

% Compl_anti_mono
tff(fact_917_Compl__subset__Compl__iff,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] :
      ( aa(set(A),$o,ord_less_eq(set(A),aa(set(A),set(A),uminus_uminus(set(A)),A2)),aa(set(A),set(A),uminus_uminus(set(A)),B2))
    <=> aa(set(A),$o,ord_less_eq(set(A),B2),A2) ) ).

% Compl_subset_Compl_iff
tff(fact_918_add__leE,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,plus_plus(nat,M),K)),Na)
     => ~ ( aa(nat,$o,ord_less_eq(nat,M),Na)
         => ~ aa(nat,$o,ord_less_eq(nat,K),Na) ) ) ).

% add_leE
tff(fact_919_le__add1,axiom,
    ! [Na: nat,M: nat] : aa(nat,$o,ord_less_eq(nat,Na),aa(nat,nat,plus_plus(nat,Na),M)) ).

% le_add1
tff(fact_920_le__add2,axiom,
    ! [Na: nat,M: nat] : aa(nat,$o,ord_less_eq(nat,Na),aa(nat,nat,plus_plus(nat,M),Na)) ).

% le_add2
tff(fact_921_le__refl,axiom,
    ! [Na: nat] : aa(nat,$o,ord_less_eq(nat,Na),Na) ).

% le_refl
tff(fact_922_add__leD1,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,plus_plus(nat,M),K)),Na)
     => aa(nat,$o,ord_less_eq(nat,M),Na) ) ).

% add_leD1
tff(fact_923_add__leD2,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,plus_plus(nat,M),K)),Na)
     => aa(nat,$o,ord_less_eq(nat,K),Na) ) ).

% add_leD2
tff(fact_924_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => ( aa(nat,$o,ord_less_eq(nat,J),K)
       => aa(nat,$o,ord_less_eq(nat,I),K) ) ) ).

% le_trans
tff(fact_925_eq__imp__le,axiom,
    ! [M: nat,Na: nat] :
      ( ( M = Na )
     => aa(nat,$o,ord_less_eq(nat,M),Na) ) ).

% eq_imp_le
tff(fact_926_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),L)
     => ? [N: nat] : L = aa(nat,nat,plus_plus(nat,K),N) ) ).

% le_Suc_ex
tff(fact_927_le__antisym,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),Na)
     => ( aa(nat,$o,ord_less_eq(nat,Na),M)
       => ( M = Na ) ) ) ).

% le_antisym
tff(fact_928_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => ( aa(nat,$o,ord_less_eq(nat,K),L)
       => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,plus_plus(nat,I),K)),aa(nat,nat,plus_plus(nat,J),L)) ) ) ).

% add_le_mono
tff(fact_929_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,plus_plus(nat,I),K)),aa(nat,nat,plus_plus(nat,J),K)) ) ).

% add_le_mono1
tff(fact_930_nat__le__linear,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),Na)
      | aa(nat,$o,ord_less_eq(nat,Na),M) ) ).

% nat_le_linear
tff(fact_931_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => aa(nat,$o,ord_less_eq(nat,I),aa(nat,nat,plus_plus(nat,J),M)) ) ).

% trans_le_add1
tff(fact_932_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => aa(nat,$o,ord_less_eq(nat,I),aa(nat,nat,plus_plus(nat,M),J)) ) ).

% trans_le_add2
tff(fact_933_nat__le__iff__add,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),Na)
    <=> ? [K3: nat] : Na = aa(nat,nat,plus_plus(nat,M),K3) ) ).

% nat_le_iff_add
tff(fact_934_complete__real,axiom,
    ! [S: set(real)] :
      ( ? [X2: real] : member(real,X2,S)
     => ( ? [Z3: real] :
          ! [X4: real] :
            ( member(real,X4,S)
           => aa(real,$o,ord_less_eq(real,X4),Z3) )
       => ? [Y3: real] :
            ( ! [X2: real] :
                ( member(real,X2,S)
               => aa(real,$o,ord_less_eq(real,X2),Y3) )
            & ! [Z3: real] :
                ( ! [X4: real] :
                    ( member(real,X4,S)
                   => aa(real,$o,ord_less_eq(real,X4),Z3) )
               => aa(real,$o,ord_less_eq(real,Y3),Z3) ) ) ) ) ).

% complete_real
tff(fact_935_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun(nat,$o),K: nat,B3: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [Y3: nat] :
            ( aa(nat,$o,P,Y3)
           => aa(nat,$o,ord_less_eq(nat,Y3),B3) )
       => ? [X4: nat] :
            ( aa(nat,$o,P,X4)
            & ! [Y4: nat] :
                ( aa(nat,$o,P,Y4)
               => aa(nat,$o,ord_less_eq(nat,Y4),X4) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_936_ComplD,axiom,
    ! [A: $tType,C2: A,A2: set(A)] :
      ( member(A,C2,aa(set(A),set(A),uminus_uminus(set(A)),A2))
     => ~ member(A,C2,A2) ) ).

% ComplD
tff(fact_937_in__mono,axiom,
    ! [A: $tType,A2: set(A),B2: set(A),X: A] :
      ( aa(set(A),$o,ord_less_eq(set(A),A2),B2)
     => ( member(A,X,A2)
       => member(A,X,B2) ) ) ).

% in_mono
tff(fact_938_subsetD,axiom,
    ! [A: $tType,A2: set(A),B2: set(A),C2: A] :
      ( aa(set(A),$o,ord_less_eq(set(A),A2),B2)
     => ( member(A,C2,A2)
       => member(A,C2,B2) ) ) ).

% subsetD
tff(fact_939_equalityE,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] :
      ( ( A2 = B2 )
     => ~ ( aa(set(A),$o,ord_less_eq(set(A),A2),B2)
         => ~ aa(set(A),$o,ord_less_eq(set(A),B2),A2) ) ) ).

% equalityE
tff(fact_940_subset__eq,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] :
      ( aa(set(A),$o,ord_less_eq(set(A),A2),B2)
    <=> ! [X3: A] :
          ( member(A,X3,A2)
         => member(A,X3,B2) ) ) ).

% subset_eq
tff(fact_941_equalityD1,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] :
      ( ( A2 = B2 )
     => aa(set(A),$o,ord_less_eq(set(A),A2),B2) ) ).

% equalityD1
tff(fact_942_Set_OequalityD2,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] :
      ( ( A2 = B2 )
     => aa(set(A),$o,ord_less_eq(set(A),B2),A2) ) ).

% Set.equalityD2
tff(fact_943_subset__iff,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] :
      ( aa(set(A),$o,ord_less_eq(set(A),A2),B2)
    <=> ! [T: A] :
          ( member(A,T,A2)
         => member(A,T,B2) ) ) ).

% subset_iff
tff(fact_944_subset__refl,axiom,
    ! [A: $tType,A2: set(A)] : aa(set(A),$o,ord_less_eq(set(A),A2),A2) ).

% subset_refl
tff(fact_945_Collect__mono,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ! [X4: A] :
          ( aa(A,$o,P,X4)
         => aa(A,$o,Q,X4) )
     => aa(set(A),$o,ord_less_eq(set(A),collect(A,P)),collect(A,Q)) ) ).

% Collect_mono
tff(fact_946_subset__trans,axiom,
    ! [A: $tType,A2: set(A),B2: set(A),C3: set(A)] :
      ( aa(set(A),$o,ord_less_eq(set(A),A2),B2)
     => ( aa(set(A),$o,ord_less_eq(set(A),B2),C3)
       => aa(set(A),$o,ord_less_eq(set(A),A2),C3) ) ) ).

% subset_trans
tff(fact_947_set__eq__subset,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] :
      ( ( A2 = B2 )
    <=> ( aa(set(A),$o,ord_less_eq(set(A),A2),B2)
        & aa(set(A),$o,ord_less_eq(set(A),B2),A2) ) ) ).

% set_eq_subset
tff(fact_948_uminus__set__def,axiom,
    ! [A: $tType,A2: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A2) = collect(A,aa(fun(A,$o),fun(A,$o),uminus_uminus(fun(A,$o)),aTP_Lamp_aa(set(A),fun(A,$o),A2))) ).

% uminus_set_def
tff(fact_949_less__eq__set__def,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] :
      ( aa(set(A),$o,ord_less_eq(set(A),A2),B2)
    <=> aa(fun(A,$o),$o,ord_less_eq(fun(A,$o),aTP_Lamp_aa(set(A),fun(A,$o),A2)),aTP_Lamp_aa(set(A),fun(A,$o),B2)) ) ).

% less_eq_set_def
tff(fact_950_Collect__mono__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( aa(set(A),$o,ord_less_eq(set(A),collect(A,P)),collect(A,Q))
    <=> ! [X3: A] :
          ( aa(A,$o,P,X3)
         => aa(A,$o,Q,X3) ) ) ).

% Collect_mono_iff
tff(fact_951_double__complement,axiom,
    ! [A: $tType,A2: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A2)) = A2 ).

% double_complement
tff(fact_952_verit__la__generic,axiom,
    ! [A3: int,X: int] :
      ( aa(int,$o,ord_less_eq(int,A3),X)
      | ( A3 = X )
      | aa(int,$o,ord_less_eq(int,X),A3) ) ).

% verit_la_generic
tff(fact_953_pred__subset__eq,axiom,
    ! [A: $tType,R: set(A),S: set(A)] :
      ( aa(fun(A,$o),$o,ord_less_eq(fun(A,$o),aTP_Lamp_aa(set(A),fun(A,$o),R)),aTP_Lamp_aa(set(A),fun(A,$o),S))
    <=> aa(set(A),$o,ord_less_eq(set(A),R),S) ) ).

% pred_subset_eq
tff(fact_954_unat__cong,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ( X = Y )
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),X) = aa(word(A),nat,semiring_1_unsigned(A,nat),Y) ) ) ) ).

% unat_cong
tff(fact_955_olen__add__eqv,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Y),aa(word(A),word(A),plus_plus(word(A),Y),X))
        <=> aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),plus_plus(word(A),Y),X)) ) ) ).

% olen_add_eqv
tff(fact_956_le__no__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),B3: word(A),A3: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),X),B3)
         => ( aa(word(A),$o,ord_less_eq(word(A),A3),aa(word(A),word(A),plus_plus(word(A),A3),B3))
           => aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),plus_plus(word(A),A3),B3)) ) ) ) ).

% le_no_overflow
tff(fact_957_word__le__nat__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),A3),B3)
        <=> aa(nat,$o,ord_less_eq(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),A3)),aa(word(A),nat,semiring_1_unsigned(A,nat),B3)) ) ) ).

% word_le_nat_alt
tff(fact_958_word__random,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),X6: word(A),X: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),P2),aa(word(A),word(A),plus_plus(word(A),P2),X6))
         => ( aa(word(A),$o,ord_less_eq(word(A),X),X6)
           => aa(word(A),$o,ord_less_eq(word(A),P2),aa(word(A),word(A),plus_plus(word(A),P2),X)) ) ) ) ).

% word_random
tff(fact_959_unat__plus__simple,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),plus_plus(word(A),X),Y))
        <=> ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),plus_plus(word(A),X),Y)) = aa(nat,nat,plus_plus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) ) ) ).

% unat_plus_simple
tff(fact_960_word__unat__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V2: word(A),W: word(A)] :
          ( ( V2 = W )
        <=> ( aa(word(A),nat,semiring_1_unsigned(A,nat),V2) = aa(word(A),nat,semiring_1_unsigned(A,nat),W) ) ) ) ).

% word_unat_eq_iff
tff(fact_961_word__eq__unatI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V2: word(A),W: word(A)] :
          ( ( aa(word(A),nat,semiring_1_unsigned(A,nat),V2) = aa(word(A),nat,semiring_1_unsigned(A,nat),W) )
         => ( V2 = W ) ) ) ).

% word_eq_unatI
tff(fact_962_word__sub__mono2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A),C2: word(A),D2: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),plus_plus(word(A),A3),B3)),aa(word(A),word(A),plus_plus(word(A),C2),D2))
         => ( aa(word(A),$o,ord_less_eq(word(A),C2),A3)
           => ( aa(word(A),$o,ord_less_eq(word(A),B3),aa(word(A),word(A),plus_plus(word(A),A3),B3))
             => ( aa(word(A),$o,ord_less_eq(word(A),D2),aa(word(A),word(A),plus_plus(word(A),C2),D2))
               => aa(word(A),$o,ord_less_eq(word(A),B3),D2) ) ) ) ) ) ).

% word_sub_mono2
tff(fact_963_word__le__plus__either,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A),Z: word(A)] :
          ( ( aa(word(A),$o,ord_less_eq(word(A),X),Y)
            | aa(word(A),$o,ord_less_eq(word(A),X),Z) )
         => ( aa(word(A),$o,ord_less_eq(word(A),Y),aa(word(A),word(A),plus_plus(word(A),Y),Z))
           => aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),plus_plus(word(A),Y),Z)) ) ) ) ).

% word_le_plus_either
tff(fact_964_word__plus__mcs__3,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V2: word(A),W: word(A),X: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),V2),W)
         => ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),plus_plus(word(A),W),X))
           => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),plus_plus(word(A),V2),X)),aa(word(A),word(A),plus_plus(word(A),W),X)) ) ) ) ).

% word_plus_mcs_3
tff(fact_965_word__plus__mcs__4,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V2: word(A),X: word(A),W: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),plus_plus(word(A),V2),X)),aa(word(A),word(A),plus_plus(word(A),W),X))
         => ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),plus_plus(word(A),V2),X))
           => aa(word(A),$o,ord_less_eq(word(A),V2),W) ) ) ) ).

% word_plus_mcs_4
tff(fact_966_word__plus__mono__right,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Z: word(A),X: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Y),Z)
         => ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),plus_plus(word(A),X),Z))
           => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),plus_plus(word(A),X),Y)),aa(word(A),word(A),plus_plus(word(A),X),Z)) ) ) ) ).

% word_plus_mono_right
tff(fact_967_word__add__le__iff2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),I),aa(word(A),word(A),plus_plus(word(A),I),K))
         => ( aa(word(A),$o,ord_less_eq(word(A),J),aa(word(A),word(A),plus_plus(word(A),J),K))
           => ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),plus_plus(word(A),I),K)),aa(word(A),word(A),plus_plus(word(A),J),K))
            <=> aa(word(A),$o,ord_less_eq(word(A),I),J) ) ) ) ) ).

% word_add_le_iff2
tff(fact_968_word__plus__mcs__4_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),V2: word(A),W: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),plus_plus(word(A),X),V2)),aa(word(A),word(A),plus_plus(word(A),X),W))
         => ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),plus_plus(word(A),X),V2))
           => aa(word(A),$o,ord_less_eq(word(A),V2),W) ) ) ) ).

% word_plus_mcs_4'
tff(fact_969_word__plus__mono__right2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A),C2: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),A3),aa(word(A),word(A),plus_plus(word(A),A3),B3))
         => ( aa(word(A),$o,ord_less_eq(word(A),C2),B3)
           => aa(word(A),$o,ord_less_eq(word(A),A3),aa(word(A),word(A),plus_plus(word(A),A3),C2)) ) ) ) ).

% word_plus_mono_right2
tff(fact_970_le__ucast__ucast__le,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [X: word(A),Y: word(B)] :
          ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Y))
         => aa(word(B),$o,ord_less_eq(word(B),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),X)),Y) ) ) ).

% le_ucast_ucast_le
tff(fact_971_word__add__increasing,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),W: word(A),X: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),plus_plus(word(A),P2),W)),X)
         => ( aa(word(A),$o,ord_less_eq(word(A),P2),aa(word(A),word(A),plus_plus(word(A),P2),W))
           => aa(word(A),$o,ord_less_eq(word(A),P2),X) ) ) ) ).

% word_add_increasing
tff(fact_972_word__plus__mono__left,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Z: word(A),X: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Y),Z)
         => ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),plus_plus(word(A),X),Z))
           => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),plus_plus(word(A),Y),X)),aa(word(A),word(A),plus_plus(word(A),Z),X)) ) ) ) ).

% word_plus_mono_left
tff(fact_973_Collect__neg__eq,axiom,
    ! [A: $tType,P: fun(A,$o)] : collect(A,aTP_Lamp_al(fun(A,$o),fun(A,$o),P)) = aa(set(A),set(A),uminus_uminus(set(A)),collect(A,P)) ).

% Collect_neg_eq
tff(fact_974_Compl__eq,axiom,
    ! [A: $tType,A2: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A2) = collect(A,aTP_Lamp_am(set(A),fun(A,$o),A2)) ).

% Compl_eq
tff(fact_975_constraint__expand,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Lower: word(A),Upper: word(A)] :
          ( member(word(A),X,collect(word(A),aa(word(A),fun(word(A),$o),aTP_Lamp_an(word(A),fun(word(A),fun(word(A),$o)),Lower),Upper)))
        <=> ( aa(word(A),$o,ord_less_eq(word(A),Lower),X)
            & aa(word(A),$o,ord_less_eq(word(A),X),Upper) ) ) ) ).

% constraint_expand
tff(fact_976_Collect__subset,axiom,
    ! [A: $tType,A2: set(A),P: fun(A,$o)] : aa(set(A),$o,ord_less_eq(set(A),collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ao(set(A),fun(fun(A,$o),fun(A,$o)),A2),P))),A2) ).

% Collect_subset
tff(fact_977_ucast__nat__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [X: word(B)] : aa(nat,word(A),semiring_1_of_nat(word(A)),aa(word(B),nat,semiring_1_unsigned(B,nat),X)) = aa(word(B),word(A),semiring_1_unsigned(B,word(A)),X) ) ).

% ucast_nat_def
tff(fact_978_ucast__1,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ( aa(word(B),word(A),semiring_1_unsigned(B,word(A)),one_one(word(B))) = one_one(word(A)) ) ) ).

% ucast_1
tff(fact_979_ucast__0__I,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [X: word(A)] :
          ( ( X = zero_zero(word(A)) )
         => ( aa(word(A),word(B),semiring_1_unsigned(A,word(B)),X) = zero_zero(word(B)) ) ) ) ).

% ucast_0_I
tff(fact_980_ucast__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ( aa(word(B),word(A),semiring_1_unsigned(B,word(A)),zero_zero(word(B))) = zero_zero(word(A)) ) ) ).

% ucast_0
tff(fact_981_word__unat__Rep__inject1,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(A)] :
          ( ( aa(word(A),nat,semiring_1_unsigned(A,nat),X) = aa(word(B),nat,semiring_1_unsigned(B,nat),one_one(word(B))) )
        <=> ( X = one_one(word(A)) ) ) ) ).

% word_unat_Rep_inject1
tff(fact_982_un__ui__le,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [A3: word(A),B3: word(B)] :
          ( aa(nat,$o,ord_less_eq(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),A3)),aa(word(B),nat,semiring_1_unsigned(B,nat),B3))
        <=> aa(int,$o,ord_less_eq(int,aa(word(A),int,semiring_1_unsigned(A,int),A3)),aa(word(B),int,semiring_1_unsigned(B,int),B3)) ) ) ).

% un_ui_le
tff(fact_983_le__unat__uoi,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: nat,Z: word(A)] :
          ( aa(nat,$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_984_word__unat__less__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: nat] :
          ( aa(word(A),$o,ord_less_eq(word(A),A3),aa(nat,word(A),semiring_1_of_nat(word(A)),B3))
         => aa(nat,$o,ord_less_eq(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),A3)),B3) ) ) ).

% word_unat_less_le
tff(fact_985_word__of__nat__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,X: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,Na),aa(word(A),nat,semiring_1_unsigned(A,nat),X))
         => aa(word(A),$o,ord_less_eq(word(A),aa(nat,word(A),semiring_1_of_nat(word(A)),Na)),X) ) ) ).

% word_of_nat_le
tff(fact_986_word__arith__nat__add,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(word(A),word(A),plus_plus(word(A),A3),B3) = aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,plus_plus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),A3)),aa(word(A),nat,semiring_1_unsigned(A,nat),B3))) ) ).

% word_arith_nat_add
tff(fact_987_unat__eq__zero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( ( aa(word(A),nat,semiring_1_unsigned(A,nat),X) = zero_zero(nat) )
        <=> ( X = zero_zero(word(A)) ) ) ) ).

% unat_eq_zero
tff(fact_988_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_989_unat__mono,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),A3),B3)
         => aa(nat,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),A3)),aa(word(A),nat,semiring_1_unsigned(A,nat),B3)) ) ) ).

% unat_mono
tff(fact_990_word__less__nat__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),A3),B3)
        <=> aa(nat,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),A3)),aa(word(A),nat,semiring_1_unsigned(A,nat),B3)) ) ) ).

% word_less_nat_alt
tff(fact_991_unat__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),nat,semiring_1_unsigned(A,nat),one_one(word(A))) = one_one(nat) ) ) ).

% unat_1
tff(fact_992_uint__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),W) = aa(nat,int,semiring_1_of_nat(int),aa(word(A),nat,semiring_1_unsigned(A,nat),W)) ) ).

% uint_nat
tff(fact_993_unat__of__nat__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A),C2: nat] :
          ( aa(word(A),$o,ord_less(word(A),A3),B3)
         => ( ( aa(word(A),nat,semiring_1_unsigned(A,nat),B3) = C2 )
           => aa(word(A),$o,ord_less(word(A),A3),aa(nat,word(A),semiring_1_of_nat(word(A)),C2)) ) ) ) ).

% unat_of_nat_less
tff(fact_994_unat__1__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),one_one(word(A))),X)
        <=> aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(word(A),nat,semiring_1_unsigned(A,nat),X)) ) ) ).

% unat_1_0
tff(fact_995_unat__max__word__pos,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(nat,$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_996_unat__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(word(A),nat,semiring_1_unsigned(A,nat),X))
        <=> ( X != zero_zero(word(A)) ) ) ) ).

% unat_gt_0
tff(fact_997_word__overflow__unat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),plus_plus(word(A),X),one_one(word(A)))) = aa(nat,nat,plus_plus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),X)),one_one(nat)) )
          | ( aa(word(A),word(A),plus_plus(word(A),X),one_one(word(A))) = zero_zero(word(A)) ) ) ) ).

% word_overflow_unat
tff(fact_998_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_999_word__of__nat__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,X: word(A)] :
          ( aa(nat,$o,ord_less(nat,Na),aa(word(A),nat,semiring_1_unsigned(A,nat),X))
         => aa(word(A),$o,ord_less(word(A),aa(nat,word(A),semiring_1_of_nat(word(A)),Na)),X) ) ) ).

% word_of_nat_less
tff(fact_1000_unat__less__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Na: nat] :
          ( aa(word(A),$o,ord_less(word(A),X),aa(nat,word(A),semiring_1_of_nat(word(A)),Na))
         => aa(nat,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),X)),Na) ) ) ).

% unat_less_helper
tff(fact_1001_linordered__field__no__ub,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X2: A] :
        ? [X_12: A] : aa(A,$o,ord_less(A,X2),X_12) ) ).

% linordered_field_no_ub
tff(fact_1002_linordered__field__no__lb,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X2: A] :
        ? [Y3: A] : aa(A,$o,ord_less(A,Y3),X2) ) ).

% linordered_field_no_lb
tff(fact_1003_lt__plus__1__le__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,MaxBound: word(A),X: word(A)] :
          ( aa(nat,$o,ord_less(nat,Na),aa(word(A),nat,semiring_1_unsigned(A,nat),MaxBound))
         => ( aa(word(A),$o,ord_less(word(A),X),aa(word(A),word(A),plus_plus(word(A),one_one(word(A))),aa(nat,word(A),semiring_1_of_nat(word(A)),Na)))
          <=> aa(word(A),$o,ord_less_eq(word(A),X),aa(nat,word(A),semiring_1_of_nat(word(A)),Na)) ) ) ) ).

% lt_plus_1_le_word
tff(fact_1004_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B3: nat,K: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),B3)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
       => ( ( archimedean_ceiling(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B3)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(int,int,plus_plus(int,aa(nat,int,semiring_1_of_nat(int),Na)),one_one(int)) )
        <=> ( aa(nat,$o,ord_less(nat,aa(nat,nat,power_power(nat,B3),Na)),K)
            & aa(nat,$o,ord_less_eq(nat,K),aa(nat,nat,power_power(nat,B3),aa(nat,nat,plus_plus(nat,Na),one_one(nat)))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_1005_ceiling__log__nat__eq__if,axiom,
    ! [B3: nat,Na: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,power_power(nat,B3),Na)),K)
     => ( aa(nat,$o,ord_less_eq(nat,K),aa(nat,nat,power_power(nat,B3),aa(nat,nat,plus_plus(nat,Na),one_one(nat))))
       => ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),B3)
         => ( archimedean_ceiling(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B3)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(int,int,plus_plus(int,aa(nat,int,semiring_1_of_nat(int),Na)),one_one(int)) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_1006_floor__log__nat__eq__powr__iff,axiom,
    ! [B3: nat,K: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),B3)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
       => ( ( archim6421214686448440834_floor(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B3)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(nat,int,semiring_1_of_nat(int),Na) )
        <=> ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,power_power(nat,B3),Na)),K)
            & aa(nat,$o,ord_less(nat,K),aa(nat,nat,power_power(nat,B3),aa(nat,nat,plus_plus(nat,Na),one_one(nat)))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_1007_ex__has__greatest__nat__lemma,axiom,
    ! [A: $tType,P: fun(A,$o),K: A,F2: fun(A,nat),Na: nat] :
      ( aa(A,$o,P,K)
     => ( ! [X4: A] :
            ( aa(A,$o,P,X4)
           => ? [Y4: A] :
                ( aa(A,$o,P,Y4)
                & ~ aa(nat,$o,ord_less_eq(nat,aa(A,nat,F2,Y4)),aa(A,nat,F2,X4)) ) )
       => ? [Y3: A] :
            ( aa(A,$o,P,Y3)
            & ~ aa(nat,$o,ord_less(nat,aa(A,nat,F2,Y3)),aa(nat,nat,plus_plus(nat,aa(A,nat,F2,K)),Na)) ) ) ) ).

% ex_has_greatest_nat_lemma
tff(fact_1008_nat__geq__1__eq__neqz,axiom,
    ! [X: nat] :
      ( aa(nat,$o,ord_less_eq(nat,one_one(nat)),X)
    <=> ( X != zero_zero(nat) ) ) ).

% nat_geq_1_eq_neqz
tff(fact_1009_ucast__id,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(word(A),word(A),semiring_1_unsigned(A,word(A)),W) = W ) ).

% ucast_id
tff(fact_1010_predicate1I,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ! [X4: A] :
          ( aa(A,$o,P,X4)
         => aa(A,$o,Q,X4) )
     => aa(fun(A,$o),$o,ord_less_eq(fun(A,$o),P),Q) ) ).

% predicate1I
tff(fact_1011_floor__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archim6421214686448440834_floor(A,zero_zero(A)) = zero_zero(int) ) ) ).

% floor_zero
tff(fact_1012_floor__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num] : archim6421214686448440834_floor(A,aa(num,A,numeral_numeral(A),V2)) = aa(num,int,numeral_numeral(int),V2) ) ).

% floor_numeral
tff(fact_1013_ceiling__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_ceiling(A,zero_zero(A)) = zero_zero(int) ) ) ).

% ceiling_zero
tff(fact_1014_ceiling__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num] : archimedean_ceiling(A,aa(num,A,numeral_numeral(A),V2)) = aa(num,int,numeral_numeral(int),V2) ) ).

% ceiling_numeral
tff(fact_1015_floor__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archim6421214686448440834_floor(A,one_one(A)) = one_one(int) ) ) ).

% floor_one
tff(fact_1016_ceiling__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_ceiling(A,one_one(A)) = one_one(int) ) ) ).

% ceiling_one
tff(fact_1017_floor__of__nat,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Na: nat] : archim6421214686448440834_floor(A,aa(nat,A,semiring_1_of_nat(A),Na)) = aa(nat,int,semiring_1_of_nat(int),Na) ) ).

% floor_of_nat
tff(fact_1018_ceiling__of__nat,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Na: nat] : archimedean_ceiling(A,aa(nat,A,semiring_1_of_nat(A),Na)) = aa(nat,int,semiring_1_of_nat(int),Na) ) ).

% ceiling_of_nat
tff(fact_1019_zero__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,ord_less_eq(int,zero_zero(int)),archim6421214686448440834_floor(A,X))
        <=> aa(A,$o,ord_less_eq(A,zero_zero(A)),X) ) ) ).

% zero_le_floor
tff(fact_1020_numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( aa(int,$o,ord_less_eq(int,aa(num,int,numeral_numeral(int),V2)),archim6421214686448440834_floor(A,X))
        <=> aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),V2)),X) ) ) ).

% numeral_le_floor
tff(fact_1021_floor__less__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,ord_less(int,archim6421214686448440834_floor(A,X)),zero_zero(int))
        <=> aa(A,$o,ord_less(A,X),zero_zero(A)) ) ) ).

% floor_less_zero
tff(fact_1022_floor__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( aa(int,$o,ord_less(int,archim6421214686448440834_floor(A,X)),aa(num,int,numeral_numeral(int),V2))
        <=> aa(A,$o,ord_less(A,X),aa(num,A,numeral_numeral(A),V2)) ) ) ).

% floor_less_numeral
tff(fact_1023_ceiling__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,ord_less_eq(int,archimedean_ceiling(A,X)),zero_zero(int))
        <=> aa(A,$o,ord_less_eq(A,X),zero_zero(A)) ) ) ).

% ceiling_le_zero
tff(fact_1024_ceiling__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( aa(int,$o,ord_less_eq(int,archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V2))
        <=> aa(A,$o,ord_less_eq(A,X),aa(num,A,numeral_numeral(A),V2)) ) ) ).

% ceiling_le_numeral
tff(fact_1025_zero__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,ord_less(int,zero_zero(int)),archim6421214686448440834_floor(A,X))
        <=> aa(A,$o,ord_less_eq(A,one_one(A)),X) ) ) ).

% zero_less_floor
tff(fact_1026_floor__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,ord_less_eq(int,archim6421214686448440834_floor(A,X)),zero_zero(int))
        <=> aa(A,$o,ord_less(A,X),one_one(A)) ) ) ).

% floor_le_zero
tff(fact_1027_zero__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,ord_less(int,zero_zero(int)),archimedean_ceiling(A,X))
        <=> aa(A,$o,ord_less(A,zero_zero(A)),X) ) ) ).

% zero_less_ceiling
tff(fact_1028_numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( aa(int,$o,ord_less(int,aa(num,int,numeral_numeral(int),V2)),archimedean_ceiling(A,X))
        <=> aa(A,$o,ord_less(A,aa(num,A,numeral_numeral(A),V2)),X) ) ) ).

% numeral_less_ceiling
tff(fact_1029_one__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,ord_less_eq(int,one_one(int)),archim6421214686448440834_floor(A,X))
        <=> aa(A,$o,ord_less_eq(A,one_one(A)),X) ) ) ).

% one_le_floor
tff(fact_1030_ceiling__less__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,ord_less(int,archimedean_ceiling(A,X)),one_one(int))
        <=> aa(A,$o,ord_less_eq(A,X),zero_zero(A)) ) ) ).

% ceiling_less_one
tff(fact_1031_one__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,ord_less_eq(int,one_one(int)),archimedean_ceiling(A,X))
        <=> aa(A,$o,ord_less(A,zero_zero(A)),X) ) ) ).

% one_le_ceiling
tff(fact_1032_ceiling__add__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] : archimedean_ceiling(A,aa(A,A,plus_plus(A,X),aa(num,A,numeral_numeral(A),V2))) = aa(int,int,plus_plus(int,archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V2)) ) ).

% ceiling_add_numeral
tff(fact_1033_floor__less__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,ord_less(int,archim6421214686448440834_floor(A,X)),one_one(int))
        <=> aa(A,$o,ord_less(A,X),one_one(A)) ) ) ).

% floor_less_one
tff(fact_1034_floor__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num] : archim6421214686448440834_floor(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2)) ) ).

% floor_neg_numeral
tff(fact_1035_ceiling__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,ord_less_eq(int,archimedean_ceiling(A,X)),one_one(int))
        <=> aa(A,$o,ord_less_eq(A,X),one_one(A)) ) ) ).

% ceiling_le_one
tff(fact_1036_one__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,ord_less(int,one_one(int)),archimedean_ceiling(A,X))
        <=> aa(A,$o,ord_less(A,one_one(A)),X) ) ) ).

% one_less_ceiling
tff(fact_1037_ceiling__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num] : archimedean_ceiling(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2)) ) ).

% ceiling_neg_numeral
tff(fact_1038_ceiling__add__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_ceiling(A,aa(A,A,plus_plus(A,X),one_one(A))) = aa(int,int,plus_plus(int,archimedean_ceiling(A,X)),one_one(int)) ) ).

% ceiling_add_one
tff(fact_1039_floor__numeral__power,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: num,Na: nat] : archim6421214686448440834_floor(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),X)),Na)) = aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),Na) ) ).

% floor_numeral_power
tff(fact_1040_ceiling__numeral__power,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: num,Na: nat] : archimedean_ceiling(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),X)),Na)) = aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),Na) ) ).

% ceiling_numeral_power
tff(fact_1041_ceiling__less__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,ord_less(int,archimedean_ceiling(A,X)),zero_zero(int))
        <=> aa(A,$o,ord_less_eq(A,X),aa(A,A,uminus_uminus(A),one_one(A))) ) ) ).

% ceiling_less_zero
tff(fact_1042_zero__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,ord_less_eq(int,zero_zero(int)),archimedean_ceiling(A,X))
        <=> aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),one_one(A))),X) ) ) ).

% zero_le_ceiling
tff(fact_1043_numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( aa(int,$o,ord_less(int,aa(num,int,numeral_numeral(int),V2)),archim6421214686448440834_floor(A,X))
        <=> aa(A,$o,ord_less_eq(A,aa(A,A,plus_plus(A,aa(num,A,numeral_numeral(A),V2)),one_one(A))),X) ) ) ).

% numeral_less_floor
tff(fact_1044_floor__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( aa(int,$o,ord_less_eq(int,archim6421214686448440834_floor(A,X)),aa(num,int,numeral_numeral(int),V2))
        <=> aa(A,$o,ord_less(A,X),aa(A,A,plus_plus(A,aa(num,A,numeral_numeral(A),V2)),one_one(A))) ) ) ).

% floor_le_numeral
tff(fact_1045_one__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,ord_less(int,one_one(int)),archim6421214686448440834_floor(A,X))
        <=> aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),bit0(one2))),X) ) ) ).

% one_less_floor
tff(fact_1046_floor__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,ord_less_eq(int,archim6421214686448440834_floor(A,X)),one_one(int))
        <=> aa(A,$o,ord_less(A,X),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% floor_le_one
tff(fact_1047_neg__numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( aa(int,$o,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),archim6421214686448440834_floor(A,X))
        <=> aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),X) ) ) ).

% neg_numeral_le_floor
tff(fact_1048_floor__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( aa(int,$o,ord_less(int,archim6421214686448440834_floor(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2)))
        <=> aa(A,$o,ord_less(A,X),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) ) ) ).

% floor_less_neg_numeral
tff(fact_1049_ceiling__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( aa(int,$o,ord_less_eq(int,archimedean_ceiling(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2)))
        <=> aa(A,$o,ord_less_eq(A,X),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) ) ) ).

% ceiling_le_neg_numeral
tff(fact_1050_neg__numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( aa(int,$o,ord_less(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),archimedean_ceiling(A,X))
        <=> aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),X) ) ) ).

% neg_numeral_less_ceiling
tff(fact_1051_neg__numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( aa(int,$o,ord_less(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),archim6421214686448440834_floor(A,X))
        <=> aa(A,$o,ord_less_eq(A,aa(A,A,plus_plus(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),one_one(A))),X) ) ) ).

% neg_numeral_less_floor
tff(fact_1052_floor__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( aa(int,$o,ord_less_eq(int,archim6421214686448440834_floor(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2)))
        <=> aa(A,$o,ord_less(A,X),aa(A,A,plus_plus(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),one_one(A))) ) ) ).

% floor_le_neg_numeral
tff(fact_1053_predicate1D,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),X: A] :
      ( aa(fun(A,$o),$o,ord_less_eq(fun(A,$o),P),Q)
     => ( aa(A,$o,P,X)
       => aa(A,$o,Q,X) ) ) ).

% predicate1D
tff(fact_1054_rev__predicate1D,axiom,
    ! [A: $tType,P: fun(A,$o),X: A,Q: fun(A,$o)] :
      ( aa(A,$o,P,X)
     => ( aa(fun(A,$o),$o,ord_less_eq(fun(A,$o),P),Q)
       => aa(A,$o,Q,X) ) ) ).

% rev_predicate1D
tff(fact_1055_floor__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(int,$o,ord_less_eq(int,archim6421214686448440834_floor(A,X)),archimedean_ceiling(A,X)) ) ).

% floor_le_ceiling
tff(fact_1056_ceiling__minus,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_ceiling(A,aa(A,A,uminus_uminus(A),X)) = aa(int,int,uminus_uminus(int),archim6421214686448440834_floor(A,X)) ) ).

% ceiling_minus
tff(fact_1057_floor__minus,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archim6421214686448440834_floor(A,aa(A,A,uminus_uminus(A),X)) = aa(int,int,uminus_uminus(int),archimedean_ceiling(A,X)) ) ).

% floor_minus
tff(fact_1058_ceiling__def,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_ceiling(A,X) = aa(int,int,uminus_uminus(int),archim6421214686448440834_floor(A,aa(A,A,uminus_uminus(A),X))) ) ).

% ceiling_def
tff(fact_1059_floor__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,X),Y)
         => aa(int,$o,ord_less_eq(int,archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y)) ) ) ).

% floor_mono
tff(fact_1060_floor__less__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( aa(int,$o,ord_less(int,archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y))
         => aa(A,$o,ord_less(A,X),Y) ) ) ).

% floor_less_cancel
tff(fact_1061_ceiling__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,ord_less_eq(A,Y),X)
         => aa(int,$o,ord_less_eq(int,archimedean_ceiling(A,Y)),archimedean_ceiling(A,X)) ) ) ).

% ceiling_mono
tff(fact_1062_ceiling__less__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( aa(int,$o,ord_less(int,archimedean_ceiling(A,X)),archimedean_ceiling(A,Y))
         => aa(A,$o,ord_less(A,X),Y) ) ) ).

% ceiling_less_cancel
tff(fact_1063_le__floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] : aa(int,$o,ord_less_eq(int,aa(int,int,plus_plus(int,archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y))),archim6421214686448440834_floor(A,aa(A,A,plus_plus(A,X),Y))) ) ).

% le_floor_add
tff(fact_1064_ceiling__add__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] : aa(int,$o,ord_less_eq(int,archimedean_ceiling(A,aa(A,A,plus_plus(A,X),Y))),aa(int,int,plus_plus(int,archimedean_ceiling(A,X)),archimedean_ceiling(A,Y))) ) ).

% ceiling_add_le
tff(fact_1065_memb__imp__not__empty,axiom,
    ! [A: $tType,X: A,S: set(A)] :
      ( member(A,X,S)
     => ( S != bot_bot(set(A)) ) ) ).

% memb_imp_not_empty
tff(fact_1066_set__notEmptyE,axiom,
    ! [A: $tType,S: set(A)] :
      ( ( S != bot_bot(set(A)) )
     => ~ ! [X4: A] : ~ member(A,X4,S) ) ).

% set_notEmptyE
tff(fact_1067_one__add__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(int,int,plus_plus(int,archim6421214686448440834_floor(A,X)),one_one(int)) = archim6421214686448440834_floor(A,aa(A,A,plus_plus(A,X),one_one(A))) ) ).

% one_add_floor
tff(fact_1068_exists__leI,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( ( ! [N4: nat] :
            ( aa(nat,$o,ord_less(nat,N4),Na)
           => ~ aa(nat,$o,P,N4) )
       => aa(nat,$o,P,Na) )
     => ? [N5: nat] :
          ( aa(nat,$o,ord_less_eq(nat,N5),Na)
          & aa(nat,$o,P,N5) ) ) ).

% exists_leI
tff(fact_1069_Lattices__Big_Oex__has__greatest__nat,axiom,
    ! [A: $tType,P: fun(A,$o),K: A,F2: fun(A,nat),B3: nat] :
      ( aa(A,$o,P,K)
     => ( ! [Y3: A] :
            ( aa(A,$o,P,Y3)
           => aa(nat,$o,ord_less(nat,aa(A,nat,F2,Y3)),B3) )
       => ? [X4: A] :
            ( aa(A,$o,P,X4)
            & ! [Y4: A] :
                ( aa(A,$o,P,Y4)
               => aa(nat,$o,ord_less_eq(nat,aa(A,nat,F2,Y4)),aa(A,nat,F2,X4)) ) ) ) ) ).

% Lattices_Big.ex_has_greatest_nat
tff(fact_1070_floor__log__nat__eq__if,axiom,
    ! [B3: nat,Na: nat,K: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,power_power(nat,B3),Na)),K)
     => ( aa(nat,$o,ord_less(nat,K),aa(nat,nat,power_power(nat,B3),aa(nat,nat,plus_plus(nat,Na),one_one(nat))))
       => ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),B3)
         => ( archim6421214686448440834_floor(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B3)),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_1071_ceiling__log__eq__powr__iff,axiom,
    ! [X: real,B3: real,K: nat] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less(real,one_one(real)),B3)
       => ( ( archimedean_ceiling(real,aa(real,real,log(B3),X)) = aa(int,int,plus_plus(int,aa(nat,int,semiring_1_of_nat(int),K)),one_one(int)) )
        <=> ( aa(real,$o,ord_less(real,powr(real,B3,aa(nat,real,semiring_1_of_nat(real),K))),X)
            & aa(real,$o,ord_less_eq(real,X),powr(real,B3,aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,plus_plus(nat,K),one_one(nat))))) ) ) ) ) ).

% ceiling_log_eq_powr_iff
tff(fact_1072_floor__log2__div2,axiom,
    ! [Na: nat] :
      ( aa(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,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_1073_dbl__inc__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_dbl_inc(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K))) = aa(A,A,uminus_uminus(A),neg_numeral_dbl_dec(A,aa(num,A,numeral_numeral(A),K))) ) ).

% dbl_inc_simps(1)
tff(fact_1074_dbl__dec__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_dbl_dec(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K))) = aa(A,A,uminus_uminus(A),neg_numeral_dbl_inc(A,aa(num,A,numeral_numeral(A),K))) ) ).

% dbl_dec_simps(1)
tff(fact_1075_neg__numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( aa(int,$o,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),archimedean_ceiling(A,X))
        <=> aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),one_one(A))),X) ) ) ).

% neg_numeral_le_ceiling
tff(fact_1076_ceiling__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( aa(int,$o,ord_less(int,archimedean_ceiling(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2)))
        <=> aa(A,$o,ord_less_eq(A,X),aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),one_one(A))) ) ) ).

% ceiling_less_neg_numeral
tff(fact_1077_power__le__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,W: num] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),W))),zero_zero(A))
        <=> ( aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(num,nat,numeral_numeral(nat),W))
            & ( ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W))
                & aa(A,$o,ord_less_eq(A,A3),zero_zero(A)) )
              | ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W))
                & ( A3 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_1078_word__2p__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] :
          ( aa(nat,$o,ord_less(nat,Na),size_size(word(A),W))
         => ( aa(word(A),$o,ord_less(word(A),W),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))
          <=> aa(int,$o,ord_less(int,aa(word(A),int,semiring_1_unsigned(A,int),W)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ) ) ).

% word_2p_lem
tff(fact_1079_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,Na: nat,A3: int] :
          ( aa(A,$o,ord_less(A,aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),Na)),aa(int,A,ring_1_of_int(A),A3))
        <=> aa(int,$o,ord_less(int,aa(nat,int,power_power(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),Na)),A3) ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_1080_minus__apply,axiom,
    ! [A: $tType,B: $tType] :
      ( minus(A)
     => ! [A2: fun(B,A),B2: fun(B,A),X: B] : aa(B,A,aa(fun(B,A),fun(B,A),minus_minus(fun(B,A),A2),B2),X) = aa(A,A,minus_minus(A,aa(B,A,A2,X)),aa(B,A,B2,X)) ) ).

% minus_apply
tff(fact_1081_diff__self__eq__0,axiom,
    ! [M: nat] : aa(nat,nat,minus_minus(nat,M),M) = zero_zero(nat) ).

% diff_self_eq_0
tff(fact_1082_diff__0__eq__0,axiom,
    ! [Na: nat] : aa(nat,nat,minus_minus(nat,zero_zero(nat)),Na) = zero_zero(nat) ).

% diff_0_eq_0
tff(fact_1083_Diff__cancel,axiom,
    ! [A: $tType,A2: set(A)] : aa(set(A),set(A),minus_minus(set(A),A2),A2) = bot_bot(set(A)) ).

% Diff_cancel
tff(fact_1084_empty__Diff,axiom,
    ! [A: $tType,A2: set(A)] : aa(set(A),set(A),minus_minus(set(A),bot_bot(set(A))),A2) = bot_bot(set(A)) ).

% empty_Diff
tff(fact_1085_Diff__empty,axiom,
    ! [A: $tType,A2: set(A)] : aa(set(A),set(A),minus_minus(set(A),A2),bot_bot(set(A))) = A2 ).

% Diff_empty
tff(fact_1086_diff__diff__left,axiom,
    ! [I: nat,J: nat,K: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,I),J)),K) = aa(nat,nat,minus_minus(nat,I),aa(nat,nat,plus_plus(nat,J),K)) ).

% diff_diff_left
tff(fact_1087_diff__diff__cancel,axiom,
    ! [I: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),Na)
     => ( aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,minus_minus(nat,Na),I)) = I ) ) ).

% diff_diff_cancel
tff(fact_1088_insert__Diff1,axiom,
    ! [A: $tType,X: A,B2: set(A),A2: set(A)] :
      ( member(A,X,B2)
     => ( aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),insert(A,X),A2)),B2) = aa(set(A),set(A),minus_minus(set(A),A2),B2) ) ) ).

% insert_Diff1
tff(fact_1089_Diff__insert0,axiom,
    ! [A: $tType,X: A,A2: set(A),B2: set(A)] :
      ( ~ member(A,X,A2)
     => ( aa(set(A),set(A),minus_minus(set(A),A2),aa(set(A),set(A),insert(A,X),B2)) = aa(set(A),set(A),minus_minus(set(A),A2),B2) ) ) ).

% Diff_insert0
tff(fact_1090_of__int__eq__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [W: int,Z: int] :
          ( ( aa(int,A,ring_1_of_int(A),W) = aa(int,A,ring_1_of_int(A),Z) )
        <=> ( W = Z ) ) ) ).

% of_int_eq_iff
tff(fact_1091_Un__Diff__cancel2,axiom,
    ! [A: $tType,B2: set(A),A2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),minus_minus(set(A),B2),A2)),A2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B2),A2) ).

% Un_Diff_cancel2
tff(fact_1092_Un__Diff__cancel,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),aa(set(A),set(A),minus_minus(set(A),B2),A2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),B2) ).

% Un_Diff_cancel
tff(fact_1093_idiff__0__right,axiom,
    ! [Na: extended_enat] : aa(extended_enat,extended_enat,minus_minus(extended_enat,Na),zero_zero(extended_enat)) = Na ).

% idiff_0_right
tff(fact_1094_idiff__0,axiom,
    ! [Na: extended_enat] : aa(extended_enat,extended_enat,minus_minus(extended_enat,zero_zero(extended_enat)),Na) = zero_zero(extended_enat) ).

% idiff_0
tff(fact_1095_diff__self,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,minus_minus(A,A3),A3) = zero_zero(A) ) ).

% diff_self
tff(fact_1096_diff__0__right,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,minus_minus(A,A3),zero_zero(A)) = A3 ) ).

% diff_0_right
tff(fact_1097_zero__diff,axiom,
    ! [A: $tType] :
      ( comm_monoid_diff(A)
     => ! [A3: A] : aa(A,A,minus_minus(A,zero_zero(A)),A3) = zero_zero(A) ) ).

% zero_diff
tff(fact_1098_diff__zero,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A] : aa(A,A,minus_minus(A,A3),zero_zero(A)) = A3 ) ).

% diff_zero
tff(fact_1099_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A] : aa(A,A,minus_minus(A,A3),A3) = zero_zero(A) ) ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1100_divide__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) = zero_zero(A) )
        <=> ( ( A3 = zero_zero(A) )
            | ( B3 = zero_zero(A) ) ) ) ) ).

% divide_eq_0_iff
tff(fact_1101_divide__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B3) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A3 = B3 ) ) ) ) ).

% divide_cancel_left
tff(fact_1102_divide__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,C2: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A3 = B3 ) ) ) ) ).

% divide_cancel_right
tff(fact_1103_bits__div__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),zero_zero(A)),A3) = zero_zero(A) ) ).

% bits_div_0
tff(fact_1104_bits__div__by__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),zero_zero(A)) = zero_zero(A) ) ).

% bits_div_by_0
tff(fact_1105_division__ring__divide__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),zero_zero(A)) = zero_zero(A) ) ).

% division_ring_divide_zero
tff(fact_1106_div__0,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),zero_zero(A)),A3) = zero_zero(A) ) ).

% div_0
tff(fact_1107_div__by__0,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),zero_zero(A)) = zero_zero(A) ) ).

% div_by_0
tff(fact_1108_add__diff__cancel__right_H,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A3: A,B3: A] : aa(A,A,minus_minus(A,aa(A,A,plus_plus(A,A3),B3)),B3) = A3 ) ).

% add_diff_cancel_right'
tff(fact_1109_add__diff__cancel__right,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A3: A,C2: A,B3: A] : aa(A,A,minus_minus(A,aa(A,A,plus_plus(A,A3),C2)),aa(A,A,plus_plus(A,B3),C2)) = aa(A,A,minus_minus(A,A3),B3) ) ).

% add_diff_cancel_right
tff(fact_1110_add__diff__cancel__left_H,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A3: A,B3: A] : aa(A,A,minus_minus(A,aa(A,A,plus_plus(A,A3),B3)),A3) = B3 ) ).

% add_diff_cancel_left'
tff(fact_1111_add__diff__cancel__left,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [C2: A,A3: A,B3: A] : aa(A,A,minus_minus(A,aa(A,A,plus_plus(A,C2),A3)),aa(A,A,plus_plus(A,C2),B3)) = aa(A,A,minus_minus(A,A3),B3) ) ).

% add_diff_cancel_left
tff(fact_1112_diff__add__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] : aa(A,A,plus_plus(A,aa(A,A,minus_minus(A,A3),B3)),B3) = A3 ) ).

% diff_add_cancel
tff(fact_1113_add__diff__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] : aa(A,A,minus_minus(A,aa(A,A,plus_plus(A,A3),B3)),B3) = A3 ) ).

% add_diff_cancel
tff(fact_1114_bits__div__by__1,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),one_one(A)) = A3 ) ).

% bits_div_by_1
tff(fact_1115_div__by__1,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),one_one(A)) = A3 ) ).

% div_by_1
tff(fact_1116_dvd__0__right,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] : aa(A,$o,dvd_dvd(A,A3),zero_zero(A)) ) ).

% dvd_0_right
tff(fact_1117_dvd__0__left__iff,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] :
          ( aa(A,$o,dvd_dvd(A,zero_zero(A)),A3)
        <=> ( A3 = zero_zero(A) ) ) ) ).

% dvd_0_left_iff
tff(fact_1118_minus__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] : aa(A,A,uminus_uminus(A),aa(A,A,minus_minus(A,A3),B3)) = aa(A,A,minus_minus(A,B3),A3) ) ).

% minus_diff_eq
tff(fact_1119_zero__less__diff,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(nat,nat,minus_minus(nat,Na),M))
    <=> aa(nat,$o,ord_less(nat,M),Na) ) ).

% zero_less_diff
tff(fact_1120_dvd__add__triv__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,A3),aa(A,A,plus_plus(A,B3),A3))
        <=> aa(A,$o,dvd_dvd(A,A3),B3) ) ) ).

% dvd_add_triv_right_iff
tff(fact_1121_dvd__add__triv__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,A3),aa(A,A,plus_plus(A,A3),B3))
        <=> aa(A,$o,dvd_dvd(A,A3),B3) ) ) ).

% dvd_add_triv_left_iff
tff(fact_1122_diff__is__0__eq_H,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),Na)
     => ( aa(nat,nat,minus_minus(nat,M),Na) = zero_zero(nat) ) ) ).

% diff_is_0_eq'
tff(fact_1123_diff__is__0__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,minus_minus(nat,M),Na) = zero_zero(nat) )
    <=> aa(nat,$o,ord_less_eq(nat,M),Na) ) ).

% diff_is_0_eq
tff(fact_1124_div__dvd__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A3),B3)
         => ( aa(A,$o,dvd_dvd(A,A3),C2)
           => ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3))
            <=> aa(A,$o,dvd_dvd(A,B3),C2) ) ) ) ) ).

% div_dvd_div
tff(fact_1125_Diff__eq__empty__iff,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] :
      ( ( aa(set(A),set(A),minus_minus(set(A),A2),B2) = bot_bot(set(A)) )
    <=> aa(set(A),$o,ord_less_eq(set(A),A2),B2) ) ).

% Diff_eq_empty_iff
tff(fact_1126_minus__dvd__iff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,uminus_uminus(A),X)),Y)
        <=> aa(A,$o,dvd_dvd(A,X),Y) ) ) ).

% minus_dvd_iff
tff(fact_1127_dvd__minus__iff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,dvd_dvd(A,X),aa(A,A,uminus_uminus(A),Y))
        <=> aa(A,$o,dvd_dvd(A,X),Y) ) ) ).

% dvd_minus_iff
tff(fact_1128_insert__Diff__single,axiom,
    ! [A: $tType,A3: A,A2: set(A)] : aa(set(A),set(A),insert(A,A3),aa(set(A),set(A),minus_minus(set(A),A2),aa(set(A),set(A),insert(A,A3),bot_bot(set(A))))) = aa(set(A),set(A),insert(A,A3),A2) ).

% insert_Diff_single
tff(fact_1129_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),J)
     => ( aa(nat,nat,plus_plus(nat,I),aa(nat,nat,minus_minus(nat,J),K)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,plus_plus(nat,I),J)),K) ) ) ).

% Nat.add_diff_assoc
tff(fact_1130_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),J)
     => ( aa(nat,nat,plus_plus(nat,aa(nat,nat,minus_minus(nat,J),K)),I) = aa(nat,nat,minus_minus(nat,aa(nat,nat,plus_plus(nat,J),I)),K) ) ) ).

% Nat.add_diff_assoc2
tff(fact_1131_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),J)
     => ( aa(nat,nat,minus_minus(nat,I),aa(nat,nat,minus_minus(nat,J),K)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,plus_plus(nat,I),K)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_1132_of__int__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [W: int,Z: int] : aa(int,A,ring_1_of_int(A),aa(int,int,minus_minus(int,W),Z)) = aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),W)),aa(int,A,ring_1_of_int(A),Z)) ) ).

% of_int_diff
tff(fact_1133_word__of__int__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: num] : aa(int,word(A),ring_1_of_int(word(A)),aa(num,int,numeral_numeral(int),Bin)) = aa(num,word(A),numeral_numeral(word(A)),Bin) ) ).

% word_of_int_numeral
tff(fact_1134_nat__dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( aa(nat,$o,dvd_dvd(nat,M),one_one(nat))
    <=> ( M = one_one(nat) ) ) ).

% nat_dvd_1_iff_1
tff(fact_1135_powr__0,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Z: A] : powr(A,zero_zero(A),Z) = zero_zero(A) ) ).

% powr_0
tff(fact_1136_powr__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [W: A,Z: A] :
          ( ( powr(A,W,Z) = zero_zero(A) )
        <=> ( W = zero_zero(A) ) ) ) ).

% powr_eq_0_iff
tff(fact_1137_word__of__int__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(int,word(A),ring_1_of_int(word(A)),zero_zero(int)) = zero_zero(word(A)) ) ) ).

% word_of_int_0
tff(fact_1138_powr__one__eq__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [A3: A] : powr(A,one_one(A),A3) = one_one(A) ) ).

% powr_one_eq_one
tff(fact_1139_word__of__int__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(int,word(A),ring_1_of_int(word(A)),one_one(int)) = one_one(word(A)) ) ) ).

% word_of_int_1
tff(fact_1140_of__int__floor__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X)) = X )
        <=> ? [N3: int] : X = aa(int,A,ring_1_of_int(A),N3) ) ) ).

% of_int_floor_cancel
tff(fact_1141_floor__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int] : archim6421214686448440834_floor(A,aa(int,A,ring_1_of_int(A),Z)) = Z ) ).

% floor_of_int
tff(fact_1142_of__int__ceiling__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X)) = X )
        <=> ? [N3: int] : X = aa(int,A,ring_1_of_int(A),N3) ) ) ).

% of_int_ceiling_cancel
tff(fact_1143_ceiling__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int] : archimedean_ceiling(A,aa(int,A,ring_1_of_int(A),Z)) = Z ) ).

% ceiling_of_int
tff(fact_1144_Compl__Diff__eq,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),minus_minus(set(A),A2),B2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A2)),B2) ).

% Compl_Diff_eq
tff(fact_1145_dbl__dec__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl_dec(A,one_one(A)) = one_one(A) ) ) ).

% dbl_dec_simps(3)
tff(fact_1146_diff__ge__0__iff__ge,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,minus_minus(A,A3),B3))
        <=> aa(A,$o,ord_less_eq(A,B3),A3) ) ) ).

% diff_ge_0_iff_ge
tff(fact_1147_zero__comp__diff__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,minus_minus(A,A3),B3))
        <=> aa(A,$o,ord_less_eq(A,B3),A3) ) ) ).

% zero_comp_diff_simps(1)
tff(fact_1148_diff__gt__0__iff__gt,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,minus_minus(A,A3),B3))
        <=> aa(A,$o,ord_less(A,B3),A3) ) ) ).

% diff_gt_0_iff_gt
tff(fact_1149_zero__comp__diff__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,minus_minus(A,A3),B3))
        <=> aa(A,$o,ord_less(A,B3),A3) ) ) ).

% zero_comp_diff_simps(2)
tff(fact_1150_diff__add__zero,axiom,
    ! [A: $tType] :
      ( comm_monoid_diff(A)
     => ! [A3: A,B3: A] : aa(A,A,minus_minus(A,A3),aa(A,A,plus_plus(A,A3),B3)) = zero_zero(A) ) ).

% diff_add_zero
tff(fact_1151_diff__numeral__special_I9_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,minus_minus(A,one_one(A)),one_one(A)) = zero_zero(A) ) ) ).

% diff_numeral_special(9)
tff(fact_1152_le__add__diff__inverse2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
         => ( aa(A,A,plus_plus(A,aa(A,A,minus_minus(A,A3),B3)),B3) = A3 ) ) ) ).

% le_add_diff_inverse2
tff(fact_1153_le__add__diff__inverse,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
         => ( aa(A,A,plus_plus(A,B3),aa(A,A,minus_minus(A,A3),B3)) = A3 ) ) ) ).

% le_add_diff_inverse
tff(fact_1154_divide__eq__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) = one_one(A) )
        <=> ( ( B3 != zero_zero(A) )
            & ( A3 = B3 ) ) ) ) ).

% divide_eq_1_iff
tff(fact_1155_one__eq__divide__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B3: A] :
          ( ( one_one(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) )
        <=> ( ( B3 != zero_zero(A) )
            & ( A3 = B3 ) ) ) ) ).

% one_eq_divide_iff
tff(fact_1156_divide__self,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),A3) = one_one(A) ) ) ) ).

% divide_self
tff(fact_1157_divide__self__if,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),A3) = $ite(A3 = zero_zero(A),zero_zero(A),one_one(A)) ) ).

% divide_self_if
tff(fact_1158_divide__eq__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3) = one_one(A) )
        <=> ( ( A3 != zero_zero(A) )
            & ( A3 = B3 ) ) ) ) ).

% divide_eq_eq_1
tff(fact_1159_eq__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,A3: A] :
          ( ( one_one(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3) )
        <=> ( ( A3 != zero_zero(A) )
            & ( A3 = B3 ) ) ) ) ).

% eq_divide_eq_1
tff(fact_1160_one__divide__eq__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% one_divide_eq_0_iff
tff(fact_1161_zero__eq__1__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% zero_eq_1_divide_iff
tff(fact_1162_div__self,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),A3) = one_one(A) ) ) ) ).

% div_self
tff(fact_1163_diff__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,minus_minus(A,zero_zero(A)),A3) = aa(A,A,uminus_uminus(A),A3) ) ).

% diff_0
tff(fact_1164_verit__minus__simplify_I3_J,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [B3: A] : aa(A,A,minus_minus(A,zero_zero(A)),B3) = aa(A,A,uminus_uminus(A),B3) ) ).

% verit_minus_simplify(3)
tff(fact_1165_uminus__add__conv__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,B3: A] : aa(A,A,plus_plus(A,aa(A,A,uminus_uminus(A),A3)),B3) = aa(A,A,minus_minus(A,B3),A3) ) ).

% uminus_add_conv_diff
tff(fact_1166_diff__minus__eq__add,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] : aa(A,A,minus_minus(A,A3),aa(A,A,uminus_uminus(A),B3)) = aa(A,A,plus_plus(A,A3),B3) ) ).

% diff_minus_eq_add
tff(fact_1167_divide__minus1,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),X),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),X) ) ).

% divide_minus1
tff(fact_1168_div__add,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,C2),A3)
         => ( aa(A,$o,dvd_dvd(A,C2),B3)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,plus_plus(A,A3),B3)),C2) = aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)) ) ) ) ) ).

% div_add
tff(fact_1169_unit__div__1__div__1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] :
          ( aa(A,$o,dvd_dvd(A,A3),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)) = A3 ) ) ) ).

% unit_div_1_div_1
tff(fact_1170_unit__div__1__unit,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] :
          ( aa(A,$o,dvd_dvd(A,A3),one_one(A))
         => aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)),one_one(A)) ) ) ).

% unit_div_1_unit
tff(fact_1171_unit__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,A3),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,B3),one_one(A))
           => aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),one_one(A)) ) ) ) ).

% unit_div
tff(fact_1172_div__diff,axiom,
    ! [A: $tType] :
      ( idom_modulo(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,C2),A3)
         => ( aa(A,$o,dvd_dvd(A,C2),B3)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,A3),B3)),C2) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)) ) ) ) ) ).

% div_diff
tff(fact_1173_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_1174_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_1175_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_1176_of__int__eq__numeral__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Z: int,Na: num] :
          ( ( aa(int,A,ring_1_of_int(A),Z) = aa(num,A,numeral_numeral(A),Na) )
        <=> ( Z = aa(num,int,numeral_numeral(int),Na) ) ) ) ).

% of_int_eq_numeral_iff
tff(fact_1177_of__int__numeral,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: num] : aa(int,A,ring_1_of_int(A),aa(num,int,numeral_numeral(int),K)) = aa(num,A,numeral_numeral(A),K) ) ).

% of_int_numeral
tff(fact_1178_of__int__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W: int,Z: int] :
          ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),W)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,ord_less_eq(int,W),Z) ) ) ).

% of_int_le_iff
tff(fact_1179_of__int__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W: int,Z: int] :
          ( aa(A,$o,ord_less(A,aa(int,A,ring_1_of_int(A),W)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,ord_less(int,W),Z) ) ) ).

% of_int_less_iff
tff(fact_1180_of__int__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ( aa(int,A,ring_1_of_int(A),one_one(int)) = one_one(A) ) ) ).

% of_int_1
tff(fact_1181_of__int__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Z: int] :
          ( ( aa(int,A,ring_1_of_int(A),Z) = one_one(A) )
        <=> ( Z = one_one(int) ) ) ) ).

% of_int_eq_1_iff
tff(fact_1182_powr__zero__eq__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [X: A] :
          powr(A,X,zero_zero(A)) = $ite(X = zero_zero(A),zero_zero(A),one_one(A)) ) ).

% powr_zero_eq_one
tff(fact_1183_of__int__add,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [W: int,Z: int] : aa(int,A,ring_1_of_int(A),aa(int,int,plus_plus(int,W),Z)) = aa(A,A,plus_plus(A,aa(int,A,ring_1_of_int(A),W)),aa(int,A,ring_1_of_int(A),Z)) ) ).

% of_int_add
tff(fact_1184_of__int__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: int] : aa(int,A,ring_1_of_int(A),aa(int,int,uminus_uminus(int),Z)) = aa(A,A,uminus_uminus(A),aa(int,A,ring_1_of_int(A),Z)) ) ).

% of_int_minus
tff(fact_1185_of__int__of__nat__eq,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Na: nat] : aa(int,A,ring_1_of_int(A),aa(nat,int,semiring_1_of_nat(int),Na)) = aa(nat,A,semiring_1_of_nat(A),Na) ) ).

% of_int_of_nat_eq
tff(fact_1186_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [X: int,B3: int,W: nat] :
          ( ( aa(int,A,ring_1_of_int(A),X) = aa(nat,A,power_power(A,aa(int,A,ring_1_of_int(A),B3)),W) )
        <=> ( X = aa(nat,int,power_power(int,B3),W) ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_1187_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [B3: int,W: nat,X: int] :
          ( ( aa(nat,A,power_power(A,aa(int,A,ring_1_of_int(A),B3)),W) = aa(int,A,ring_1_of_int(A),X) )
        <=> ( aa(nat,int,power_power(int,B3),W) = X ) ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_1188_of__int__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: int,Na: nat] : aa(int,A,ring_1_of_int(A),aa(nat,int,power_power(int,Z),Na)) = aa(nat,A,power_power(A,aa(int,A,ring_1_of_int(A),Z)),Na) ) ).

% of_int_power
tff(fact_1189_zle__diff1__eq,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,ord_less_eq(int,W),aa(int,int,minus_minus(int,Z),one_one(int)))
    <=> aa(int,$o,ord_less(int,W),Z) ) ).

% zle_diff1_eq
tff(fact_1190_word__size__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(nat,$o,ord_less(nat,zero_zero(nat)),size_size(word(A),W)) ) ).

% word_size_gt_0
tff(fact_1191_Word_Oof__int__uint,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_1(A) )
     => ! [W: word(B)] : aa(int,A,ring_1_of_int(A),aa(word(B),int,semiring_1_unsigned(B,int),W)) = aa(word(B),A,semiring_1_unsigned(B,A),W) ) ).

% Word.of_int_uint
tff(fact_1192_floor__diff__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z: int] : archim6421214686448440834_floor(A,aa(A,A,minus_minus(A,X),aa(int,A,ring_1_of_int(A),Z))) = aa(int,int,minus_minus(int,archim6421214686448440834_floor(A,X)),Z) ) ).

% floor_diff_of_int
tff(fact_1193_ceiling__diff__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z: int] : archimedean_ceiling(A,aa(A,A,minus_minus(A,X),aa(int,A,ring_1_of_int(A),Z))) = aa(int,int,minus_minus(int,archimedean_ceiling(A,X)),Z) ) ).

% ceiling_diff_of_int
tff(fact_1194_powr__gt__zero,axiom,
    ! [X: real,A3: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),powr(real,X,A3))
    <=> ( X != zero_zero(real) ) ) ).

% powr_gt_zero
tff(fact_1195_powr__nonneg__iff,axiom,
    ! [A3: real,X: real] :
      ( aa(real,$o,ord_less_eq(real,powr(real,A3,X)),zero_zero(real))
    <=> ( A3 = zero_zero(real) ) ) ).

% powr_nonneg_iff
tff(fact_1196_powr__less__cancel__iff,axiom,
    ! [X: real,A3: real,B3: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),X)
     => ( aa(real,$o,ord_less(real,powr(real,X,A3)),powr(real,X,B3))
      <=> aa(real,$o,ord_less(real,A3),B3) ) ) ).

% powr_less_cancel_iff
tff(fact_1197_divide__le__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)),zero_zero(A))
        <=> aa(A,$o,ord_less_eq(A,A3),zero_zero(A)) ) ) ).

% divide_le_0_1_iff
tff(fact_1198_zero__le__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3))
        <=> aa(A,$o,ord_less_eq(A,zero_zero(A)),A3) ) ) ).

% zero_le_divide_1_iff
tff(fact_1199_zero__less__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3))
        <=> aa(A,$o,ord_less(A,zero_zero(A)),A3) ) ) ).

% zero_less_divide_1_iff
tff(fact_1200_less__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less(A,one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3))
          <=> aa(A,$o,ord_less(A,A3),B3) ) ) ) ).

% less_divide_eq_1_pos
tff(fact_1201_less__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),zero_zero(A))
         => ( aa(A,$o,ord_less(A,one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3))
          <=> aa(A,$o,ord_less(A,B3),A3) ) ) ) ).

% less_divide_eq_1_neg
tff(fact_1202_divide__less__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3)),one_one(A))
          <=> aa(A,$o,ord_less(A,B3),A3) ) ) ) ).

% divide_less_eq_1_pos
tff(fact_1203_divide__less__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),zero_zero(A))
         => ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3)),one_one(A))
          <=> aa(A,$o,ord_less(A,A3),B3) ) ) ) ).

% divide_less_eq_1_neg
tff(fact_1204_divide__less__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)),zero_zero(A))
        <=> aa(A,$o,ord_less(A,A3),zero_zero(A)) ) ) ).

% divide_less_0_1_iff
tff(fact_1205_diff__numeral__special_I12_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(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_1206_floor__diff__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] : archim6421214686448440834_floor(A,aa(A,A,minus_minus(A,X),aa(num,A,numeral_numeral(A),V2))) = aa(int,int,minus_minus(int,archim6421214686448440834_floor(A,X)),aa(num,int,numeral_numeral(int),V2)) ) ).

% floor_diff_numeral
tff(fact_1207_ceiling__diff__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] : archimedean_ceiling(A,aa(A,A,minus_minus(A,X),aa(num,A,numeral_numeral(A),V2))) = aa(int,int,minus_minus(int,archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V2)) ) ).

% ceiling_diff_numeral
tff(fact_1208_floor__diff__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archim6421214686448440834_floor(A,aa(A,A,minus_minus(A,X),one_one(A))) = aa(int,int,minus_minus(int,archim6421214686448440834_floor(A,X)),one_one(int)) ) ).

% floor_diff_one
tff(fact_1209_diff__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,Na: num] : aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),Na)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,plus_plus(num,M),Na))) ) ).

% diff_numeral_simps(3)
tff(fact_1210_diff__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,Na: num] : aa(A,A,minus_minus(A,aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))) = aa(num,A,numeral_numeral(A),aa(num,num,plus_plus(num,M),Na)) ) ).

% diff_numeral_simps(2)
tff(fact_1211_ceiling__diff__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_ceiling(A,aa(A,A,minus_minus(A,X),one_one(A))) = aa(int,int,minus_minus(int,archimedean_ceiling(A,X)),one_one(int)) ) ).

% ceiling_diff_one
tff(fact_1212_ceiling__add__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z: int] : archimedean_ceiling(A,aa(A,A,plus_plus(A,X),aa(int,A,ring_1_of_int(A),Z))) = aa(int,int,plus_plus(int,archimedean_ceiling(A,X)),Z) ) ).

% ceiling_add_of_int
tff(fact_1213_floor__uminus__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int] : archim6421214686448440834_floor(A,aa(A,A,uminus_uminus(A),aa(int,A,ring_1_of_int(A),Z))) = aa(int,int,uminus_uminus(int),Z) ) ).

% floor_uminus_of_int
tff(fact_1214_powr__eq__one__iff,axiom,
    ! [A3: real,X: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A3)
     => ( ( powr(real,A3,X) = one_one(real) )
      <=> ( X = zero_zero(real) ) ) ) ).

% powr_eq_one_iff
tff(fact_1215_powr__one,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => ( powr(real,X,one_one(real)) = X ) ) ).

% powr_one
tff(fact_1216_powr__one__gt__zero__iff,axiom,
    ! [X: real] :
      ( ( powr(real,X,one_one(real)) = X )
    <=> aa(real,$o,ord_less_eq(real,zero_zero(real)),X) ) ).

% powr_one_gt_zero_iff
tff(fact_1217_word__of__int__neg__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: num] : aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Bin))) = aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Bin)) ) ).

% word_of_int_neg_numeral
tff(fact_1218_powr__le__cancel__iff,axiom,
    ! [X: real,A3: real,B3: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),X)
     => ( aa(real,$o,ord_less_eq(real,powr(real,X,A3)),powr(real,X,B3))
      <=> aa(real,$o,ord_less_eq(real,A3),B3) ) ) ).

% powr_le_cancel_iff
tff(fact_1219_numeral__powr__numeral__real,axiom,
    ! [M: num,Na: num] : powr(real,aa(num,real,numeral_numeral(real),M),aa(num,real,numeral_numeral(real),Na)) = aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),M)),aa(num,nat,numeral_numeral(nat),Na)) ).

% numeral_powr_numeral_real
tff(fact_1220_word__le__sub1__numberof,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: num] :
          ( ( aa(num,word(A),numeral_numeral(word(A)),W) != zero_zero(word(A)) )
         => ( aa(word(A),$o,ord_less_eq(word(A),one_one(word(A))),aa(num,word(A),numeral_numeral(word(A)),W))
          <=> aa(word(A),$o,ord_less_eq(word(A),zero_zero(word(A))),aa(word(A),word(A),minus_minus(word(A),aa(num,word(A),numeral_numeral(word(A)),W)),one_one(word(A)))) ) ) ) ).

% word_le_sub1_numberof
tff(fact_1221_word__of__int__neg__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,uminus_uminus(int),one_one(int))) = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ).

% word_of_int_neg_1
tff(fact_1222_word__less__sub1__numberof,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: num] :
          ( ( aa(num,word(A),numeral_numeral(word(A)),W) != zero_zero(word(A)) )
         => ( aa(word(A),$o,ord_less(word(A),one_one(word(A))),aa(num,word(A),numeral_numeral(word(A)),W))
          <=> aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),aa(word(A),word(A),minus_minus(word(A),aa(num,word(A),numeral_numeral(word(A)),W)),one_one(word(A)))) ) ) ) ).

% word_less_sub1_numberof
tff(fact_1223_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_1224_le__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less_eq(A,one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3))
          <=> aa(A,$o,ord_less_eq(A,A3),B3) ) ) ) ).

% le_divide_eq_1_pos
tff(fact_1225_le__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),zero_zero(A))
         => ( aa(A,$o,ord_less_eq(A,one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3))
          <=> aa(A,$o,ord_less_eq(A,B3),A3) ) ) ) ).

% le_divide_eq_1_neg
tff(fact_1226_divide__le__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3)),one_one(A))
          <=> aa(A,$o,ord_less_eq(A,B3),A3) ) ) ) ).

% divide_le_eq_1_pos
tff(fact_1227_divide__le__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),zero_zero(A))
         => ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3)),one_one(A))
          <=> aa(A,$o,ord_less_eq(A,A3),B3) ) ) ) ).

% divide_le_eq_1_neg
tff(fact_1228_even__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,plus_plus(A,A3),B3))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
          <=> aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),B3) ) ) ) ).

% even_add
tff(fact_1229_odd__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A,B3: A] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,plus_plus(A,A3),B3))
        <=> ~ ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
            <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),B3) ) ) ) ).

% odd_add
tff(fact_1230_of__int__0__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,ord_less_eq(int,zero_zero(int)),Z) ) ) ).

% of_int_0_le_iff
tff(fact_1231_of__int__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),Z)),zero_zero(A))
        <=> aa(int,$o,ord_less_eq(int,Z),zero_zero(int)) ) ) ).

% of_int_le_0_iff
tff(fact_1232_of__int__le__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int,Na: num] :
          ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),Z)),aa(num,A,numeral_numeral(A),Na))
        <=> aa(int,$o,ord_less_eq(int,Z),aa(num,int,numeral_numeral(int),Na)) ) ) ).

% of_int_le_numeral_iff
tff(fact_1233_of__int__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: num,Z: int] :
          ( aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),Na)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,ord_less_eq(int,aa(num,int,numeral_numeral(int),Na)),Z) ) ) ).

% of_int_numeral_le_iff
tff(fact_1234_of__int__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,ord_less(A,aa(int,A,ring_1_of_int(A),Z)),zero_zero(A))
        <=> aa(int,$o,ord_less(int,Z),zero_zero(int)) ) ) ).

% of_int_less_0_iff
tff(fact_1235_of__int__0__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,ord_less(int,zero_zero(int)),Z) ) ) ).

% of_int_0_less_iff
tff(fact_1236_of__int__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: num,Z: int] :
          ( aa(A,$o,ord_less(A,aa(num,A,numeral_numeral(A),Na)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,ord_less(int,aa(num,int,numeral_numeral(int),Na)),Z) ) ) ).

% of_int_numeral_less_iff
tff(fact_1237_of__int__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int,Na: num] :
          ( aa(A,$o,ord_less(A,aa(int,A,ring_1_of_int(A),Z)),aa(num,A,numeral_numeral(A),Na))
        <=> aa(int,$o,ord_less(int,Z),aa(num,int,numeral_numeral(int),Na)) ) ) ).

% of_int_less_numeral_iff
tff(fact_1238_of__int__1__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,ord_less_eq(A,one_one(A)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,ord_less_eq(int,one_one(int)),Z) ) ) ).

% of_int_1_le_iff
tff(fact_1239_of__int__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),Z)),one_one(A))
        <=> aa(int,$o,ord_less_eq(int,Z),one_one(int)) ) ) ).

% of_int_le_1_iff
tff(fact_1240_of__int__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,ord_less(A,aa(int,A,ring_1_of_int(A),Z)),one_one(A))
        <=> aa(int,$o,ord_less(int,Z),one_one(int)) ) ) ).

% of_int_less_1_iff
tff(fact_1241_of__int__1__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,ord_less(A,one_one(A)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,ord_less(int,one_one(int)),Z) ) ) ).

% of_int_1_less_iff
tff(fact_1242_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Y: int,X: num,Na: nat] :
          ( ( aa(int,A,ring_1_of_int(A),Y) = aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),X)),Na) )
        <=> ( Y = aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),Na) ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_1243_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [X: num,Na: nat,Y: int] :
          ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),X)),Na) = aa(int,A,ring_1_of_int(A),Y) )
        <=> ( aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),Na) = Y ) ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_1244_of__int__le__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B3: int,W: nat,X: int] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,power_power(A,aa(int,A,ring_1_of_int(A),B3)),W)),aa(int,A,ring_1_of_int(A),X))
        <=> aa(int,$o,ord_less_eq(int,aa(nat,int,power_power(int,B3),W)),X) ) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_1245_of__int__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: int,B3: int,W: nat] :
          ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),X)),aa(nat,A,power_power(A,aa(int,A,ring_1_of_int(A),B3)),W))
        <=> aa(int,$o,ord_less_eq(int,X),aa(nat,int,power_power(int,B3),W)) ) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_1246_of__int__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: int,B3: int,W: nat] :
          ( aa(A,$o,ord_less(A,aa(int,A,ring_1_of_int(A),X)),aa(nat,A,power_power(A,aa(int,A,ring_1_of_int(A),B3)),W))
        <=> aa(int,$o,ord_less(int,X),aa(nat,int,power_power(int,B3),W)) ) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_1247_of__int__less__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B3: int,W: nat,X: int] :
          ( aa(A,$o,ord_less(A,aa(nat,A,power_power(A,aa(int,A,ring_1_of_int(A),B3)),W)),aa(int,A,ring_1_of_int(A),X))
        <=> aa(int,$o,ord_less(int,aa(nat,int,power_power(int,B3),W)),X) ) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_1248_powr__log__cancel,axiom,
    ! [A3: real,X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
     => ( ( A3 != one_one(real) )
       => ( aa(real,$o,ord_less(real,zero_zero(real)),X)
         => ( powr(real,A3,aa(real,real,log(A3),X)) = X ) ) ) ) ).

% powr_log_cancel
tff(fact_1249_log__powr__cancel,axiom,
    ! [A3: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
     => ( ( A3 != one_one(real) )
       => ( aa(real,real,log(A3),powr(real,A3,Y)) = Y ) ) ) ).

% log_powr_cancel
tff(fact_1250_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_1251_diff__numeral__special_I10_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),one_one(A))),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% diff_numeral_special(10)
tff(fact_1252_diff__numeral__special_I11_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,minus_minus(A,one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(num,A,numeral_numeral(A),bit0(one2)) ) ) ).

% diff_numeral_special(11)
tff(fact_1253_minus__1__div__2__eq,axiom,
    ! [A: $tType] :
      ( euclid8789492081693882211th_nat(A)
     => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% minus_1_div_2_eq
tff(fact_1254_even__plus__one__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,plus_plus(A,A3),one_one(A)))
        <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3) ) ) ).

% even_plus_one_iff
tff(fact_1255_even__diff,axiom,
    ! [A: $tType] :
      ( ring_parity(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,minus_minus(A,A3),B3))
        <=> aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,plus_plus(A,A3),B3)) ) ) ).

% even_diff
tff(fact_1256_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Na: nat,A3: A] :
          ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
         => ( aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),A3)),Na) = aa(nat,A,power_power(A,A3),Na) ) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_1257_power__minus__odd,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Na: nat,A3: A] :
          ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
         => ( aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),A3)),Na) = aa(A,A,uminus_uminus(A),aa(nat,A,power_power(A,A3),Na)) ) ) ) ).

% power_minus_odd
tff(fact_1258_even__diff__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,M),Na))
    <=> ( aa(nat,$o,ord_less(nat,M),Na)
        | aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,plus_plus(nat,M),Na)) ) ) ).

% even_diff_nat
tff(fact_1259_diff__numeral__special_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,plus_plus(num,M),one2))) ) ).

% diff_numeral_special(4)
tff(fact_1260_diff__numeral__special_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Na: num] : aa(A,A,minus_minus(A,one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))) = aa(num,A,numeral_numeral(A),aa(num,num,plus_plus(num,one2),Na)) ) ).

% diff_numeral_special(3)
tff(fact_1261_powr__numeral,axiom,
    ! [X: real,Na: num] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => ( powr(real,X,aa(num,real,numeral_numeral(real),Na)) = aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),Na)) ) ) ).

% powr_numeral
tff(fact_1262_even__succ__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,plus_plus(A,A3),one_one(A))),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ) ).

% even_succ_div_two
tff(fact_1263_odd__succ__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,plus_plus(A,A3),one_one(A))),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))),one_one(A)) ) ) ) ).

% odd_succ_div_two
tff(fact_1264_even__succ__div__2,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,plus_plus(A,one_one(A)),A3)),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ) ).

% even_succ_div_2
tff(fact_1265_even__power,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,power_power(A,A3),Na))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
            & aa(nat,$o,ord_less(nat,zero_zero(nat)),Na) ) ) ) ).

% even_power
tff(fact_1266_zero__le__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,W: num] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),W)))
        <=> ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W))
            | ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W))
              & aa(A,$o,ord_less_eq(A,zero_zero(A)),A3) ) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_1267_power__less__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,W: num] :
          ( aa(A,$o,ord_less(A,aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),W))),zero_zero(A))
        <=> ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W))
            & aa(A,$o,ord_less(A,A3),zero_zero(A)) ) ) ) ).

% power_less_zero_eq_numeral
tff(fact_1268_power__less__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,ord_less(A,aa(nat,A,power_power(A,A3),Na)),zero_zero(A))
        <=> ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
            & aa(A,$o,ord_less(A,A3),zero_zero(A)) ) ) ) ).

% power_less_zero_eq
tff(fact_1269_neg__one__even__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Na: nat] :
          ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
         => ( aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),Na) = one_one(A) ) ) ) ).

% neg_one_even_power
tff(fact_1270_neg__one__odd__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Na: nat] :
          ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
         => ( aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),Na) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% neg_one_odd_power
tff(fact_1271_even__of__nat,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Na: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,semiring_1_of_nat(A),Na))
        <=> aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) ) ) ).

% even_of_nat
tff(fact_1272_ceiling__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V2: num] :
          ( aa(int,$o,ord_less(int,archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V2))
        <=> aa(A,$o,ord_less_eq(A,X),aa(A,A,minus_minus(A,aa(num,A,numeral_numeral(A),V2)),one_one(A))) ) ) ).

% ceiling_less_numeral
tff(fact_1273_numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,X: A] :
          ( aa(int,$o,ord_less_eq(int,aa(num,int,numeral_numeral(int),V2)),archimedean_ceiling(A,X))
        <=> aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,aa(num,A,numeral_numeral(A),V2)),one_one(A))),X) ) ) ).

% numeral_le_ceiling
tff(fact_1274_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: int,X: num,Na: nat] :
          ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),A3)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),X)),Na))
        <=> aa(int,$o,ord_less_eq(int,A3),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),Na)) ) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_1275_numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,Na: nat,A3: int] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),X)),Na)),aa(int,A,ring_1_of_int(A),A3))
        <=> aa(int,$o,ord_less_eq(int,aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),Na)),A3) ) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_1276_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: int,X: num,Na: nat] :
          ( aa(A,$o,ord_less(A,aa(int,A,ring_1_of_int(A),A3)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),X)),Na))
        <=> aa(int,$o,ord_less(int,A3),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),Na)) ) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_1277_numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,Na: nat,A3: int] :
          ( aa(A,$o,ord_less(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),X)),Na)),aa(int,A,ring_1_of_int(A),A3))
        <=> aa(int,$o,ord_less(int,aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),Na)),A3) ) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_1278_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Y: int,X: num,Na: nat] :
          ( ( aa(int,A,ring_1_of_int(A),Y) = aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),Na) )
        <=> ( Y = aa(nat,int,power_power(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),Na) ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_1279_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [X: num,Na: nat,Y: int] :
          ( ( aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),Na) = aa(int,A,ring_1_of_int(A),Y) )
        <=> ( aa(nat,int,power_power(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),Na) = Y ) ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_1280_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Na: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,minus_minus(A,aa(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_1281_zero__less__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,W: num] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),W)))
        <=> ( ( aa(num,nat,numeral_numeral(nat),W) = zero_zero(nat) )
            | ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W))
              & ( A3 != zero_zero(A) ) )
            | ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W))
              & aa(A,$o,ord_less(A,zero_zero(A)),A3) ) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_1282_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: int,X: num,Na: nat] :
          ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),A3)),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),Na))
        <=> aa(int,$o,ord_less_eq(int,A3),aa(nat,int,power_power(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),Na)) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_1283_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,Na: nat,A3: int] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),Na)),aa(int,A,ring_1_of_int(A),A3))
        <=> aa(int,$o,ord_less_eq(int,aa(nat,int,power_power(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),Na)),A3) ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_1284_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: int,X: num,Na: nat] :
          ( aa(A,$o,ord_less(A,aa(int,A,ring_1_of_int(A),A3)),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),Na))
        <=> aa(int,$o,ord_less(int,A3),aa(nat,int,power_power(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),Na)) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_1285_even__succ__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
         => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,plus_plus(A,one_one(A)),A3)),aa(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),A3),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) ) ) ) ) ).

% even_succ_div_exp
tff(fact_1286_dvd__diffD,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,dvd_dvd(nat,K),aa(nat,nat,minus_minus(nat,M),Na))
     => ( aa(nat,$o,dvd_dvd(nat,K),Na)
       => ( aa(nat,$o,ord_less_eq(nat,Na),M)
         => aa(nat,$o,dvd_dvd(nat,K),M) ) ) ) ).

% dvd_diffD
tff(fact_1287_dvd__diffD1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,dvd_dvd(nat,K),aa(nat,nat,minus_minus(nat,M),Na))
     => ( aa(nat,$o,dvd_dvd(nat,K),M)
       => ( aa(nat,$o,ord_less_eq(nat,Na),M)
         => aa(nat,$o,dvd_dvd(nat,K),Na) ) ) ) ).

% dvd_diffD1
tff(fact_1288_less__eq__dvd__minus,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),Na)
     => ( aa(nat,$o,dvd_dvd(nat,M),Na)
      <=> aa(nat,$o,dvd_dvd(nat,M),aa(nat,nat,minus_minus(nat,Na),M)) ) ) ).

% less_eq_dvd_minus
tff(fact_1289_word__sub__wi,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(word(A),word(A),minus_minus(word(A),A3),B3) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),A3)),aa(word(A),int,semiring_1_unsigned(A,int),B3))) ) ).

% word_sub_wi
tff(fact_1290_dvd__neg__div,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,dvd_dvd(A,B3),A3)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A3)),B3) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)) ) ) ) ).

% dvd_neg_div
tff(fact_1291_dvd__div__neg,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,dvd_dvd(A,B3),A3)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,uminus_uminus(A),B3)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)) ) ) ) ).

% dvd_div_neg
tff(fact_1292_floor__divide__of__int__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [K: int,L: int] : archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(int,A,ring_1_of_int(A),K)),aa(int,A,ring_1_of_int(A),L))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) ) ).

% floor_divide_of_int_eq
tff(fact_1293_size__neq__size__imp__neq,axiom,
    ! [A: $tType] :
      ( size(A)
     => ! [X: A,Y: A] :
          ( ( size_size(A,X) != size_size(A,Y) )
         => ( X != Y ) ) ) ).

% size_neq_size_imp_neq
tff(fact_1294_div__div__div__same,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [D2: A,B3: A,A3: A] :
          ( aa(A,$o,dvd_dvd(A,D2),B3)
         => ( aa(A,$o,dvd_dvd(A,B3),A3)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),D2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),D2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) ) ) ) ) ).

% div_div_div_same
tff(fact_1295_dvd__div__eq__cancel,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A,C2: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2) )
         => ( aa(A,$o,dvd_dvd(A,C2),A3)
           => ( aa(A,$o,dvd_dvd(A,C2),B3)
             => ( A3 = B3 ) ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_1296_dvd__div__eq__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,C2),A3)
         => ( aa(A,$o,dvd_dvd(A,C2),B3)
           => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2) )
            <=> ( A3 = B3 ) ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_1297_dvd__trans,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A3),B3)
         => ( aa(A,$o,dvd_dvd(A,B3),C2)
           => aa(A,$o,dvd_dvd(A,A3),C2) ) ) ) ).

% dvd_trans
tff(fact_1298_dvd__refl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A] : aa(A,$o,dvd_dvd(A,A3),A3) ) ).

% dvd_refl
tff(fact_1299_dvd__diff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y: A,Z: A] :
          ( aa(A,$o,dvd_dvd(A,X),Y)
         => ( aa(A,$o,dvd_dvd(A,X),Z)
           => aa(A,$o,dvd_dvd(A,X),aa(A,A,minus_minus(A,Y),Z)) ) ) ) ).

% dvd_diff
tff(fact_1300_fun__diff__def,axiom,
    ! [B: $tType,A: $tType] :
      ( minus(B)
     => ! [A2: fun(A,B),B2: fun(A,B),X2: A] : aa(A,B,aa(fun(A,B),fun(A,B),minus_minus(fun(A,B),A2),B2),X2) = aa(B,B,minus_minus(B,aa(A,B,A2,X2)),aa(A,B,B2,X2)) ) ).

% fun_diff_def
tff(fact_1301_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,dvd_dvd(nat,K),M)
     => ( aa(nat,$o,dvd_dvd(nat,K),Na)
       => aa(nat,$o,dvd_dvd(nat,K),aa(nat,nat,minus_minus(nat,M),Na)) ) ) ).

% dvd_diff_nat
tff(fact_1302_dvd__antisym,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,dvd_dvd(nat,M),Na)
     => ( aa(nat,$o,dvd_dvd(nat,Na),M)
       => ( M = Na ) ) ) ).

% dvd_antisym
tff(fact_1303_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A3: A,C2: A,B3: A] : aa(A,A,minus_minus(A,aa(A,A,minus_minus(A,A3),C2)),B3) = aa(A,A,minus_minus(A,aa(A,A,minus_minus(A,A3),B3)),C2) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_1304_diff__eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A,C2: A,D2: A] :
          ( ( aa(A,A,minus_minus(A,A3),B3) = aa(A,A,minus_minus(A,C2),D2) )
         => ( ( A3 = B3 )
          <=> ( C2 = D2 ) ) ) ) ).

% diff_eq_diff_eq
tff(fact_1305_dvd__minus__self,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,dvd_dvd(nat,M),aa(nat,nat,minus_minus(nat,Na),M))
    <=> ( aa(nat,$o,ord_less(nat,Na),M)
        | aa(nat,$o,dvd_dvd(nat,M),Na) ) ) ).

% dvd_minus_self
tff(fact_1306_div__power,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A,Na: nat] :
          ( aa(A,$o,dvd_dvd(A,B3),A3)
         => ( aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),Na) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,power_power(A,A3),Na)),aa(nat,A,power_power(A,B3),Na)) ) ) ) ).

% div_power
tff(fact_1307_dvd__div__unit__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B3),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B3))
          <=> aa(A,$o,dvd_dvd(A,A3),C2) ) ) ) ).

% dvd_div_unit_iff
tff(fact_1308_div__unit__dvd__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B3),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),C2)
          <=> aa(A,$o,dvd_dvd(A,A3),C2) ) ) ) ).

% div_unit_dvd_iff
tff(fact_1309_unit__div__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A3),one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3) )
          <=> ( B3 = C2 ) ) ) ) ).

% unit_div_cancel
tff(fact_1310_dvd__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,dvd_dvd(A,B3),A3)
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) = zero_zero(A) )
          <=> ( A3 = zero_zero(A) ) ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_1311_ceiling__divide__eq__div,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: int,B3: int] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(int,A,ring_1_of_int(A),A3)),aa(int,A,ring_1_of_int(A),B3))) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),A3)),B3)) ) ).

% ceiling_divide_eq_div
tff(fact_1312_unit__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,dvd_dvd(A,B3),one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) = zero_zero(A) )
          <=> ( A3 = zero_zero(A) ) ) ) ) ).

% unit_div_eq_0_iff
tff(fact_1313_of__nat__diff,axiom,
    ! [A: $tType] :
      ( semiring_1_cancel(A)
     => ! [Na: nat,M: nat] :
          ( aa(nat,$o,ord_less_eq(nat,Na),M)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,M),Na)) = aa(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_1314_uint__sub__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A)] :
          ( aa(int,$o,ord_less_eq(int,aa(word(A),int,semiring_1_unsigned(A,int),Y)),aa(word(A),int,semiring_1_unsigned(A,int),X))
        <=> ( aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),minus_minus(word(A),X),Y)) = aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ) ) ).

% uint_sub_lem
tff(fact_1315_uint__sub__ge,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] : aa(int,$o,ord_less_eq(int,aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Y))),aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),minus_minus(word(A),X),Y))) ) ).

% uint_sub_ge
tff(fact_1316_uint__minus__simple__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),minus_minus(word(A),X),Y)),X)
        <=> ( aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),minus_minus(word(A),X),Y)) = aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ) ) ).

% uint_minus_simple_iff
tff(fact_1317_uint__minus__simple__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Y),X)
        <=> ( aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),minus_minus(word(A),X),Y)) = aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ) ) ).

% uint_minus_simple_alt
tff(fact_1318_size__0__same_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),V2: word(A)] :
          ( ( size_size(word(A),W) = zero_zero(nat) )
         => ( W = V2 ) ) ) ).

% size_0_same'
tff(fact_1319_lens__not__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : size_size(word(A),W) != zero_zero(nat) ) ).

% lens_not_0
tff(fact_1320_size__0__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),V2: word(A)] :
          ( ( size_size(word(A),W) = zero_zero(nat) )
         => ( V2 = W ) ) ) ).

% size_0_eq
tff(fact_1321_unat__sub,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: word(A),A3: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),B3),A3)
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),minus_minus(word(A),A3),B3)) = aa(nat,nat,minus_minus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),A3)),aa(word(A),nat,semiring_1_unsigned(A,nat),B3)) ) ) ) ).

% unat_sub
tff(fact_1322_ex__le__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [Z2: int] : aa(A,$o,ord_less_eq(A,X),aa(int,A,ring_1_of_int(A),Z2)) ) ).

% ex_le_of_int
tff(fact_1323_ex__of__int__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [Z2: int] : aa(A,$o,ord_less(A,aa(int,A,ring_1_of_int(A),Z2)),X) ) ).

% ex_of_int_less
tff(fact_1324_ex__less__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [Z2: int] : aa(A,$o,ord_less(A,X),aa(int,A,ring_1_of_int(A),Z2)) ) ).

% ex_less_of_int
tff(fact_1325_dvd__field__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,A3),B3)
        <=> ( ( A3 = zero_zero(A) )
           => ( B3 = zero_zero(A) ) ) ) ) ).

% dvd_field_iff
tff(fact_1326_dvd__0__left,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] :
          ( aa(A,$o,dvd_dvd(A,zero_zero(A)),A3)
         => ( A3 = zero_zero(A) ) ) ) ).

% dvd_0_left
tff(fact_1327_eq__iff__diff__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = B3 )
        <=> ( aa(A,A,minus_minus(A,A3),B3) = zero_zero(A) ) ) ) ).

% eq_iff_diff_eq_0
tff(fact_1328_diff__eq__diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A,C2: A,D2: A] :
          ( ( aa(A,A,minus_minus(A,A3),B3) = aa(A,A,minus_minus(A,C2),D2) )
         => ( aa(A,$o,ord_less_eq(A,A3),B3)
          <=> aa(A,$o,ord_less_eq(A,C2),D2) ) ) ) ).

% diff_eq_diff_less_eq
tff(fact_1329_diff__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => aa(A,$o,ord_less_eq(A,aa(A,A,minus_minus(A,A3),C2)),aa(A,A,minus_minus(A,B3),C2)) ) ) ).

% diff_right_mono
tff(fact_1330_diff__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
         => aa(A,$o,ord_less_eq(A,aa(A,A,minus_minus(A,C2),A3)),aa(A,A,minus_minus(A,C2),B3)) ) ) ).

% diff_left_mono
tff(fact_1331_diff__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A,D2: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,D2),C2)
           => aa(A,$o,ord_less_eq(A,aa(A,A,minus_minus(A,A3),C2)),aa(A,A,minus_minus(A,B3),D2)) ) ) ) ).

% diff_mono
tff(fact_1332_dvd__add__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A3),B3)
         => ( aa(A,$o,dvd_dvd(A,A3),aa(A,A,plus_plus(A,B3),C2))
          <=> aa(A,$o,dvd_dvd(A,A3),C2) ) ) ) ).

% dvd_add_right_iff
tff(fact_1333_dvd__add__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,A3),C2)
         => ( aa(A,$o,dvd_dvd(A,A3),aa(A,A,plus_plus(A,B3),C2))
          <=> aa(A,$o,dvd_dvd(A,A3),B3) ) ) ) ).

% dvd_add_left_iff
tff(fact_1334_dvd__add,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A3),B3)
         => ( aa(A,$o,dvd_dvd(A,A3),C2)
           => aa(A,$o,dvd_dvd(A,A3),aa(A,A,plus_plus(A,B3),C2)) ) ) ) ).

% dvd_add
tff(fact_1335_one__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A] : aa(A,$o,dvd_dvd(A,one_one(A)),A3) ) ).

% one_dvd
tff(fact_1336_unit__imp__dvd,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,dvd_dvd(A,B3),one_one(A))
         => aa(A,$o,dvd_dvd(A,B3),A3) ) ) ).

% unit_imp_dvd
tff(fact_1337_dvd__unit__imp__unit,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,A3),B3)
         => ( aa(A,$o,dvd_dvd(A,B3),one_one(A))
           => aa(A,$o,dvd_dvd(A,A3),one_one(A)) ) ) ) ).

% dvd_unit_imp_unit
tff(fact_1338_diff__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,A3),C2)),aa(A,A,minus_minus(A,B3),C2)) ) ) ).

% diff_strict_right_mono
tff(fact_1339_diff__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,ord_less(A,B3),A3)
         => aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,C2),A3)),aa(A,A,minus_minus(A,C2),B3)) ) ) ).

% diff_strict_left_mono
tff(fact_1340_diff__eq__diff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A,C2: A,D2: A] :
          ( ( aa(A,A,minus_minus(A,A3),B3) = aa(A,A,minus_minus(A,C2),D2) )
         => ( aa(A,$o,ord_less(A,A3),B3)
          <=> aa(A,$o,ord_less(A,C2),D2) ) ) ) ).

% diff_eq_diff_less
tff(fact_1341_diff__strict__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A,D2: A,C2: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(A,$o,ord_less(A,D2),C2)
           => aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,A3),C2)),aa(A,A,minus_minus(A,B3),D2)) ) ) ) ).

% diff_strict_mono
tff(fact_1342_diff__diff__eq,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,minus_minus(A,aa(A,A,minus_minus(A,A3),B3)),C2) = aa(A,A,minus_minus(A,A3),aa(A,A,plus_plus(A,B3),C2)) ) ).

% diff_diff_eq
tff(fact_1343_add__diff__add,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,C2: A,B3: A,D2: A] : aa(A,A,minus_minus(A,aa(A,A,plus_plus(A,A3),C2)),aa(A,A,plus_plus(A,B3),D2)) = aa(A,A,plus_plus(A,aa(A,A,minus_minus(A,A3),B3)),aa(A,A,minus_minus(A,C2),D2)) ) ).

% add_diff_add
tff(fact_1344_add__implies__diff,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [C2: A,B3: A,A3: A] :
          ( ( aa(A,A,plus_plus(A,C2),B3) = A3 )
         => ( C2 = aa(A,A,minus_minus(A,A3),B3) ) ) ) ).

% add_implies_diff
tff(fact_1345_diff__add__eq__diff__diff__swap,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,minus_minus(A,A3),aa(A,A,plus_plus(A,B3),C2)) = aa(A,A,minus_minus(A,aa(A,A,minus_minus(A,A3),C2)),B3) ) ).

% diff_add_eq_diff_diff_swap
tff(fact_1346_diff__add__eq,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,plus_plus(A,aa(A,A,minus_minus(A,A3),B3)),C2) = aa(A,A,minus_minus(A,aa(A,A,plus_plus(A,A3),C2)),B3) ) ).

% diff_add_eq
tff(fact_1347_diff__diff__eq2,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,minus_minus(A,A3),aa(A,A,minus_minus(A,B3),C2)) = aa(A,A,minus_minus(A,aa(A,A,plus_plus(A,A3),C2)),B3) ) ).

% diff_diff_eq2
tff(fact_1348_add__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,plus_plus(A,A3),aa(A,A,minus_minus(A,B3),C2)) = aa(A,A,minus_minus(A,aa(A,A,plus_plus(A,A3),B3)),C2) ) ).

% add_diff_eq
tff(fact_1349_eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,C2: A,B3: A] :
          ( ( A3 = aa(A,A,minus_minus(A,C2),B3) )
        <=> ( aa(A,A,plus_plus(A,A3),B3) = C2 ) ) ) ).

% eq_diff_eq
tff(fact_1350_diff__eq__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( ( aa(A,A,minus_minus(A,A3),B3) = C2 )
        <=> ( A3 = aa(A,A,plus_plus(A,C2),B3) ) ) ) ).

% diff_eq_eq
tff(fact_1351_group__cancel_Osub1,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,K: A,A3: A,B3: A] :
          ( ( A2 = aa(A,A,plus_plus(A,K),A3) )
         => ( aa(A,A,minus_minus(A,A2),B3) = aa(A,A,plus_plus(A,K),aa(A,A,minus_minus(A,A3),B3)) ) ) ) ).

% group_cancel.sub1
tff(fact_1352_dvd__power__same,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [X: A,Y: A,Na: nat] :
          ( aa(A,$o,dvd_dvd(A,X),Y)
         => aa(A,$o,dvd_dvd(A,aa(nat,A,power_power(A,X),Na)),aa(nat,A,power_power(A,Y),Na)) ) ) ).

% dvd_power_same
tff(fact_1353_minus__diff__minus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,B3: A] : aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),A3)),aa(A,A,uminus_uminus(A),B3)) = aa(A,A,uminus_uminus(A),aa(A,A,minus_minus(A,A3),B3)) ) ).

% minus_diff_minus
tff(fact_1354_minus__diff__commute,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [B3: A,A3: A] : aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),B3)),A3) = aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),A3)),B3) ) ).

% minus_diff_commute
tff(fact_1355_power__divide,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B3: A,Na: nat] : aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),Na) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,power_power(A,A3),Na)),aa(nat,A,power_power(A,B3),Na)) ) ).

% power_divide
tff(fact_1356_powr__minus__divide,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [X: A,A3: A] : powr(A,X,aa(A,A,uminus_uminus(A),A3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),powr(A,X,A3)) ) ).

% powr_minus_divide
tff(fact_1357_even__of__int__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(int,A,ring_1_of_int(A),K))
        <=> aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K) ) ) ).

% even_of_int_iff
tff(fact_1358_diffs0__imp__equal,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,minus_minus(nat,M),Na) = zero_zero(nat) )
     => ( ( aa(nat,nat,minus_minus(nat,Na),M) = zero_zero(nat) )
       => ( M = Na ) ) ) ).

% diffs0_imp_equal
tff(fact_1359_minus__nat_Odiff__0,axiom,
    ! [M: nat] : aa(nat,nat,minus_minus(nat,M),zero_zero(nat)) = M ).

% minus_nat.diff_0
tff(fact_1360_Diff__mono,axiom,
    ! [A: $tType,A2: set(A),C3: set(A),D3: set(A),B2: set(A)] :
      ( aa(set(A),$o,ord_less_eq(set(A),A2),C3)
     => ( aa(set(A),$o,ord_less_eq(set(A),D3),B2)
       => aa(set(A),$o,ord_less_eq(set(A),aa(set(A),set(A),minus_minus(set(A),A2),B2)),aa(set(A),set(A),minus_minus(set(A),C3),D3)) ) ) ).

% Diff_mono
tff(fact_1361_Diff__subset,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] : aa(set(A),$o,ord_less_eq(set(A),aa(set(A),set(A),minus_minus(set(A),A2),B2)),A2) ).

% Diff_subset
tff(fact_1362_double__diff,axiom,
    ! [A: $tType,A2: set(A),B2: set(A),C3: set(A)] :
      ( aa(set(A),$o,ord_less_eq(set(A),A2),B2)
     => ( aa(set(A),$o,ord_less_eq(set(A),B2),C3)
       => ( aa(set(A),set(A),minus_minus(set(A),B2),aa(set(A),set(A),minus_minus(set(A),C3),A2)) = A2 ) ) ) ).

% double_diff
tff(fact_1363_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,J),K)
     => aa(nat,$o,ord_less(nat,aa(nat,nat,minus_minus(nat,J),Na)),K) ) ).

% less_imp_diff_less
tff(fact_1364_diff__less__mono2,axiom,
    ! [M: nat,Na: nat,L: nat] :
      ( aa(nat,$o,ord_less(nat,M),Na)
     => ( aa(nat,$o,ord_less(nat,M),L)
       => aa(nat,$o,ord_less(nat,aa(nat,nat,minus_minus(nat,L),Na)),aa(nat,nat,minus_minus(nat,L),M)) ) ) ).

% diff_less_mono2
tff(fact_1365_diff__diff__less,axiom,
    ! [I: nat,M: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,I),aa(nat,nat,minus_minus(nat,M),aa(nat,nat,minus_minus(nat,M),Na)))
    <=> ( aa(nat,$o,ord_less(nat,I),M)
        & aa(nat,$o,ord_less(nat,I),Na) ) ) ).

% diff_diff_less
tff(fact_1366_insert__Diff__if,axiom,
    ! [A: $tType,X: A,A2: set(A),B2: set(A)] :
      aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),insert(A,X),A2)),B2) = $ite(member(A,X,B2),aa(set(A),set(A),minus_minus(set(A),A2),B2),aa(set(A),set(A),insert(A,X),aa(set(A),set(A),minus_minus(set(A),A2),B2))) ).

% insert_Diff_if
tff(fact_1367_minus__int__code_I1_J,axiom,
    ! [K: int] : aa(int,int,minus_minus(int,K),zero_zero(int)) = K ).

% minus_int_code(1)
tff(fact_1368_word__of__int__uint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(int,word(A),ring_1_of_int(word(A)),aa(word(A),int,semiring_1_unsigned(A,int),W)) = W ) ).

% word_of_int_uint
tff(fact_1369_More__Word_Oof__int__uint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(int,word(A),ring_1_of_int(word(A)),aa(word(A),int,semiring_1_unsigned(A,int),X)) = X ) ).

% More_Word.of_int_uint
tff(fact_1370_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,Na: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,plus_plus(nat,K),M)),aa(nat,nat,plus_plus(nat,K),Na)) = aa(nat,nat,minus_minus(nat,M),Na) ).

% Nat.diff_cancel
tff(fact_1371_diff__cancel2,axiom,
    ! [M: nat,K: nat,Na: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,plus_plus(nat,M),K)),aa(nat,nat,plus_plus(nat,Na),K)) = aa(nat,nat,minus_minus(nat,M),Na) ).

% diff_cancel2
tff(fact_1372_diff__add__inverse,axiom,
    ! [Na: nat,M: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,plus_plus(nat,Na),M)),Na) = M ).

% diff_add_inverse
tff(fact_1373_diff__add__inverse2,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,plus_plus(nat,M),Na)),Na) = M ).

% diff_add_inverse2
tff(fact_1374_eq__diff__iff,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),M)
     => ( aa(nat,$o,ord_less_eq(nat,K),Na)
       => ( ( aa(nat,nat,minus_minus(nat,M),K) = aa(nat,nat,minus_minus(nat,Na),K) )
        <=> ( M = Na ) ) ) ) ).

% eq_diff_iff
tff(fact_1375_le__diff__iff,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),M)
     => ( aa(nat,$o,ord_less_eq(nat,K),Na)
       => ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,M),K)),aa(nat,nat,minus_minus(nat,Na),K))
        <=> aa(nat,$o,ord_less_eq(nat,M),Na) ) ) ) ).

% le_diff_iff
tff(fact_1376_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),M)
     => ( aa(nat,$o,ord_less_eq(nat,K),Na)
       => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,M),K)),aa(nat,nat,minus_minus(nat,Na),K)) = aa(nat,nat,minus_minus(nat,M),Na) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_1377_diff__le__mono,axiom,
    ! [M: nat,Na: nat,L: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),Na)
     => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,M),L)),aa(nat,nat,minus_minus(nat,Na),L)) ) ).

% diff_le_mono
tff(fact_1378_diff__le__self,axiom,
    ! [M: nat,Na: nat] : aa(nat,$o,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,M),Na)),M) ).

% diff_le_self
tff(fact_1379_le__diff__iff_H,axiom,
    ! [A3: nat,C2: nat,B3: nat] :
      ( aa(nat,$o,ord_less_eq(nat,A3),C2)
     => ( aa(nat,$o,ord_less_eq(nat,B3),C2)
       => ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,C2),A3)),aa(nat,nat,minus_minus(nat,C2),B3))
        <=> aa(nat,$o,ord_less_eq(nat,B3),A3) ) ) ) ).

% le_diff_iff'
tff(fact_1380_diff__le__mono2,axiom,
    ! [M: nat,Na: nat,L: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),Na)
     => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,L),Na)),aa(nat,nat,minus_minus(nat,L),M)) ) ).

% diff_le_mono2
tff(fact_1381_Un__Diff,axiom,
    ! [A: $tType,A2: set(A),B2: set(A),C3: set(A)] : aa(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)),A2),B2)),C3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),minus_minus(set(A),A2),C3)),aa(set(A),set(A),minus_minus(set(A),B2),C3)) ).

% Un_Diff
tff(fact_1382_set__diff__diff__left,axiom,
    ! [A: $tType,A2: set(A),B2: set(A),C3: set(A)] : aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),minus_minus(set(A),A2),B2)),C3) = aa(set(A),set(A),minus_minus(set(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B2),C3)) ).

% set_diff_diff_left
tff(fact_1383_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M4: nat,N: nat] : Z != aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),M4)),aa(nat,int,semiring_1_of_nat(int),N)) ).

% int_diff_cases
tff(fact_1384_word__sub__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Y),X)
         => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),minus_minus(word(A),X),Y)),X) ) ) ).

% word_sub_le
tff(fact_1385_word__sub__le__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),minus_minus(word(A),X),Y)),X)
        <=> aa(word(A),$o,ord_less_eq(word(A),Y),X) ) ) ).

% word_sub_le_iff
tff(fact_1386_word__le__minus__mono,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),C2: word(A),D2: word(A),B3: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),A3),C2)
         => ( aa(word(A),$o,ord_less_eq(word(A),D2),B3)
           => ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),minus_minus(word(A),A3),B3)),A3)
             => ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),minus_minus(word(A),C2),D2)),C2)
               => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),minus_minus(word(A),A3),B3)),aa(word(A),word(A),minus_minus(word(A),C2),D2)) ) ) ) ) ) ).

% word_le_minus_mono
tff(fact_1387_word__le__minus__cancel,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A),Z: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),minus_minus(word(A),Y),X)),aa(word(A),word(A),minus_minus(word(A),Z),X))
         => ( aa(word(A),$o,ord_less_eq(word(A),X),Z)
           => aa(word(A),$o,ord_less_eq(word(A),Y),Z) ) ) ) ).

% word_le_minus_cancel
tff(fact_1388_word__le__minus__mono__left,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Z: word(A),X: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Y),Z)
         => ( aa(word(A),$o,ord_less_eq(word(A),X),Y)
           => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),minus_minus(word(A),Y),X)),aa(word(A),word(A),minus_minus(word(A),Z),X)) ) ) ) ).

% word_le_minus_mono_left
tff(fact_1389_word__le__imp__diff__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A),Na: word(A),M: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),K),Na)
         => ( aa(word(A),$o,ord_less_eq(word(A),Na),M)
           => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),minus_minus(word(A),Na),K)),M) ) ) ) ).

% word_le_imp_diff_le
tff(fact_1390_word__le__minus__mono__right,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Z: word(A),Y: word(A),X: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Z),Y)
         => ( aa(word(A),$o,ord_less_eq(word(A),Y),X)
           => ( aa(word(A),$o,ord_less_eq(word(A),Z),X)
             => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),minus_minus(word(A),X),Y)),aa(word(A),word(A),minus_minus(word(A),X),Z)) ) ) ) ) ).

% word_le_minus_mono_right
tff(fact_1391_sub__wrap__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Z: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),X),aa(word(A),word(A),minus_minus(word(A),X),Z))
        <=> aa(word(A),$o,ord_less(word(A),X),Z) ) ) ).

% sub_wrap_lt
tff(fact_1392_word__sub__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),X),aa(word(A),word(A),minus_minus(word(A),X),Y))
        <=> aa(word(A),$o,ord_less(word(A),X),Y) ) ) ).

% word_sub_less_iff
tff(fact_1393_word__less__minus__mono,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),C2: word(A),D2: word(A),B3: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),A3),C2)
         => ( aa(word(A),$o,ord_less(word(A),D2),B3)
           => ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),A3),B3)),A3)
             => ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),C2),D2)),C2)
               => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),A3),B3)),aa(word(A),word(A),minus_minus(word(A),C2),D2)) ) ) ) ) ) ).

% word_less_minus_mono
tff(fact_1394_psubset__imp__ex__mem,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] :
      ( aa(set(A),$o,ord_less(set(A),A2),B2)
     => ? [B5: A] : member(A,B5,aa(set(A),set(A),minus_minus(set(A),B2),A2)) ) ).

% psubset_imp_ex_mem
tff(fact_1395_dbl__dec__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [X: A] : neg_numeral_dbl_dec(A,X) = aa(A,A,minus_minus(A,aa(A,A,plus_plus(A,X),X)),one_one(A)) ) ).

% dbl_dec_def
tff(fact_1396_bit__eq__rec,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = B3 )
        <=> ( ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
            <=> aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),B3) )
            & ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ) ) ).

% bit_eq_rec
tff(fact_1397_of__int__ceiling__diff__one__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R3: A] : aa(A,$o,ord_less_eq(A,aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,R3))),one_one(A))),R3) ) ).

% of_int_ceiling_diff_one_le
tff(fact_1398_power__diff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A,Na: nat,M: nat] :
          ( ( A3 != zero_zero(A) )
         => ( aa(nat,$o,ord_less_eq(nat,Na),M)
           => ( aa(nat,A,power_power(A,A3),aa(nat,nat,minus_minus(nat,M),Na)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,power_power(A,A3),M)),aa(nat,A,power_power(A,A3),Na)) ) ) ) ) ).

% power_diff
tff(fact_1399_subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B3: A] :
          ( aa(set(A),$o,ord_less_eq(set(A),collect(A,aTP_Lamp_ap(A,fun(A,$o),A3))),collect(A,aTP_Lamp_ap(A,fun(A,$o),B3)))
        <=> aa(A,$o,dvd_dvd(A,A3),B3) ) ) ).

% subset_divisors_dvd
tff(fact_1400_int__ops_I6_J,axiom,
    ! [A3: nat,B3: nat] :
      aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,minus_minus(nat,A3),B3)) = $ite(aa(int,$o,ord_less(int,aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B3)),zero_zero(int),aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B3))) ).

% int_ops(6)
tff(fact_1401_floor__divide__of__nat__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [M: nat,Na: nat] : archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Na))) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)) ) ).

% floor_divide_of_nat_eq
tff(fact_1402_real__of__int__floor__gt__diff__one,axiom,
    ! [R3: real] : aa(real,$o,ord_less(real,aa(real,real,minus_minus(real,R3),one_one(real))),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R3))) ).

% real_of_int_floor_gt_diff_one
tff(fact_1403_real__of__int__floor__ge__diff__one,axiom,
    ! [R3: real] : aa(real,$o,ord_less_eq(real,aa(real,real,minus_minus(real,R3),one_one(real))),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R3))) ).

% real_of_int_floor_ge_diff_one
tff(fact_1404_strict__subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B3: A] :
          ( aa(set(A),$o,ord_less(set(A),collect(A,aTP_Lamp_ap(A,fun(A,$o),A3))),collect(A,aTP_Lamp_ap(A,fun(A,$o),B3)))
        <=> ( aa(A,$o,dvd_dvd(A,A3),B3)
            & ~ aa(A,$o,dvd_dvd(A,B3),A3) ) ) ) ).

% strict_subset_divisors_dvd
tff(fact_1405_even__mask__div__iff_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,Na: nat] :
          ( aa(A,$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,minus_minus(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M)),one_one(A))),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)))
        <=> aa(nat,$o,ord_less_eq(nat,M),Na) ) ) ).

% even_mask_div_iff'
tff(fact_1406_ceiling__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X))),one_one(A))),X)
          & aa(A,$o,ord_less_eq(A,X),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X))) ) ) ).

% ceiling_correct
tff(fact_1407_ceiling__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),Z)),one_one(A))),X)
         => ( aa(A,$o,ord_less_eq(A,X),aa(int,A,ring_1_of_int(A),Z))
           => ( archimedean_ceiling(A,X) = Z ) ) ) ) ).

% ceiling_unique
tff(fact_1408_ceiling__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,A3: int] :
          ( ( archimedean_ceiling(A,X) = A3 )
        <=> ( aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),A3)),one_one(A))),X)
            & aa(A,$o,ord_less_eq(A,X),aa(int,A,ring_1_of_int(A),A3)) ) ) ) ).

% ceiling_eq_iff
tff(fact_1409_ceiling__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,$o),Ta: A] :
          ( aa(int,$o,P,archimedean_ceiling(A,Ta))
        <=> ! [I4: int] :
              ( ( aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),I4)),one_one(A))),Ta)
                & aa(A,$o,ord_less_eq(A,Ta),aa(int,A,ring_1_of_int(A),I4)) )
             => aa(int,$o,P,I4) ) ) ) ).

% ceiling_split
tff(fact_1410_ceiling__less__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z: int] :
          ( aa(int,$o,ord_less(int,archimedean_ceiling(A,X)),Z)
        <=> aa(A,$o,ord_less_eq(A,X),aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),Z)),one_one(A))) ) ) ).

% ceiling_less_iff
tff(fact_1411_le__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] :
          ( aa(int,$o,ord_less_eq(int,Z),archimedean_ceiling(A,X))
        <=> aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),Z)),one_one(A))),X) ) ) ).

% le_ceiling_iff
tff(fact_1412_powr__non__neg,axiom,
    ! [A3: real,X: real] : ~ aa(real,$o,ord_less(real,powr(real,A3,X)),zero_zero(real)) ).

% powr_non_neg
tff(fact_1413_powr__less__mono2__neg,axiom,
    ! [A3: real,X: real,Y: real] :
      ( aa(real,$o,ord_less(real,A3),zero_zero(real))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),X)
       => ( aa(real,$o,ord_less(real,X),Y)
         => aa(real,$o,ord_less(real,powr(real,Y,A3)),powr(real,X,A3)) ) ) ) ).

% powr_less_mono2_neg
tff(fact_1414_powr__mono2,axiom,
    ! [A3: real,X: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),A3)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
       => ( aa(real,$o,ord_less_eq(real,X),Y)
         => aa(real,$o,ord_less_eq(real,powr(real,X,A3)),powr(real,Y,A3)) ) ) ) ).

% powr_mono2
tff(fact_1415_powr__ge__pzero,axiom,
    ! [X: real,Y: real] : aa(real,$o,ord_less_eq(real,zero_zero(real)),powr(real,X,Y)) ).

% powr_ge_pzero
tff(fact_1416_powr__less__mono,axiom,
    ! [A3: real,B3: real,X: real] :
      ( aa(real,$o,ord_less(real,A3),B3)
     => ( aa(real,$o,ord_less(real,one_one(real)),X)
       => aa(real,$o,ord_less(real,powr(real,X,A3)),powr(real,X,B3)) ) ) ).

% powr_less_mono
tff(fact_1417_powr__less__cancel,axiom,
    ! [X: real,A3: real,B3: real] :
      ( aa(real,$o,ord_less(real,powr(real,X,A3)),powr(real,X,B3))
     => ( aa(real,$o,ord_less(real,one_one(real)),X)
       => aa(real,$o,ord_less(real,A3),B3) ) ) ).

% powr_less_cancel
tff(fact_1418_powr__mono,axiom,
    ! [A3: real,B3: real,X: real] :
      ( aa(real,$o,ord_less_eq(real,A3),B3)
     => ( aa(real,$o,ord_less_eq(real,one_one(real)),X)
       => aa(real,$o,ord_less_eq(real,powr(real,X,A3)),powr(real,X,B3)) ) ) ).

% powr_mono
tff(fact_1419_not__is__unit__0,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ~ aa(A,$o,dvd_dvd(A,zero_zero(A)),one_one(A)) ) ).

% not_is_unit_0
tff(fact_1420_le__iff__diff__le__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
        <=> aa(A,$o,ord_less_eq(A,aa(A,A,minus_minus(A,A3),B3)),zero_zero(A)) ) ) ).

% le_iff_diff_le_0
tff(fact_1421_divide__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),zero_zero(A))
        <=> ( ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
              & aa(A,$o,ord_less_eq(A,B3),zero_zero(A)) )
            | ( aa(A,$o,ord_less_eq(A,A3),zero_zero(A))
              & aa(A,$o,ord_less_eq(A,zero_zero(A)),B3) ) ) ) ) ).

% divide_le_0_iff
tff(fact_1422_divide__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)) ) ) ) ).

% divide_right_mono
tff(fact_1423_zero__le__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3))
        <=> ( ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
              & aa(A,$o,ord_less_eq(A,zero_zero(A)),B3) )
            | ( aa(A,$o,ord_less_eq(A,A3),zero_zero(A))
              & aa(A,$o,ord_less_eq(A,B3),zero_zero(A)) ) ) ) ) ).

% zero_le_divide_iff
tff(fact_1424_divide__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),X)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Y)
           => aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)) ) ) ) ).

% divide_nonneg_nonneg
tff(fact_1425_divide__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),X)
         => ( aa(A,$o,ord_less_eq(A,Y),zero_zero(A))
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),zero_zero(A)) ) ) ) ).

% divide_nonneg_nonpos
tff(fact_1426_divide__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,X),zero_zero(A))
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Y)
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),zero_zero(A)) ) ) ) ).

% divide_nonpos_nonneg
tff(fact_1427_divide__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,X),zero_zero(A))
         => ( aa(A,$o,ord_less_eq(A,Y),zero_zero(A))
           => aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)) ) ) ) ).

% divide_nonpos_nonpos
tff(fact_1428_divide__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,C2),zero_zero(A))
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)) ) ) ) ).

% divide_right_mono_neg
tff(fact_1429_less__iff__diff__less__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
        <=> aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,A3),B3)),zero_zero(A)) ) ) ).

% less_iff_diff_less_0
tff(fact_1430_divide__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,ord_less(A,B3),A3)
         => ( aa(A,$o,ord_less(A,C2),zero_zero(A))
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)) ) ) ) ).

% divide_strict_right_mono_neg
tff(fact_1431_divide__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)) ) ) ) ).

% divide_strict_right_mono
tff(fact_1432_zero__less__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
              & aa(A,$o,ord_less(A,zero_zero(A)),B3) )
            | ( aa(A,$o,ord_less(A,A3),zero_zero(A))
              & aa(A,$o,ord_less(A,B3),zero_zero(A)) ) ) ) ) ).

% zero_less_divide_iff
tff(fact_1433_divide__less__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
             => aa(A,$o,ord_less(A,A3),B3) )
            & ( aa(A,$o,ord_less(A,C2),zero_zero(A))
             => aa(A,$o,ord_less(A,B3),A3) )
            & ( C2 != zero_zero(A) ) ) ) ) ).

% divide_less_cancel
tff(fact_1434_divide__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),zero_zero(A))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
              & aa(A,$o,ord_less(A,B3),zero_zero(A)) )
            | ( aa(A,$o,ord_less(A,A3),zero_zero(A))
              & aa(A,$o,ord_less(A,zero_zero(A)),B3) ) ) ) ) ).

% divide_less_0_iff
tff(fact_1435_divide__pos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),X)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),Y)
           => aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)) ) ) ) ).

% divide_pos_pos
tff(fact_1436_divide__pos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),X)
         => ( aa(A,$o,ord_less(A,Y),zero_zero(A))
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),zero_zero(A)) ) ) ) ).

% divide_pos_neg
tff(fact_1437_divide__neg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less(A,X),zero_zero(A))
         => ( aa(A,$o,ord_less(A,zero_zero(A)),Y)
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),zero_zero(A)) ) ) ) ).

% divide_neg_pos
tff(fact_1438_divide__neg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less(A,X),zero_zero(A))
         => ( aa(A,$o,ord_less(A,Y),zero_zero(A))
           => aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)) ) ) ) ).

% divide_neg_neg
tff(fact_1439_unat__minus__one,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( ( W != zero_zero(word(A)) )
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),minus_minus(word(A),W),one_one(word(A)))) = aa(nat,nat,minus_minus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),W)),one_one(nat)) ) ) ) ).

% unat_minus_one
tff(fact_1440_add__le__add__imp__diff__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: A,K: A,Na: A,J: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,plus_plus(A,I),K)),Na)
         => ( aa(A,$o,ord_less_eq(A,Na),aa(A,A,plus_plus(A,J),K))
           => ( aa(A,$o,ord_less_eq(A,aa(A,A,plus_plus(A,I),K)),Na)
             => ( aa(A,$o,ord_less_eq(A,Na),aa(A,A,plus_plus(A,J),K))
               => aa(A,$o,ord_less_eq(A,aa(A,A,minus_minus(A,Na),K)),J) ) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1441_add__le__imp__le__diff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: A,K: A,Na: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,plus_plus(A,I),K)),Na)
         => aa(A,$o,ord_less_eq(A,I),aa(A,A,minus_minus(A,Na),K)) ) ) ).

% add_le_imp_le_diff
tff(fact_1442_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,A3),B3)
           => ( ( aa(A,A,minus_minus(A,B3),A3) = C2 )
            <=> ( B3 = aa(A,A,plus_plus(A,C2),A3) ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_1443_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,A,plus_plus(A,A3),aa(A,A,minus_minus(A,B3),A3)) = B3 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_1444_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,A,minus_minus(A,C2),aa(A,A,minus_minus(A,B3),A3)) = aa(A,A,minus_minus(A,aa(A,A,plus_plus(A,C2),A3)),B3) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_1445_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,A,minus_minus(A,aa(A,A,plus_plus(A,B3),C2)),A3) = aa(A,A,plus_plus(A,aa(A,A,minus_minus(A,B3),A3)),C2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_1446_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,A,plus_plus(A,aa(A,A,minus_minus(A,B3),A3)),C2) = aa(A,A,minus_minus(A,aa(A,A,plus_plus(A,B3),C2)),A3) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_1447_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,A,minus_minus(A,aa(A,A,plus_plus(A,C2),B3)),A3) = aa(A,A,plus_plus(A,C2),aa(A,A,minus_minus(A,B3),A3)) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_1448_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,A,plus_plus(A,C2),aa(A,A,minus_minus(A,B3),A3)) = aa(A,A,minus_minus(A,aa(A,A,plus_plus(A,C2),B3)),A3) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_1449_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,C2),aa(A,A,minus_minus(A,B3),A3))
          <=> aa(A,$o,ord_less_eq(A,aa(A,A,plus_plus(A,C2),A3)),B3) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_1450_le__add__diff,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => aa(A,$o,ord_less_eq(A,C2),aa(A,A,minus_minus(A,aa(A,A,plus_plus(A,B3),C2)),A3)) ) ) ).

% le_add_diff
tff(fact_1451_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,A,plus_plus(A,aa(A,A,minus_minus(A,B3),A3)),A3) = B3 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add
tff(fact_1452_le__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),aa(A,A,minus_minus(A,C2),B3))
        <=> aa(A,$o,ord_less_eq(A,aa(A,A,plus_plus(A,A3),B3)),C2) ) ) ).

% le_diff_eq
tff(fact_1453_diff__le__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,minus_minus(A,A3),B3)),C2)
        <=> aa(A,$o,ord_less_eq(A,A3),aa(A,A,plus_plus(A,C2),B3)) ) ) ).

% diff_le_eq
tff(fact_1454_right__inverse__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B3: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) = one_one(A) )
          <=> ( A3 = B3 ) ) ) ) ).

% right_inverse_eq
tff(fact_1455_diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,A3),B3)),C2)
        <=> aa(A,$o,ord_less(A,A3),aa(A,A,plus_plus(A,C2),B3)) ) ) ).

% diff_less_eq
tff(fact_1456_less__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),aa(A,A,minus_minus(A,C2),B3))
        <=> aa(A,$o,ord_less(A,aa(A,A,plus_plus(A,A3),B3)),C2) ) ) ).

% less_diff_eq
tff(fact_1457_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,B3: A] :
          ( ~ aa(A,$o,ord_less(A,A3),B3)
         => ( aa(A,A,plus_plus(A,B3),aa(A,A,minus_minus(A,A3),B3)) = A3 ) ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1458_divide__numeral__1,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),one2)) = A3 ) ).

% divide_numeral_1
tff(fact_1459_diff__shunt__var,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,minus_minus(A,X),Y) = bot_bot(A) )
        <=> aa(A,$o,ord_less_eq(A,X),Y) ) ) ).

% diff_shunt_var
tff(fact_1460_nonzero__minus__divide__right,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B3: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,uminus_uminus(A),B3)) ) ) ) ).

% nonzero_minus_divide_right
tff(fact_1461_nonzero__minus__divide__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B3: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,uminus_uminus(A),B3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) ) ) ) ).

% nonzero_minus_divide_divide
tff(fact_1462_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,B3: A] : aa(A,A,minus_minus(A,A3),B3) = aa(A,A,plus_plus(A,A3),aa(A,A,uminus_uminus(A),B3)) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_1463_diff__conv__add__uminus,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] : aa(A,A,minus_minus(A,A3),B3) = aa(A,A,plus_plus(A,A3),aa(A,A,uminus_uminus(A),B3)) ) ).

% diff_conv_add_uminus
tff(fact_1464_group__cancel_Osub2,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [B2: A,K: A,B3: A,A3: A] :
          ( ( B2 = aa(A,A,plus_plus(A,K),B3) )
         => ( aa(A,A,minus_minus(A,A3),B2) = aa(A,A,plus_plus(A,aa(A,A,uminus_uminus(A),K)),aa(A,A,minus_minus(A,A3),B3)) ) ) ) ).

% group_cancel.sub2
tff(fact_1465_of__int__floor__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X))),X) ) ).

% of_int_floor_le
tff(fact_1466_power__one__over,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,Na: nat] : aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)),Na) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,power_power(A,A3),Na)) ) ).

% power_one_over
tff(fact_1467_dvd__power__le,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [X: A,Y: A,Na: nat,M: nat] :
          ( aa(A,$o,dvd_dvd(A,X),Y)
         => ( aa(nat,$o,ord_less_eq(nat,Na),M)
           => aa(A,$o,dvd_dvd(A,aa(nat,A,power_power(A,X),Na)),aa(nat,A,power_power(A,Y),M)) ) ) ) ).

% dvd_power_le
tff(fact_1468_power__le__dvd,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,Na: nat,B3: A,M: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(nat,A,power_power(A,A3),Na)),B3)
         => ( aa(nat,$o,ord_less_eq(nat,M),Na)
           => aa(A,$o,dvd_dvd(A,aa(nat,A,power_power(A,A3),M)),B3) ) ) ) ).

% power_le_dvd
tff(fact_1469_le__imp__power__dvd,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [M: nat,Na: nat,A3: A] :
          ( aa(nat,$o,ord_less_eq(nat,M),Na)
         => aa(A,$o,dvd_dvd(A,aa(nat,A,power_power(A,A3),M)),aa(nat,A,power_power(A,A3),Na)) ) ) ).

% le_imp_power_dvd
tff(fact_1470_le__of__int__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(A,$o,ord_less_eq(A,X),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X))) ) ).

% le_of_int_ceiling
tff(fact_1471_nat__dvd__not__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
     => ( aa(nat,$o,ord_less(nat,M),Na)
       => ~ aa(nat,$o,dvd_dvd(nat,Na),M) ) ) ).

% nat_dvd_not_less
tff(fact_1472_even__mask__div__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Na: nat] :
          ( aa(A,$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,minus_minus(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M)),one_one(A))),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)))
        <=> ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na) = zero_zero(A) )
            | aa(nat,$o,ord_less_eq(nat,M),Na) ) ) ) ).

% even_mask_div_iff
tff(fact_1473_word__numeral__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: num] : aa(num,word(A),numeral_numeral(word(A)),B3) = aa(int,word(A),ring_1_of_int(word(A)),aa(num,int,numeral_numeral(int),B3)) ) ).

% word_numeral_alt
tff(fact_1474_subset__minus__empty,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] :
      ( aa(set(A),$o,ord_less_eq(set(A),A2),B2)
     => ( aa(set(A),set(A),minus_minus(set(A),A2),B2) = bot_bot(set(A)) ) ) ).

% subset_minus_empty
tff(fact_1475_unat__sub__if__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),minus_minus(word(A),X),Y)) = $ite(aa(nat,$o,ord_less_eq(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(nat,nat,minus_minus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),aa(nat,nat,minus_minus(nat,aa(nat,nat,plus_plus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),size_size(word(A),X)))),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))) ) ).

% unat_sub_if_size
tff(fact_1476_Diff__insert__absorb,axiom,
    ! [A: $tType,X: A,A2: set(A)] :
      ( ~ member(A,X,A2)
     => ( aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),insert(A,X),A2)),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))) = A2 ) ) ).

% Diff_insert_absorb
tff(fact_1477_Diff__insert2,axiom,
    ! [A: $tType,A2: set(A),A3: A,B2: set(A)] : aa(set(A),set(A),minus_minus(set(A),A2),aa(set(A),set(A),insert(A,A3),B2)) = aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),minus_minus(set(A),A2),aa(set(A),set(A),insert(A,A3),bot_bot(set(A))))),B2) ).

% Diff_insert2
tff(fact_1478_insert__Diff,axiom,
    ! [A: $tType,A3: A,A2: set(A)] :
      ( member(A,A3,A2)
     => ( aa(set(A),set(A),insert(A,A3),aa(set(A),set(A),minus_minus(set(A),A2),aa(set(A),set(A),insert(A,A3),bot_bot(set(A))))) = A2 ) ) ).

% insert_Diff
tff(fact_1479_Diff__insert,axiom,
    ! [A: $tType,A2: set(A),A3: A,B2: set(A)] : aa(set(A),set(A),minus_minus(set(A),A2),aa(set(A),set(A),insert(A,A3),B2)) = aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),minus_minus(set(A),A2),B2)),aa(set(A),set(A),insert(A,A3),bot_bot(set(A)))) ).

% Diff_insert
tff(fact_1480_insert__minus__eq,axiom,
    ! [A: $tType,X: A,Y: A,A2: set(A)] :
      ( ( X != Y )
     => ( aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),insert(A,X),A2)),aa(set(A),set(A),insert(A,Y),bot_bot(set(A)))) = aa(set(A),set(A),insert(A,X),aa(set(A),set(A),minus_minus(set(A),A2),aa(set(A),set(A),insert(A,Y),bot_bot(set(A))))) ) ) ).

% insert_minus_eq
tff(fact_1481_set__minus__singleton__eq,axiom,
    ! [A: $tType,X: A,X5: set(A)] :
      ( ~ member(A,X,X5)
     => ( aa(set(A),set(A),minus_minus(set(A),X5),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))) = X5 ) ) ).

% set_minus_singleton_eq
tff(fact_1482_diff__less,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
       => aa(nat,$o,ord_less(nat,aa(nat,nat,minus_minus(nat,M),Na)),M) ) ) ).

% diff_less
tff(fact_1483_subset__Diff__insert,axiom,
    ! [A: $tType,A2: set(A),B2: set(A),X: A,C3: set(A)] :
      ( aa(set(A),$o,ord_less_eq(set(A),A2),aa(set(A),set(A),minus_minus(set(A),B2),aa(set(A),set(A),insert(A,X),C3)))
    <=> ( aa(set(A),$o,ord_less_eq(set(A),A2),aa(set(A),set(A),minus_minus(set(A),B2),C3))
        & ~ member(A,X,A2) ) ) ).

% subset_Diff_insert
tff(fact_1484_diff__add__0,axiom,
    ! [Na: nat,M: nat] : aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,plus_plus(nat,Na),M)) = zero_zero(nat) ).

% diff_add_0
tff(fact_1485_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,I),aa(nat,nat,minus_minus(nat,J),K))
    <=> aa(nat,$o,ord_less(nat,aa(nat,nat,plus_plus(nat,I),K)),J) ) ).

% less_diff_conv
tff(fact_1486_add__diff__inverse__nat,axiom,
    ! [M: nat,Na: nat] :
      ( ~ aa(nat,$o,ord_less(nat,M),Na)
     => ( aa(nat,nat,plus_plus(nat,Na),aa(nat,nat,minus_minus(nat,M),Na)) = M ) ) ).

% add_diff_inverse_nat
tff(fact_1487_less__diff__iff,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),M)
     => ( aa(nat,$o,ord_less_eq(nat,K),Na)
       => ( aa(nat,$o,ord_less(nat,aa(nat,nat,minus_minus(nat,M),K)),aa(nat,nat,minus_minus(nat,Na),K))
        <=> aa(nat,$o,ord_less(nat,M),Na) ) ) ) ).

% less_diff_iff
tff(fact_1488_diff__less__mono,axiom,
    ! [A3: nat,B3: nat,C2: nat] :
      ( aa(nat,$o,ord_less(nat,A3),B3)
     => ( aa(nat,$o,ord_less_eq(nat,C2),A3)
       => aa(nat,$o,ord_less(nat,aa(nat,nat,minus_minus(nat,A3),C2)),aa(nat,nat,minus_minus(nat,B3),C2)) ) ) ).

% diff_less_mono
tff(fact_1489_uint__sub__if__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),minus_minus(word(A),X),Y)) = $ite(aa(int,$o,ord_less_eq(int,aa(word(A),int,semiring_1_unsigned(A,int),Y)),aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Y)),aa(int,int,plus_plus(int,aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Y))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),size_size(word(A),X)))) ) ).

% uint_sub_if_size
tff(fact_1490_wi__hom__add,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: int,B3: int] : aa(word(A),word(A),plus_plus(word(A),aa(int,word(A),ring_1_of_int(word(A)),A3)),aa(int,word(A),ring_1_of_int(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,plus_plus(int,A3),B3)) ) ).

% wi_hom_add
tff(fact_1491_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,J),K)),I)
    <=> aa(nat,$o,ord_less_eq(nat,J),aa(nat,nat,plus_plus(nat,I),K)) ) ).

% le_diff_conv
tff(fact_1492_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),J)
     => ( aa(nat,$o,ord_less_eq(nat,I),aa(nat,nat,minus_minus(nat,J),K))
      <=> aa(nat,$o,ord_less_eq(nat,aa(nat,nat,plus_plus(nat,I),K)),J) ) ) ).

% Nat.le_diff_conv2
tff(fact_1493_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),J)
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,plus_plus(nat,I),J)),K) = aa(nat,nat,plus_plus(nat,I),aa(nat,nat,minus_minus(nat,J),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_1494_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),J)
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,plus_plus(nat,J),I)),K) = aa(nat,nat,plus_plus(nat,aa(nat,nat,minus_minus(nat,J),K)),I) ) ) ).

% Nat.diff_add_assoc2
tff(fact_1495_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => ( ( aa(nat,nat,minus_minus(nat,J),I) = K )
      <=> ( J = aa(nat,nat,plus_plus(nat,K),I) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_1496_Diff__partition,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] :
      ( aa(set(A),$o,ord_less_eq(set(A),A2),B2)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),aa(set(A),set(A),minus_minus(set(A),B2),A2)) = B2 ) ) ).

% Diff_partition
tff(fact_1497_Diff__subset__conv,axiom,
    ! [A: $tType,A2: set(A),B2: set(A),C3: set(A)] :
      ( aa(set(A),$o,ord_less_eq(set(A),aa(set(A),set(A),minus_minus(set(A),A2),B2)),C3)
    <=> aa(set(A),$o,ord_less_eq(set(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B2),C3)) ) ).

% Diff_subset_conv
tff(fact_1498_minus__int__code_I2_J,axiom,
    ! [L: int] : aa(int,int,minus_minus(int,zero_zero(int)),L) = aa(int,int,uminus_uminus(int),L) ).

% minus_int_code(2)
tff(fact_1499_wi__hom__neg,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: int] : aa(word(A),word(A),uminus_uminus(word(A)),aa(int,word(A),ring_1_of_int(word(A)),A3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,uminus_uminus(int),A3)) ) ).

% wi_hom_neg
tff(fact_1500_int__le__induct,axiom,
    ! [I: int,K: int,P: fun(int,$o)] :
      ( aa(int,$o,ord_less_eq(int,I),K)
     => ( aa(int,$o,P,K)
       => ( ! [I3: int] :
              ( aa(int,$o,ord_less_eq(int,I3),K)
             => ( aa(int,$o,P,I3)
               => aa(int,$o,P,aa(int,int,minus_minus(int,I3),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_le_induct
tff(fact_1501_int__less__induct,axiom,
    ! [I: int,K: int,P: fun(int,$o)] :
      ( aa(int,$o,ord_less(int,I),K)
     => ( aa(int,$o,P,aa(int,int,minus_minus(int,K),one_one(int)))
       => ( ! [I3: int] :
              ( aa(int,$o,ord_less(int,I3),K)
             => ( aa(int,$o,P,I3)
               => aa(int,$o,P,aa(int,int,minus_minus(int,I3),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_less_induct
tff(fact_1502_le__step__down__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),Na: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),I),Na)
         => ( ( I != Na )
           => aa(word(A),$o,ord_less_eq(word(A),I),aa(word(A),word(A),minus_minus(word(A),Na),one_one(word(A)))) ) ) ) ).

% le_step_down_word
tff(fact_1503_le__step__down__word__2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),X),Y)
         => ( ( X != Y )
           => aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),minus_minus(word(A),Y),one_one(word(A)))) ) ) ) ).

% le_step_down_word_2
tff(fact_1504_ucast__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [W: word(B)] : aa(word(B),word(A),semiring_1_unsigned(B,word(A)),W) = aa(int,word(A),ring_1_of_int(word(A)),aa(word(B),int,semiring_1_unsigned(B,int),W)) ) ).

% ucast_eq
tff(fact_1505_word__of__int__power__hom,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: int,Na: nat] : aa(nat,word(A),power_power(word(A),aa(int,word(A),ring_1_of_int(word(A)),A3)),Na) = aa(int,word(A),ring_1_of_int(word(A)),aa(nat,int,power_power(int,A3),Na)) ) ).

% word_of_int_power_hom
tff(fact_1506_word__diff__ls_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),W: word(A),Xa2: word(A),X: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Y),aa(word(A),word(A),plus_plus(word(A),W),Xa2))
         => ( aa(word(A),$o,ord_less_eq(word(A),X),Y)
           => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),minus_minus(word(A),Y),X)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),plus_plus(word(A),W),Xa2)),X)) ) ) ) ).

% word_diff_ls(4)
tff(fact_1507_word__diff__ls_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A),W: word(A),Xa2: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),minus_minus(word(A),Y),X)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),plus_plus(word(A),W),Xa2)),X))
         => ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),plus_plus(word(A),W),Xa2))
           => aa(word(A),$o,ord_less_eq(word(A),Y),aa(word(A),word(A),plus_plus(word(A),W),Xa2)) ) ) ) ).

% word_diff_ls(3)
tff(fact_1508_Word_Oword__l__diffs_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xa2: word(A),Z: word(A),X: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),plus_plus(word(A),W),Xa2)),Z)
         => ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),plus_plus(word(A),W),Xa2))
           => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),plus_plus(word(A),W),Xa2)),X)),aa(word(A),word(A),minus_minus(word(A),Z),X)) ) ) ) ).

% Word.word_l_diffs(4)
tff(fact_1509_Word_Oword__l__diffs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xa2: word(A),X: word(A),Z: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),plus_plus(word(A),W),Xa2)),X)),aa(word(A),word(A),minus_minus(word(A),Z),X))
         => ( aa(word(A),$o,ord_less_eq(word(A),X),Z)
           => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),plus_plus(word(A),W),Xa2)),Z) ) ) ) ).

% Word.word_l_diffs(3)
tff(fact_1510_word__plus__mcs_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V2: word(A),Xb: word(A),W: word(A),Xa2: word(A),X: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),plus_plus(word(A),V2),Xb)),aa(word(A),word(A),plus_plus(word(A),W),Xa2))
         => ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),plus_plus(word(A),V2),Xb))
           => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),plus_plus(word(A),V2),Xb)),X)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),plus_plus(word(A),W),Xa2)),X)) ) ) ) ).

% word_plus_mcs(4)
tff(fact_1511_word__plus__mcs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V2: word(A),Xb: word(A),X: word(A),W: word(A),Xa2: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),plus_plus(word(A),V2),Xb)),X)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),plus_plus(word(A),W),Xa2)),X))
         => ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),plus_plus(word(A),W),Xa2))
           => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),plus_plus(word(A),V2),Xb)),aa(word(A),word(A),plus_plus(word(A),W),Xa2)) ) ) ) ).

% word_plus_mcs(3)
tff(fact_1512_More__Word_Oword__l__diffs_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),X: word(A),Z: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),plus_plus(word(A),W),X)),Z)
         => ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),plus_plus(word(A),W),X))
           => aa(word(A),$o,ord_less_eq(word(A),W),aa(word(A),word(A),minus_minus(word(A),Z),X)) ) ) ) ).

% More_Word.word_l_diffs(4)
tff(fact_1513_More__Word_Oword__l__diffs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Z: word(A),X: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),W),aa(word(A),word(A),minus_minus(word(A),Z),X))
         => ( aa(word(A),$o,ord_less_eq(word(A),X),Z)
           => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),plus_plus(word(A),W),X)),Z) ) ) ) ).

% More_Word.word_l_diffs(3)
tff(fact_1514_word__diff__ls_H_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),W: word(A),X: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Y),aa(word(A),word(A),plus_plus(word(A),W),X))
         => ( aa(word(A),$o,ord_less_eq(word(A),X),Y)
           => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),minus_minus(word(A),Y),X)),W) ) ) ) ).

% word_diff_ls'(4)
tff(fact_1515_word__diff__ls_H_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A),W: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),minus_minus(word(A),Y),X)),W)
         => ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),plus_plus(word(A),W),X))
           => aa(word(A),$o,ord_less_eq(word(A),Y),aa(word(A),word(A),plus_plus(word(A),W),X)) ) ) ) ).

% word_diff_ls'(3)
tff(fact_1516_word__l__diffs_H_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),X: word(A),Z: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),plus_plus(word(A),W),X)),Z)
         => ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),plus_plus(word(A),W),X))
           => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),plus_plus(word(A),W),X)),X)),aa(word(A),word(A),minus_minus(word(A),Z),X)) ) ) ) ).

% word_l_diffs'(4)
tff(fact_1517_word__l__diffs_H_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),X: word(A),Z: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),plus_plus(word(A),W),X)),X)),aa(word(A),word(A),minus_minus(word(A),Z),X))
         => ( aa(word(A),$o,ord_less_eq(word(A),X),Z)
           => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),plus_plus(word(A),W),X)),Z) ) ) ) ).

% word_l_diffs'(3)
tff(fact_1518_word__diff__ls_H_H_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),W: word(A),X: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Y),aa(word(A),word(A),plus_plus(word(A),W),X))
         => ( aa(word(A),$o,ord_less_eq(word(A),X),Y)
           => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),minus_minus(word(A),Y),X)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),plus_plus(word(A),W),X)),X)) ) ) ) ).

% word_diff_ls''(4)
tff(fact_1519_word__diff__ls_H_H_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A),W: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),minus_minus(word(A),Y),X)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),plus_plus(word(A),W),X)),X))
         => ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),plus_plus(word(A),W),X))
           => aa(word(A),$o,ord_less_eq(word(A),Y),aa(word(A),word(A),plus_plus(word(A),W),X)) ) ) ) ).

% word_diff_ls''(3)
tff(fact_1520_le__plus,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [C2: word(A),B3: word(A),A3: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),C2),aa(word(A),word(A),minus_minus(word(A),B3),A3))
         => ( aa(word(A),$o,ord_less_eq(word(A),A3),B3)
           => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),plus_plus(word(A),A3),C2)),B3) ) ) ) ).

% le_plus
tff(fact_1521_le__plus_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A),C2: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),A3),B3)
         => ( aa(word(A),$o,ord_less_eq(word(A),C2),aa(word(A),word(A),minus_minus(word(A),B3),A3))
           => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),plus_plus(word(A),A3),C2)),B3) ) ) ) ).

% le_plus'
tff(fact_1522_le__minus_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),C2: word(A),B3: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),plus_plus(word(A),A3),C2)),B3)
         => ( aa(word(A),$o,ord_less_eq(word(A),A3),aa(word(A),word(A),plus_plus(word(A),A3),C2))
           => aa(word(A),$o,ord_less_eq(word(A),C2),aa(word(A),word(A),minus_minus(word(A),B3),A3)) ) ) ) ).

% le_minus'
tff(fact_1523_plus__minus__no__overflow__ab,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Ab: word(A),C2: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),minus_minus(word(A),Ab),C2))
         => ( aa(word(A),$o,ord_less_eq(word(A),C2),Ab)
           => aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),plus_plus(word(A),X),C2)) ) ) ) ).

% plus_minus_no_overflow_ab
tff(fact_1524_word__less__cases,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),X),Y)
         => ( ( X = aa(word(A),word(A),minus_minus(word(A),Y),one_one(word(A))) )
            | aa(word(A),$o,ord_less(word(A),X),aa(word(A),word(A),minus_minus(word(A),Y),one_one(word(A)))) ) ) ) ).

% word_less_cases
tff(fact_1525_word__less__minus__cancel,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A),Z: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Y),X)),aa(word(A),word(A),minus_minus(word(A),Z),X))
         => ( aa(word(A),$o,ord_less_eq(word(A),X),Z)
           => aa(word(A),$o,ord_less(word(A),Y),Z) ) ) ) ).

% word_less_minus_cancel
tff(fact_1526_word__less__minus__mono__left,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Z: word(A),X: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Y),Z)
         => ( aa(word(A),$o,ord_less_eq(word(A),X),Y)
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Y),X)),aa(word(A),word(A),minus_minus(word(A),Z),X)) ) ) ) ).

% word_less_minus_mono_left
tff(fact_1527_word__less__imp__diff__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A),Na: word(A),M: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),K),Na)
         => ( aa(word(A),$o,ord_less(word(A),Na),M)
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Na),K)),M) ) ) ) ).

% word_less_imp_diff_less
tff(fact_1528_minus__real__def,axiom,
    ! [X: real,Y: real] : aa(real,real,minus_minus(real,X),Y) = aa(real,real,plus_plus(real,X),aa(real,real,uminus_uminus(real),Y)) ).

% minus_real_def
tff(fact_1529_uno__simps_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Z: word(A),Na: nat] : 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),divide_divide(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Z)),Na))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Z)),Na) ) ).

% uno_simps(2)
tff(fact_1530_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y: extended_enat,X: extended_enat] :
      ( aa(extended_enat,$o,ord_less_eq(extended_enat,Z),Y)
     => ( aa(extended_enat,extended_enat,plus_plus(extended_enat,X),aa(extended_enat,extended_enat,minus_minus(extended_enat,Y),Z)) = aa(extended_enat,extended_enat,minus_minus(extended_enat,aa(extended_enat,extended_enat,plus_plus(extended_enat,X),Y)),Z) ) ) ).

% add_diff_assoc_enat
tff(fact_1531_powr__less__mono2,axiom,
    ! [A3: real,X: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
       => ( aa(real,$o,ord_less(real,X),Y)
         => aa(real,$o,ord_less(real,powr(real,X,A3)),powr(real,Y,A3)) ) ) ) ).

% powr_less_mono2
tff(fact_1532_powr__mono2_H,axiom,
    ! [A3: real,X: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,A3),zero_zero(real))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),X)
       => ( aa(real,$o,ord_less_eq(real,X),Y)
         => aa(real,$o,ord_less_eq(real,powr(real,Y,A3)),powr(real,X,A3)) ) ) ) ).

% powr_mono2'
tff(fact_1533_powr__inj,axiom,
    ! [A3: real,X: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
     => ( ( A3 != one_one(real) )
       => ( ( powr(real,A3,X) = powr(real,A3,Y) )
        <=> ( X = Y ) ) ) ) ).

% powr_inj
tff(fact_1534_gr__one__powr,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),X)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
       => aa(real,$o,ord_less(real,one_one(real)),powr(real,X,Y)) ) ) ).

% gr_one_powr
tff(fact_1535_powr__le1,axiom,
    ! [A3: real,X: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),A3)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
       => ( aa(real,$o,ord_less_eq(real,X),one_one(real))
         => aa(real,$o,ord_less_eq(real,powr(real,X,A3)),one_one(real)) ) ) ) ).

% powr_le1
tff(fact_1536_powr__mono__both,axiom,
    ! [A3: real,B3: real,X: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),A3)
     => ( aa(real,$o,ord_less_eq(real,A3),B3)
       => ( aa(real,$o,ord_less_eq(real,one_one(real)),X)
         => ( aa(real,$o,ord_less_eq(real,X),Y)
           => aa(real,$o,ord_less_eq(real,powr(real,X,A3)),powr(real,Y,B3)) ) ) ) ) ).

% powr_mono_both
tff(fact_1537_ge__one__powr__ge__zero,axiom,
    ! [X: real,A3: real] :
      ( aa(real,$o,ord_less_eq(real,one_one(real)),X)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),A3)
       => aa(real,$o,ord_less_eq(real,one_one(real)),powr(real,X,A3)) ) ) ).

% ge_one_powr_ge_zero
tff(fact_1538_even__numeral,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Na: num] : aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(num,A,numeral_numeral(A),bit0(Na))) ) ).

% even_numeral
tff(fact_1539_div__positive,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),B3)
         => ( aa(A,$o,ord_less_eq(A,B3),A3)
           => aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)) ) ) ) ).

% div_positive
tff(fact_1540_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less(A,A3),B3)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) = zero_zero(A) ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_1541_divide__nonpos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,X),zero_zero(A))
         => ( aa(A,$o,ord_less(A,zero_zero(A)),Y)
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),zero_zero(A)) ) ) ) ).

% divide_nonpos_pos
tff(fact_1542_divide__nonpos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,X),zero_zero(A))
         => ( aa(A,$o,ord_less(A,Y),zero_zero(A))
           => aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)) ) ) ) ).

% divide_nonpos_neg
tff(fact_1543_divide__nonneg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),X)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),Y)
           => aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)) ) ) ) ).

% divide_nonneg_pos
tff(fact_1544_divide__nonneg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),X)
         => ( aa(A,$o,ord_less(A,Y),zero_zero(A))
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),zero_zero(A)) ) ) ) ).

% divide_nonneg_neg
tff(fact_1545_divide__le__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
             => aa(A,$o,ord_less_eq(A,A3),B3) )
            & ( aa(A,$o,ord_less(A,C2),zero_zero(A))
             => aa(A,$o,ord_less_eq(A,B3),A3) ) ) ) ) ).

% divide_le_cancel
tff(fact_1546_frac__less2,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A,W: A,Z: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),X)
         => ( aa(A,$o,ord_less_eq(A,X),Y)
           => ( aa(A,$o,ord_less(A,zero_zero(A)),W)
             => ( aa(A,$o,ord_less(A,W),Z)
               => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),W)) ) ) ) ) ) ).

% frac_less2
tff(fact_1547_frac__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A,W: A,Z: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),X)
         => ( aa(A,$o,ord_less(A,X),Y)
           => ( aa(A,$o,ord_less(A,zero_zero(A)),W)
             => ( aa(A,$o,ord_less_eq(A,W),Z)
               => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),W)) ) ) ) ) ) ).

% frac_less
tff(fact_1548_frac__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,X: A,W: A,Z: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Y)
         => ( aa(A,$o,ord_less_eq(A,X),Y)
           => ( aa(A,$o,ord_less(A,zero_zero(A)),W)
             => ( aa(A,$o,ord_less_eq(A,W),Z)
               => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),W)) ) ) ) ) ) ).

% frac_le
tff(fact_1549_less__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
              & aa(A,$o,ord_less(A,A3),B3) )
            | ( aa(A,$o,ord_less(A,A3),zero_zero(A))
              & aa(A,$o,ord_less(A,B3),A3) ) ) ) ) ).

% less_divide_eq_1
tff(fact_1550_divide__less__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3)),one_one(A))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
              & aa(A,$o,ord_less(A,B3),A3) )
            | ( aa(A,$o,ord_less(A,A3),zero_zero(A))
              & aa(A,$o,ord_less(A,A3),B3) )
            | ( A3 = zero_zero(A) ) ) ) ) ).

% divide_less_eq_1
tff(fact_1551_less__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => aa(A,$o,ord_less(A,A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,plus_plus(A,A3),B3)),aa(A,A,plus_plus(A,one_one(A)),one_one(A)))) ) ) ).

% less_half_sum
tff(fact_1552_gt__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,plus_plus(A,A3),B3)),aa(A,A,plus_plus(A,one_one(A)),one_one(A)))),B3) ) ) ).

% gt_half_sum
tff(fact_1553_is__unit__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(nat,A,power_power(A,A3),Na)),one_one(A))
        <=> ( aa(A,$o,dvd_dvd(A,A3),one_one(A))
            | ( Na = zero_zero(nat) ) ) ) ) ).

% is_unit_power_iff
tff(fact_1554_divide__eq__minus__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> ( ( B3 != zero_zero(A) )
            & ( A3 = aa(A,A,uminus_uminus(A),B3) ) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_1555_le__floor__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] :
          ( aa(int,$o,ord_less_eq(int,Z),archim6421214686448440834_floor(A,X))
        <=> aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),Z)),X) ) ) ).

% le_floor_iff
tff(fact_1556_floor__less__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z: int] :
          ( aa(int,$o,ord_less(int,archim6421214686448440834_floor(A,X)),Z)
        <=> aa(A,$o,ord_less(A,X),aa(int,A,ring_1_of_int(A),Z)) ) ) ).

% floor_less_iff
tff(fact_1557_ceiling__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z: int] :
          ( aa(int,$o,ord_less_eq(int,archimedean_ceiling(A,X)),Z)
        <=> aa(A,$o,ord_less_eq(A,X),aa(int,A,ring_1_of_int(A),Z)) ) ) ).

% ceiling_le_iff
tff(fact_1558_ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,A3: int] :
          ( aa(A,$o,ord_less_eq(A,X),aa(int,A,ring_1_of_int(A),A3))
         => aa(int,$o,ord_less_eq(int,archimedean_ceiling(A,X)),A3) ) ) ).

% ceiling_le
tff(fact_1559_less__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] :
          ( aa(int,$o,ord_less(int,Z),archimedean_ceiling(A,X))
        <=> aa(A,$o,ord_less(A,aa(int,A,ring_1_of_int(A),Z)),X) ) ) ).

% less_ceiling_iff
tff(fact_1560_floor__add__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z: int] : aa(int,int,plus_plus(int,archim6421214686448440834_floor(A,X)),Z) = archim6421214686448440834_floor(A,aa(A,A,plus_plus(A,X),aa(int,A,ring_1_of_int(A),Z))) ) ).

% floor_add_int
tff(fact_1561_int__add__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] : aa(int,int,plus_plus(int,Z),archim6421214686448440834_floor(A,X)) = archim6421214686448440834_floor(A,aa(A,A,plus_plus(A,aa(int,A,ring_1_of_int(A),Z)),X)) ) ).

% int_add_floor
tff(fact_1562_dvd__imp__le,axiom,
    ! [K: nat,Na: nat] :
      ( aa(nat,$o,dvd_dvd(nat,K),Na)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
       => aa(nat,$o,ord_less_eq(nat,K),Na) ) ) ).

% dvd_imp_le
tff(fact_1563_floor__power,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Na: nat] :
          ( ( X = aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X)) )
         => ( archim6421214686448440834_floor(A,aa(nat,A,power_power(A,X),Na)) = aa(nat,int,power_power(int,archim6421214686448440834_floor(A,X)),Na) ) ) ) ).

% floor_power
tff(fact_1564_Diff__single__insert,axiom,
    ! [A: $tType,A2: set(A),X: A,B2: set(A)] :
      ( aa(set(A),$o,ord_less_eq(set(A),aa(set(A),set(A),minus_minus(set(A),A2),aa(set(A),set(A),insert(A,X),bot_bot(set(A))))),B2)
     => aa(set(A),$o,ord_less_eq(set(A),A2),aa(set(A),set(A),insert(A,X),B2)) ) ).

% Diff_single_insert
tff(fact_1565_subset__insert__iff,axiom,
    ! [A: $tType,A2: set(A),X: A,B2: set(A)] :
      ( aa(set(A),$o,ord_less_eq(set(A),A2),aa(set(A),set(A),insert(A,X),B2))
    <=> $ite(member(A,X,A2),aa(set(A),$o,ord_less_eq(set(A),aa(set(A),set(A),minus_minus(set(A),A2),aa(set(A),set(A),insert(A,X),bot_bot(set(A))))),B2),aa(set(A),$o,ord_less_eq(set(A),A2),B2)) ) ).

% subset_insert_iff
tff(fact_1566_nat__diff__split,axiom,
    ! [P: fun(nat,$o),A3: nat,B3: nat] :
      ( aa(nat,$o,P,aa(nat,nat,minus_minus(nat,A3),B3))
    <=> ( ( aa(nat,$o,ord_less(nat,A3),B3)
         => aa(nat,$o,P,zero_zero(nat)) )
        & ! [D4: nat] :
            ( ( A3 = aa(nat,nat,plus_plus(nat,B3),D4) )
           => aa(nat,$o,P,D4) ) ) ) ).

% nat_diff_split
tff(fact_1567_nat__diff__split__asm,axiom,
    ! [P: fun(nat,$o),A3: nat,B3: nat] :
      ( aa(nat,$o,P,aa(nat,nat,minus_minus(nat,A3),B3))
    <=> ~ ( ( aa(nat,$o,ord_less(nat,A3),B3)
            & ~ aa(nat,$o,P,zero_zero(nat)) )
          | ? [D4: nat] :
              ( ( A3 = aa(nat,nat,plus_plus(nat,B3),D4) )
              & ~ aa(nat,$o,P,D4) ) ) ) ).

% nat_diff_split_asm
tff(fact_1568_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),J)
     => ( aa(nat,$o,ord_less(nat,aa(nat,nat,minus_minus(nat,J),K)),I)
      <=> aa(nat,$o,ord_less(nat,J),aa(nat,nat,plus_plus(nat,I),K)) ) ) ).

% less_diff_conv2
tff(fact_1569_le__minus,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & order(A) )
     => ! [Y: A,X: A,A3: word(B),C2: word(B),B3: word(B)] :
          ( aa(A,$o,ord_less_eq(A,Y),X)
         => ( aa(word(B),$o,ord_less_eq(word(B),aa(word(B),word(B),plus_plus(word(B),A3),C2)),B3)
           => ( aa(word(B),$o,ord_less_eq(word(B),A3),aa(word(B),word(B),plus_plus(word(B),A3),C2))
             => aa(word(B),$o,ord_less_eq(word(B),C2),aa(word(B),word(B),minus_minus(word(B),B3),A3)) ) ) ) ) ).

% le_minus
tff(fact_1570_word__add__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(word(A),word(A),plus_plus(word(A),A3),B3) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,plus_plus(int,aa(word(A),int,semiring_1_unsigned(A,int),A3)),aa(word(A),int,semiring_1_unsigned(A,int),B3))) ) ).

% word_add_def
tff(fact_1571_word__minus__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : aa(word(A),word(A),uminus_uminus(word(A)),A3) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,uminus_uminus(int),aa(word(A),int,semiring_1_unsigned(A,int),A3))) ) ).

% word_minus_def
tff(fact_1572_remove__subset,axiom,
    ! [A: $tType,X: A,S: set(A)] :
      ( member(A,X,S)
     => aa(set(A),$o,ord_less(set(A),aa(set(A),set(A),minus_minus(set(A),S),aa(set(A),set(A),insert(A,X),bot_bot(set(A))))),S) ) ).

% remove_subset
tff(fact_1573_less__1__helper,axiom,
    ! [Na: int,M: int] :
      ( aa(int,$o,ord_less_eq(int,Na),M)
     => aa(int,$o,ord_less(int,aa(int,int,minus_minus(int,Na),one_one(int))),M) ) ).

% less_1_helper
tff(fact_1574_int__induct,axiom,
    ! [P: fun(int,$o),K: int,I: int] :
      ( aa(int,$o,P,K)
     => ( ! [I3: int] :
            ( aa(int,$o,ord_less_eq(int,K),I3)
           => ( aa(int,$o,P,I3)
             => aa(int,$o,P,aa(int,int,plus_plus(int,I3),one_one(int))) ) )
       => ( ! [I3: int] :
              ( aa(int,$o,ord_less_eq(int,I3),K)
             => ( aa(int,$o,P,I3)
               => aa(int,$o,P,aa(int,int,minus_minus(int,I3),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_induct
tff(fact_1575_no__ulen__sub,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),minus_minus(word(A),X),Y)),X)
        <=> aa(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),X)) ) ) ).

% no_ulen_sub
tff(fact_1576_word__arith__power__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),Na: nat] : aa(nat,word(A),power_power(word(A),A3),Na) = aa(int,word(A),ring_1_of_int(word(A)),aa(nat,int,power_power(int,aa(word(A),int,semiring_1_unsigned(A,int),A3)),Na)) ) ).

% word_arith_power_alt
tff(fact_1577_Compl__insert,axiom,
    ! [A: $tType,X: A,A2: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),insert(A,X),A2)) = aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),uminus_uminus(set(A)),A2)),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))) ).

% Compl_insert
tff(fact_1578_word__le__sub1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( ( X != zero_zero(word(A)) )
         => ( aa(word(A),$o,ord_less_eq(word(A),one_one(word(A))),X)
          <=> aa(word(A),$o,ord_less_eq(word(A),zero_zero(word(A))),aa(word(A),word(A),minus_minus(word(A),X),one_one(word(A)))) ) ) ) ).

% word_le_sub1
tff(fact_1579_word__sub__1__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( ( X != zero_zero(word(A)) )
         => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),minus_minus(word(A),X),one_one(word(A)))),X) ) ) ).

% word_sub_1_le
tff(fact_1580_word__must__wrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Na: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),minus_minus(word(A),Na),one_one(word(A))))
         => ( aa(word(A),$o,ord_less_eq(word(A),Na),X)
           => ( Na = zero_zero(word(A)) ) ) ) ) ).

% word_must_wrap
tff(fact_1581_word__minus__one__le__leq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),X),one_one(word(A)))),Y)
         => aa(word(A),$o,ord_less_eq(word(A),X),Y) ) ) ).

% word_minus_one_le_leq
tff(fact_1582_word__le__minus__one__leq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),X),Y)
         => aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),minus_minus(word(A),Y),one_one(word(A)))) ) ) ).

% word_le_minus_one_leq
tff(fact_1583_plus__minus__not__NULL__ab,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Ab: word(A),C2: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),minus_minus(word(A),Ab),C2))
         => ( aa(word(A),$o,ord_less_eq(word(A),C2),Ab)
           => ( ( C2 != zero_zero(word(A)) )
             => ( aa(word(A),word(A),plus_plus(word(A),X),C2) != zero_zero(word(A)) ) ) ) ) ) ).

% plus_minus_not_NULL_ab
tff(fact_1584_gt0__iff__gem1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),X)
        <=> aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),X),one_one(word(A)))),X) ) ) ).

% gt0_iff_gem1
tff(fact_1585_word__less__sub1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( ( X != zero_zero(word(A)) )
         => ( aa(word(A),$o,ord_less(word(A),one_one(word(A))),X)
          <=> aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),aa(word(A),word(A),minus_minus(word(A),X),one_one(word(A)))) ) ) ) ).

% word_less_sub1
tff(fact_1586_sub__wrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Z: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),minus_minus(word(A),X),Z))
        <=> ( ( Z = zero_zero(word(A)) )
            | aa(word(A),$o,ord_less(word(A),X),Z) ) ) ) ).

% sub_wrap
tff(fact_1587_word__diff__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A),M: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),Na)
         => ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),M)
           => ( aa(word(A),$o,ord_less_eq(word(A),Na),M)
             => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),M),Na)),M) ) ) ) ) ).

% word_diff_less
tff(fact_1588_plus__minus__no__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Ab: word(A),C2: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),X),aa(word(A),word(A),minus_minus(word(A),Ab),C2))
         => ( aa(word(A),$o,ord_less_eq(word(A),C2),Ab)
           => aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),plus_plus(word(A),X),C2)) ) ) ) ).

% plus_minus_no_overflow
tff(fact_1589_word__less__sub__right,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A),Z: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),X),aa(word(A),word(A),plus_plus(word(A),Y),Z))
         => ( aa(word(A),$o,ord_less_eq(word(A),Y),X)
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),X),Y)),Z) ) ) ) ).

% word_less_sub_right
tff(fact_1590_word__less__add__right,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A),Z: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),X),aa(word(A),word(A),minus_minus(word(A),Y),Z))
         => ( aa(word(A),$o,ord_less_eq(word(A),Z),Y)
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),plus_plus(word(A),X),Z)),Y) ) ) ) ).

% word_less_add_right
tff(fact_1591_word__diff__ls_H_H_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A),W: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Y),X)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),plus_plus(word(A),W),X)),X))
         => ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),plus_plus(word(A),W),X))
           => aa(word(A),$o,ord_less(word(A),Y),aa(word(A),word(A),plus_plus(word(A),W),X)) ) ) ) ).

% word_diff_ls''(1)
tff(fact_1592_word__diff__ls_H_H_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),W: word(A),X: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Y),aa(word(A),word(A),plus_plus(word(A),W),X))
         => ( aa(word(A),$o,ord_less_eq(word(A),X),Y)
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Y),X)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),plus_plus(word(A),W),X)),X)) ) ) ) ).

% word_diff_ls''(2)
tff(fact_1593_word__l__diffs_H_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),X: word(A),Z: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),plus_plus(word(A),W),X)),X)),aa(word(A),word(A),minus_minus(word(A),Z),X))
         => ( aa(word(A),$o,ord_less_eq(word(A),X),Z)
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),plus_plus(word(A),W),X)),Z) ) ) ) ).

% word_l_diffs'(1)
tff(fact_1594_word__l__diffs_H_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),X: word(A),Z: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),plus_plus(word(A),W),X)),Z)
         => ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),plus_plus(word(A),W),X))
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),plus_plus(word(A),W),X)),X)),aa(word(A),word(A),minus_minus(word(A),Z),X)) ) ) ) ).

% word_l_diffs'(2)
tff(fact_1595_word__diff__ls_H_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A),W: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Y),X)),W)
         => ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),plus_plus(word(A),W),X))
           => aa(word(A),$o,ord_less(word(A),Y),aa(word(A),word(A),plus_plus(word(A),W),X)) ) ) ) ).

% word_diff_ls'(1)
tff(fact_1596_word__diff__ls_H_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),W: word(A),X: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Y),aa(word(A),word(A),plus_plus(word(A),W),X))
         => ( aa(word(A),$o,ord_less_eq(word(A),X),Y)
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Y),X)),W) ) ) ) ).

% word_diff_ls'(2)
tff(fact_1597_More__Word_Oword__l__diffs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Z: word(A),X: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),W),aa(word(A),word(A),minus_minus(word(A),Z),X))
         => ( aa(word(A),$o,ord_less_eq(word(A),X),Z)
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),plus_plus(word(A),W),X)),Z) ) ) ) ).

% More_Word.word_l_diffs(1)
tff(fact_1598_More__Word_Oword__l__diffs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),X: word(A),Z: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),plus_plus(word(A),W),X)),Z)
         => ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),plus_plus(word(A),W),X))
           => aa(word(A),$o,ord_less(word(A),W),aa(word(A),word(A),minus_minus(word(A),Z),X)) ) ) ) ).

% More_Word.word_l_diffs(2)
tff(fact_1599_word__plus__mcs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V2: word(A),Xb: word(A),X: word(A),W: word(A),Xa2: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),plus_plus(word(A),V2),Xb)),X)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),plus_plus(word(A),W),Xa2)),X))
         => ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),plus_plus(word(A),W),Xa2))
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),plus_plus(word(A),V2),Xb)),aa(word(A),word(A),plus_plus(word(A),W),Xa2)) ) ) ) ).

% word_plus_mcs(1)
tff(fact_1600_word__plus__mcs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V2: word(A),Xb: word(A),W: word(A),Xa2: word(A),X: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),plus_plus(word(A),V2),Xb)),aa(word(A),word(A),plus_plus(word(A),W),Xa2))
         => ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),plus_plus(word(A),V2),Xb))
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),plus_plus(word(A),V2),Xb)),X)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),plus_plus(word(A),W),Xa2)),X)) ) ) ) ).

% word_plus_mcs(2)
tff(fact_1601_Word_Oword__l__diffs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xa2: word(A),X: word(A),Z: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),plus_plus(word(A),W),Xa2)),X)),aa(word(A),word(A),minus_minus(word(A),Z),X))
         => ( aa(word(A),$o,ord_less_eq(word(A),X),Z)
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),plus_plus(word(A),W),Xa2)),Z) ) ) ) ).

% Word.word_l_diffs(1)
tff(fact_1602_Word_Oword__l__diffs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xa2: word(A),Z: word(A),X: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),plus_plus(word(A),W),Xa2)),Z)
         => ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),plus_plus(word(A),W),Xa2))
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),plus_plus(word(A),W),Xa2)),X)),aa(word(A),word(A),minus_minus(word(A),Z),X)) ) ) ) ).

% Word.word_l_diffs(2)
tff(fact_1603_word__diff__ls_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A),W: word(A),Xa2: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Y),X)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),plus_plus(word(A),W),Xa2)),X))
         => ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),plus_plus(word(A),W),Xa2))
           => aa(word(A),$o,ord_less(word(A),Y),aa(word(A),word(A),plus_plus(word(A),W),Xa2)) ) ) ) ).

% word_diff_ls(1)
tff(fact_1604_word__diff__ls_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),W: word(A),Xa2: word(A),X: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Y),aa(word(A),word(A),plus_plus(word(A),W),Xa2))
         => ( aa(word(A),$o,ord_less_eq(word(A),X),Y)
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Y),X)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),plus_plus(word(A),W),Xa2)),X)) ) ) ) ).

% word_diff_ls(2)
tff(fact_1605_unat__plus__if__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),plus_plus(word(A),X),Y)) = $ite(aa(nat,$o,ord_less(nat,aa(nat,nat,plus_plus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),size_size(word(A),X))),aa(nat,nat,plus_plus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),aa(nat,nat,minus_minus(nat,aa(nat,nat,plus_plus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),size_size(word(A),X)))) ) ).

% unat_plus_if_size
tff(fact_1606_uint__plus__if__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),plus_plus(word(A),X),Y)) = $ite(aa(int,$o,ord_less(int,aa(int,int,plus_plus(int,aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Y))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),size_size(word(A),X))),aa(int,int,plus_plus(int,aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Y)),aa(int,int,minus_minus(int,aa(int,int,plus_plus(int,aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Y))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),size_size(word(A),X)))) ) ).

% uint_plus_if_size
tff(fact_1607_of__int__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z)
         => aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(int,A,ring_1_of_int(A),Z)) ) ) ).

% of_int_nonneg
tff(fact_1608_powr__realpow,axiom,
    ! [X: real,Na: nat] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ( powr(real,X,aa(nat,real,semiring_1_of_nat(real),Na)) = aa(nat,real,power_power(real,X),Na) ) ) ).

% powr_realpow
tff(fact_1609_of__int__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(int,$o,ord_less(int,zero_zero(int)),Z)
         => aa(A,$o,ord_less(A,zero_zero(A)),aa(int,A,ring_1_of_int(A),Z)) ) ) ).

% of_int_pos
tff(fact_1610_powr__less__iff,axiom,
    ! [B3: real,X: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),B3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),X)
       => ( aa(real,$o,ord_less(real,powr(real,B3,Y)),X)
        <=> aa(real,$o,ord_less(real,Y),aa(real,real,log(B3),X)) ) ) ) ).

% powr_less_iff
tff(fact_1611_less__powr__iff,axiom,
    ! [B3: real,X: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),B3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),X)
       => ( aa(real,$o,ord_less(real,X),powr(real,B3,Y))
        <=> aa(real,$o,ord_less(real,aa(real,real,log(B3),X)),Y) ) ) ) ).

% less_powr_iff
tff(fact_1612_log__less__iff,axiom,
    ! [B3: real,X: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),B3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),X)
       => ( aa(real,$o,ord_less(real,aa(real,real,log(B3),X)),Y)
        <=> aa(real,$o,ord_less(real,X),powr(real,B3,Y)) ) ) ) ).

% log_less_iff
tff(fact_1613_less__log__iff,axiom,
    ! [B3: real,X: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),B3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),X)
       => ( aa(real,$o,ord_less(real,Y),aa(real,real,log(B3),X))
        <=> aa(real,$o,ord_less(real,powr(real,B3,Y)),X) ) ) ) ).

% less_log_iff
tff(fact_1614_floor__log__eq__powr__iff,axiom,
    ! [X: real,B3: real,K: int] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less(real,one_one(real)),B3)
       => ( ( archim6421214686448440834_floor(real,aa(real,real,log(B3),X)) = K )
        <=> ( aa(real,$o,ord_less_eq(real,powr(real,B3,aa(int,real,ring_1_of_int(real),K))),X)
            & aa(real,$o,ord_less(real,X),powr(real,B3,aa(int,real,ring_1_of_int(real),aa(int,int,plus_plus(int,K),one_one(int))))) ) ) ) ) ).

% floor_log_eq_powr_iff
tff(fact_1615_floor__exists,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [Z2: int] :
          ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),Z2)),X)
          & aa(A,$o,ord_less(A,X),aa(int,A,ring_1_of_int(A),aa(int,int,plus_plus(int,Z2),one_one(int)))) ) ) ).

% floor_exists
tff(fact_1616_floor__exists1,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [X4: int] :
          ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),X4)),X)
          & aa(A,$o,ord_less(A,X),aa(int,A,ring_1_of_int(A),aa(int,int,plus_plus(int,X4),one_one(int))))
          & ! [Y4: int] :
              ( ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),Y4)),X)
                & aa(A,$o,ord_less(A,X),aa(int,A,ring_1_of_int(A),aa(int,int,plus_plus(int,Y4),one_one(int)))) )
             => ( Y4 = X4 ) ) ) ) ).

% floor_exists1
tff(fact_1617_of__int__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: num] : aa(int,A,ring_1_of_int(A),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K)) ) ).

% of_int_neg_numeral
tff(fact_1618_even__zero,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),zero_zero(A)) ) ).

% even_zero
tff(fact_1619_of__int__ceiling__le__add__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R3: A] : aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,R3))),aa(A,A,plus_plus(A,R3),one_one(A))) ) ).

% of_int_ceiling_le_add_one
tff(fact_1620_odd__even__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A,B3: A] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
         => ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),B3)
           => aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,plus_plus(A,A3),B3)) ) ) ) ).

% odd_even_add
tff(fact_1621_odd__one,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),one_one(A)) ) ).

% odd_one
tff(fact_1622_even__minus,axiom,
    ! [A: $tType] :
      ( ring_parity(A)
     => ! [A3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,uminus_uminus(A),A3))
        <=> aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3) ) ) ).

% even_minus
tff(fact_1623_le__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
              & aa(A,$o,ord_less_eq(A,A3),B3) )
            | ( aa(A,$o,ord_less(A,A3),zero_zero(A))
              & aa(A,$o,ord_less_eq(A,B3),A3) ) ) ) ) ).

% le_divide_eq_1
tff(fact_1624_divide__le__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3)),one_one(A))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
              & aa(A,$o,ord_less_eq(A,B3),A3) )
            | ( aa(A,$o,ord_less(A,A3),zero_zero(A))
              & aa(A,$o,ord_less_eq(A,A3),B3) )
            | ( A3 = zero_zero(A) ) ) ) ) ).

% divide_le_eq_1
tff(fact_1625_of__nat__less__of__int__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat,X: int] :
          ( aa(A,$o,ord_less(A,aa(nat,A,semiring_1_of_nat(A),Na)),aa(int,A,ring_1_of_int(A),X))
        <=> aa(int,$o,ord_less(int,aa(nat,int,semiring_1_of_nat(int),Na)),X) ) ) ).

% of_nat_less_of_int_iff
tff(fact_1626_field__sum__of__halves,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] : aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),aa(num,A,numeral_numeral(A),bit0(one2)))) = X ) ).

% field_sum_of_halves
tff(fact_1627_dvd__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [X: A,M: nat,Na: nat] :
          ( ( X != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,aa(nat,A,power_power(A,X),M)),aa(nat,A,power_power(A,X),Na))
          <=> ( aa(A,$o,dvd_dvd(A,X),one_one(A))
              | aa(nat,$o,ord_less_eq(nat,M),Na) ) ) ) ) ).

% dvd_power_iff
tff(fact_1628_dvd__power,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Na: nat,X: A] :
          ( ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
            | ( X = one_one(A) ) )
         => aa(A,$o,dvd_dvd(A,X),aa(nat,A,power_power(A,X),Na)) ) ) ).

% dvd_power
tff(fact_1629_power2__commute,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y: A] : aa(nat,A,power_power(A,aa(A,A,minus_minus(A,X),Y)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,power_power(A,aa(A,A,minus_minus(A,Y),X)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% power2_commute
tff(fact_1630_int__le__real__less,axiom,
    ! [Na: int,M: int] :
      ( aa(int,$o,ord_less_eq(int,Na),M)
    <=> aa(real,$o,ord_less(real,aa(int,real,ring_1_of_int(real),Na)),aa(real,real,plus_plus(real,aa(int,real,ring_1_of_int(real),M)),one_one(real))) ) ).

% int_le_real_less
tff(fact_1631_int__less__real__le,axiom,
    ! [Na: int,M: int] :
      ( aa(int,$o,ord_less(int,Na),M)
    <=> aa(real,$o,ord_less_eq(real,aa(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_1632_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,Na: nat] :
      ( aa(nat,$o,dvd_dvd(nat,aa(nat,nat,power_power(nat,I),M)),aa(nat,nat,power_power(nat,I),Na))
     => ( aa(nat,$o,ord_less(nat,one_one(nat)),I)
       => aa(nat,$o,ord_less_eq(nat,M),Na) ) ) ).

% power_dvd_imp_le
tff(fact_1633_word__neg__numeral__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: num] : aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3))) ) ).

% word_neg_numeral_alt
tff(fact_1634_ceiling__altdef,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          archimedean_ceiling(A,X) = $ite(X = aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,X),aa(int,int,plus_plus(int,archim6421214686448440834_floor(A,X)),one_one(int))) ) ).

% ceiling_altdef
tff(fact_1635_psubset__insert__iff,axiom,
    ! [A: $tType,A2: set(A),X: A,B2: set(A)] :
      ( aa(set(A),$o,ord_less(set(A),A2),aa(set(A),set(A),insert(A,X),B2))
    <=> $ite(
          member(A,X,B2),
          aa(set(A),$o,ord_less(set(A),A2),B2),
          $ite(member(A,X,A2),aa(set(A),$o,ord_less(set(A),aa(set(A),set(A),minus_minus(set(A),A2),aa(set(A),set(A),insert(A,X),bot_bot(set(A))))),B2),aa(set(A),$o,ord_less_eq(set(A),A2),B2)) ) ) ).

% psubset_insert_iff
tff(fact_1636_measure__unat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A)] :
          ( ( P2 != zero_zero(word(A)) )
         => aa(nat,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),minus_minus(word(A),P2),one_one(word(A))))),aa(word(A),nat,semiring_1_unsigned(A,nat),P2)) ) ) ).

% measure_unat
tff(fact_1637_real__of__int__floor__add__one__gt,axiom,
    ! [R3: real] : aa(real,$o,ord_less(real,R3),aa(real,real,plus_plus(real,aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R3))),one_one(real))) ).

% real_of_int_floor_add_one_gt
tff(fact_1638_floor__eq,axiom,
    ! [Na: int,X: real] :
      ( aa(real,$o,ord_less(real,aa(int,real,ring_1_of_int(real),Na)),X)
     => ( aa(real,$o,ord_less(real,X),aa(real,real,plus_plus(real,aa(int,real,ring_1_of_int(real),Na)),one_one(real)))
       => ( archim6421214686448440834_floor(real,X) = Na ) ) ) ).

% floor_eq
tff(fact_1639_real__of__int__floor__add__one__ge,axiom,
    ! [R3: real] : aa(real,$o,ord_less_eq(real,R3),aa(real,real,plus_plus(real,aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R3))),one_one(real))) ).

% real_of_int_floor_add_one_ge
tff(fact_1640_word__sub__plus__one__nonzero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N6: word(A),Na: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),N6),Na)
         => ( ( N6 != zero_zero(word(A)) )
           => ( aa(word(A),word(A),plus_plus(word(A),aa(word(A),word(A),minus_minus(word(A),Na),N6)),one_one(word(A))) != zero_zero(word(A)) ) ) ) ) ).

% word_sub_plus_one_nonzero
tff(fact_1641_word__leq__minus__one__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A)] :
          ( ( Y != zero_zero(word(A)) )
         => ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),minus_minus(word(A),Y),one_one(word(A))))
           => aa(word(A),$o,ord_less(word(A),X),Y) ) ) ) ).

% word_leq_minus_one_le
tff(fact_1642_word__leq__le__minus__one,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),X),Y)
         => ( ( X != zero_zero(word(A)) )
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),X),one_one(word(A)))),Y) ) ) ) ).

% word_leq_le_minus_one
tff(fact_1643_le__m1__iff__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),X)
        <=> ( aa(word(A),$o,ord_less_eq(word(A),Y),aa(word(A),word(A),minus_minus(word(A),X),one_one(word(A))))
          <=> aa(word(A),$o,ord_less(word(A),Y),X) ) ) ) ).

% le_m1_iff_lt
tff(fact_1644_less__1__simp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A),M: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Na),one_one(word(A)))),M)
        <=> ( aa(word(A),$o,ord_less_eq(word(A),Na),M)
            & ( Na != zero_zero(word(A)) ) ) ) ) ).

% less_1_simp
tff(fact_1645_word__less__nowrapI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Z: word(A),K: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),X),aa(word(A),word(A),minus_minus(word(A),Z),K))
         => ( aa(word(A),$o,ord_less_eq(word(A),K),Z)
           => ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),K)
             => aa(word(A),$o,ord_less(word(A),X),aa(word(A),word(A),plus_plus(word(A),X),K)) ) ) ) ) ).

% word_less_nowrapI
tff(fact_1646_plus__minus__not__NULL,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Ab: word(A),C2: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),X),aa(word(A),word(A),minus_minus(word(A),Ab),C2))
         => ( aa(word(A),$o,ord_less_eq(word(A),C2),Ab)
           => ( ( C2 != zero_zero(word(A)) )
             => ( aa(word(A),word(A),plus_plus(word(A),X),C2) != zero_zero(word(A)) ) ) ) ) ) ).

% plus_minus_not_NULL
tff(fact_1647_word__less__nowrapI_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Z: word(A),K: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),minus_minus(word(A),Z),K))
         => ( aa(word(A),$o,ord_less_eq(word(A),K),Z)
           => ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),K)
             => aa(word(A),$o,ord_less(word(A),X),aa(word(A),word(A),plus_plus(word(A),X),K)) ) ) ) ) ).

% word_less_nowrapI'
tff(fact_1648_ceiling__diff__floor__le__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(int,$o,ord_less_eq(int,aa(int,int,minus_minus(int,archimedean_ceiling(A,X)),archim6421214686448440834_floor(A,X))),one_one(int)) ) ).

% ceiling_diff_floor_le_1
tff(fact_1649_powr__le__iff,axiom,
    ! [B3: real,X: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),B3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),X)
       => ( aa(real,$o,ord_less_eq(real,powr(real,B3,Y)),X)
        <=> aa(real,$o,ord_less_eq(real,Y),aa(real,real,log(B3),X)) ) ) ) ).

% powr_le_iff
tff(fact_1650_le__powr__iff,axiom,
    ! [B3: real,X: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),B3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),X)
       => ( aa(real,$o,ord_less_eq(real,X),powr(real,B3,Y))
        <=> aa(real,$o,ord_less_eq(real,aa(real,real,log(B3),X)),Y) ) ) ) ).

% le_powr_iff
tff(fact_1651_log__le__iff,axiom,
    ! [B3: real,X: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),B3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),X)
       => ( aa(real,$o,ord_less_eq(real,aa(real,real,log(B3),X)),Y)
        <=> aa(real,$o,ord_less_eq(real,X),powr(real,B3,Y)) ) ) ) ).

% log_le_iff
tff(fact_1652_le__log__iff,axiom,
    ! [B3: real,X: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),B3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),X)
       => ( aa(real,$o,ord_less_eq(real,Y),aa(real,real,log(B3),X))
        <=> aa(real,$o,ord_less_eq(real,powr(real,B3,Y)),X) ) ) ) ).

% le_log_iff
tff(fact_1653_floor__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] :
          ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),Z)),X)
         => ( aa(A,$o,ord_less(A,X),aa(A,A,plus_plus(A,aa(int,A,ring_1_of_int(A),Z)),one_one(A)))
           => ( archim6421214686448440834_floor(A,X) = Z ) ) ) ) ).

% floor_unique
tff(fact_1654_floor__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,A3: int] :
          ( ( archim6421214686448440834_floor(A,X) = A3 )
        <=> ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),A3)),X)
            & aa(A,$o,ord_less(A,X),aa(A,A,plus_plus(A,aa(int,A,ring_1_of_int(A),A3)),one_one(A))) ) ) ) ).

% floor_eq_iff
tff(fact_1655_floor__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,$o),Ta: A] :
          ( aa(int,$o,P,archim6421214686448440834_floor(A,Ta))
        <=> ! [I4: int] :
              ( ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),I4)),Ta)
                & aa(A,$o,ord_less(A,Ta),aa(A,A,plus_plus(A,aa(int,A,ring_1_of_int(A),I4)),one_one(A))) )
             => aa(int,$o,P,I4) ) ) ) ).

% floor_split
tff(fact_1656_less__floor__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] :
          ( aa(int,$o,ord_less(int,Z),archim6421214686448440834_floor(A,X))
        <=> aa(A,$o,ord_less_eq(A,aa(A,A,plus_plus(A,aa(int,A,ring_1_of_int(A),Z)),one_one(A))),X) ) ) ).

% less_floor_iff
tff(fact_1657_floor__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z: int] :
          ( aa(int,$o,ord_less_eq(int,archim6421214686448440834_floor(A,X)),Z)
        <=> aa(A,$o,ord_less(A,X),aa(A,A,plus_plus(A,aa(int,A,ring_1_of_int(A),Z)),one_one(A))) ) ) ).

% floor_le_iff
tff(fact_1658_half__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))))
        <=> aa(A,$o,ord_less(A,zero_zero(A)),A3) ) ) ).

% half_gt_zero_iff
tff(fact_1659_half__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ).

% half_gt_zero
tff(fact_1660_field__less__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less(A,X),Y)
         => aa(A,$o,ord_less(A,X),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,plus_plus(A,X),Y)),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ).

% field_less_half_sum
tff(fact_1661_floor__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X))),X)
          & aa(A,$o,ord_less(A,X),aa(int,A,ring_1_of_int(A),aa(int,int,plus_plus(int,archim6421214686448440834_floor(A,X)),one_one(int)))) ) ) ).

% floor_correct
tff(fact_1662_div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,M: nat,Na: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M))),aa(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),A3),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,plus_plus(nat,M),Na))) ) ).

% div_exp_eq
tff(fact_1663_power__mono__odd,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat,A3: A,B3: A] :
          ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
         => ( aa(A,$o,ord_less_eq(A,A3),B3)
           => aa(A,$o,ord_less_eq(A,aa(nat,A,power_power(A,A3),Na)),aa(nat,A,power_power(A,B3),Na)) ) ) ) ).

% power_mono_odd
tff(fact_1664_inverse__of__nat__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Na: nat,M: nat] :
          ( aa(nat,$o,ord_less_eq(nat,Na),M)
         => ( ( Na != zero_zero(nat) )
           => aa(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_1665_odd__pos,axiom,
    ! [Na: nat] :
      ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => aa(nat,$o,ord_less(nat,zero_zero(nat)),Na) ) ).

% odd_pos
tff(fact_1666_uminus__power__if,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A,Na: nat] :
          aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),A3)),Na) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na),aa(nat,A,power_power(A,A3),Na),aa(A,A,uminus_uminus(A),aa(nat,A,power_power(A,A3),Na))) ) ).

% uminus_power_if
tff(fact_1667_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Na: nat,M: nat] :
          ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na) != zero_zero(A) )
         => ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,minus_minus(nat,Na),M)) != zero_zero(A) ) ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_1668_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
     => ( aa(nat,$o,dvd_dvd(nat,aa(nat,nat,power_power(nat,K),M)),aa(nat,nat,power_power(nat,K),Na))
      <=> aa(nat,$o,ord_less_eq(nat,M),Na) ) ) ).

% dvd_power_iff_le
tff(fact_1669_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
     => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,M),Na)),aa(nat,nat,minus_minus(nat,aa(nat,nat,power_power(nat,K),M)),aa(nat,nat,power_power(nat,K),Na))) ) ).

% diff_le_diff_pow
tff(fact_1670_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,ord_less_eq(nat,K),Na)
         => ( aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,plus_plus(nat,Na),K)) = aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,minus_minus(nat,Na),K)) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_1671_floor__eq2,axiom,
    ! [Na: int,X: real] :
      ( aa(real,$o,ord_less_eq(real,aa(int,real,ring_1_of_int(real),Na)),X)
     => ( aa(real,$o,ord_less(real,X),aa(real,real,plus_plus(real,aa(int,real,ring_1_of_int(real),Na)),one_one(real)))
       => ( archim6421214686448440834_floor(real,X) = Na ) ) ) ).

% floor_eq2
tff(fact_1672_zero__le__even__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat,A3: A] :
          ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
         => aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,power_power(A,A3),Na)) ) ) ).

% zero_le_even_power
tff(fact_1673_zero__le__odd__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat,A3: A] :
          ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,power_power(A,A3),Na))
          <=> aa(A,$o,ord_less_eq(A,zero_zero(A)),A3) ) ) ) ).

% zero_le_odd_power
tff(fact_1674_zero__le__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,power_power(A,A3),Na))
        <=> ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
            | ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
              & aa(A,$o,ord_less_eq(A,zero_zero(A)),A3) ) ) ) ) ).

% zero_le_power_eq
tff(fact_1675_minus__one__power__iff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Na: nat] :
          aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),Na) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na),one_one(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% minus_one_power_iff
tff(fact_1676_word__of__int__2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] : aa(int,word(A),ring_1_of_int(word(A)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) = aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na) ) ).

% word_of_int_2p
tff(fact_1677_ceiling__log2__div2,axiom,
    ! [Na: nat] :
      ( aa(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,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,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(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_1678_ceiling__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Na: int,X: A] :
          ( aa(A,$o,ord_less(A,aa(int,A,ring_1_of_int(A),Na)),X)
         => ( aa(A,$o,ord_less_eq(A,X),aa(A,A,plus_plus(A,aa(int,A,ring_1_of_int(A),Na)),one_one(A)))
           => ( archimedean_ceiling(A,X) = aa(int,int,plus_plus(int,Na),one_one(int)) ) ) ) ) ).

% ceiling_eq
tff(fact_1679_zero__less__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(nat,A,power_power(A,A3),Na))
        <=> ( ( Na = zero_zero(nat) )
            | ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
              & ( A3 != zero_zero(A) ) )
            | ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
              & aa(A,$o,ord_less(A,zero_zero(A)),A3) ) ) ) ) ).

% zero_less_power_eq
tff(fact_1680_odd__word__imp__even__next,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,semiring_1_unsigned(A,nat),X))
         => ( ( aa(word(A),word(A),plus_plus(word(A),X),one_one(word(A))) = zero_zero(word(A)) )
            | aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),plus_plus(word(A),X),one_one(word(A))))) ) ) ) ).

% odd_word_imp_even_next
tff(fact_1681_even__word__imp__odd__next,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,semiring_1_unsigned(A,nat),X))
         => ( ( aa(word(A),word(A),plus_plus(word(A),X),one_one(word(A))) = zero_zero(word(A)) )
            | ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),plus_plus(word(A),X),one_one(word(A))))) ) ) ) ).

% even_word_imp_odd_next
tff(fact_1682_uint__range__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(word(A),int,semiring_1_unsigned(A,int),W))
          & aa(int,$o,ord_less(int,aa(word(A),int,semiring_1_unsigned(A,int),W)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),size_size(word(A),W))) ) ) ).

% uint_range_size
tff(fact_1683_power__le__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,power_power(A,A3),Na)),zero_zero(A))
        <=> ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
            & ( ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
                & aa(A,$o,ord_less_eq(A,A3),zero_zero(A)) )
              | ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
                & ( A3 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_1684_no__plus__overflow__unat__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),plus_plus(word(A),X),Y))
        <=> aa(nat,$o,ord_less(nat,aa(nat,nat,plus_plus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),size_size(word(A),X))) ) ) ).

% no_plus_overflow_unat_size
tff(fact_1685_no__plus__overflow__uint__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),plus_plus(word(A),X),Y))
        <=> aa(int,$o,ord_less(int,aa(int,int,plus_plus(int,aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Y))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),size_size(word(A),X))) ) ) ).

% no_plus_overflow_uint_size
tff(fact_1686_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_1687_add__self__div__2,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,plus_plus(nat,M),M)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = M ).

% add_self_div_2
tff(fact_1688_div__eq__dividend__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa(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_1689_pow__divides__pow__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Na: nat,A3: A,B3: A] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
         => ( aa(A,$o,dvd_dvd(A,aa(nat,A,power_power(A,A3),Na)),aa(nat,A,power_power(A,B3),Na))
          <=> aa(A,$o,dvd_dvd(A,A3),B3) ) ) ) ).

% pow_divides_pow_iff
tff(fact_1690_div__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa(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_1691_VEBT__internal_Olog__ceil__idem,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,one_one(real)),X)
     => ( archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),X)) = 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,X)))) ) ) ).

% VEBT_internal.log_ceil_idem
tff(fact_1692_div__minus1__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),A3) ) ).

% div_minus1_right
tff(fact_1693_less__two__pow__divI,axiom,
    ! [X: nat,Na: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,Na),M)))
     => ( aa(nat,$o,ord_less_eq(nat,M),Na)
       => aa(nat,$o,ord_less(nat,X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M))) ) ) ).

% less_two_pow_divI
tff(fact_1694_less__two__pow__divD,axiom,
    ! [X: nat,Na: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,X),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M)))
     => ( aa(nat,$o,ord_less_eq(nat,M),Na)
        & aa(nat,$o,ord_less(nat,X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,Na),M))) ) ) ).

% less_two_pow_divD
tff(fact_1695_Diff__idemp,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] : aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),minus_minus(set(A),A2),B2)),B2) = aa(set(A),set(A),minus_minus(set(A),A2),B2) ).

% Diff_idemp
tff(fact_1696_Diff__iff,axiom,
    ! [A: $tType,C2: A,A2: set(A),B2: set(A)] :
      ( member(A,C2,aa(set(A),set(A),minus_minus(set(A),A2),B2))
    <=> ( member(A,C2,A2)
        & ~ member(A,C2,B2) ) ) ).

% Diff_iff
tff(fact_1697_DiffI,axiom,
    ! [A: $tType,C2: A,A2: set(A),B2: set(A)] :
      ( member(A,C2,A2)
     => ( ~ member(A,C2,B2)
       => member(A,C2,aa(set(A),set(A),minus_minus(set(A),A2),B2)) ) ) ).

% DiffI
tff(fact_1698_div__of__0__id,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),zero_zero(word(A))),Na) = zero_zero(word(A)) ) ).

% div_of_0_id
tff(fact_1699_zdiv__numeral__Bit0,axiom,
    ! [V2: num,W: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),bit0(V2))),aa(num,int,numeral_numeral(int),bit0(W))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),V2)),aa(num,int,numeral_numeral(int),W)) ).

% zdiv_numeral_Bit0
tff(fact_1700_int__dvd__int__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa(int,$o,dvd_dvd(int,aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),Na))
    <=> aa(nat,$o,dvd_dvd(nat,M),Na) ) ).

% int_dvd_int_iff
tff(fact_1701_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,ord_less_eq(int,K),zero_zero(int))
     => ( aa(int,$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_1702_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
     => ( aa(int,$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_1703_int__div__same__is__1,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),A3)
     => ( ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3) = A3 )
      <=> ( B3 = one_one(int) ) ) ) ).

% int_div_same_is_1
tff(fact_1704_word__div__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),A3))),aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),B3)))) ) ).

% word_div_no
tff(fact_1705_floor__divide__eq__div__numeral,axiom,
    ! [A3: num,B3: num] : archim6421214686448440834_floor(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),A3)),aa(num,real,numeral_numeral(real),B3))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),A3)),aa(num,int,numeral_numeral(int),B3)) ).

% floor_divide_eq_div_numeral
tff(fact_1706_int__div__minus__is__minus1,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less(int,A3),zero_zero(int))
     => ( ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3) = aa(int,int,uminus_uminus(int),A3) )
      <=> ( B3 = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% int_div_minus_is_minus1
tff(fact_1707_ceiling__divide__eq__div__numeral,axiom,
    ! [A3: num,B3: num] : archimedean_ceiling(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),A3)),aa(num,real,numeral_numeral(real),B3))) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3))),aa(num,int,numeral_numeral(int),B3))) ).

% ceiling_divide_eq_div_numeral
tff(fact_1708_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa(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,ord_less_eq(int,zero_zero(int)),K) ) ).

% half_nonnegative_int_iff
tff(fact_1709_half__negative__int__iff,axiom,
    ! [K: int] :
      ( aa(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,ord_less(int,K),zero_zero(int)) ) ).

% half_negative_int_iff
tff(fact_1710_floor__one__divide__eq__div__numeral,axiom,
    ! [B3: num] : archim6421214686448440834_floor(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),B3))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),one_one(int)),aa(num,int,numeral_numeral(int),B3)) ).

% floor_one_divide_eq_div_numeral
tff(fact_1711_floor__minus__divide__eq__div__numeral,axiom,
    ! [A3: num,B3: num] : archim6421214686448440834_floor(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),A3)),aa(num,real,numeral_numeral(real),B3)))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3))),aa(num,int,numeral_numeral(int),B3)) ).

% floor_minus_divide_eq_div_numeral
tff(fact_1712_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [A3: num,B3: num] : archimedean_ceiling(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),A3)),aa(num,real,numeral_numeral(real),B3)))) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),A3)),aa(num,int,numeral_numeral(int),B3))) ).

% ceiling_minus_divide_eq_div_numeral
tff(fact_1713_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [B3: num] : archim6421214686448440834_floor(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),B3)))) = 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),B3)) ).

% floor_minus_one_divide_eq_div_numeral
tff(fact_1714_real__of__int__div4,axiom,
    ! [Na: int,X: int] : aa(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),X))),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),X))) ).

% real_of_int_div4
tff(fact_1715_word__div__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),A3),B3) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(word(A),int,semiring_1_unsigned(A,int),A3)),aa(word(A),int,semiring_1_unsigned(A,int),B3))) ) ).

% word_div_def
tff(fact_1716_uint__div__distrib,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V2: word(A),W: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),V2),W)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(word(A),int,semiring_1_unsigned(A,int),V2)),aa(word(A),int,semiring_1_unsigned(A,int),W)) ) ).

% uint_div_distrib
tff(fact_1717_uint__div,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: 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)),divide_divide(word(A)),X),Y)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ).

% uint_div
tff(fact_1718_set__diff__eq,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] : aa(set(A),set(A),minus_minus(set(A),A2),B2) = collect(A,aa(set(A),fun(A,$o),aTP_Lamp_aq(set(A),fun(set(A),fun(A,$o)),A2),B2)) ).

% set_diff_eq
tff(fact_1719_word__of__int__Ex,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
        ? [Y3: int] : X = aa(int,word(A),ring_1_of_int(word(A)),Y3) ) ).

% word_of_int_Ex
tff(fact_1720_wi__hom__sub,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: int,B3: int] : aa(word(A),word(A),minus_minus(word(A),aa(int,word(A),ring_1_of_int(word(A)),A3)),aa(int,word(A),ring_1_of_int(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,minus_minus(int,A3),B3)) ) ).

% wi_hom_sub
tff(fact_1721_zdvd__zdiffD,axiom,
    ! [K: int,M: int,Na: int] :
      ( aa(int,$o,dvd_dvd(int,K),aa(int,int,minus_minus(int,M),Na))
     => ( aa(int,$o,dvd_dvd(int,K),Na)
       => aa(int,$o,dvd_dvd(int,K),M) ) ) ).

% zdvd_zdiffD
tff(fact_1722_minus__set__def,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] : aa(set(A),set(A),minus_minus(set(A),A2),B2) = collect(A,aa(fun(A,$o),fun(A,$o),minus_minus(fun(A,$o),aTP_Lamp_aa(set(A),fun(A,$o),A2)),aTP_Lamp_aa(set(A),fun(A,$o),B2))) ).

% minus_set_def
tff(fact_1723_DiffD2,axiom,
    ! [A: $tType,C2: A,A2: set(A),B2: set(A)] :
      ( member(A,C2,aa(set(A),set(A),minus_minus(set(A),A2),B2))
     => ~ member(A,C2,B2) ) ).

% DiffD2
tff(fact_1724_DiffD1,axiom,
    ! [A: $tType,C2: A,A2: set(A),B2: set(A)] :
      ( member(A,C2,aa(set(A),set(A),minus_minus(set(A),A2),B2))
     => member(A,C2,A2) ) ).

% DiffD1
tff(fact_1725_DiffE,axiom,
    ! [A: $tType,C2: A,A2: set(A),B2: set(A)] :
      ( member(A,C2,aa(set(A),set(A),minus_minus(set(A),A2),B2))
     => ~ ( member(A,C2,A2)
         => member(A,C2,B2) ) ) ).

% DiffE
tff(fact_1726_real__of__int__div,axiom,
    ! [D2: int,Na: int] :
      ( aa(int,$o,dvd_dvd(int,D2),Na)
     => ( aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),divide_divide(int),Na),D2)) = 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),D2)) ) ) ).

% real_of_int_div
tff(fact_1727_diff__commute,axiom,
    ! [I: nat,J: nat,K: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,I),J)),K) = aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,I),K)),J) ).

% diff_commute
tff(fact_1728_int__div__sub__1,axiom,
    ! [M: int,Na: int] :
      ( aa(int,$o,ord_less_eq(int,one_one(int)),M)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,minus_minus(int,Na),one_one(int))),M) = $ite(aa(int,$o,dvd_dvd(int,M),Na),aa(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_1729_real__of__int__div2,axiom,
    ! [Na: int,X: int] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(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),X))),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),divide_divide(int),Na),X)))) ).

% real_of_int_div2
tff(fact_1730_real__of__int__div3,axiom,
    ! [Na: int,X: int] : aa(real,$o,ord_less_eq(real,aa(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),X))),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),divide_divide(int),Na),X)))),one_one(real)) ).

% real_of_int_div3
tff(fact_1731_length__induct,axiom,
    ! [A: $tType,P: fun(list(A),$o),Xsa: list(A)] :
      ( ! [Xs: list(A)] :
          ( ! [Ys2: list(A)] :
              ( aa(nat,$o,ord_less(nat,size_size(list(A),Ys2)),size_size(list(A),Xs))
             => aa(list(A),$o,P,Ys2) )
         => aa(list(A),$o,P,Xs) )
     => aa(list(A),$o,P,Xsa) ) ).

% length_induct
tff(fact_1732_word__div__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Na),one_one(word(A))) = Na ) ).

% word_div_1
tff(fact_1733_div__by__0__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),X),zero_zero(word(A))) = zero_zero(word(A)) ) ).

% div_by_0_word
tff(fact_1734_floor__divide__real__eq__div,axiom,
    ! [B3: int,A3: real] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),B3)
     => ( archim6421214686448440834_floor(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),A3),aa(int,real,ring_1_of_int(real),B3))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),archim6421214686448440834_floor(real,A3)),B3) ) ) ).

% floor_divide_real_eq_div
tff(fact_1735_zdiv__le__dividend,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),A3)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),B3)
       => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3)),A3) ) ) ).

% zdiv_le_dividend
tff(fact_1736_zdvd__antisym__nonneg,axiom,
    ! [M: int,Na: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),M)
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Na)
       => ( aa(int,$o,dvd_dvd(int,M),Na)
         => ( aa(int,$o,dvd_dvd(int,Na),M)
           => ( M = Na ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_1737_zdvd__not__zless,axiom,
    ! [M: int,Na: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),M)
     => ( aa(int,$o,ord_less(int,M),Na)
       => ~ aa(int,$o,dvd_dvd(int,Na),M) ) ) ).

% zdvd_not_zless
tff(fact_1738_pos__imp__zdiv__neg__iff,axiom,
    ! [B3: int,A3: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),B3)
     => ( aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3)),zero_zero(int))
      <=> aa(int,$o,ord_less(int,A3),zero_zero(int)) ) ) ).

% pos_imp_zdiv_neg_iff
tff(fact_1739_neg__imp__zdiv__neg__iff,axiom,
    ! [B3: int,A3: int] :
      ( aa(int,$o,ord_less(int,B3),zero_zero(int))
     => ( aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3)),zero_zero(int))
      <=> aa(int,$o,ord_less(int,zero_zero(int)),A3) ) ) ).

% neg_imp_zdiv_neg_iff
tff(fact_1740_div__neg__pos__less0,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less(int,A3),zero_zero(int))
     => ( aa(int,$o,ord_less(int,zero_zero(int)),B3)
       => aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3)),zero_zero(int)) ) ) ).

% div_neg_pos_less0
tff(fact_1741_zdiv__int,axiom,
    ! [A3: nat,B3: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A3),B3)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B3)) ).

% zdiv_int
tff(fact_1742_unat__div__distrib,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V2: word(A),W: word(A)] : 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)),V2),W)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),V2)),aa(word(A),nat,semiring_1_unsigned(A,nat),W)) ) ).

% unat_div_distrib
tff(fact_1743_unat__div,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: 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)),divide_divide(word(A)),X),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) ).

% unat_div
tff(fact_1744_div__word__self,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( ( W != zero_zero(word(A)) )
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),W),W) = one_one(word(A)) ) ) ) ).

% div_word_self
tff(fact_1745_word__div__lt__eq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),X),Y)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),X),Y) = zero_zero(word(A)) ) ) ) ).

% word_div_lt_eq_0
tff(fact_1746_word__less__div,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),X),Y) = zero_zero(word(A)) )
         => ( ( Y = zero_zero(word(A)) )
            | aa(word(A),$o,ord_less(word(A),X),Y) ) ) ) ).

% word_less_div
tff(fact_1747_word__div__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),V2: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),W),V2)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),W),V2) = zero_zero(word(A)) ) ) ) ).

% word_div_less
tff(fact_1748_length__pos__if__in__set,axiom,
    ! [A: $tType,X: A,Xsa: list(A)] :
      ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
     => aa(nat,$o,ord_less(nat,zero_zero(nat)),size_size(list(A),Xsa)) ) ).

% length_pos_if_in_set
tff(fact_1749_zdvd__imp__le,axiom,
    ! [Z: int,Na: int] :
      ( aa(int,$o,dvd_dvd(int,Z),Na)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),Na)
       => aa(int,$o,ord_less_eq(int,Z),Na) ) ) ).

% zdvd_imp_le
tff(fact_1750_zdiv__mono1,axiom,
    ! [A3: int,A5: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,A3),A5)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),B3)
       => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A5),B3)) ) ) ).

% zdiv_mono1
tff(fact_1751_zdiv__mono2,axiom,
    ! [A3: int,B6: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),A3)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),B6)
       => ( aa(int,$o,ord_less_eq(int,B6),B3)
         => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B6)) ) ) ) ).

% zdiv_mono2
tff(fact_1752_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,$o,ord_less_eq(int,zero_zero(int)),I)
          & aa(int,$o,ord_less(int,I),K) )
        | ( aa(int,$o,ord_less_eq(int,I),zero_zero(int))
          & aa(int,$o,ord_less(int,K),I) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_1753_zdiv__mono1__neg,axiom,
    ! [A3: int,A5: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,A3),A5)
     => ( aa(int,$o,ord_less(int,B3),zero_zero(int))
       => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A5),B3)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3)) ) ) ).

% zdiv_mono1_neg
tff(fact_1754_zdiv__mono2__neg,axiom,
    ! [A3: int,B6: int,B3: int] :
      ( aa(int,$o,ord_less(int,A3),zero_zero(int))
     => ( aa(int,$o,ord_less(int,zero_zero(int)),B6)
       => ( aa(int,$o,ord_less_eq(int,B6),B3)
         => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B6)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3)) ) ) ) ).

% zdiv_mono2_neg
tff(fact_1755_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa(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,ord_less_eq(int,zero_zero(int)),K)
          & aa(int,$o,ord_less_eq(int,zero_zero(int)),L) )
        | ( aa(int,$o,ord_less(int,K),zero_zero(int))
          & aa(int,$o,ord_less(int,L),zero_zero(int)) ) ) ) ).

% div_int_pos_iff
tff(fact_1756_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,ord_less_eq(int,L),K)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),L)
       => aa(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_1757_div__nonneg__neg__le0,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),A3)
     => ( aa(int,$o,ord_less(int,B3),zero_zero(int))
       => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3)),zero_zero(int)) ) ) ).

% div_nonneg_neg_le0
tff(fact_1758_div__nonpos__pos__le0,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,A3),zero_zero(int))
     => ( aa(int,$o,ord_less(int,zero_zero(int)),B3)
       => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3)),zero_zero(int)) ) ) ).

% div_nonpos_pos_le0
tff(fact_1759_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),K)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),I),K))
      <=> aa(int,$o,ord_less_eq(int,K),I) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_1760_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B3: int,A3: int] :
      ( aa(int,$o,ord_less(int,B3),zero_zero(int))
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3))
      <=> aa(int,$o,ord_less_eq(int,A3),zero_zero(int)) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_1761_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B3: int,A3: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),B3)
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3))
      <=> aa(int,$o,ord_less_eq(int,zero_zero(int)),A3) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_1762_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),A3)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3))
      <=> ( aa(int,$o,ord_less_eq(int,B3),A3)
          & aa(int,$o,ord_less(int,zero_zero(int)),B3) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_1763_int__div__less__self,axiom,
    ! [X: int,K: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),X)
     => ( aa(int,$o,ord_less(int,one_one(int)),K)
       => aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),X),K)),X) ) ) ).

% int_div_less_self
tff(fact_1764_real__of__nat__div4,axiom,
    ! [Na: nat,X: nat] : aa(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),X))),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),X))) ).

% real_of_nat_div4
tff(fact_1765_div__less__dividend__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Na: word(A)] :
          ( ( X != zero_zero(word(A)) )
         => ( ( Na != one_one(word(A)) )
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),X),Na)),X) ) ) ) ).

% div_less_dividend_word
tff(fact_1766_powr__divide,axiom,
    ! [X: real,Y: real,A3: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
       => ( powr(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),X),Y),A3) = aa(real,real,aa(real,fun(real,real),divide_divide(real),powr(real,X,A3)),powr(real,Y,A3)) ) ) ) ).

% powr_divide
tff(fact_1767_real__of__nat__div,axiom,
    ! [D2: nat,Na: nat] :
      ( aa(nat,$o,dvd_dvd(nat,D2),Na)
     => ( aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),D2)) = 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),D2)) ) ) ).

% real_of_nat_div
tff(fact_1768_word__arith__nat__div,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),A3),B3) = aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A3)),aa(word(A),nat,semiring_1_unsigned(A,nat),B3))) ) ).

% word_arith_nat_div
tff(fact_1769_log__base__powr,axiom,
    ! [A3: real,B3: real,X: real] :
      ( ( A3 != zero_zero(real) )
     => ( aa(real,real,log(powr(real,A3,B3)),X) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,log(A3),X)),B3) ) ) ).

% log_base_powr
tff(fact_1770_num_Osize_I4_J,axiom,
    size_size(num,one2) = zero_zero(nat) ).

% num.size(4)
tff(fact_1771_even__diff__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,minus_minus(int,K),L))
    <=> aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,plus_plus(int,K),L)) ) ).

% even_diff_iff
tff(fact_1772_verit__less__mono__div__int2,axiom,
    ! [A2: int,B2: int,Na: int] :
      ( aa(int,$o,ord_less_eq(int,A2),B2)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),aa(int,int,uminus_uminus(int),Na))
       => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),B2),Na)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),Na)) ) ) ).

% verit_less_mono_div_int2
tff(fact_1773_div__eq__minus1,axiom,
    ! [B3: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),B3)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),one_one(int))),B3) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ).

% div_eq_minus1
tff(fact_1774_real__of__nat__div2,axiom,
    ! [Na: nat,X: nat] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(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),X))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),X)))) ).

% real_of_nat_div2
tff(fact_1775_real__of__nat__div3,axiom,
    ! [Na: nat,X: nat] : aa(real,$o,ord_less_eq(real,aa(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),X))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),X)))),one_one(real)) ).

% real_of_nat_div3
tff(fact_1776_log__base__change,axiom,
    ! [A3: real,B3: real,X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
     => ( ( A3 != one_one(real) )
       => ( aa(real,real,log(B3),X) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,log(A3),X)),aa(real,real,log(A3),B3)) ) ) ) ).

% log_base_change
tff(fact_1777_log__divide,axiom,
    ! [A3: real,X: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
     => ( ( A3 != one_one(real) )
       => ( aa(real,$o,ord_less(real,zero_zero(real)),X)
         => ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
           => ( aa(real,real,log(A3),aa(real,real,aa(real,fun(real,real),divide_divide(real),X),Y)) = aa(real,real,minus_minus(real,aa(real,real,log(A3),X)),aa(real,real,log(A3),Y)) ) ) ) ) ) ).

% log_divide
tff(fact_1778_word__div__sub,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Y),X)
         => ( aa(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),minus_minus(word(A),X),Y)),Y) = aa(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)),X),Y)),one_one(word(A))) ) ) ) ) ).

% word_div_sub
tff(fact_1779_axxdiv2,axiom,
    ! [X: int] :
      ( ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,plus_plus(int,aa(int,int,plus_plus(int,one_one(int)),X)),X)),aa(num,int,numeral_numeral(int),bit0(one2))) = X )
      & ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,plus_plus(int,aa(int,int,plus_plus(int,zero_zero(int)),X)),X)),aa(num,int,numeral_numeral(int),bit0(one2))) = X ) ) ).

% axxdiv2
tff(fact_1780_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),K)
     => ( aa(int,$o,ord_less_eq(int,aa(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_1781_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),L)
     => ( aa(int,$o,ord_less_eq(int,L),K)
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,minus_minus(int,K),L)),L)),one_one(int)) ) ) ) ).

% div_pos_geq
tff(fact_1782_powr__neg__one,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ( powr(real,X,aa(real,real,uminus_uminus(real),one_one(real))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),X) ) ) ).

% powr_neg_one
tff(fact_1783_log__base__pow,axiom,
    ! [A3: real,Na: nat,X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
     => ( aa(real,real,log(aa(nat,real,power_power(real,A3),Na)),X) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,log(A3),X)),aa(nat,real,semiring_1_of_nat(real),Na)) ) ) ).

% log_base_pow
tff(fact_1784_minus__log__eq__powr,axiom,
    ! [B3: real,X: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),B3)
     => ( ( B3 != one_one(real) )
       => ( aa(real,$o,ord_less(real,zero_zero(real)),X)
         => ( aa(real,real,minus_minus(real,Y),aa(real,real,log(B3),X)) = aa(real,real,log(B3),aa(real,real,aa(real,fun(real,real),divide_divide(real),powr(real,B3,Y)),X)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_1785_minus__1__div__exp__eq__int,axiom,
    ! [Na: nat] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) = aa(int,int,uminus_uminus(int),one_one(int)) ).

% minus_1_div_exp_eq_int
tff(fact_1786_word__less__two__pow__divD,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Na: nat,M: nat] :
          ( aa(word(A),$o,ord_less(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)))
         => ( aa(nat,$o,ord_less_eq(nat,M),Na)
            & aa(word(A),$o,ord_less(word(A),X),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,Na),M))) ) ) ) ).

% word_less_two_pow_divD
tff(fact_1787_Euclid__induct,axiom,
    ! [P: fun(nat,fun(nat,$o)),A3: nat,B3: nat] :
      ( ! [A4: nat,B5: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),P,A4),B5)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),P,B5),A4) )
     => ( ! [A4: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,A4),zero_zero(nat))
       => ( ! [A4: nat,B5: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),P,A4),B5)
             => aa(nat,$o,aa(nat,fun(nat,$o),P,A4),aa(nat,nat,plus_plus(nat,A4),B5)) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P,A3),B3) ) ) ) ).

% Euclid_induct
tff(fact_1788_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A3: nat] :
      ( aa(nat,$o,dvd_dvd(nat,zero_zero(nat)),A3)
     => ( A3 = zero_zero(nat) ) ) ).

% gcd_nat.extremum_uniqueI
tff(fact_1789_gcd__nat_Onot__eq__extremum,axiom,
    ! [A3: nat] :
      ( ( A3 != zero_zero(nat) )
    <=> ( aa(nat,$o,dvd_dvd(nat,A3),zero_zero(nat))
        & ( A3 != zero_zero(nat) ) ) ) ).

% gcd_nat.not_eq_extremum
tff(fact_1790_gcd__nat_Oextremum__unique,axiom,
    ! [A3: nat] :
      ( aa(nat,$o,dvd_dvd(nat,zero_zero(nat)),A3)
    <=> ( A3 = zero_zero(nat) ) ) ).

% gcd_nat.extremum_unique
tff(fact_1791_gcd__nat_Oextremum__strict,axiom,
    ! [A3: nat] :
      ~ ( aa(nat,$o,dvd_dvd(nat,zero_zero(nat)),A3)
        & ( zero_zero(nat) != A3 ) ) ).

% gcd_nat.extremum_strict
tff(fact_1792_gcd__nat_Oextremum,axiom,
    ! [A3: nat] : aa(nat,$o,dvd_dvd(nat,A3),zero_zero(nat)) ).

% gcd_nat.extremum
tff(fact_1793_powr__neg__numeral,axiom,
    ! [X: real,Na: num] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ( powr(real,X,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,power_power(real,X),aa(num,nat,numeral_numeral(nat),Na))) ) ) ).

% powr_neg_numeral
tff(fact_1794_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,ord_less(nat,M),Na)
        | ( Na = zero_zero(nat) ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_1795_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,Na: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Na)) ) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_1796_dvd__pos__nat,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(nat,$o,dvd_dvd(nat,M),Na)
       => aa(nat,$o,ord_less(nat,zero_zero(nat)),M) ) ) ).

% dvd_pos_nat
tff(fact_1797_of__nat__dvd__iff,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,Na: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Na))
        <=> aa(nat,$o,dvd_dvd(nat,M),Na) ) ) ).

% of_nat_dvd_iff
tff(fact_1798_numeral__Bit0__div__2,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Na: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),bit0(Na))),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(num,A,numeral_numeral(A),Na) ) ).

% numeral_Bit0_div_2
tff(fact_1799_div__add__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B3: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,plus_plus(A,A3),B3)),B3) = aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),one_one(A)) ) ) ) ).

% div_add_self2
tff(fact_1800_div__add__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B3: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,plus_plus(A,B3),A3)),B3) = aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),one_one(A)) ) ) ) ).

% div_add_self1
tff(fact_1801_div__greater__zero__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na))
    <=> ( aa(nat,$o,ord_less_eq(nat,Na),M)
        & aa(nat,$o,ord_less(nat,zero_zero(nat)),Na) ) ) ).

% div_greater_zero_iff
tff(fact_1802_div__le__mono2,axiom,
    ! [M: nat,Na: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
     => ( aa(nat,$o,ord_less_eq(nat,M),Na)
       => aa(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_1803_div__less__dividend,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,one_one(nat)),Na)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
       => aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)),M) ) ) ).

% div_less_dividend
tff(fact_1804_power__diff__power__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,M: nat,Na: nat] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,power_power(A,A3),M)),aa(nat,A,power_power(A,A3),Na)) = $ite(aa(nat,$o,ord_less_eq(nat,Na),M),aa(nat,A,power_power(A,A3),aa(nat,nat,minus_minus(nat,M),Na)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,power_power(A,A3),aa(nat,nat,minus_minus(nat,Na),M)))) ) ) ) ).

% power_diff_power_eq
tff(fact_1805_power__sub,axiom,
    ! [Na: nat,M: nat,A3: nat] :
      ( aa(nat,$o,ord_less_eq(nat,Na),M)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),A3)
       => ( aa(nat,nat,power_power(nat,A3),aa(nat,nat,minus_minus(nat,M),Na)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,power_power(nat,A3),M)),aa(nat,nat,power_power(nat,A3),Na)) ) ) ) ).

% power_sub
tff(fact_1806_power__minus__is__div,axiom,
    ! [B3: nat,A3: nat] :
      ( aa(nat,$o,ord_less_eq(nat,B3),A3)
     => ( aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,A3),B3)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),A3)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),B3)) ) ) ).

% power_minus_is_div
tff(fact_1807_two__pow__div__gt__le,axiom,
    ! [V2: nat,Na: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,V2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M)))
     => aa(nat,$o,ord_less_eq(nat,M),Na) ) ).

% two_pow_div_gt_le
tff(fact_1808_real__average__minus__second,axiom,
    ! [B3: real,A3: real] : aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,plus_plus(real,B3),A3)),aa(num,real,numeral_numeral(real),bit0(one2)))),A3) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,B3),A3)),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% real_average_minus_second
tff(fact_1809_real__average__minus__first,axiom,
    ! [A3: real,B3: real] : aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,plus_plus(real,A3),B3)),aa(num,real,numeral_numeral(real),bit0(one2)))),A3) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,B3),A3)),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% real_average_minus_first
tff(fact_1810_zdiff__int__split,axiom,
    ! [P: fun(int,$o),X: nat,Y: nat] :
      ( aa(int,$o,P,aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,minus_minus(nat,X),Y)))
    <=> ( ( aa(nat,$o,ord_less_eq(nat,Y),X)
         => aa(int,$o,P,aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),X)),aa(nat,int,semiring_1_of_nat(int),Y))) )
        & ( aa(nat,$o,ord_less(nat,X),Y)
         => aa(int,$o,P,zero_zero(int)) ) ) ) ).

% zdiff_int_split
tff(fact_1811_VEBT__internal_Opow__sum,axiom,
    ! [A3: nat,B3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,plus_plus(nat,A3),B3))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),A3)) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),B3) ).

% VEBT_internal.pow_sum
tff(fact_1812_artanh__def,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [X: A] : aa(A,A,artanh(A),X) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,ln_ln(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,plus_plus(A,one_one(A)),X)),aa(A,A,minus_minus(A,one_one(A)),X)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% artanh_def
tff(fact_1813_div2__even__ext__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
     => ( ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),X)
        <=> aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Y) )
       => ( X = Y ) ) ) ).

% div2_even_ext_nat
tff(fact_1814_ln__inj__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
       => ( ( aa(real,real,ln_ln(real),X) = aa(real,real,ln_ln(real),Y) )
        <=> ( X = Y ) ) ) ) ).

% ln_inj_iff
tff(fact_1815_ln__less__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
       => ( aa(real,$o,ord_less(real,aa(real,real,ln_ln(real),X)),aa(real,real,ln_ln(real),Y))
        <=> aa(real,$o,ord_less(real,X),Y) ) ) ) ).

% ln_less_cancel_iff
tff(fact_1816_ln__le__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
       => ( aa(real,$o,ord_less_eq(real,aa(real,real,ln_ln(real),X)),aa(real,real,ln_ln(real),Y))
        <=> aa(real,$o,ord_less_eq(real,X),Y) ) ) ) ).

% ln_le_cancel_iff
tff(fact_1817_ln__eq__zero__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ( ( aa(real,real,ln_ln(real),X) = zero_zero(real) )
      <=> ( X = one_one(real) ) ) ) ).

% ln_eq_zero_iff
tff(fact_1818_ln__gt__zero__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,ln_ln(real),X))
      <=> aa(real,$o,ord_less(real,one_one(real)),X) ) ) ).

% ln_gt_zero_iff
tff(fact_1819_ln__less__zero__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less(real,aa(real,real,ln_ln(real),X)),zero_zero(real))
      <=> aa(real,$o,ord_less(real,X),one_one(real)) ) ) ).

% ln_less_zero_iff
tff(fact_1820_ln__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,ln_ln(A),one_one(A)) = zero_zero(A) ) ) ).

% ln_one
tff(fact_1821_ln__le__zero__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less_eq(real,aa(real,real,ln_ln(real),X)),zero_zero(real))
      <=> aa(real,$o,ord_less_eq(real,X),one_one(real)) ) ) ).

% ln_le_zero_iff
tff(fact_1822_ln__ge__zero__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,ln_ln(real),X))
      <=> aa(real,$o,ord_less_eq(real,one_one(real)),X) ) ) ).

% ln_ge_zero_iff
tff(fact_1823_neq__if__length__neq,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] :
      ( ( size_size(list(A),Xsa) != size_size(list(A),Ys) )
     => ( Xsa != Ys ) ) ).

% neq_if_length_neq
tff(fact_1824_Ex__list__of__length,axiom,
    ! [A: $tType,Na: nat] :
    ? [Xs: list(A)] : size_size(list(A),Xs) = Na ).

% Ex_list_of_length
tff(fact_1825_ln__less__self,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => aa(real,$o,ord_less(real,aa(real,real,ln_ln(real),X)),X) ) ).

% ln_less_self
tff(fact_1826_ln__bound,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => aa(real,$o,ord_less_eq(real,aa(real,real,ln_ln(real),X)),X) ) ).

% ln_bound
tff(fact_1827_ln__gt__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),X)
     => aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,ln_ln(real),X)) ) ).

% ln_gt_zero
tff(fact_1828_ln__less__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less(real,X),one_one(real))
       => aa(real,$o,ord_less(real,aa(real,real,ln_ln(real),X)),zero_zero(real)) ) ) ).

% ln_less_zero
tff(fact_1829_ln__gt__zero__imp__gt__one,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,ln_ln(real),X))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),X)
       => aa(real,$o,ord_less(real,one_one(real)),X) ) ) ).

% ln_gt_zero_imp_gt_one
tff(fact_1830_ln__ge__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,one_one(real)),X)
     => aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,ln_ln(real),X)) ) ).

% ln_ge_zero
tff(fact_1831_ln__div,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ( aa(real,$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),X),Y)) = aa(real,real,minus_minus(real,aa(real,real,ln_ln(real),X)),aa(real,real,ln_ln(real),Y)) ) ) ) ).

% ln_div
tff(fact_1832_pinf_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P3: fun(A,$o),Q: fun(A,$o),Q2: fun(A,$o)] :
          ( ? [Z3: A] :
            ! [X4: A] :
              ( aa(A,$o,ord_less(A,Z3),X4)
             => ( aa(A,$o,P,X4)
              <=> aa(A,$o,P3,X4) ) )
         => ( ? [Z3: A] :
              ! [X4: A] :
                ( aa(A,$o,ord_less(A,Z3),X4)
               => ( aa(A,$o,Q,X4)
                <=> aa(A,$o,Q2,X4) ) )
           => ? [Z2: A] :
              ! [X2: A] :
                ( aa(A,$o,ord_less(A,Z2),X2)
               => ( ( aa(A,$o,P,X2)
                    & aa(A,$o,Q,X2) )
                <=> ( aa(A,$o,P3,X2)
                    & aa(A,$o,Q2,X2) ) ) ) ) ) ) ).

% pinf(1)
tff(fact_1833_pinf_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P3: fun(A,$o),Q: fun(A,$o),Q2: fun(A,$o)] :
          ( ? [Z3: A] :
            ! [X4: A] :
              ( aa(A,$o,ord_less(A,Z3),X4)
             => ( aa(A,$o,P,X4)
              <=> aa(A,$o,P3,X4) ) )
         => ( ? [Z3: A] :
              ! [X4: A] :
                ( aa(A,$o,ord_less(A,Z3),X4)
               => ( aa(A,$o,Q,X4)
                <=> aa(A,$o,Q2,X4) ) )
           => ? [Z2: A] :
              ! [X2: A] :
                ( aa(A,$o,ord_less(A,Z2),X2)
               => ( ( aa(A,$o,P,X2)
                    | aa(A,$o,Q,X2) )
                <=> ( aa(A,$o,P3,X2)
                    | aa(A,$o,Q2,X2) ) ) ) ) ) ) ).

% pinf(2)
tff(fact_1834_pinf_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( aa(A,$o,ord_less(A,Z2),X2)
         => ( X2 != Ta ) ) ) ).

% pinf(3)
tff(fact_1835_pinf_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( aa(A,$o,ord_less(A,Z2),X2)
         => ( X2 != Ta ) ) ) ).

% pinf(4)
tff(fact_1836_pinf_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( aa(A,$o,ord_less(A,Z2),X2)
         => ~ aa(A,$o,ord_less(A,X2),Ta) ) ) ).

% pinf(5)
tff(fact_1837_pinf_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( aa(A,$o,ord_less(A,Z2),X2)
         => aa(A,$o,ord_less(A,Ta),X2) ) ) ).

% pinf(7)
tff(fact_1838_pinf_I11_J,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [F3: B] :
        ? [Z2: A] :
        ! [X2: A] :
          ( aa(A,$o,ord_less(A,Z2),X2)
         => ( F3 = F3 ) ) ) ).

% pinf(11)
tff(fact_1839_minf_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P3: fun(A,$o),Q: fun(A,$o),Q2: fun(A,$o)] :
          ( ? [Z3: A] :
            ! [X4: A] :
              ( aa(A,$o,ord_less(A,X4),Z3)
             => ( aa(A,$o,P,X4)
              <=> aa(A,$o,P3,X4) ) )
         => ( ? [Z3: A] :
              ! [X4: A] :
                ( aa(A,$o,ord_less(A,X4),Z3)
               => ( aa(A,$o,Q,X4)
                <=> aa(A,$o,Q2,X4) ) )
           => ? [Z2: A] :
              ! [X2: A] :
                ( aa(A,$o,ord_less(A,X2),Z2)
               => ( ( aa(A,$o,P,X2)
                    & aa(A,$o,Q,X2) )
                <=> ( aa(A,$o,P3,X2)
                    & aa(A,$o,Q2,X2) ) ) ) ) ) ) ).

% minf(1)
tff(fact_1840_minf_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P3: fun(A,$o),Q: fun(A,$o),Q2: fun(A,$o)] :
          ( ? [Z3: A] :
            ! [X4: A] :
              ( aa(A,$o,ord_less(A,X4),Z3)
             => ( aa(A,$o,P,X4)
              <=> aa(A,$o,P3,X4) ) )
         => ( ? [Z3: A] :
              ! [X4: A] :
                ( aa(A,$o,ord_less(A,X4),Z3)
               => ( aa(A,$o,Q,X4)
                <=> aa(A,$o,Q2,X4) ) )
           => ? [Z2: A] :
              ! [X2: A] :
                ( aa(A,$o,ord_less(A,X2),Z2)
               => ( ( aa(A,$o,P,X2)
                    | aa(A,$o,Q,X2) )
                <=> ( aa(A,$o,P3,X2)
                    | aa(A,$o,Q2,X2) ) ) ) ) ) ) ).

% minf(2)
tff(fact_1841_minf_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( aa(A,$o,ord_less(A,X2),Z2)
         => ( X2 != Ta ) ) ) ).

% minf(3)
tff(fact_1842_minf_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( aa(A,$o,ord_less(A,X2),Z2)
         => ( X2 != Ta ) ) ) ).

% minf(4)
tff(fact_1843_minf_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( aa(A,$o,ord_less(A,X2),Z2)
         => aa(A,$o,ord_less(A,X2),Ta) ) ) ).

% minf(5)
tff(fact_1844_minf_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( aa(A,$o,ord_less(A,X2),Z2)
         => ~ aa(A,$o,ord_less(A,Ta),X2) ) ) ).

% minf(7)
tff(fact_1845_minf_I11_J,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [F3: B] :
        ? [Z2: A] :
        ! [X2: A] :
          ( aa(A,$o,ord_less(A,X2),Z2)
         => ( F3 = F3 ) ) ) ).

% minf(11)
tff(fact_1846_ln__ge__zero__imp__ge__one,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,ln_ln(real),X))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),X)
       => aa(real,$o,ord_less_eq(real,one_one(real)),X) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_1847_ln__add__one__self__le__self,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => aa(real,$o,ord_less_eq(real,aa(real,real,ln_ln(real),aa(real,real,plus_plus(real,one_one(real)),X))),X) ) ).

% ln_add_one_self_le_self
tff(fact_1848_ln__diff__le,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
       => aa(real,$o,ord_less_eq(real,aa(real,real,minus_minus(real,aa(real,real,ln_ln(real),X)),aa(real,real,ln_ln(real),Y))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,X),Y)),Y)) ) ) ).

% ln_diff_le
tff(fact_1849_ln__eq__minus__one,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ( ( aa(real,real,ln_ln(real),X) = aa(real,real,minus_minus(real,X),one_one(real)) )
       => ( X = one_one(real) ) ) ) ).

% ln_eq_minus_one
tff(fact_1850_ln__le__minus__one,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => aa(real,$o,ord_less_eq(real,aa(real,real,ln_ln(real),X)),aa(real,real,minus_minus(real,X),one_one(real))) ) ).

% ln_le_minus_one
tff(fact_1851_ln__add__one__self__le__self2,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),X)
     => aa(real,$o,ord_less_eq(real,aa(real,real,ln_ln(real),aa(real,real,plus_plus(real,one_one(real)),X))),X) ) ).

% ln_add_one_self_le_self2
tff(fact_1852_ln__powr__bound,axiom,
    ! [X: real,A3: real] :
      ( aa(real,$o,ord_less_eq(real,one_one(real)),X)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
       => aa(real,$o,ord_less_eq(real,aa(real,real,ln_ln(real),X)),aa(real,real,aa(real,fun(real,real),divide_divide(real),powr(real,X,A3)),A3)) ) ) ).

% ln_powr_bound
tff(fact_1853_ln__one__minus__pos__upper__bound,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less(real,X),one_one(real))
       => aa(real,$o,ord_less_eq(real,aa(real,real,ln_ln(real),aa(real,real,minus_minus(real,one_one(real)),X))),aa(real,real,uminus_uminus(real),X)) ) ) ).

% ln_one_minus_pos_upper_bound
tff(fact_1854_pinf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( aa(A,$o,ord_less(A,Z2),X2)
         => ~ aa(A,$o,ord_less_eq(A,X2),Ta) ) ) ).

% pinf(6)
tff(fact_1855_pinf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( aa(A,$o,ord_less(A,Z2),X2)
         => aa(A,$o,ord_less_eq(A,Ta),X2) ) ) ).

% pinf(8)
tff(fact_1856_minf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( aa(A,$o,ord_less(A,X2),Z2)
         => aa(A,$o,ord_less_eq(A,X2),Ta) ) ) ).

% minf(6)
tff(fact_1857_minf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( aa(A,$o,ord_less(A,X2),Z2)
         => ~ aa(A,$o,ord_less_eq(A,Ta),X2) ) ) ).

% minf(8)
tff(fact_1858_imp__le__cong,axiom,
    ! [X: int,X6: int,P: $o,P3: $o] :
      ( ( X = X6 )
     => ( ( aa(int,$o,ord_less_eq(int,zero_zero(int)),X6)
         => ( (P)
          <=> (P3) ) )
       => ( ( aa(int,$o,ord_less_eq(int,zero_zero(int)),X)
           => (P) )
        <=> ( aa(int,$o,ord_less_eq(int,zero_zero(int)),X6)
           => (P3) ) ) ) ) ).

% imp_le_cong
tff(fact_1859_conj__le__cong,axiom,
    ! [X: int,X6: int,P: $o,P3: $o] :
      ( ( X = X6 )
     => ( ( aa(int,$o,ord_less_eq(int,zero_zero(int)),X6)
         => ( (P)
          <=> (P3) ) )
       => ( ( aa(int,$o,ord_less_eq(int,zero_zero(int)),X)
            & (P) )
        <=> ( aa(int,$o,ord_less_eq(int,zero_zero(int)),X6)
            & (P3) ) ) ) ) ).

% conj_le_cong
tff(fact_1860_ln__one__plus__pos__lower__bound,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less_eq(real,X),one_one(real))
       => aa(real,$o,ord_less_eq(real,aa(real,real,minus_minus(real,X),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(real,real,ln_ln(real),aa(real,real,plus_plus(real,one_one(real)),X))) ) ) ).

% ln_one_plus_pos_lower_bound
tff(fact_1861_pinf_I9_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D2: A,Sa: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( aa(A,$o,ord_less(A,Z2),X2)
         => ( aa(A,$o,dvd_dvd(A,D2),aa(A,A,plus_plus(A,X2),Sa))
          <=> aa(A,$o,dvd_dvd(A,D2),aa(A,A,plus_plus(A,X2),Sa)) ) ) ) ).

% pinf(9)
tff(fact_1862_pinf_I10_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D2: A,Sa: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( aa(A,$o,ord_less(A,Z2),X2)
         => ( ~ aa(A,$o,dvd_dvd(A,D2),aa(A,A,plus_plus(A,X2),Sa))
          <=> ~ aa(A,$o,dvd_dvd(A,D2),aa(A,A,plus_plus(A,X2),Sa)) ) ) ) ).

% pinf(10)
tff(fact_1863_minf_I9_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D2: A,Sa: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( aa(A,$o,ord_less(A,X2),Z2)
         => ( aa(A,$o,dvd_dvd(A,D2),aa(A,A,plus_plus(A,X2),Sa))
          <=> aa(A,$o,dvd_dvd(A,D2),aa(A,A,plus_plus(A,X2),Sa)) ) ) ) ).

% minf(9)
tff(fact_1864_minf_I10_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D2: A,Sa: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( aa(A,$o,ord_less(A,X2),Z2)
         => ( ~ aa(A,$o,dvd_dvd(A,D2),aa(A,A,plus_plus(A,X2),Sa))
          <=> ~ aa(A,$o,dvd_dvd(A,D2),aa(A,A,plus_plus(A,X2),Sa)) ) ) ) ).

% minf(10)
tff(fact_1865_ln__2__less__1,axiom,
    aa(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_1866_Bolzano,axiom,
    ! [A3: real,B3: real,P: fun(real,fun(real,$o))] :
      ( aa(real,$o,ord_less_eq(real,A3),B3)
     => ( ! [A4: real,B5: real,C5: real] :
            ( aa(real,$o,aa(real,fun(real,$o),P,A4),B5)
           => ( aa(real,$o,aa(real,fun(real,$o),P,B5),C5)
             => ( aa(real,$o,ord_less_eq(real,A4),B5)
               => ( aa(real,$o,ord_less_eq(real,B5),C5)
                 => aa(real,$o,aa(real,fun(real,$o),P,A4),C5) ) ) ) )
       => ( ! [X4: real] :
              ( aa(real,$o,ord_less_eq(real,A3),X4)
             => ( aa(real,$o,ord_less_eq(real,X4),B3)
               => ? [D5: real] :
                    ( aa(real,$o,ord_less(real,zero_zero(real)),D5)
                    & ! [A4: real,B5: real] :
                        ( ( aa(real,$o,ord_less_eq(real,A4),X4)
                          & aa(real,$o,ord_less_eq(real,X4),B5)
                          & aa(real,$o,ord_less(real,aa(real,real,minus_minus(real,B5),A4)),D5) )
                       => aa(real,$o,aa(real,fun(real,$o),P,A4),B5) ) ) ) )
         => aa(real,$o,aa(real,fun(real,$o),P,A3),B3) ) ) ) ).

% Bolzano
tff(fact_1867_round__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: int] :
          ( aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(int,A,ring_1_of_int(A),Y))
         => ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),Y)),aa(A,A,plus_plus(A,X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2)))))
           => ( archimedean_round(A,X) = Y ) ) ) ) ).

% round_unique
tff(fact_1868_even__word__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( even_word(A) = dvd_dvd(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))) ) ) ).

% even_word_def
tff(fact_1869_arcosh__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [X: A] : aa(A,A,arcosh(A),X) = aa(A,A,ln_ln(A),aa(A,A,plus_plus(A,X),powr(A,aa(A,A,minus_minus(A,aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)),aa(real,A,real_Vector_of_real(A),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))))) ) ).

% arcosh_def
tff(fact_1870_ln__one__minus__pos__lower__bound,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less_eq(real,X),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,ord_less_eq(real,aa(real,real,minus_minus(real,aa(real,real,uminus_uminus(real),X)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(real,real,ln_ln(real),aa(real,real,minus_minus(real,one_one(real)),X))) ) ) ).

% ln_one_minus_pos_lower_bound
tff(fact_1871_tanh__ln__real,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ( aa(real,real,tanh(real),aa(real,real,ln_ln(real),X)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))),aa(real,real,plus_plus(real,aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))) ) ) ).

% tanh_ln_real
tff(fact_1872_mult__cancel__right,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [A3: A,C2: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A3 = B3 ) ) ) ) ).

% mult_cancel_right
tff(fact_1873_mult__cancel__left,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C2: A,A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A3 = B3 ) ) ) ) ).

% mult_cancel_left
tff(fact_1874_mult__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3) = zero_zero(A) )
        <=> ( ( A3 = zero_zero(A) )
            | ( B3 = zero_zero(A) ) ) ) ) ).

% mult_eq_0_iff
tff(fact_1875_mult__zero__right,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),zero_zero(A)) = zero_zero(A) ) ).

% mult_zero_right
tff(fact_1876_mult__zero__left,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),zero_zero(A)),A3) = zero_zero(A) ) ).

% mult_zero_left
tff(fact_1877_numeral__times__numeral,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [M: num,Na: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Na)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),Na)) ) ).

% numeral_times_numeral
tff(fact_1878_mult__numeral__left__semiring__numeral,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [V2: num,W: num,Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V2),W))),Z) ) ).

% mult_numeral_left_semiring_numeral
tff(fact_1879_mult__1,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),one_one(A)),A3) = A3 ) ).

% mult_1
tff(fact_1880_mult_Oright__neutral,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),one_one(A)) = A3 ) ).

% mult.right_neutral
tff(fact_1881_mult__minus__left,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),A3)),B3) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) ) ).

% mult_minus_left
tff(fact_1882_minus__mult__minus,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,uminus_uminus(A),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3) ) ).

% minus_mult_minus
tff(fact_1883_mult__minus__right,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),B3)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) ) ).

% mult_minus_right
tff(fact_1884_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_1885_of__int__mult,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [W: int,Z: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),W)),aa(int,A,ring_1_of_int(A),Z)) ) ).

% of_int_mult
tff(fact_1886_real__divide__square__eq,axiom,
    ! [R3: real,A3: real] : aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),R3),A3)),aa(real,real,aa(real,fun(real,real),times_times(real),R3),R3)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),A3),R3) ).

% real_divide_square_eq
tff(fact_1887_tanh__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ( aa(A,A,tanh(A),zero_zero(A)) = zero_zero(A) ) ) ).

% tanh_0
tff(fact_1888_tanh__real__zero__iff,axiom,
    ! [X: real] :
      ( ( aa(real,real,tanh(real),X) = zero_zero(real) )
    <=> ( X = zero_zero(real) ) ) ).

% tanh_real_zero_iff
tff(fact_1889_tanh__real__less__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,tanh(real),X)),aa(real,real,tanh(real),Y))
    <=> aa(real,$o,ord_less(real,X),Y) ) ).

% tanh_real_less_iff
tff(fact_1890_round__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Na: int] : archimedean_round(A,aa(int,A,ring_1_of_int(A),Na)) = Na ) ).

% round_of_int
tff(fact_1891_sum__squares__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)) = zero_zero(A) )
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_1892_mult__cancel__right2,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [A3: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = C2 )
        <=> ( ( C2 = zero_zero(A) )
            | ( A3 = one_one(A) ) ) ) ) ).

% mult_cancel_right2
tff(fact_1893_mult__cancel__right1,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C2: A,B3: A] :
          ( ( C2 = aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( B3 = one_one(A) ) ) ) ) ).

% mult_cancel_right1
tff(fact_1894_mult__cancel__left2,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C2: A,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3) = C2 )
        <=> ( ( C2 = zero_zero(A) )
            | ( A3 = one_one(A) ) ) ) ) ).

% mult_cancel_left2
tff(fact_1895_mult__cancel__left1,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C2: A,B3: A] :
          ( ( C2 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3) )
        <=> ( ( C2 = zero_zero(A) )
            | ( B3 = one_one(A) ) ) ) ) ).

% mult_cancel_left1
tff(fact_1896_distrib__right__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [A3: A,B3: A,V2: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,plus_plus(A,A3),B3)),aa(num,A,numeral_numeral(A),V2)) = aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),V2))),aa(A,A,aa(A,fun(A,A),times_times(A),B3),aa(num,A,numeral_numeral(A),V2))) ) ).

% distrib_right_numeral
tff(fact_1897_distrib__left__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [V2: num,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),aa(A,A,plus_plus(A,B3),C2)) = aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),C2)) ) ).

% distrib_left_numeral
tff(fact_1898_div__mult__mult1__if,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A3: A,B3: A] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) = $ite(C2 = zero_zero(A),zero_zero(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)) ) ).

% div_mult_mult1_if
tff(fact_1899_div__mult__mult2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A3: A,B3: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) ) ) ) ).

% div_mult_mult2
tff(fact_1900_div__mult__mult1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A3: A,B3: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) ) ) ) ).

% div_mult_mult1
tff(fact_1901_mult__divide__mult__cancel__left__if,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A3: A,B3: A] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) = $ite(C2 = zero_zero(A),zero_zero(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_1902_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A3: A,B3: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_1903_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A3: A,B3: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_1904_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A3: A,B3: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_1905_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A3: A,B3: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_1906_nonzero__mult__div__cancel__left,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),A3) = B3 ) ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1907_nonzero__mult__div__cancel__right,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B3: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),B3) = A3 ) ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1908_left__diff__distrib__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ring(A) )
     => ! [A3: A,B3: A,V2: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,A3),B3)),aa(num,A,numeral_numeral(A),V2)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),V2))),aa(A,A,aa(A,fun(A,A),times_times(A),B3),aa(num,A,numeral_numeral(A),V2))) ) ).

% left_diff_distrib_numeral
tff(fact_1909_right__diff__distrib__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ring(A) )
     => ! [V2: num,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),aa(A,A,minus_minus(A,B3),C2)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),C2)) ) ).

% right_diff_distrib_numeral
tff(fact_1910_mult__neg__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M: num,Na: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),Na)) ) ).

% mult_neg_numeral_simps(1)
tff(fact_1911_mult__neg__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M: num,Na: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),Na)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),Na))) ) ).

% mult_neg_numeral_simps(2)
tff(fact_1912_mult__neg__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M: num,Na: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),Na))) ) ).

% mult_neg_numeral_simps(3)
tff(fact_1913_semiring__norm_I170_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [V2: num,W: num,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V2),W)))),Y) ) ).

% semiring_norm(170)
tff(fact_1914_semiring__norm_I171_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [V2: num,W: num,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V2),W)))),Y) ) ).

% semiring_norm(171)
tff(fact_1915_semiring__norm_I172_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [V2: num,W: num,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V2),W))),Y) ) ).

% semiring_norm(172)
tff(fact_1916_mult__minus1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),one_one(A))),Z) = aa(A,A,uminus_uminus(A),Z) ) ).

% mult_minus1
tff(fact_1917_mult__minus1__right,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),Z),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),Z) ) ).

% mult_minus1_right
tff(fact_1918_dvd__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
        <=> ( ( C2 = zero_zero(A) )
            | aa(A,$o,dvd_dvd(A,A3),B3) ) ) ) ).

% dvd_mult_cancel_left
tff(fact_1919_dvd__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))
        <=> ( ( C2 = zero_zero(A) )
            | aa(A,$o,dvd_dvd(A,A3),B3) ) ) ) ).

% dvd_mult_cancel_right
tff(fact_1920_dvd__times__left__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A,C2: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))
          <=> aa(A,$o,dvd_dvd(A,B3),C2) ) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_1921_dvd__times__right__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A,C2: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3))
          <=> aa(A,$o,dvd_dvd(A,B3),C2) ) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_1922_dvd__add__times__triv__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A3),aa(A,A,plus_plus(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)))
        <=> aa(A,$o,dvd_dvd(A,A3),B3) ) ) ).

% dvd_add_times_triv_right_iff
tff(fact_1923_dvd__add__times__triv__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,A3),aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),B3))
        <=> aa(A,$o,dvd_dvd(A,A3),B3) ) ) ).

% dvd_add_times_triv_left_iff
tff(fact_1924_unit__prod,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,A3),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,B3),one_one(A))
           => aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),one_one(A)) ) ) ) ).

% unit_prod
tff(fact_1925_dvd__div__mult__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3)),A3) = B3 ) ) ) ).

% dvd_div_mult_self
tff(fact_1926_dvd__mult__div__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3)) = B3 ) ) ) ).

% dvd_mult_div_cancel
tff(fact_1927_not__real__square__gt__zero,axiom,
    ! [X: real] :
      ( ~ aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,aa(real,fun(real,real),times_times(real),X),X))
    <=> ( X = zero_zero(real) ) ) ).

% not_real_square_gt_zero
tff(fact_1928_round__0,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_round(A,zero_zero(A)) = zero_zero(int) ) ) ).

% round_0
tff(fact_1929_round__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Na: num] : archimedean_round(A,aa(num,A,numeral_numeral(A),Na)) = aa(num,int,numeral_numeral(int),Na) ) ).

% round_numeral
tff(fact_1930_tanh__real__pos__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,tanh(real),X))
    <=> aa(real,$o,ord_less(real,zero_zero(real)),X) ) ).

% tanh_real_pos_iff
tff(fact_1931_tanh__real__neg__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,tanh(real),X)),zero_zero(real))
    <=> aa(real,$o,ord_less(real,X),zero_zero(real)) ) ).

% tanh_real_neg_iff
tff(fact_1932_tanh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,tanh(real),X)),zero_zero(real))
    <=> aa(real,$o,ord_less_eq(real,X),zero_zero(real)) ) ).

% tanh_real_nonpos_iff
tff(fact_1933_tanh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,tanh(real),X))
    <=> aa(real,$o,ord_less_eq(real,zero_zero(real)),X) ) ).

% tanh_real_nonneg_iff
tff(fact_1934_round__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_round(A,one_one(A)) = one_one(int) ) ) ).

% round_1
tff(fact_1935_round__of__nat,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Na: nat] : archimedean_round(A,aa(nat,A,semiring_1_of_nat(A),Na)) = aa(nat,int,semiring_1_of_nat(int),Na) ) ).

% round_of_nat
tff(fact_1936_divide__eq__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B3: A,W: num,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),aa(num,A,numeral_numeral(A),W)) = A3 )
        <=> $ite(aa(num,A,numeral_numeral(A),W) != zero_zero(A),B3 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W)),A3 = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_1937_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A,W: num] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),aa(num,A,numeral_numeral(A),W)) )
        <=> $ite(aa(num,A,numeral_numeral(A),W) != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W)) = B3,A3 = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_1938_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,W: num] :
          ( aa(A,$o,ord_less_eq(A,A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),aa(num,A,numeral_numeral(A),W)))
        <=> aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W))),B3) ) ) ).

% le_divide_eq_numeral1(1)
tff(fact_1939_divide__le__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,W: num,A3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),aa(num,A,numeral_numeral(A),W))),A3)
        <=> aa(A,$o,ord_less_eq(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W))) ) ) ).

% divide_le_eq_numeral1(1)
tff(fact_1940_divide__less__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,W: num,A3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),aa(num,A,numeral_numeral(A),W))),A3)
        <=> aa(A,$o,ord_less(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W))) ) ) ).

% divide_less_eq_numeral1(1)
tff(fact_1941_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,W: num] :
          ( aa(A,$o,ord_less(A,A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),aa(num,A,numeral_numeral(A),W)))
        <=> aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W))),B3) ) ) ).

% less_divide_eq_numeral1(1)
tff(fact_1942_div__mult__self4,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B3: A,C2: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)),A3)),B3) = aa(A,A,plus_plus(A,C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)) ) ) ) ).

% div_mult_self4
tff(fact_1943_div__mult__self3,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B3: A,C2: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)),A3)),B3) = aa(A,A,plus_plus(A,C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)) ) ) ) ).

% div_mult_self3
tff(fact_1944_div__mult__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B3: A,A3: A,C2: A] :
          ( ( B3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,plus_plus(A,A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))),B3) = aa(A,A,plus_plus(A,C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)) ) ) ) ).

% div_mult_self2
tff(fact_1945_div__mult__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B3: A,A3: A,C2: A] :
          ( ( B3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,plus_plus(A,A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))),B3) = aa(A,A,plus_plus(A,C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)) ) ) ) ).

% div_mult_self1
tff(fact_1946_nonzero__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [B3: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3) ) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_1947_nonzero__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B3) ) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_1948_left__minus__one__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Na: nat,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),Na)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),Na)),A3)) = A3 ) ).

% left_minus_one_mult_self
tff(fact_1949_minus__one__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Na: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),Na)),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),Na)) = one_one(A) ) ).

% minus_one_mult_self
tff(fact_1950_unit__div__mult__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,A3),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3)),A3) = B3 ) ) ) ).

% unit_div_mult_self
tff(fact_1951_unit__mult__div__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,A3),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),B3),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3) ) ) ) ).

% unit_mult_div_div
tff(fact_1952_power__add__numeral,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,M: num,Na: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),M))),aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),Na))) = aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),aa(num,num,plus_plus(num,M),Na))) ) ).

% power_add_numeral
tff(fact_1953_power__add__numeral2,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,M: num,Na: num,B3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),M))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),Na))),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),aa(num,num,plus_plus(num,M),Na)))),B3) ) ).

% power_add_numeral2
tff(fact_1954_divide__eq__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B3: A,W: num,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) = A3 )
        <=> $ite(aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) != zero_zero(A),B3 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),A3 = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_1955_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A,W: num] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) )
        <=> $ite(aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) = B3,A3 = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_1956_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,W: num] :
          ( aa(A,$o,ord_less_eq(A,A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))))
        <=> aa(A,$o,ord_less_eq(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))) ) ) ).

% le_divide_eq_numeral1(2)
tff(fact_1957_divide__le__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,W: num,A3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),A3)
        <=> aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),B3) ) ) ).

% divide_le_eq_numeral1(2)
tff(fact_1958_divide__less__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,W: num,A3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),A3)
        <=> aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),B3) ) ) ).

% divide_less_eq_numeral1(2)
tff(fact_1959_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,W: num] :
          ( aa(A,$o,ord_less(A,A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))))
        <=> aa(A,$o,ord_less(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))) ) ) ).

% less_divide_eq_numeral1(2)
tff(fact_1960_even__mult__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
            | aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),B3) ) ) ) ).

% even_mult_iff
tff(fact_1961_round__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Na: num] : archimedean_round(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na)) ) ).

% round_neg_numeral
tff(fact_1962_odd__two__times__div__two__succ,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
         => ( aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))))),one_one(A)) = A3 ) ) ) ).

% odd_two_times_div_two_succ
tff(fact_1963_mult__commute__abs,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [C2: A] : aTP_Lamp_ar(A,fun(A,A),C2) = aa(A,fun(A,A),times_times(A),C2) ) ).

% mult_commute_abs
tff(fact_1964_mult_Oassoc,axiom,
    ! [A: $tType] :
      ( semigroup_mult(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ).

% mult.assoc
tff(fact_1965_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3) = aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3) ) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_1966_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [B3: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_1967_mult__right__cancel,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C2: A,A3: A,B3: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2) )
          <=> ( A3 = B3 ) ) ) ) ).

% mult_right_cancel
tff(fact_1968_mult__left__cancel,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C2: A,A3: A,B3: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3) )
          <=> ( A3 = B3 ) ) ) ) ).

% mult_left_cancel
tff(fact_1969_no__zero__divisors,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B3 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3) != zero_zero(A) ) ) ) ) ).

% no_zero_divisors
tff(fact_1970_divisors__zero,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3) = zero_zero(A) )
         => ( ( A3 = zero_zero(A) )
            | ( B3 = zero_zero(A) ) ) ) ) ).

% divisors_zero
tff(fact_1971_mult__not__zero,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3) != zero_zero(A) )
         => ( ( A3 != zero_zero(A) )
            & ( B3 != zero_zero(A) ) ) ) ) ).

% mult_not_zero
tff(fact_1972_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,plus_plus(A,A3),B3)),C2) = aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ).

% ring_class.ring_distribs(2)
tff(fact_1973_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,plus_plus(A,B3),C2)) = aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ).

% ring_class.ring_distribs(1)
tff(fact_1974_comm__semiring__class_Odistrib,axiom,
    ! [A: $tType] :
      ( comm_semiring(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,plus_plus(A,A3),B3)),C2) = aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ).

% comm_semiring_class.distrib
tff(fact_1975_distrib__left,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,plus_plus(A,B3),C2)) = aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ).

% distrib_left
tff(fact_1976_distrib__right,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,plus_plus(A,A3),B3)),C2) = aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ).

% distrib_right
tff(fact_1977_combine__common__factor,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A3: A,E2: A,B3: A,C2: A] : aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),E2)),aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),E2)),C2)) = aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,plus_plus(A,A3),B3)),E2)),C2) ) ).

% combine_common_factor
tff(fact_1978_mult_Ocomm__neutral,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),one_one(A)) = A3 ) ).

% mult.comm_neutral
tff(fact_1979_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),one_one(A)),A3) = A3 ) ).

% comm_monoid_mult_class.mult_1
tff(fact_1980_left__diff__distrib,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,A3),B3)),C2) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ).

% left_diff_distrib
tff(fact_1981_right__diff__distrib,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,minus_minus(A,B3),C2)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ).

% right_diff_distrib
tff(fact_1982_left__diff__distrib_H,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [B3: A,C2: A,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,B3),C2)),A3) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)) ) ).

% left_diff_distrib'
tff(fact_1983_right__diff__distrib_H,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,minus_minus(A,B3),C2)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ).

% right_diff_distrib'
tff(fact_1984_power__commutes,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,Na: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,A3),Na)),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,power_power(A,A3),Na)) ) ).

% power_commutes
tff(fact_1985_power__mult__distrib,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B3: A,Na: nat] : aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),Na) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,A3),Na)),aa(nat,A,power_power(A,B3),Na)) ) ).

% power_mult_distrib
tff(fact_1986_power__commuting__commutes,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [X: A,Y: A,Na: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),X),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),X) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,X),Na)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(nat,A,power_power(A,X),Na)) ) ) ) ).

% power_commuting_commutes
tff(fact_1987_square__eq__iff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),B3),B3) )
        <=> ( ( A3 = B3 )
            | ( A3 = aa(A,A,uminus_uminus(A),B3) ) ) ) ) ).

% square_eq_iff
tff(fact_1988_minus__mult__commute,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),A3)),B3) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),B3)) ) ).

% minus_mult_commute
tff(fact_1989_mult__of__nat__commute,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [X: nat,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),X)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(nat,A,semiring_1_of_nat(A),X)) ) ).

% mult_of_nat_commute
tff(fact_1990_dvdE,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,dvd_dvd(A,B3),A3)
         => ~ ! [K2: A] : A3 != aa(A,A,aa(A,fun(A,A),times_times(A),B3),K2) ) ) ).

% dvdE
tff(fact_1991_dvdI,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [A3: A,B3: A,K: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),B3),K) )
         => aa(A,$o,dvd_dvd(A,B3),A3) ) ) ).

% dvdI
tff(fact_1992_dvd__def,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,dvd_dvd(A,B3),A3)
        <=> ? [K3: A] : A3 = aa(A,A,aa(A,fun(A,A),times_times(A),B3),K3) ) ) ).

% dvd_def
tff(fact_1993_dvd__mult,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,A3),C2)
         => aa(A,$o,dvd_dvd(A,A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ) ).

% dvd_mult
tff(fact_1994_dvd__mult2,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A3),B3)
         => aa(A,$o,dvd_dvd(A,A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ) ).

% dvd_mult2
tff(fact_1995_dvd__mult__left,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),C2)
         => aa(A,$o,dvd_dvd(A,A3),C2) ) ) ).

% dvd_mult_left
tff(fact_1996_dvd__triv__left,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B3: A] : aa(A,$o,dvd_dvd(A,A3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) ) ).

% dvd_triv_left
tff(fact_1997_mult__dvd__mono,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B3: A,C2: A,D2: A] :
          ( aa(A,$o,dvd_dvd(A,A3),B3)
         => ( aa(A,$o,dvd_dvd(A,C2),D2)
           => aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),D2)) ) ) ) ).

% mult_dvd_mono
tff(fact_1998_dvd__mult__right,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),C2)
         => aa(A,$o,dvd_dvd(A,B3),C2) ) ) ).

% dvd_mult_right
tff(fact_1999_dvd__triv__right,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B3: A] : aa(A,$o,dvd_dvd(A,A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3)) ) ).

% dvd_triv_right
tff(fact_2000_mult__of__int__commute,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: int,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),X)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(int,A,ring_1_of_int(A),X)) ) ).

% mult_of_int_commute
tff(fact_2001_lambda__zero,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ( aTP_Lamp_as(A,A) = aa(A,fun(A,A),times_times(A),zero_zero(A)) ) ) ).

% lambda_zero
tff(fact_2002_lambda__one,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ( aTP_Lamp_at(A,A) = aa(A,fun(A,A),times_times(A),one_one(A)) ) ) ).

% lambda_one
tff(fact_2003_le__mult__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B3)
           => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(A,A3)),archim6421214686448440834_floor(A,B3))),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))) ) ) ) ).

% le_mult_floor
tff(fact_2004_mult__ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B3)
           => aa(int,$o,ord_less_eq(int,archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(A,A3)),archimedean_ceiling(A,B3))) ) ) ) ).

% mult_ceiling_le
tff(fact_2005_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordere2520102378445227354miring(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2006_zero__le__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))
        <=> ( ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
              & aa(A,$o,ord_less_eq(A,zero_zero(A)),B3) )
            | ( aa(A,$o,ord_less_eq(A,A3),zero_zero(A))
              & aa(A,$o,ord_less_eq(A,B3),zero_zero(A)) ) ) ) ) ).

% zero_le_mult_iff
tff(fact_2007_mult__nonneg__nonpos2,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less_eq(A,B3),zero_zero(A))
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3)),zero_zero(A)) ) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2008_mult__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),zero_zero(A))
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B3)
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),zero_zero(A)) ) ) ) ).

% mult_nonpos_nonneg
tff(fact_2009_mult__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less_eq(A,B3),zero_zero(A))
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),zero_zero(A)) ) ) ) ).

% mult_nonneg_nonpos
tff(fact_2010_mult__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B3)
           => aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) ) ) ) ).

% mult_nonneg_nonneg
tff(fact_2011_split__mult__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B3: A] :
          ( ( ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
              & aa(A,$o,ord_less_eq(A,B3),zero_zero(A)) )
            | ( aa(A,$o,ord_less_eq(A,A3),zero_zero(A))
              & aa(A,$o,ord_less_eq(A,zero_zero(A)),B3) ) )
         => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),zero_zero(A)) ) ) ).

% split_mult_neg_le
tff(fact_2012_mult__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),zero_zero(A))
        <=> ( ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
              & aa(A,$o,ord_less_eq(A,B3),zero_zero(A)) )
            | ( aa(A,$o,ord_less_eq(A,A3),zero_zero(A))
              & aa(A,$o,ord_less_eq(A,zero_zero(A)),B3) ) ) ) ) ).

% mult_le_0_iff
tff(fact_2013_mult__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ) ) ).

% mult_right_mono
tff(fact_2014_mult__right__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
         => ( aa(A,$o,ord_less_eq(A,C2),zero_zero(A))
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ) ) ).

% mult_right_mono_neg
tff(fact_2015_mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) ) ) ) ).

% mult_left_mono
tff(fact_2016_mult__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),zero_zero(A))
         => ( aa(A,$o,ord_less_eq(A,B3),zero_zero(A))
           => aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) ) ) ) ).

% mult_nonpos_nonpos
tff(fact_2017_mult__left__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
         => ( aa(A,$o,ord_less_eq(A,C2),zero_zero(A))
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) ) ) ) ).

% mult_left_mono_neg
tff(fact_2018_split__mult__pos__le,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,B3: A] :
          ( ( ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
              & aa(A,$o,ord_less_eq(A,zero_zero(A)),B3) )
            | ( aa(A,$o,ord_less_eq(A,A3),zero_zero(A))
              & aa(A,$o,ord_less_eq(A,B3),zero_zero(A)) ) )
         => aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) ) ) ).

% split_mult_pos_le
tff(fact_2019_zero__le__square,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A3: A] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3)) ) ).

% zero_le_square
tff(fact_2020_mult__mono_H,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A,B3: A,C2: A,D2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,C2),D2)
           => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
             => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
               => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),D2)) ) ) ) ) ) ).

% mult_mono'
tff(fact_2021_mult__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A,B3: A,C2: A,D2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,C2),D2)
           => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B3)
             => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
               => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),D2)) ) ) ) ) ) ).

% mult_mono
tff(fact_2022_mult__neg__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),zero_zero(A))
         => ( aa(A,$o,ord_less(A,B3),zero_zero(A))
           => aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) ) ) ) ).

% mult_neg_neg
tff(fact_2023_not__square__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A3: A] : ~ aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3)),zero_zero(A)) ) ).

% not_square_less_zero
tff(fact_2024_mult__less__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),zero_zero(A))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
              & aa(A,$o,ord_less(A,B3),zero_zero(A)) )
            | ( aa(A,$o,ord_less(A,A3),zero_zero(A))
              & aa(A,$o,ord_less(A,zero_zero(A)),B3) ) ) ) ) ).

% mult_less_0_iff
tff(fact_2025_mult__neg__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),zero_zero(A))
         => ( aa(A,$o,ord_less(A,zero_zero(A)),B3)
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),zero_zero(A)) ) ) ) ).

% mult_neg_pos
tff(fact_2026_mult__pos__neg,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less(A,B3),zero_zero(A))
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),zero_zero(A)) ) ) ) ).

% mult_pos_neg
tff(fact_2027_mult__pos__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),B3)
           => aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) ) ) ) ).

% mult_pos_pos
tff(fact_2028_mult__pos__neg2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less(A,B3),zero_zero(A))
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3)),zero_zero(A)) ) ) ) ).

% mult_pos_neg2
tff(fact_2029_zero__less__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
              & aa(A,$o,ord_less(A,zero_zero(A)),B3) )
            | ( aa(A,$o,ord_less(A,A3),zero_zero(A))
              & aa(A,$o,ord_less(A,B3),zero_zero(A)) ) ) ) ) ).

% zero_less_mult_iff
tff(fact_2030_zero__less__mult__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))
         => ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
           => aa(A,$o,ord_less(A,zero_zero(A)),B3) ) ) ) ).

% zero_less_mult_pos
tff(fact_2031_zero__less__mult__pos2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3))
         => ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
           => aa(A,$o,ord_less(A,zero_zero(A)),B3) ) ) ) ).

% zero_less_mult_pos2
tff(fact_2032_mult__less__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,C2),zero_zero(A))
         => ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
          <=> aa(A,$o,ord_less(A,B3),A3) ) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2033_mult__less__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
         => ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
          <=> aa(A,$o,ord_less(A,A3),B3) ) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2034_mult__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,ord_less(A,B3),A3)
         => ( aa(A,$o,ord_less(A,C2),zero_zero(A))
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) ) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2035_mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) ) ) ) ).

% mult_strict_left_mono
tff(fact_2036_mult__less__cancel__left__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
              & aa(A,$o,ord_less(A,A3),B3) )
            | ( aa(A,$o,ord_less(A,C2),zero_zero(A))
              & aa(A,$o,ord_less(A,B3),A3) ) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2037_mult__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,ord_less(A,B3),A3)
         => ( aa(A,$o,ord_less(A,C2),zero_zero(A))
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2038_mult__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ) ) ).

% mult_strict_right_mono
tff(fact_2039_mult__less__cancel__right__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
              & aa(A,$o,ord_less(A,A3),B3) )
            | ( aa(A,$o,ord_less(A,C2),zero_zero(A))
              & aa(A,$o,ord_less(A,B3),A3) ) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2040_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord2810124833399127020strict(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2041_mult__numeral__1,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),one2)),A3) = A3 ) ).

% mult_numeral_1
tff(fact_2042_mult__numeral__1__right,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),one2)) = A3 ) ).

% mult_numeral_1_right
tff(fact_2043_less__1__mult,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: A,Na: A] :
          ( aa(A,$o,ord_less(A,one_one(A)),M)
         => ( aa(A,$o,ord_less(A,one_one(A)),Na)
           => aa(A,$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_2044_frac__eq__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,X: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z) )
            <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),X),Z) = aa(A,A,aa(A,fun(A,A),times_times(A),W),Y) ) ) ) ) ) ).

% frac_eq_eq
tff(fact_2045_divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B3: A,C2: A,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2) = A3 )
        <=> $ite(C2 != zero_zero(A),B3 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),A3 = zero_zero(A)) ) ) ).

% divide_eq_eq
tff(fact_2046_eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A,C2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2) )
        <=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = B3,A3 = zero_zero(A)) ) ) ).

% eq_divide_eq
tff(fact_2047_divide__eq__imp,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,B3: A,A3: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( B3 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2) = A3 ) ) ) ) ).

% divide_eq_imp
tff(fact_2048_eq__divide__imp,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,A3: A,B3: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = B3 )
           => ( A3 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2) ) ) ) ) ).

% eq_divide_imp
tff(fact_2049_nonzero__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,B3: A,A3: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2) = A3 )
          <=> ( B3 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) ) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_2050_nonzero__eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,A3: A,B3: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( A3 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = B3 ) ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_2051_square__diff__square__factored,axiom,
    ! [A: $tType] :
      ( comm_ring(A)
     => ! [X: A,Y: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,plus_plus(A,X),Y)),aa(A,A,minus_minus(A,X),Y)) ) ).

% square_diff_square_factored
tff(fact_2052_eq__add__iff2,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,E2: A,C2: A,B3: A,D2: A] :
          ( ( aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),E2)),C2) = aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),E2)),D2) )
        <=> ( C2 = aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,B3),A3)),E2)),D2) ) ) ) ).

% eq_add_iff2
tff(fact_2053_eq__add__iff1,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,E2: A,C2: A,B3: A,D2: A] :
          ( ( aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),E2)),C2) = aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),E2)),D2) )
        <=> ( aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,A3),B3)),E2)),C2) = D2 ) ) ) ).

% eq_add_iff1
tff(fact_2054_mult__diff__mult,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [X: A,Y: A,A3: A,B3: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) = aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(A,A,minus_minus(A,Y),B3))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,X),A3)),B3)) ) ).

% mult_diff_mult
tff(fact_2055_numeral__times__minus__swap,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [W: num,X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),aa(A,A,uminus_uminus(A),X)) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) ) ).

% numeral_times_minus_swap
tff(fact_2056_left__right__inverse__power,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [X: A,Y: A,Na: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),X),Y) = one_one(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,X),Na)),aa(nat,A,power_power(A,Y),Na)) = one_one(A) ) ) ) ).

% left_right_inverse_power
tff(fact_2057_square__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [X: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),X),X) = one_one(A) )
        <=> ( ( X = one_one(A) )
            | ( X = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).

% square_eq_1_iff
tff(fact_2058_is__unit__mult__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),one_one(A))
        <=> ( aa(A,$o,dvd_dvd(A,A3),one_one(A))
            & aa(A,$o,dvd_dvd(A,B3),one_one(A)) ) ) ) ).

% is_unit_mult_iff
tff(fact_2059_dvd__mult__unit__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B3),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
          <=> aa(A,$o,dvd_dvd(A,A3),C2) ) ) ) ).

% dvd_mult_unit_iff
tff(fact_2060_mult__unit__dvd__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B3),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),C2)
          <=> aa(A,$o,dvd_dvd(A,A3),C2) ) ) ) ).

% mult_unit_dvd_iff
tff(fact_2061_dvd__mult__unit__iff_H,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B3),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))
          <=> aa(A,$o,dvd_dvd(A,A3),C2) ) ) ) ).

% dvd_mult_unit_iff'
tff(fact_2062_mult__unit__dvd__iff_H,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A3),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),C2)
          <=> aa(A,$o,dvd_dvd(A,B3),C2) ) ) ) ).

% mult_unit_dvd_iff'
tff(fact_2063_unit__mult__left__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A3),one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) )
          <=> ( B3 = C2 ) ) ) ) ).

% unit_mult_left_cancel
tff(fact_2064_unit__mult__right__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A3),one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3) )
          <=> ( B3 = C2 ) ) ) ) ).

% unit_mult_right_cancel
tff(fact_2065_power__add,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,M: nat,Na: nat] : aa(nat,A,power_power(A,A3),aa(nat,nat,plus_plus(nat,M),Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,A3),M)),aa(nat,A,power_power(A,A3),Na)) ) ).

% power_add
tff(fact_2066_div__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A,M: nat,Na: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Na))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,semiring_1_of_nat(A),M))),aa(nat,A,semiring_1_of_nat(A),Na)) ) ).

% div_mult2_eq'
tff(fact_2067_dvd__div__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,dvd_dvd(A,C2),B3)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3)),C2) ) ) ) ).

% dvd_div_mult
tff(fact_2068_div__mult__swap,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,dvd_dvd(A,C2),B3)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),C2) ) ) ) ).

% div_mult_swap
tff(fact_2069_div__div__eq__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,dvd_dvd(A,C2),B3)
         => ( aa(A,$o,dvd_dvd(A,B3),A3)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),C2) ) ) ) ) ).

% div_div_eq_right
tff(fact_2070_dvd__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,C2: A,A3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)),A3)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),C2) ) ) ) ).

% dvd_div_mult2_eq
tff(fact_2071_dvd__mult__imp__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B3)
         => aa(A,$o,dvd_dvd(A,A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)) ) ) ).

% dvd_mult_imp_div
tff(fact_2072_div__mult__div__if__dvd,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A,D2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B3),A3)
         => ( aa(A,$o,dvd_dvd(A,D2),C2)
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),D2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),D2)) ) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_2073_real__minus__mult__self__le,axiom,
    ! [U: real,X: real] : aa(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),X),X)) ).

% real_minus_mult_self_le
tff(fact_2074_ln__powr,axiom,
    ! [X: real,Y: real] :
      ( ( X != zero_zero(real) )
     => ( aa(real,real,ln_ln(real),powr(real,X,Y)) = aa(real,real,aa(real,fun(real,real),times_times(real),Y),aa(real,real,ln_ln(real),X)) ) ) ).

% ln_powr
tff(fact_2075_minus__power__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [A3: A,Na: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),A3)),Na)),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),A3)),Na)) = aa(nat,A,power_power(A,A3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ) ).

% minus_power_mult_self
tff(fact_2076_tanh__real__lt__1,axiom,
    ! [X: real] : aa(real,$o,ord_less(real,aa(real,real,tanh(real),X)),one_one(real)) ).

% tanh_real_lt_1
tff(fact_2077_round__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,X),Y)
         => aa(int,$o,ord_less_eq(int,archimedean_round(A,X)),archimedean_round(A,Y)) ) ) ).

% round_mono
tff(fact_2078_mult__le__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
             => aa(A,$o,ord_less_eq(A,A3),B3) )
            & ( aa(A,$o,ord_less(A,C2),zero_zero(A))
             => aa(A,$o,ord_less_eq(A,B3),A3) ) ) ) ) ).

% mult_le_cancel_left
tff(fact_2079_mult__le__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
             => aa(A,$o,ord_less_eq(A,A3),B3) )
            & ( aa(A,$o,ord_less(A,C2),zero_zero(A))
             => aa(A,$o,ord_less_eq(A,B3),A3) ) ) ) ) ).

% mult_le_cancel_right
tff(fact_2080_mult__left__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
           => aa(A,$o,ord_less(A,A3),B3) ) ) ) ).

% mult_left_less_imp_less
tff(fact_2081_mult__strict__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A,C2: A,D2: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(A,$o,ord_less(A,C2),D2)
           => ( aa(A,$o,ord_less(A,zero_zero(A)),B3)
             => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
               => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),D2)) ) ) ) ) ) ).

% mult_strict_mono
tff(fact_2082_mult__less__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
        <=> ( ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
             => aa(A,$o,ord_less(A,A3),B3) )
            & ( aa(A,$o,ord_less_eq(A,C2),zero_zero(A))
             => aa(A,$o,ord_less(A,B3),A3) ) ) ) ) ).

% mult_less_cancel_left
tff(fact_2083_mult__right__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
           => aa(A,$o,ord_less(A,A3),B3) ) ) ) ).

% mult_right_less_imp_less
tff(fact_2084_mult__strict__mono_H,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A,C2: A,D2: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(A,$o,ord_less(A,C2),D2)
           => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
             => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
               => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),D2)) ) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2085_mult__less__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))
        <=> ( ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
             => aa(A,$o,ord_less(A,A3),B3) )
            & ( aa(A,$o,ord_less_eq(A,C2),zero_zero(A))
             => aa(A,$o,ord_less(A,B3),A3) ) ) ) ) ).

% mult_less_cancel_right
tff(fact_2086_mult__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,C2),zero_zero(A))
         => ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
          <=> aa(A,$o,ord_less_eq(A,B3),A3) ) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2087_mult__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
         => ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
          <=> aa(A,$o,ord_less_eq(A,A3),B3) ) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2088_mult__left__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
         => ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
           => aa(A,$o,ord_less_eq(A,A3),B3) ) ) ) ).

% mult_left_le_imp_le
tff(fact_2089_mult__right__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))
         => ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
           => aa(A,$o,ord_less_eq(A,A3),B3) ) ) ) ).

% mult_right_le_imp_le
tff(fact_2090_mult__le__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A,C2: A,D2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less(A,C2),D2)
           => ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
             => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
               => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),D2)) ) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2091_mult__less__le__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A,C2: A,D2: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,C2),D2)
           => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
             => ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
               => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),D2)) ) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2092_sum__squares__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A))
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2093_sum__squares__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [X: A,Y: A] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))) ) ).

% sum_squares_ge_zero
tff(fact_2094_mult__left__le,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [C2: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,C2),one_one(A))
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),A3) ) ) ) ).

% mult_left_le
tff(fact_2095_mult__le__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),one_one(A))
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B3)
           => ( aa(A,$o,ord_less_eq(A,B3),one_one(A))
             => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),one_one(A)) ) ) ) ) ).

% mult_le_one
tff(fact_2096_mult__right__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),X)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Y)
           => ( aa(A,$o,ord_less_eq(A,Y),one_one(A))
             => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Y)),X) ) ) ) ) ).

% mult_right_le_one_le
tff(fact_2097_mult__left__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),X)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Y)
           => ( aa(A,$o,ord_less_eq(A,Y),one_one(A))
             => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),Y),X)),X) ) ) ) ) ).

% mult_left_le_one_le
tff(fact_2098_not__sum__squares__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [X: A,Y: A] : ~ aa(A,$o,ord_less(A,aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A)) ) ).

% not_sum_squares_lt_zero
tff(fact_2099_sum__squares__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)))
        <=> ( ( X != zero_zero(A) )
            | ( Y != zero_zero(A) ) ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2100_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),C2) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_2101_divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,C2: A,A3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)),A3)
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,ord_less(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B3),aa(A,$o,ord_less(A,zero_zero(A)),A3)) ) ) ) ).

% divide_less_eq
tff(fact_2102_less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less(A,A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2))
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B3),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,ord_less(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,$o,ord_less(A,A3),zero_zero(A))) ) ) ) ).

% less_divide_eq
tff(fact_2103_neg__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,C2),zero_zero(A))
         => ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)),A3)
          <=> aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B3) ) ) ) ).

% neg_divide_less_eq
tff(fact_2104_neg__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,C2),zero_zero(A))
         => ( aa(A,$o,ord_less(A,A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2))
          <=> aa(A,$o,ord_less(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).

% neg_less_divide_eq
tff(fact_2105_pos__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
         => ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)),A3)
          <=> aa(A,$o,ord_less(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).

% pos_divide_less_eq
tff(fact_2106_pos__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
         => ( aa(A,$o,ord_less(A,A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2))
          <=> aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B3) ) ) ) ).

% pos_less_divide_eq
tff(fact_2107_mult__imp__div__pos__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,X: A,Z: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Y)
         => ( aa(A,$o,ord_less(A,X),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),Z) ) ) ) ).

% mult_imp_div_pos_less
tff(fact_2108_mult__imp__less__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,X: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Y)
         => ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)),X)
           => aa(A,$o,ord_less(A,Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)) ) ) ) ).

% mult_imp_less_div_pos
tff(fact_2109_divide__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,ord_less(A,B3),A3)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
           => ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))
             => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B3)) ) ) ) ) ).

% divide_strict_left_mono
tff(fact_2110_divide__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(A,$o,ord_less(A,C2),zero_zero(A))
           => ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))
             => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B3)) ) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_2111_divide__eq__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B3: A,C2: A,W: num] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2) = aa(num,A,numeral_numeral(A),W) )
        <=> $ite(C2 != zero_zero(A),B3 = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2),aa(num,A,numeral_numeral(A),W) = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_2112_eq__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W: num,B3: A,C2: A] :
          ( ( aa(num,A,numeral_numeral(A),W) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2) )
        <=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2) = B3,aa(num,A,numeral_numeral(A),W) = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_2113_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E2: A,C2: A,B3: A,D2: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),E2)),C2)),aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),E2)),D2))
        <=> aa(A,$o,ord_less_eq(A,aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,A3),B3)),E2)),C2)),D2) ) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_2114_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E2: A,C2: A,B3: A,D2: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),E2)),C2)),aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),E2)),D2))
        <=> aa(A,$o,ord_less_eq(A,C2),aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,B3),A3)),E2)),D2)) ) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_2115_divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z)),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,plus_plus(A,X),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),Z) ) ) ) ).

% divide_add_eq_iff
tff(fact_2116_add__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,plus_plus(A,X),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,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)),Y)),Z) ) ) ) ).

% add_divide_eq_iff
tff(fact_2117_add__num__frac,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,X: A] :
          ( ( Y != zero_zero(A) )
         => ( aa(A,A,plus_plus(A,Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,plus_plus(A,X),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))),Y) ) ) ) ).

% add_num_frac
tff(fact_2118_add__frac__num,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,X: A,Z: A] :
          ( ( Y != zero_zero(A) )
         => ( aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),Z) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,plus_plus(A,X),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))),Y) ) ) ) ).

% add_frac_num
tff(fact_2119_add__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,X: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)) ) ) ) ) ).

% add_frac_eq
tff(fact_2120_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A,Z: A] :
          aa(A,A,plus_plus(A,A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),Z)) = $ite(Z = zero_zero(A),A3,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),Z)),B3)),Z)) ) ).

% add_divide_eq_if_simps(1)
tff(fact_2121_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,Z: A,B3: A] :
          aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),Z)),B3) = $ite(Z = zero_zero(A),B3,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,plus_plus(A,A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),Z))),Z)) ) ).

% add_divide_eq_if_simps(2)
tff(fact_2122_less__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E2: A,C2: A,B3: A,D2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),E2)),C2)),aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),E2)),D2))
        <=> aa(A,$o,ord_less(A,aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,A3),B3)),E2)),C2)),D2) ) ) ).

% less_add_iff1
tff(fact_2123_less__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E2: A,C2: A,B3: A,D2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),E2)),C2)),aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),E2)),D2))
        <=> aa(A,$o,ord_less(A,C2),aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,B3),A3)),E2)),D2)) ) ) ).

% less_add_iff2
tff(fact_2124_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A,Z: A] :
          aa(A,A,minus_minus(A,A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),Z)) = $ite(Z = zero_zero(A),A3,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),Z)),B3)),Z)) ) ).

% add_divide_eq_if_simps(4)
tff(fact_2125_diff__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,X: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)) ) ) ) ) ).

% diff_frac_eq
tff(fact_2126_diff__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,minus_minus(A,X),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,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)),Y)),Z) ) ) ) ).

% diff_divide_eq_iff
tff(fact_2127_divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z)),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,X),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),Z) ) ) ) ).

% divide_diff_eq_iff
tff(fact_2128_square__diff__one__factored,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),one_one(A)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,plus_plus(A,X),one_one(A))),aa(A,A,minus_minus(A,X),one_one(A))) ) ).

% square_diff_one_factored
tff(fact_2129_mult__1s__ring__1_I2_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [B3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),B3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),one2))) = aa(A,A,uminus_uminus(A),B3) ) ).

% mult_1s_ring_1(2)
tff(fact_2130_mult__1s__ring__1_I1_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [B3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),one2))),B3) = aa(A,A,uminus_uminus(A),B3) ) ).

% mult_1s_ring_1(1)
tff(fact_2131_power__gt1__lemma,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,ord_less(A,one_one(A)),A3)
         => aa(A,$o,ord_less(A,one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,power_power(A,A3),Na))) ) ) ).

% power_gt1_lemma
tff(fact_2132_power__less__power__Suc,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,ord_less(A,one_one(A)),A3)
         => aa(A,$o,ord_less(A,aa(nat,A,power_power(A,A3),Na)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,power_power(A,A3),Na))) ) ) ).

% power_less_power_Suc
tff(fact_2133_ex__less__of__nat__mult,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),X)
         => ? [N: nat] : aa(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)),X)) ) ) ).

% ex_less_of_nat_mult
tff(fact_2134_nonzero__neg__divide__eq__eq2,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B3: A,C2: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( ( C2 = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3) = aa(A,A,uminus_uminus(A),A3) ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_2135_nonzero__neg__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B3: A,A3: A,C2: A] :
          ( ( B3 != zero_zero(A) )
         => ( ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)) = C2 )
          <=> ( aa(A,A,uminus_uminus(A),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3) ) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_2136_minus__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B3: A,C2: A,A3: A] :
          ( ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)) = A3 )
        <=> $ite(C2 != zero_zero(A),aa(A,A,uminus_uminus(A),B3) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),A3 = zero_zero(A)) ) ) ).

% minus_divide_eq_eq
tff(fact_2137_eq__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A,C2: A] :
          ( ( A3 = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)) )
        <=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = aa(A,A,uminus_uminus(A),B3),A3 = zero_zero(A)) ) ) ).

% eq_minus_divide_eq
tff(fact_2138_unity__coeff__ex,axiom,
    ! [A: $tType] :
      ( ( dvd(A)
        & semiring_0(A) )
     => ! [P: fun(A,$o),L: A] :
          ( ? [X3: A] : aa(A,$o,P,aa(A,A,aa(A,fun(A,A),times_times(A),L),X3))
        <=> ? [X3: A] :
              ( aa(A,$o,dvd_dvd(A,L),aa(A,A,plus_plus(A,X3),zero_zero(A)))
              & aa(A,$o,P,X3) ) ) ) ).

% unity_coeff_ex
tff(fact_2139_unit__dvdE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,A3),one_one(A))
         => ~ ( ( A3 != zero_zero(A) )
             => ! [C5: A] : B3 != aa(A,A,aa(A,fun(A,A),times_times(A),A3),C5) ) ) ) ).

% unit_dvdE
tff(fact_2140_power__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A,Na: nat] : aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),A3)),Na) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),Na)),aa(nat,A,power_power(A,A3),Na)) ) ).

% power_minus
tff(fact_2141_dvd__div__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,C2: A,B3: A,D2: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( C2 != zero_zero(A) )
           => ( aa(A,$o,dvd_dvd(A,A3),B3)
             => ( aa(A,$o,dvd_dvd(A,C2),D2)
               => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),D2),C2) )
                <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),D2) ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_2142_dvd__div__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B3: A,A3: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,C2),B3)
           => ( aa(A,$o,dvd_dvd(A,A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2))
            <=> aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B3) ) ) ) ) ).

% dvd_div_iff_mult
tff(fact_2143_div__dvd__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A,C2: A] :
          ( ( B3 != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,B3),A3)
           => ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),C2)
            <=> aa(A,$o,dvd_dvd(A,A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) ) ) ) ) ).

% div_dvd_iff_mult
tff(fact_2144_dvd__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A,C2: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,A3),B3)
           => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3) = C2 )
            <=> ( B3 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3) ) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_2145_is__unit__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,C2: A,A3: A] :
          ( aa(A,$o,dvd_dvd(A,B3),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,C2),one_one(A))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),C2) ) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_2146_unit__div__mult__swap,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,C2),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),C2) ) ) ) ).

% unit_div_mult_swap
tff(fact_2147_unit__div__commute,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B3),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B3) ) ) ) ).

% unit_div_commute
tff(fact_2148_div__mult__unit2,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,dvd_dvd(A,C2),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,B3),A3)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),C2) ) ) ) ) ).

% div_mult_unit2
tff(fact_2149_unit__eq__div2,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B3),one_one(A))
         => ( ( A3 = aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B3) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3) = C2 ) ) ) ) ).

% unit_eq_div2
tff(fact_2150_unit__eq__div1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B3),one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) = C2 )
          <=> ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3) ) ) ) ) ).

% unit_eq_div1
tff(fact_2151_floor__le__round,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(int,$o,ord_less_eq(int,archim6421214686448440834_floor(A,X)),archimedean_round(A,X)) ) ).

% floor_le_round
tff(fact_2152_ceiling__ge__round,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(int,$o,ord_less_eq(int,archimedean_round(A,X)),archimedean_ceiling(A,X)) ) ).

% ceiling_ge_round
tff(fact_2153_ln__mult,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ( aa(real,$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),X),Y)) = aa(real,real,plus_plus(real,aa(real,real,ln_ln(real),X)),aa(real,real,ln_ln(real),Y)) ) ) ) ).

% ln_mult
tff(fact_2154_reals__Archimedean3,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ! [Y4: real] :
        ? [N: nat] : aa(real,$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)),X)) ) ).

% reals_Archimedean3
tff(fact_2155_powr__mult,axiom,
    ! [X: real,Y: real,A3: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
       => ( powr(real,aa(real,real,aa(real,fun(real,real),times_times(real),X),Y),A3) = aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,X,A3)),powr(real,Y,A3)) ) ) ) ).

% powr_mult
tff(fact_2156_log__powr,axiom,
    ! [X: real,B3: real,Y: real] :
      ( ( X != zero_zero(real) )
     => ( aa(real,real,log(B3),powr(real,X,Y)) = aa(real,real,aa(real,fun(real,real),times_times(real),Y),aa(real,real,log(B3),X)) ) ) ).

% log_powr
tff(fact_2157_power__numeral__even,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Z: A,W: num] :
          aa(nat,A,power_power(A,Z),aa(num,nat,numeral_numeral(nat),bit0(W))) = $let(
            w: A,
            w:= aa(nat,A,power_power(A,Z),aa(num,nat,numeral_numeral(nat),W)),
            aa(A,A,aa(A,fun(A,A),times_times(A),w),w) ) ) ).

% power_numeral_even
tff(fact_2158_tanh__real__gt__neg1,axiom,
    ! [X: real] : aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),aa(real,real,tanh(real),X)) ).

% tanh_real_gt_neg1
tff(fact_2159_mult__le__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
             => aa(A,$o,ord_less_eq(A,one_one(A)),B3) )
            & ( aa(A,$o,ord_less(A,C2),zero_zero(A))
             => aa(A,$o,ord_less_eq(A,B3),one_one(A)) ) ) ) ) ).

% mult_le_cancel_left1
tff(fact_2160_mult__le__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),C2)
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
             => aa(A,$o,ord_less_eq(A,A3),one_one(A)) )
            & ( aa(A,$o,ord_less(A,C2),zero_zero(A))
             => aa(A,$o,ord_less_eq(A,one_one(A)),A3) ) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2161_mult__le__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,C2),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
             => aa(A,$o,ord_less_eq(A,one_one(A)),B3) )
            & ( aa(A,$o,ord_less(A,C2),zero_zero(A))
             => aa(A,$o,ord_less_eq(A,B3),one_one(A)) ) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2162_mult__le__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),C2)
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
             => aa(A,$o,ord_less_eq(A,A3),one_one(A)) )
            & ( aa(A,$o,ord_less(A,C2),zero_zero(A))
             => aa(A,$o,ord_less_eq(A,one_one(A)),A3) ) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2163_mult__less__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B3: A] :
          ( aa(A,$o,ord_less(A,C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
        <=> ( ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
             => aa(A,$o,ord_less(A,one_one(A)),B3) )
            & ( aa(A,$o,ord_less_eq(A,C2),zero_zero(A))
             => aa(A,$o,ord_less(A,B3),one_one(A)) ) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2164_mult__less__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),C2)
        <=> ( ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
             => aa(A,$o,ord_less(A,A3),one_one(A)) )
            & ( aa(A,$o,ord_less_eq(A,C2),zero_zero(A))
             => aa(A,$o,ord_less(A,one_one(A)),A3) ) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2165_mult__less__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B3: A] :
          ( aa(A,$o,ord_less(A,C2),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))
        <=> ( ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
             => aa(A,$o,ord_less(A,one_one(A)),B3) )
            & ( aa(A,$o,ord_less_eq(A,C2),zero_zero(A))
             => aa(A,$o,ord_less(A,B3),one_one(A)) ) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2166_mult__less__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,C2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),C2)
        <=> ( ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
             => aa(A,$o,ord_less(A,A3),one_one(A)) )
            & ( aa(A,$o,ord_less_eq(A,C2),zero_zero(A))
             => aa(A,$o,ord_less(A,one_one(A)),A3) ) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2167_field__le__mult__one__interval,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( ! [Z2: A] :
              ( aa(A,$o,ord_less(A,zero_zero(A)),Z2)
             => ( aa(A,$o,ord_less(A,Z2),one_one(A))
               => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),Z2),X)),Y) ) )
         => aa(A,$o,ord_less_eq(A,X),Y) ) ) ).

% field_le_mult_one_interval
tff(fact_2168_convex__bound__le,axiom,
    ! [A: $tType] :
      ( linord6961819062388156250ring_1(A)
     => ! [X: A,A3: A,Y: A,U: A,V2: A] :
          ( aa(A,$o,ord_less_eq(A,X),A3)
         => ( aa(A,$o,ord_less_eq(A,Y),A3)
           => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),U)
             => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),V2)
               => ( ( aa(A,A,plus_plus(A,U),V2) = one_one(A) )
                 => aa(A,$o,ord_less_eq(A,aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),U),X)),aa(A,A,aa(A,fun(A,A),times_times(A),V2),Y))),A3) ) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2169_divide__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,C2),zero_zero(A))
           => ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))
             => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B3)) ) ) ) ) ).

% divide_left_mono_neg
tff(fact_2170_mult__imp__le__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,X: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Y)
         => ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)),X)
           => aa(A,$o,ord_less_eq(A,Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)) ) ) ) ).

% mult_imp_le_div_pos
tff(fact_2171_mult__imp__div__pos__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,X: A,Z: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Y)
         => ( aa(A,$o,ord_less_eq(A,X),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),Z) ) ) ) ).

% mult_imp_div_pos_le
tff(fact_2172_pos__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
         => ( aa(A,$o,ord_less_eq(A,A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2))
          <=> aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B3) ) ) ) ).

% pos_le_divide_eq
tff(fact_2173_pos__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
         => ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)),A3)
          <=> aa(A,$o,ord_less_eq(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).

% pos_divide_le_eq
tff(fact_2174_neg__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,C2),zero_zero(A))
         => ( aa(A,$o,ord_less_eq(A,A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2))
          <=> aa(A,$o,ord_less_eq(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).

% neg_le_divide_eq
tff(fact_2175_neg__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,C2),zero_zero(A))
         => ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)),A3)
          <=> aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B3) ) ) ) ).

% neg_divide_le_eq
tff(fact_2176_divide__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
           => ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))
             => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B3)) ) ) ) ) ).

% divide_left_mono
tff(fact_2177_le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2))
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B3),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,ord_less_eq(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,$o,ord_less_eq(A,A3),zero_zero(A))) ) ) ) ).

% le_divide_eq
tff(fact_2178_divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,C2: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)),A3)
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,ord_less_eq(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B3),aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)) ) ) ) ).

% divide_le_eq
tff(fact_2179_divide__less__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,C2: A,W: num] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)),aa(num,A,numeral_numeral(A),W))
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,ord_less(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),B3),aa(A,$o,ord_less(A,zero_zero(A)),aa(num,A,numeral_numeral(A),W))) ) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_2180_less__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B3: A,C2: A] :
          ( aa(A,$o,ord_less(A,aa(num,A,numeral_numeral(A),W)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2))
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),B3),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,ord_less(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),aa(A,$o,ord_less(A,aa(num,A,numeral_numeral(A),W)),zero_zero(A))) ) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_2181_mult__2,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Z) = aa(A,A,plus_plus(A,Z),Z) ) ).

% mult_2
tff(fact_2182_mult__2__right,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),Z),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,plus_plus(A,Z),Z) ) ).

% mult_2_right
tff(fact_2183_left__add__twice,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [A3: A,B3: A] : aa(A,A,plus_plus(A,A3),aa(A,A,plus_plus(A,A3),B3)) = aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)),B3) ) ).

% left_add_twice
tff(fact_2184_frac__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,X: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z))
            <=> aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),zero_zero(A)) ) ) ) ) ).

% frac_le_eq
tff(fact_2185_power__Suc__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less(A,A3),one_one(A))
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,power_power(A,A3),Na))),aa(nat,A,power_power(A,A3),Na)) ) ) ) ).

% power_Suc_less
tff(fact_2186_frac__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,X: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z))
            <=> aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),zero_zero(A)) ) ) ) ) ).

% frac_less_eq
tff(fact_2187_less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less(A,A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)))
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B3)),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,$o,ord_less(A,A3),zero_zero(A))) ) ) ) ).

% less_minus_divide_eq
tff(fact_2188_minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,C2: A,A3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2))),A3)
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B3)),aa(A,$o,ord_less(A,zero_zero(A)),A3)) ) ) ) ).

% minus_divide_less_eq
tff(fact_2189_neg__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,C2),zero_zero(A))
         => ( aa(A,$o,ord_less(A,A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)))
          <=> aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).

% neg_less_minus_divide_eq
tff(fact_2190_neg__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,C2),zero_zero(A))
         => ( aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2))),A3)
          <=> aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B3)) ) ) ) ).

% neg_minus_divide_less_eq
tff(fact_2191_pos__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
         => ( aa(A,$o,ord_less(A,A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)))
          <=> aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B3)) ) ) ) ).

% pos_less_minus_divide_eq
tff(fact_2192_pos__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
         => ( aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2))),A3)
          <=> aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).

% pos_minus_divide_less_eq
tff(fact_2193_divide__eq__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B3: A,C2: A,W: num] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) )
        <=> $ite(C2 != zero_zero(A),B3 = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_2194_eq__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W: num,B3: A,C2: A] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2) )
        <=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2) = B3,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_2195_minus__divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,plus_plus(A,aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z))),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,plus_plus(A,aa(A,A,uminus_uminus(A),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),Z) ) ) ) ).

% minus_divide_add_eq_iff
tff(fact_2196_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,Z: A,B3: A] :
          aa(A,A,plus_plus(A,aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),Z))),B3) = $ite(Z = zero_zero(A),B3,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,plus_plus(A,aa(A,A,uminus_uminus(A),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),Z))),Z)) ) ).

% add_divide_eq_if_simps(3)
tff(fact_2197_evenE,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
         => ~ ! [B5: A] : A3 != aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B5) ) ) ).

% evenE
tff(fact_2198_power4__eq__xxxx,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [X: A] : aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),X)),X) ) ).

% power4_eq_xxxx
tff(fact_2199_power2__eq__square,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A] : aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3) ) ).

% power2_eq_square
tff(fact_2200_minus__divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z))),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),Z) ) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_2201_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,Z: A,B3: A] :
          aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),Z)),B3) = $ite(Z = zero_zero(A),aa(A,A,uminus_uminus(A),B3),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),Z))),Z)) ) ).

% add_divide_eq_if_simps(5)
tff(fact_2202_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,Z: A,B3: A] :
          aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),Z))),B3) = $ite(Z = zero_zero(A),aa(A,A,uminus_uminus(A),B3),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),Z))),Z)) ) ).

% add_divide_eq_if_simps(6)
tff(fact_2203_is__unitE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A3),one_one(A))
         => ~ ( ( A3 != zero_zero(A) )
             => ! [B5: A] :
                  ( ( B5 != zero_zero(A) )
                 => ( aa(A,$o,dvd_dvd(A,B5),one_one(A))
                   => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3) = B5 )
                     => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B5) = A3 )
                       => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B5) = one_one(A) )
                         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3) != aa(A,A,aa(A,fun(A,A),times_times(A),C2),B5) ) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_2204_is__unit__div__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,B3),one_one(A))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B3) ) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_2205_is__unit__div__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,B3),one_one(A))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B3) ) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_2206_ln__realpow,axiom,
    ! [X: real,Na: nat] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ( aa(real,real,ln_ln(real),aa(nat,real,power_power(real,X),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),X)) ) ) ).

% ln_realpow
tff(fact_2207_linear__plus__1__le__power,axiom,
    ! [X: real,Na: nat] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => aa(real,$o,ord_less_eq(real,aa(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)),X)),one_one(real))),aa(nat,real,power_power(real,aa(real,real,plus_plus(real,X),one_one(real))),Na)) ) ).

% linear_plus_1_le_power
tff(fact_2208_convex__bound__lt,axiom,
    ! [A: $tType] :
      ( linord715952674999750819strict(A)
     => ! [X: A,A3: A,Y: A,U: A,V2: A] :
          ( aa(A,$o,ord_less(A,X),A3)
         => ( aa(A,$o,ord_less(A,Y),A3)
           => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),U)
             => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),V2)
               => ( ( aa(A,A,plus_plus(A,U),V2) = one_one(A) )
                 => aa(A,$o,ord_less(A,aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),U),X)),aa(A,A,aa(A,fun(A,A),times_times(A),V2),Y))),A3) ) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2209_le__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),W)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2))
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),B3),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,ord_less_eq(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),W)),zero_zero(A))) ) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_2210_divide__le__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,C2: A,W: num] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)),aa(num,A,numeral_numeral(A),W))
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,ord_less_eq(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),B3),aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(num,A,numeral_numeral(A),W))) ) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_2211_scaling__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,V2: A,R3: A,Sa: A] :
          ( aa(A,$o,ord_less_eq(A,U),V2)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),R3)
           => ( aa(A,$o,ord_less_eq(A,R3),Sa)
             => aa(A,$o,ord_less_eq(A,aa(A,A,plus_plus(A,U),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),R3),aa(A,A,minus_minus(A,V2),U))),Sa))),V2) ) ) ) ) ).

% scaling_mono
tff(fact_2212_le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)))
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B3)),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,$o,ord_less_eq(A,A3),zero_zero(A))) ) ) ) ).

% le_minus_divide_eq
tff(fact_2213_minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,C2: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2))),A3)
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B3)),aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)) ) ) ) ).

% minus_divide_le_eq
tff(fact_2214_neg__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,C2),zero_zero(A))
         => ( aa(A,$o,ord_less_eq(A,A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)))
          <=> aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).

% neg_le_minus_divide_eq
tff(fact_2215_neg__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,C2),zero_zero(A))
         => ( aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2))),A3)
          <=> aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B3)) ) ) ) ).

% neg_minus_divide_le_eq
tff(fact_2216_pos__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
         => ( aa(A,$o,ord_less_eq(A,A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)))
          <=> aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B3)) ) ) ) ).

% pos_le_minus_divide_eq
tff(fact_2217_pos__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
         => ( aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2))),A3)
          <=> aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).

% pos_minus_divide_le_eq
tff(fact_2218_less__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B3: A,C2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2))
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(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),W))),C2)),B3),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,ord_less(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),zero_zero(A))) ) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_2219_divide__less__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,C2: A,W: num] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,ord_less(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),B3),aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))) ) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_2220_even__two__times__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))) = A3 ) ) ) ).

% even_two_times_div_two
tff(fact_2221_power__eq__if,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [P2: A,M: nat] :
          aa(nat,A,power_power(A,P2),M) = $ite(M = zero_zero(nat),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),P2),aa(nat,A,power_power(A,P2),aa(nat,nat,minus_minus(nat,M),one_one(nat))))) ) ).

% power_eq_if
tff(fact_2222_power__minus__mult,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Na: nat,A3: A] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,A3),aa(nat,nat,minus_minus(nat,Na),one_one(nat)))),A3) = aa(nat,A,power_power(A,A3),Na) ) ) ) ).

% power_minus_mult
tff(fact_2223_four__x__squared,axiom,
    ! [X: real] : aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),X)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% four_x_squared
tff(fact_2224_real__archimedian__rdiv__eq__0,axiom,
    ! [X: real,C2: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),C2)
       => ( ! [M4: nat] :
              ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M4)
             => aa(real,$o,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M4)),X)),C2) )
         => ( X = zero_zero(real) ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_2225_ln__powr__bound2,axiom,
    ! [X: real,A3: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),X)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
       => aa(real,$o,ord_less_eq(real,powr(real,aa(real,real,ln_ln(real),X),A3)),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,A3,A3)),X)) ) ) ).

% ln_powr_bound2
tff(fact_2226_shiftl__eq__mult,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [X: A,Na: nat] : bit_Sh4282982442137083160shiftl(A,X,Na) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) ) ).

% shiftl_eq_mult
tff(fact_2227_log__eq__div__ln__mult__log,axiom,
    ! [A3: real,B3: real,X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
     => ( ( A3 != one_one(real) )
       => ( aa(real,$o,ord_less(real,zero_zero(real)),B3)
         => ( ( B3 != one_one(real) )
           => ( aa(real,$o,ord_less(real,zero_zero(real)),X)
             => ( aa(real,real,log(A3),X) = 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),B3)),aa(real,real,ln_ln(real),A3))),aa(real,real,log(B3),X)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_2228_log__mult,axiom,
    ! [A3: real,X: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
     => ( ( A3 != one_one(real) )
       => ( aa(real,$o,ord_less(real,zero_zero(real)),X)
         => ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
           => ( aa(real,real,log(A3),aa(real,real,aa(real,fun(real,real),times_times(real),X),Y)) = aa(real,real,plus_plus(real,aa(real,real,log(A3),X)),aa(real,real,log(A3),Y)) ) ) ) ) ) ).

% log_mult
tff(fact_2229_powr__mult__base,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),X),powr(real,X,Y)) = powr(real,X,aa(real,real,plus_plus(real,one_one(real)),Y)) ) ) ).

% powr_mult_base
tff(fact_2230_log__nat__power,axiom,
    ! [X: real,B3: real,Na: nat] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ( aa(real,real,log(B3),aa(nat,real,power_power(real,X),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(B3),X)) ) ) ).

% log_nat_power
tff(fact_2231_divide__le__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,C2: A,W: num] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,ord_less_eq(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),B3),aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))) ) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_2232_le__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2))
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(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),W))),C2)),B3),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,ord_less_eq(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),zero_zero(A))) ) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_2233_power2__sum,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [X: A,Y: A] : aa(nat,A,power_power(A,aa(A,A,plus_plus(A,X),Y)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,plus_plus(A,aa(A,A,plus_plus(A,aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),X)),Y)) ) ).

% power2_sum
tff(fact_2234_oddE,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
         => ~ ! [B5: A] : A3 != aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B5)),one_one(A)) ) ) ).

% oddE
tff(fact_2235_floor__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q3: A,P2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Q3)
         => aa(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),P2),Q3)))),Q3)),P2) ) ) ).

% floor_divide_lower
tff(fact_2236_ceiling__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q3: A,P2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Q3)
         => aa(A,$o,ord_less_eq(A,P2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P2),Q3)))),Q3)) ) ) ).

% ceiling_divide_upper
tff(fact_2237_Bernoulli__inequality,axiom,
    ! [X: real,Na: nat] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),X)
     => aa(real,$o,ord_less_eq(real,aa(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)),X))),aa(nat,real,power_power(real,aa(real,real,plus_plus(real,one_one(real)),X)),Na)) ) ).

% Bernoulli_inequality
tff(fact_2238_add__log__eq__powr,axiom,
    ! [B3: real,X: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),B3)
     => ( ( B3 != one_one(real) )
       => ( aa(real,$o,ord_less(real,zero_zero(real)),X)
         => ( aa(real,real,plus_plus(real,Y),aa(real,real,log(B3),X)) = aa(real,real,log(B3),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,B3,Y)),X)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_2239_log__add__eq__powr,axiom,
    ! [B3: real,X: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),B3)
     => ( ( B3 != one_one(real) )
       => ( aa(real,$o,ord_less(real,zero_zero(real)),X)
         => ( aa(real,real,plus_plus(real,aa(real,real,log(B3),X)),Y) = aa(real,real,log(B3),aa(real,real,aa(real,fun(real,real),times_times(real),X),powr(real,B3,Y))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_2240_sum__squares__bound,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] : aa(A,$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))),X)),Y)),aa(A,A,plus_plus(A,aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% sum_squares_bound
tff(fact_2241_power2__diff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y: A] : aa(nat,A,power_power(A,aa(A,A,minus_minus(A,X),Y)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,minus_minus(A,aa(A,A,plus_plus(A,aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),X)),Y)) ) ).

% power2_diff
tff(fact_2242_floor__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q3: A,P2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Q3)
         => aa(A,$o,ord_less(A,P2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,plus_plus(A,aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P2),Q3)))),one_one(A))),Q3)) ) ) ).

% floor_divide_upper
tff(fact_2243_ceiling__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q3: A,P2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Q3)
         => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P2),Q3)))),one_one(A))),Q3)),P2) ) ) ).

% ceiling_divide_lower
tff(fact_2244_L2__set__mult__ineq__lemma,axiom,
    ! [A3: real,C2: real,B3: real,D2: real] : aa(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),A3),C2))),aa(real,real,aa(real,fun(real,real),times_times(real),B3),D2))),aa(real,real,plus_plus(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,power_power(real,A3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(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,power_power(real,B3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,C2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% L2_set_mult_ineq_lemma
tff(fact_2245_log__minus__eq__powr,axiom,
    ! [B3: real,X: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),B3)
     => ( ( B3 != one_one(real) )
       => ( aa(real,$o,ord_less(real,zero_zero(real)),X)
         => ( aa(real,real,minus_minus(real,aa(real,real,log(B3),X)),Y) = aa(real,real,log(B3),aa(real,real,aa(real,fun(real,real),times_times(real),X),powr(real,B3,aa(real,real,uminus_uminus(real),Y)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_2246_arith__geo__mean,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,X: A,Y: A] :
          ( ( aa(nat,A,power_power(A,U),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),X),Y) )
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),X)
           => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Y)
             => aa(A,$o,ord_less_eq(A,U),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,plus_plus(A,X),Y)),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ) ) ).

% arith_geo_mean
tff(fact_2247_Bernoulli__inequality__even,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => aa(real,$o,ord_less_eq(real,aa(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)),X))),aa(nat,real,power_power(real,aa(real,real,plus_plus(real,one_one(real)),X)),Na)) ) ).

% Bernoulli_inequality_even
tff(fact_2248_round__def,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_round(A,X) = archim6421214686448440834_floor(A,aa(A,A,plus_plus(A,X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% round_def
tff(fact_2249_even__mult__exp__div__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,M: nat,Na: nat] :
          ( aa(A,$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),A3),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M))),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)))
        <=> ( aa(nat,$o,ord_less(nat,Na),M)
            | ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na) = zero_zero(A) )
            | ( aa(nat,$o,ord_less_eq(nat,M),Na)
              & aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,minus_minus(nat,Na),M)))) ) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_2250_of__int__round__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),archimedean_round(A,X))),aa(A,A,plus_plus(A,X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% of_int_round_le
tff(fact_2251_of__int__round__ge,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(A,$o,ord_less_eq(A,aa(A,A,minus_minus(A,X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,X))) ) ).

% of_int_round_ge
tff(fact_2252_of__int__round__gt,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,X))) ) ).

% of_int_round_gt
tff(fact_2253_arsinh__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [X: A] : aa(A,A,arsinh(A),X) = aa(A,A,ln_ln(A),aa(A,A,plus_plus(A,X),powr(A,aa(A,A,plus_plus(A,aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)),aa(real,A,real_Vector_of_real(A),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))))) ) ).

% arsinh_def
tff(fact_2254_of__real__neg__numeral,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [W: num] : aa(real,A,real_Vector_of_real(A),aa(real,real,uminus_uminus(real),aa(num,real,numeral_numeral(real),W))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) ) ).

% of_real_neg_numeral
tff(fact_2255_of__real__of__nat__eq,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Na: nat] : aa(real,A,real_Vector_of_real(A),aa(nat,real,semiring_1_of_nat(real),Na)) = aa(nat,A,semiring_1_of_nat(A),Na) ) ).

% of_real_of_nat_eq
tff(fact_2256_of__real__power,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [X: real,Na: nat] : aa(real,A,real_Vector_of_real(A),aa(nat,real,power_power(real,X),Na)) = aa(nat,A,power_power(A,aa(real,A,real_Vector_of_real(A),X)),Na) ) ).

% of_real_power
tff(fact_2257_of__real__numeral,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [W: num] : aa(real,A,real_Vector_of_real(A),aa(num,real,numeral_numeral(real),W)) = aa(num,A,numeral_numeral(A),W) ) ).

% of_real_numeral
tff(fact_2258_of__real__eq__1__iff,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [X: real] :
          ( ( aa(real,A,real_Vector_of_real(A),X) = one_one(A) )
        <=> ( X = one_one(real) ) ) ) ).

% of_real_eq_1_iff
tff(fact_2259_of__real__1,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ( aa(real,A,real_Vector_of_real(A),one_one(real)) = one_one(A) ) ) ).

% of_real_1
tff(fact_2260_of__real__eq__0__iff,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [X: real] :
          ( ( aa(real,A,real_Vector_of_real(A),X) = zero_zero(A) )
        <=> ( X = zero_zero(real) ) ) ) ).

% of_real_eq_0_iff
tff(fact_2261_star__false__right,axiom,
    ! [P: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),bot_bot(assn)) = bot_bot(assn) ).

% star_false_right
tff(fact_2262_star__false__left,axiom,
    ! [P: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),bot_bot(assn)),P) = bot_bot(assn) ).

% star_false_left
tff(fact_2263_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_2264_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_2265_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_2266_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_2267_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_2268_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_2269_semiring__norm_I13_J,axiom,
    ! [M: num,Na: num] : aa(num,num,aa(num,fun(num,num),times_times(num),bit0(M)),bit0(Na)) = bit0(bit0(aa(num,num,aa(num,fun(num,num),times_times(num),M),Na))) ).

% semiring_norm(13)
tff(fact_2270_semiring__norm_I11_J,axiom,
    ! [M: num] : aa(num,num,aa(num,fun(num,num),times_times(num),M),one2) = M ).

% semiring_norm(11)
tff(fact_2271_semiring__norm_I12_J,axiom,
    ! [Na: num] : aa(num,num,aa(num,fun(num,num),times_times(num),one2),Na) = Na ).

% semiring_norm(12)
tff(fact_2272_norm__pre__pure__iff,axiom,
    ! [A: $tType,P: assn,B3: $o,F2: heap_Time_Heap(A),Q: fun(A,assn)] :
      ( hoare_hoare_triple(A,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),pure_assn((B3))),F2,Q)
    <=> ( (B3)
       => hoare_hoare_triple(A,P,F2,Q) ) ) ).

% norm_pre_pure_iff
tff(fact_2273_nat__0__less__mult__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na))
    <=> ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
        & aa(nat,$o,ord_less(nat,zero_zero(nat)),Na) ) ) ).

% nat_0_less_mult_iff
tff(fact_2274_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa(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,ord_less(nat,zero_zero(nat)),K)
        & aa(nat,$o,ord_less(nat,M),Na) ) ) ).

% mult_less_cancel2
tff(fact_2275_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(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,ord_less(nat,zero_zero(nat)),K)
        & aa(nat,$o,ord_less(nat,M),Na) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_2276_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_2277_of__real__0,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ( aa(real,A,real_Vector_of_real(A),zero_zero(real)) = zero_zero(A) ) ) ).

% of_real_0
tff(fact_2278_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(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,dvd_dvd(nat,M),Na) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_2279_num__double,axiom,
    ! [Na: num] : aa(num,num,aa(num,fun(num,num),times_times(num),bit0(one2)),Na) = bit0(Na) ).

% num_double
tff(fact_2280_power__mult__numeral,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,M: num,Na: num] : aa(nat,A,power_power(A,aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),M))),aa(num,nat,numeral_numeral(nat),Na)) = aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),times_times(num),M),Na))) ) ).

% power_mult_numeral
tff(fact_2281_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa(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,ord_less(nat,zero_zero(nat)),K)
       => aa(nat,$o,ord_less_eq(nat,M),Na) ) ) ).

% mult_le_cancel2
tff(fact_2282_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(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,ord_less(nat,zero_zero(nat)),K)
       => aa(nat,$o,ord_less_eq(nat,M),Na) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_2283_div__mult__self__is__m,axiom,
    ! [Na: nat,M: nat] :
      ( aa(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_2284_div__mult__self1__is__m,axiom,
    ! [Na: nat,M: nat] :
      ( aa(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_2285_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A,Na: nat] : aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),A3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) = aa(nat,A,power_power(A,A3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ) ).

% Power.ring_1_class.power_minus_even
tff(fact_2286_power__minus1__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Na: nat] : aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) = one_one(A) ) ).

% power_minus1_even
tff(fact_2287_odd__two__times__div__two__nat,axiom,
    ! [Na: nat] :
      ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => ( 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),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(nat,nat,minus_minus(nat,Na),one_one(nat)) ) ) ).

% odd_two_times_div_two_nat
tff(fact_2288_int__ops_I7_J,axiom,
    ! [A3: nat,B3: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),B3)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B3)) ).

% int_ops(7)
tff(fact_2289_Abs__fnat__hom__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: nat,B3: nat] : 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)),A3)),aa(nat,word(A),semiring_1_of_nat(word(A)),B3)) = aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),B3)) ) ).

% Abs_fnat_hom_mult
tff(fact_2290_wi__hom__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: int,B3: int] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(int,word(A),ring_1_of_int(word(A)),A3)),aa(int,word(A),ring_1_of_int(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),times_times(int),A3),B3)) ) ).

% wi_hom_mult
tff(fact_2291_word__mult__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A3),B3) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,semiring_1_unsigned(A,int),A3)),aa(word(A),int,semiring_1_unsigned(A,int),B3))) ) ).

% word_mult_def
tff(fact_2292_word__arith__nat__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A3),B3) = aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A3)),aa(word(A),nat,semiring_1_unsigned(A,nat),B3))) ) ).

% word_arith_nat_mult
tff(fact_2293_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_2294_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_2295_power__mult,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,M: nat,Na: nat] : aa(nat,A,power_power(A,A3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na)) = aa(nat,A,power_power(A,aa(nat,A,power_power(A,A3),M)),Na) ) ).

% power_mult
tff(fact_2296_add__mult__distrib,axiom,
    ! [M: nat,Na: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,plus_plus(nat,M),Na)),K) = aa(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_2297_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,plus_plus(nat,M),Na)) = aa(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_2298_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J: nat,K: nat] : aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),K)) = aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,plus_plus(nat,I),J)),U)),K) ).

% left_add_mult_distrib
tff(fact_2299_le__cube,axiom,
    ! [M: nat] : aa(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_2300_le__square,axiom,
    ! [M: nat] : aa(nat,$o,ord_less_eq(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),M)) ).

% le_square
tff(fact_2301_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => ( aa(nat,$o,ord_less_eq(nat,K),L)
       => aa(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_2302_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => aa(nat,$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_2303_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => aa(nat,$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_2304_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_2305_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_2306_times__int__code_I2_J,axiom,
    ! [L: int] : aa(int,int,aa(int,fun(int,int),times_times(int),zero_zero(int)),L) = zero_zero(int) ).

% times_int_code(2)
tff(fact_2307_times__int__code_I1_J,axiom,
    ! [K: int] : aa(int,int,aa(int,fun(int,int),times_times(int),K),zero_zero(int)) = zero_zero(int) ).

% times_int_code(1)
tff(fact_2308_diff__mult__distrib,axiom,
    ! [M: nat,Na: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,M),Na)),K) = aa(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_2309_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,minus_minus(nat,M),Na)) = aa(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_2310_int__distrib_I2_J,axiom,
    ! [W: int,Z1: int,Z22: int] : aa(int,int,aa(int,fun(int,int),times_times(int),W),aa(int,int,plus_plus(int,Z1),Z22)) = aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),times_times(int),W),Z1)),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z22)) ).

% int_distrib(2)
tff(fact_2311_int__distrib_I1_J,axiom,
    ! [Z1: int,Z22: int,W: int] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,plus_plus(int,Z1),Z22)),W) = aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),times_times(int),Z1),W)),aa(int,int,aa(int,fun(int,int),times_times(int),Z22),W)) ).

% int_distrib(1)
tff(fact_2312_int__distrib_I4_J,axiom,
    ! [W: int,Z1: int,Z22: int] : aa(int,int,aa(int,fun(int,int),times_times(int),W),aa(int,int,minus_minus(int,Z1),Z22)) = aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),W),Z1)),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z22)) ).

% int_distrib(4)
tff(fact_2313_int__distrib_I3_J,axiom,
    ! [Z1: int,Z22: int,W: int] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,minus_minus(int,Z1),Z22)),W) = aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),Z1),W)),aa(int,int,aa(int,fun(int,int),times_times(int),Z22),W)) ).

% int_distrib(3)
tff(fact_2314_assn__one__left,axiom,
    ! [P: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),one_one(assn)),P) = P ).

% assn_one_left
tff(fact_2315_imult__is__0,axiom,
    ! [M: extended_enat,Na: extended_enat] :
      ( ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),M),Na) = zero_zero(extended_enat) )
    <=> ( ( M = zero_zero(extended_enat) )
        | ( Na = zero_zero(extended_enat) ) ) ) ).

% imult_is_0
tff(fact_2316_frame__rule,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),R: assn] :
      ( hoare_hoare_triple(A,P,C2,Q)
     => hoare_hoare_triple(A,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),R),C2,aa(assn,fun(A,assn),aTP_Lamp_au(fun(A,assn),fun(assn,fun(A,assn)),Q),R)) ) ).

% frame_rule
tff(fact_2317_div__mult2__numeral__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A,K: num,L: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),L)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),K),L))) ) ).

% div_mult2_numeral_eq
tff(fact_2318_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,I),J)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
       => aa(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_2319_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,I),J)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
       => aa(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_2320_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(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_2321_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
     => ( aa(nat,$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,ord_less(nat,M),Na) ) ) ).

% nat_mult_less_cancel1
tff(fact_2322_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_2323_mlex__snd__decrI,axiom,
    ! [A3: nat,A5: nat,B3: nat,B6: nat,N2: nat] :
      ( ( A3 = A5 )
     => ( aa(nat,$o,ord_less(nat,B3),B6)
       => aa(nat,$o,ord_less(nat,aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),N2)),B3)),aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A5),N2)),B6)) ) ) ).

% mlex_snd_decrI
tff(fact_2324_mlex__fst__decrI,axiom,
    ! [A3: nat,A5: nat,B3: nat,N2: nat,B6: nat] :
      ( aa(nat,$o,ord_less(nat,A3),A5)
     => ( aa(nat,$o,ord_less(nat,B3),N2)
       => ( aa(nat,$o,ord_less(nat,B6),N2)
         => aa(nat,$o,ord_less(nat,aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),N2)),B3)),aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A5),N2)),B6)) ) ) ) ).

% mlex_fst_decrI
tff(fact_2325_mlex__bound,axiom,
    ! [A3: nat,A2: nat,B3: nat,N2: nat] :
      ( aa(nat,$o,ord_less(nat,A3),A2)
     => ( aa(nat,$o,ord_less(nat,B3),N2)
       => aa(nat,$o,ord_less(nat,aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),N2)),B3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),N2)) ) ) ).

% mlex_bound
tff(fact_2326_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),Na))
     => aa(nat,$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_2327_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K: int] :
      ( aa(int,$o,ord_less(int,I),J)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),K)
       => aa(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_2328_zmult__eq__1__iff,axiom,
    ! [M: int,Na: int] :
      ( ( aa(int,int,aa(int,fun(int,int),times_times(int),M),Na) = one_one(int) )
    <=> ( ( ( M = one_one(int) )
          & ( Na = one_one(int) ) )
        | ( ( M = aa(int,int,uminus_uminus(int),one_one(int)) )
          & ( Na = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ) ).

% zmult_eq_1_iff
tff(fact_2329_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M: int,Na: int] :
      ( ( aa(int,int,aa(int,fun(int,int),times_times(int),M),Na) = one_one(int) )
     => ( ( M = one_one(int) )
        | ( M = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
tff(fact_2330_zdvd__mono,axiom,
    ! [K: int,M: int,Ta: int] :
      ( ( K != zero_zero(int) )
     => ( aa(int,$o,dvd_dvd(int,M),Ta)
      <=> aa(int,$o,dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),K),M)),aa(int,int,aa(int,fun(int,int),times_times(int),K),Ta)) ) ) ).

% zdvd_mono
tff(fact_2331_zdvd__mult__cancel,axiom,
    ! [K: int,M: int,Na: int] :
      ( aa(int,$o,dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),K),M)),aa(int,int,aa(int,fun(int,int),times_times(int),K),Na))
     => ( ( K != zero_zero(int) )
       => aa(int,$o,dvd_dvd(int,M),Na) ) ) ).

% zdvd_mult_cancel
tff(fact_2332_zdvd__period,axiom,
    ! [A3: int,D2: int,X: int,Ta: int,C2: int] :
      ( aa(int,$o,dvd_dvd(int,A3),D2)
     => ( aa(int,$o,dvd_dvd(int,A3),aa(int,int,plus_plus(int,X),Ta))
      <=> aa(int,$o,dvd_dvd(int,A3),aa(int,int,plus_plus(int,aa(int,int,plus_plus(int,X),aa(int,int,aa(int,fun(int,int),times_times(int),C2),D2))),Ta)) ) ) ).

% zdvd_period
tff(fact_2333_zdvd__reduce,axiom,
    ! [K: int,Na: int,M: int] :
      ( aa(int,$o,dvd_dvd(int,K),aa(int,int,plus_plus(int,Na),aa(int,int,aa(int,fun(int,int),times_times(int),K),M)))
    <=> aa(int,$o,dvd_dvd(int,K),Na) ) ).

% zdvd_reduce
tff(fact_2334_div__to__mult__word__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A),Z: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Y),Z))
         => aa(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)),X),Z)),Y) ) ) ).

% div_to_mult_word_lt
tff(fact_2335_word__div__mult__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(word(A),$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)),A3),B3)),B3)),A3) ) ).

% word_div_mult_le
tff(fact_2336_norm__pre__pure__rule1,axiom,
    ! [A: $tType,B3: $o,P: assn,F2: heap_Time_Heap(A),Q: fun(A,assn)] :
      ( ( (B3)
       => hoare_hoare_triple(A,P,F2,Q) )
     => hoare_hoare_triple(A,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),pure_assn((B3))),F2,Q) ) ).

% norm_pre_pure_rule1
tff(fact_2337_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,Na: extended_enat] :
      ( aa(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,ord_less(extended_enat,zero_zero(extended_enat)),M)
        & aa(extended_enat,$o,ord_less(extended_enat,zero_zero(extended_enat)),Na) ) ) ).

% enat_0_less_mult_iff
tff(fact_2338_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
     => ( aa(nat,$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,ord_less_eq(nat,M),Na) ) ) ).

% nat_mult_le_cancel1
tff(fact_2339_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(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_2340_td__gal__lt,axiom,
    ! [C2: nat,A3: nat,B3: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),C2)
     => ( aa(nat,$o,ord_less(nat,A3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B3),C2))
      <=> aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A3),C2)),B3) ) ) ).

% td_gal_lt
tff(fact_2341_div__less__iff__less__mult,axiom,
    ! [Q3: nat,M: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Q3)
     => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Q3)),Na)
      <=> aa(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_2342_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,J),I)
     => ( ( aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M) = aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Na) )
      <=> ( aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,I),J)),U)),M) = Na ) ) ) ).

% nat_eq_add_iff1
tff(fact_2343_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => ( ( aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M) = aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Na) )
      <=> ( M = aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,J),I)),U)),Na) ) ) ) ).

% nat_eq_add_iff2
tff(fact_2344_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,J),I)
     => ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Na))
      <=> aa(nat,$o,ord_less_eq(nat,aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,I),J)),U)),M)),Na) ) ) ).

% nat_le_add_iff1
tff(fact_2345_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Na))
      <=> aa(nat,$o,ord_less_eq(nat,M),aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,J),I)),U)),Na)) ) ) ).

% nat_le_add_iff2
tff(fact_2346_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,J),I)
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Na)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,I),J)),U)),M)),Na) ) ) ).

% nat_diff_add_eq1
tff(fact_2347_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Na)) = aa(nat,nat,minus_minus(nat,M),aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,J),I)),U)),Na)) ) ) ).

% nat_diff_add_eq2
tff(fact_2348_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(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,ord_less(nat,zero_zero(nat)),K)
       => aa(nat,$o,dvd_dvd(nat,M),Na) ) ) ).

% dvd_mult_cancel
tff(fact_2349_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
     => ( aa(nat,$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,dvd_dvd(nat,M),Na) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_2350_bezout__add__strong__nat,axiom,
    ! [A3: nat,B3: nat] :
      ( ( A3 != zero_zero(nat) )
     => ? [D6: nat,X4: nat,Y3: nat] :
          ( aa(nat,$o,dvd_dvd(nat,D6),A3)
          & aa(nat,$o,dvd_dvd(nat,D6),B3)
          & ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X4) = aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B3),Y3)),D6) ) ) ) ).

% bezout_add_strong_nat
tff(fact_2351_pos__zmult__eq__1__iff,axiom,
    ! [M: int,Na: int] :
      ( aa(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_2352_uint__mult__ge0,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xa2: word(A),X: word(B)] : aa(int,$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),Xa2)),aa(word(B),int,semiring_1_unsigned(B,int),X))) ) ).

% uint_mult_ge0
tff(fact_2353_minusinfinity,axiom,
    ! [D2: int,P1: fun(int,$o),P: fun(int,$o)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D2)
     => ( ! [X4: int,K2: int] :
            ( aa(int,$o,P1,X4)
          <=> aa(int,$o,P1,aa(int,int,minus_minus(int,X4),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D2))) )
       => ( ? [Z3: int] :
            ! [X4: int] :
              ( aa(int,$o,ord_less(int,X4),Z3)
             => ( aa(int,$o,P,X4)
              <=> aa(int,$o,P1,X4) ) )
         => ( ? [X_13: int] : aa(int,$o,P1,X_13)
           => ? [X_12: int] : aa(int,$o,P,X_12) ) ) ) ) ).

% minusinfinity
tff(fact_2354_plusinfinity,axiom,
    ! [D2: int,P3: fun(int,$o),P: fun(int,$o)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D2)
     => ( ! [X4: int,K2: int] :
            ( aa(int,$o,P3,X4)
          <=> aa(int,$o,P3,aa(int,int,minus_minus(int,X4),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D2))) )
       => ( ? [Z3: int] :
            ! [X4: int] :
              ( aa(int,$o,ord_less(int,Z3),X4)
             => ( aa(int,$o,P,X4)
              <=> aa(int,$o,P3,X4) ) )
         => ( ? [X_13: int] : aa(int,$o,P3,X_13)
           => ? [X_12: int] : aa(int,$o,P,X_12) ) ) ) ) ).

% plusinfinity
tff(fact_2355_zdiv__zmult2__eq,axiom,
    ! [C2: int,A3: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),C2)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),aa(int,int,aa(int,fun(int,int),times_times(int),B3),C2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3)),C2) ) ) ).

% zdiv_zmult2_eq
tff(fact_2356_zdiv__mult__self,axiom,
    ! [M: int,A3: int,Na: int] :
      ( ( M != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,plus_plus(int,A3),aa(int,int,aa(int,fun(int,int),times_times(int),M),Na))),M) = aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),M)),Na) ) ) ).

% zdiv_mult_self
tff(fact_2357_div__lt__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),X: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),K),X))
         => ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),X)
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),X)),K) ) ) ) ).

% div_lt_mult
tff(fact_2358_More__Word_Oword__div__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [C2: word(A),A3: word(A),B3: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),C2)
         => ( aa(word(A),$o,ord_less(word(A),A3),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),B3),C2))
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),A3),C2)),B3) ) ) ) ).

% More_Word.word_div_mult
tff(fact_2359_power__even__eq,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,Na: nat] : aa(nat,A,power_power(A,A3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) = aa(nat,A,power_power(A,aa(nat,A,power_power(A,A3),Na)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% power_even_eq
tff(fact_2360_msrevs_I1_J,axiom,
    ! [Na: nat,K: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Na)),M)),Na) = aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)),K) ) ) ).

% msrevs(1)
tff(fact_2361_split__div,axiom,
    ! [P: fun(nat,$o),M: nat,Na: nat] :
      ( aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na))
    <=> ( ( ( Na = zero_zero(nat) )
         => aa(nat,$o,P,zero_zero(nat)) )
        & ( ( Na != zero_zero(nat) )
         => ! [I4: nat,J3: nat] :
              ( aa(nat,$o,ord_less(nat,J3),Na)
             => ( ( M = aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),I4)),J3) )
               => aa(nat,$o,P,I4) ) ) ) ) ) ).

% split_div
tff(fact_2362_dividend__less__div__times,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => aa(nat,$o,ord_less(nat,M),aa(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_2363_dividend__less__times__div,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => aa(nat,$o,ord_less(nat,M),aa(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_2364_td__gal,axiom,
    ! [C2: nat,B3: nat,A3: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),C2)
     => ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B3),C2)),A3)
      <=> aa(nat,$o,ord_less_eq(nat,B3),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A3),C2)) ) ) ).

% td_gal
tff(fact_2365_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q3: nat,M: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Q3)
     => ( aa(nat,$o,ord_less_eq(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),Q3))
      <=> aa(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_2366_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,plus_plus(nat,Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,M),one_one(nat))),Na))) ).

% mult_eq_if
tff(fact_2367_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => ( aa(nat,$o,ord_less(nat,aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Na))
      <=> aa(nat,$o,ord_less(nat,M),aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,J),I)),U)),Na)) ) ) ).

% nat_less_add_iff2
tff(fact_2368_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,J),I)
     => ( aa(nat,$o,ord_less(nat,aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Na))
      <=> aa(nat,$o,ord_less(nat,aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,I),J)),U)),M)),Na) ) ) ).

% nat_less_add_iff1
tff(fact_2369_nat__mult__power__less__eq,axiom,
    ! [B3: nat,A3: nat,Na: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),B3)
     => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),aa(nat,nat,power_power(nat,B3),Na))),aa(nat,nat,power_power(nat,B3),M))
      <=> aa(nat,$o,ord_less(nat,A3),aa(nat,nat,power_power(nat,B3),aa(nat,nat,minus_minus(nat,M),Na))) ) ) ).

% nat_mult_power_less_eq
tff(fact_2370_dvd__mult__cancel1,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
     => ( aa(nat,$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_2371_dvd__mult__cancel2,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
     => ( aa(nat,$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_2372_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K: nat] :
      ( aa(int,$o,ord_less(int,I),J)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
       => aa(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_2373_dvd__minus__add,axiom,
    ! [Q3: nat,Na: nat,R3: nat,M: nat] :
      ( aa(nat,$o,ord_less_eq(nat,Q3),Na)
     => ( aa(nat,$o,ord_less_eq(nat,Q3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R3),M))
       => ( aa(nat,$o,dvd_dvd(nat,M),aa(nat,nat,minus_minus(nat,Na),Q3))
        <=> aa(nat,$o,dvd_dvd(nat,M),aa(nat,nat,plus_plus(nat,Na),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R3),M)),Q3))) ) ) ) ).

% dvd_minus_add
tff(fact_2374_pos__mult__pos__ge,axiom,
    ! [X: int,Na: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),X)
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Na)
       => aa(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),X)) ) ) ).

% pos_mult_pos_ge
tff(fact_2375_incr__mult__lemma,axiom,
    ! [D2: int,P: fun(int,$o),K: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D2)
     => ( ! [X4: int] :
            ( aa(int,$o,P,X4)
           => aa(int,$o,P,aa(int,int,plus_plus(int,X4),D2)) )
       => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
         => ! [X2: int] :
              ( aa(int,$o,P,X2)
             => aa(int,$o,P,aa(int,int,plus_plus(int,X2),aa(int,int,aa(int,fun(int,int),times_times(int),K),D2))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_2376_unique__quotient__lemma__neg,axiom,
    ! [B3: int,Q4: int,R4: int,Q3: int,R3: int] :
      ( aa(int,$o,ord_less_eq(int,aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q4)),R4)),aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q3)),R3))
     => ( aa(int,$o,ord_less_eq(int,R3),zero_zero(int))
       => ( aa(int,$o,ord_less(int,B3),R3)
         => ( aa(int,$o,ord_less(int,B3),R4)
           => aa(int,$o,ord_less_eq(int,Q3),Q4) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_2377_unique__quotient__lemma,axiom,
    ! [B3: int,Q4: int,R4: int,Q3: int,R3: int] :
      ( aa(int,$o,ord_less_eq(int,aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q4)),R4)),aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q3)),R3))
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),R4)
       => ( aa(int,$o,ord_less(int,R4),B3)
         => ( aa(int,$o,ord_less(int,R3),B3)
           => aa(int,$o,ord_less_eq(int,Q4),Q3) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_2378_zdiv__mono2__neg__lemma,axiom,
    ! [B3: int,Q3: int,R3: int,B6: int,Q4: int,R4: int] :
      ( ( aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q3)),R3) = aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),times_times(int),B6),Q4)),R4) )
     => ( aa(int,$o,ord_less(int,aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),times_times(int),B6),Q4)),R4)),zero_zero(int))
       => ( aa(int,$o,ord_less(int,R3),B3)
         => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),R4)
           => ( aa(int,$o,ord_less(int,zero_zero(int)),B6)
             => ( aa(int,$o,ord_less_eq(int,B6),B3)
               => aa(int,$o,ord_less_eq(int,Q4),Q3) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_2379_zdiv__mono2__lemma,axiom,
    ! [B3: int,Q3: int,R3: int,B6: int,Q4: int,R4: int] :
      ( ( aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q3)),R3) = aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),times_times(int),B6),Q4)),R4) )
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),times_times(int),B6),Q4)),R4))
       => ( aa(int,$o,ord_less(int,R4),B6)
         => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),R3)
           => ( aa(int,$o,ord_less(int,zero_zero(int)),B6)
             => ( aa(int,$o,ord_less_eq(int,B6),B3)
               => aa(int,$o,ord_less_eq(int,Q3),Q4) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_2380_q__pos__lemma,axiom,
    ! [B6: int,Q4: int,R4: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),times_times(int),B6),Q4)),R4))
     => ( aa(int,$o,ord_less(int,R4),B6)
       => ( aa(int,$o,ord_less(int,zero_zero(int)),B6)
         => aa(int,$o,ord_less_eq(int,zero_zero(int)),Q4) ) ) ) ).

% q_pos_lemma
tff(fact_2381_decr__mult__lemma,axiom,
    ! [D2: int,P: fun(int,$o),K: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D2)
     => ( ! [X4: int] :
            ( aa(int,$o,P,X4)
           => aa(int,$o,P,aa(int,int,minus_minus(int,X4),D2)) )
       => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
         => ! [X2: int] :
              ( aa(int,$o,P,X2)
             => aa(int,$o,P,aa(int,int,minus_minus(int,X2),aa(int,int,aa(int,fun(int,int),times_times(int),K),D2))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_2382_udvd__incr__lem0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Up: int,Uq: int,Na: int,K4: word(A),N6: int] :
          ( aa(int,$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),N6),aa(word(A),int,semiring_1_unsigned(A,int),K4)) )
             => aa(int,$o,ord_less_eq(int,aa(int,int,plus_plus(int,Up),aa(word(A),int,semiring_1_unsigned(A,int),K4))),Uq) ) ) ) ) ).

% udvd_incr_lem0
tff(fact_2383_udvd__incr__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Up: int,Uq: int,Ua: int,Na: int,K4: word(A),N6: int] :
          ( aa(int,$o,ord_less(int,Up),Uq)
         => ( ( Up = aa(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,plus_plus(int,Ua),aa(int,int,aa(int,fun(int,int),times_times(int),N6),aa(word(A),int,semiring_1_unsigned(A,int),K4))) )
             => aa(int,$o,ord_less_eq(int,aa(int,int,plus_plus(int,Up),aa(word(A),int,semiring_1_unsigned(A,int),K4))),Uq) ) ) ) ) ).

% udvd_incr_lem
tff(fact_2384_udvd__incr0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),Q3: word(A),Na: int,K4: word(A),N6: int] :
          ( aa(word(A),$o,ord_less(word(A),P2),Q3)
         => ( ( aa(word(A),int,semiring_1_unsigned(A,int),P2) = 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),N6),aa(word(A),int,semiring_1_unsigned(A,int),K4)) )
             => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),plus_plus(word(A),P2),K4)),Q3) ) ) ) ) ).

% udvd_incr0
tff(fact_2385_udvd__decr0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),Q3: word(A),Na: int,K4: word(A),N6: int] :
          ( aa(word(A),$o,ord_less(word(A),P2),Q3)
         => ( ( aa(word(A),int,semiring_1_unsigned(A,int),P2) = 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),N6),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),N6),aa(word(A),int,semiring_1_unsigned(A,int),K4)) )
               => aa(word(A),$o,ord_less_eq(word(A),P2),aa(word(A),word(A),minus_minus(word(A),Q3),K4)) ) ) ) ) ) ).

% udvd_decr0
tff(fact_2386_div__le__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),X: word(A)] :
          ( aa(word(A),$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),X))
         => ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),X)
           => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),X)),K) ) ) ) ).

% div_le_mult
tff(fact_2387_int__div__pos__eq,axiom,
    ! [A3: int,B3: int,Q3: int,R3: int] :
      ( ( A3 = aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q3)),R3) )
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),R3)
       => ( aa(int,$o,ord_less(int,R3),B3)
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3) = Q3 ) ) ) ) ).

% int_div_pos_eq
tff(fact_2388_int__div__neg__eq,axiom,
    ! [A3: int,B3: int,Q3: int,R3: int] :
      ( ( A3 = aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q3)),R3) )
     => ( aa(int,$o,ord_less_eq(int,R3),zero_zero(int))
       => ( aa(int,$o,ord_less(int,B3),R3)
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3) = Q3 ) ) ) ) ).

% int_div_neg_eq
tff(fact_2389_split__zdiv,axiom,
    ! [P: fun(int,$o),Na: int,K: int] :
      ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),divide_divide(int),Na),K))
    <=> ( ( ( K = zero_zero(int) )
         => aa(int,$o,P,zero_zero(int)) )
        & ( aa(int,$o,ord_less(int,zero_zero(int)),K)
         => ! [I4: int,J3: int] :
              ( ( aa(int,$o,ord_less_eq(int,zero_zero(int)),J3)
                & aa(int,$o,ord_less(int,J3),K)
                & ( Na = aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
             => aa(int,$o,P,I4) ) )
        & ( aa(int,$o,ord_less(int,K),zero_zero(int))
         => ! [I4: int,J3: int] :
              ( ( aa(int,$o,ord_less(int,K),J3)
                & aa(int,$o,ord_less_eq(int,J3),zero_zero(int))
                & ( Na = aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
             => aa(int,$o,P,I4) ) ) ) ) ).

% split_zdiv
tff(fact_2390_udvd__incr_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),Q3: word(A),Ua: int,Na: int,K4: word(A),N6: int] :
          ( aa(word(A),$o,ord_less(word(A),P2),Q3)
         => ( ( aa(word(A),int,semiring_1_unsigned(A,int),P2) = aa(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,plus_plus(int,Ua),aa(int,int,aa(int,fun(int,int),times_times(int),N6),aa(word(A),int,semiring_1_unsigned(A,int),K4))) )
             => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),plus_plus(word(A),P2),K4)),Q3) ) ) ) ) ).

% udvd_incr'
tff(fact_2391_udvd__decr_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),Q3: word(A),Ua: int,Na: int,K4: word(A),N6: int] :
          ( aa(word(A),$o,ord_less(word(A),P2),Q3)
         => ( ( aa(word(A),int,semiring_1_unsigned(A,int),P2) = aa(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,plus_plus(int,Ua),aa(int,int,aa(int,fun(int,int),times_times(int),N6),aa(word(A),int,semiring_1_unsigned(A,int),K4))) )
             => ( ( aa(word(A),int,semiring_1_unsigned(A,int),Q3) = aa(int,int,plus_plus(int,Ua),aa(int,int,aa(int,fun(int,int),times_times(int),N6),aa(word(A),int,semiring_1_unsigned(A,int),K4))) )
               => aa(word(A),$o,ord_less_eq(word(A),P2),aa(word(A),word(A),minus_minus(word(A),Q3),K4)) ) ) ) ) ) ).

% udvd_decr'
tff(fact_2392_zero__le__even__power_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,Na: nat] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,power_power(A,A3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ) ).

% zero_le_even_power'
tff(fact_2393_nat__add__offset__less,axiom,
    ! [Y: nat,Na: nat,X: nat,M: nat,Sz: nat] :
      ( aa(nat,$o,ord_less(nat,Y),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
     => ( aa(nat,$o,ord_less(nat,X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
       => ( ( Sz = aa(nat,nat,plus_plus(nat,M),Na) )
         => aa(nat,$o,ord_less(nat,aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))),Y)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Sz)) ) ) ) ).

% nat_add_offset_less
tff(fact_2394_nat__power__less__diff,axiom,
    ! [Na: nat,Q3: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),Q3)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
     => aa(nat,$o,ord_less(nat,Q3),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,M),Na))) ) ).

% nat_power_less_diff
tff(fact_2395_nat__le__power__trans,axiom,
    ! [Na: nat,M: nat,K: nat] :
      ( aa(nat,$o,ord_less_eq(nat,Na),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,M),K)))
     => ( aa(nat,$o,ord_less_eq(nat,K),M)
       => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),K)),Na)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) ) ) ).

% nat_le_power_trans
tff(fact_2396_z1pdiv2,axiom,
    ! [B3: int] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B3)),one_one(int))),aa(num,int,numeral_numeral(int),bit0(one2))) = B3 ).

% z1pdiv2
tff(fact_2397_nat__less__power__trans,axiom,
    ! [Na: nat,M: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,Na),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,M),K)))
     => ( aa(nat,$o,ord_less_eq(nat,K),M)
       => aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),K)),Na)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) ) ) ).

% nat_less_power_trans
tff(fact_2398_int__bit__induct,axiom,
    ! [P: fun(int,$o),K: int] :
      ( aa(int,$o,P,zero_zero(int))
     => ( aa(int,$o,P,aa(int,int,uminus_uminus(int),one_one(int)))
       => ( ! [K2: int] :
              ( aa(int,$o,P,K2)
             => ( ( K2 != zero_zero(int) )
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),times_times(int),K2),aa(num,int,numeral_numeral(int),bit0(one2)))) ) )
         => ( ! [K2: int] :
                ( aa(int,$o,P,K2)
               => ( ( K2 != aa(int,int,uminus_uminus(int),one_one(int)) )
                 => aa(int,$o,P,aa(int,int,plus_plus(int,one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),K2),aa(num,int,numeral_numeral(int),bit0(one2))))) ) )
           => aa(int,$o,P,K) ) ) ) ) ).

% int_bit_induct
tff(fact_2399_power__2__mult__step__le,axiom,
    ! [N6: nat,Na: nat,K5: nat,K: nat] :
      ( aa(nat,$o,ord_less_eq(nat,N6),Na)
     => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N6)),K5)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),K))
       => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N6)),aa(nat,nat,plus_plus(nat,K5),one_one(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),K)) ) ) ).

% power_2_mult_step_le
tff(fact_2400_neg__zdiv__mult__2,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,A3),zero_zero(int))
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(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))),B3))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A3)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,plus_plus(int,B3),one_one(int))),A3) ) ) ).

% neg_zdiv_mult_2
tff(fact_2401_pos__zdiv__mult__2,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),A3)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(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))),B3))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A3)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),B3),A3) ) ) ).

% pos_zdiv_mult_2
tff(fact_2402_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,ord_less_eq(word(A),M),Na)
            & aa(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_2403_of__nat__eq__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] :
          ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),Na) = W )
        <=> ? [Q5: nat] : Na = aa(nat,nat,plus_plus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),W)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q5),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),size_size(word(A),W)))) ) ) ).

% of_nat_eq_size
tff(fact_2404_nonzero__of__real__divide,axiom,
    ! [A: $tType] :
      ( real_V7773925162809079976_field(A)
     => ! [Y: real,X: real] :
          ( ( Y != zero_zero(real) )
         => ( aa(real,A,real_Vector_of_real(A),aa(real,real,aa(real,fun(real,real),divide_divide(real),X),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(real,A,real_Vector_of_real(A),X)),aa(real,A,real_Vector_of_real(A),Y)) ) ) ) ).

% nonzero_of_real_divide
tff(fact_2405_VEBT__internal_Omulcomm,axiom,
    ! [I: nat,Va: 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,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Va),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,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Va),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),I) ).

% VEBT_internal.mulcomm
tff(fact_2406_set__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : bit_se5668285175392031749et_bit(A,zero_zero(nat),A3) = aa(A,A,plus_plus(A,one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% set_bit_0
tff(fact_2407_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))),X)
     => ( aa(real,$o,ord_less_eq(real,X),zero_zero(real))
       => aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,aa(real,real,ln_ln(real),aa(real,real,plus_plus(real,one_one(real)),X))),X))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_2408_mult__le__cancel__iff2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,X: A,Y: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Z)
         => ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),Z),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))
          <=> aa(A,$o,ord_less_eq(A,X),Y) ) ) ) ).

% mult_le_cancel_iff2
tff(fact_2409_mult__le__cancel__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,X: A,Y: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Z)
         => ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))
          <=> aa(A,$o,ord_less_eq(A,X),Y) ) ) ) ).

% mult_le_cancel_iff1
tff(fact_2410_unset__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : bit_se2638667681897837118et_bit(A,zero_zero(nat),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% unset_bit_0
tff(fact_2411_abs__idempotent,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : aa(A,A,abs_abs(A),aa(A,A,abs_abs(A),A3)) = aa(A,A,abs_abs(A),A3) ) ).

% abs_idempotent
tff(fact_2412_abs__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : aa(A,A,abs_abs(A),aa(A,A,abs_abs(A),A3)) = aa(A,A,abs_abs(A),A3) ) ).

% abs_abs
tff(fact_2413_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_2414_abs__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ( aa(A,A,abs_abs(A),zero_zero(A)) = zero_zero(A) ) ) ).

% abs_zero
tff(fact_2415_abs__eq__0,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( ( aa(A,A,abs_abs(A),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% abs_eq_0
tff(fact_2416_abs__0__eq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( ( zero_zero(A) = aa(A,A,abs_abs(A),A3) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% abs_0_eq
tff(fact_2417_abs__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: num] : aa(A,A,abs_abs(A),aa(num,A,numeral_numeral(A),Na)) = aa(num,A,numeral_numeral(A),Na) ) ).

% abs_numeral
tff(fact_2418_abs__add__abs,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] : aa(A,A,abs_abs(A),aa(A,A,plus_plus(A,aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B3))) = aa(A,A,plus_plus(A,aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B3)) ) ).

% abs_add_abs
tff(fact_2419_abs__1,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( aa(A,A,abs_abs(A),one_one(A)) = one_one(A) ) ) ).

% abs_1
tff(fact_2420_abs__mult__self__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),A3)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3) ) ).

% abs_mult_self_eq
tff(fact_2421_abs__minus__cancel,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),A3)) = aa(A,A,abs_abs(A),A3) ) ).

% abs_minus_cancel
tff(fact_2422_abs__minus,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),A3)) = aa(A,A,abs_abs(A),A3) ) ).

% abs_minus
tff(fact_2423_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_2424_abs__dvd__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: A,K: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,abs_abs(A),M)),K)
        <=> aa(A,$o,dvd_dvd(A,M),K) ) ) ).

% abs_dvd_iff
tff(fact_2425_dvd__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: A,K: A] :
          ( aa(A,$o,dvd_dvd(A,M),aa(A,A,abs_abs(A),K))
        <=> aa(A,$o,dvd_dvd(A,M),K) ) ) ).

% dvd_abs_iff
tff(fact_2426_unset__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),bit_se2638667681897837118et_bit(int,Na,K))
    <=> aa(int,$o,ord_less_eq(int,zero_zero(int)),K) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_2427_set__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),bit_se5668285175392031749et_bit(int,Na,K))
    <=> aa(int,$o,ord_less_eq(int,zero_zero(int)),K) ) ).

% set_bit_nonnegative_int_iff
tff(fact_2428_unset__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,ord_less(int,bit_se2638667681897837118et_bit(int,Na,K)),zero_zero(int))
    <=> aa(int,$o,ord_less(int,K),zero_zero(int)) ) ).

% unset_bit_negative_int_iff
tff(fact_2429_set__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,ord_less(int,bit_se5668285175392031749et_bit(int,Na,K)),zero_zero(int))
    <=> aa(int,$o,ord_less(int,K),zero_zero(int)) ) ).

% set_bit_negative_int_iff
tff(fact_2430_of__int__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: int] : aa(int,A,ring_1_of_int(A),aa(int,int,abs_abs(int),X)) = aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),X)) ) ).

% of_int_abs
tff(fact_2431_abs__of__nonneg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => ( aa(A,A,abs_abs(A),A3) = A3 ) ) ) ).

% abs_of_nonneg
tff(fact_2432_abs__le__self__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,abs_abs(A),A3)),A3)
        <=> aa(A,$o,ord_less_eq(A,zero_zero(A)),A3) ) ) ).

% abs_le_self_iff
tff(fact_2433_abs__le__zero__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,abs_abs(A),A3)),zero_zero(A))
        <=> ( A3 = zero_zero(A) ) ) ) ).

% abs_le_zero_iff
tff(fact_2434_zero__less__abs__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,abs_abs(A),A3))
        <=> ( A3 != zero_zero(A) ) ) ) ).

% zero_less_abs_iff
tff(fact_2435_abs__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: num] : aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))) = aa(num,A,numeral_numeral(A),Na) ) ).

% abs_neg_numeral
tff(fact_2436_abs__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ( aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),one_one(A))) = one_one(A) ) ) ).

% abs_neg_one
tff(fact_2437_abs__power__minus,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,Na: nat] : aa(A,A,abs_abs(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),A3)),Na)) = aa(A,A,abs_abs(A),aa(nat,A,power_power(A,A3),Na)) ) ).

% abs_power_minus
tff(fact_2438_zero__le__divide__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,abs_abs(A),B3)))
        <=> ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
            | ( B3 = zero_zero(A) ) ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_2439_divide__le__0__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,abs_abs(A),B3))),zero_zero(A))
        <=> ( aa(A,$o,ord_less_eq(A,A3),zero_zero(A))
            | ( B3 = zero_zero(A) ) ) ) ) ).

% divide_le_0_abs_iff
tff(fact_2440_abs__of__nonpos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),zero_zero(A))
         => ( aa(A,A,abs_abs(A),A3) = aa(A,A,uminus_uminus(A),A3) ) ) ) ).

% abs_of_nonpos
tff(fact_2441_artanh__minus__real,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,abs_abs(real),X)),one_one(real))
     => ( aa(real,real,artanh(real),aa(real,real,uminus_uminus(real),X)) = aa(real,real,uminus_uminus(real),aa(real,real,artanh(real),X)) ) ) ).

% artanh_minus_real
tff(fact_2442_zero__less__power__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(nat,A,power_power(A,aa(A,A,abs_abs(A),A3)),Na))
        <=> ( ( A3 != zero_zero(A) )
            | ( Na = zero_zero(nat) ) ) ) ) ).

% zero_less_power_abs_iff
tff(fact_2443_abs__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] : aa(A,A,abs_abs(A),aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% abs_power2
tff(fact_2444_power2__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] : aa(nat,A,power_power(A,aa(A,A,abs_abs(A),A3)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% power2_abs
tff(fact_2445_power__even__abs__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W: num,A3: A] :
          ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W))
         => ( aa(nat,A,power_power(A,aa(A,A,abs_abs(A),A3)),aa(num,nat,numeral_numeral(nat),W)) = aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),W)) ) ) ) ).

% power_even_abs_numeral
tff(fact_2446_square__powr__half,axiom,
    ! [X: real] : powr(real,aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2))),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,abs_abs(real),X) ).

% square_powr_half
tff(fact_2447_abs__eq__0__iff,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] :
          ( ( aa(A,A,abs_abs(A),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% abs_eq_0_iff
tff(fact_2448_abs__le__D1,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,abs_abs(A),A3)),B3)
         => aa(A,$o,ord_less_eq(A,A3),B3) ) ) ).

% abs_le_D1
tff(fact_2449_abs__ge__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : aa(A,$o,ord_less_eq(A,A3),aa(A,A,abs_abs(A),A3)) ) ).

% abs_ge_self
tff(fact_2450_abs__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ( aa(A,A,abs_abs(A),one_one(A)) = one_one(A) ) ) ).

% abs_one
tff(fact_2451_abs__mult,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A,B3: A] : aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B3)) ) ).

% abs_mult
tff(fact_2452_abs__minus__commute,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] : aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,A3),B3)) = aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,B3),A3)) ) ).

% abs_minus_commute
tff(fact_2453_power__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,Na: nat] : aa(A,A,abs_abs(A),aa(nat,A,power_power(A,A3),Na)) = aa(nat,A,power_power(A,aa(A,A,abs_abs(A),A3)),Na) ) ).

% power_abs
tff(fact_2454_abs__eq__iff,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,abs_abs(A),X) = aa(A,A,abs_abs(A),Y) )
        <=> ( ( X = Y )
            | ( X = aa(A,A,uminus_uminus(A),Y) ) ) ) ) ).

% abs_eq_iff
tff(fact_2455_dvd__if__abs__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [L: A,K: A] :
          ( ( aa(A,A,abs_abs(A),L) = aa(A,A,abs_abs(A),K) )
         => aa(A,$o,dvd_dvd(A,L),K) ) ) ).

% dvd_if_abs_eq
tff(fact_2456_abs__ge__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,abs_abs(A),A3)) ) ).

% abs_ge_zero
tff(fact_2457_abs__of__pos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => ( aa(A,A,abs_abs(A),A3) = A3 ) ) ) ).

% abs_of_pos
tff(fact_2458_abs__not__less__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : ~ aa(A,$o,ord_less(A,aa(A,A,abs_abs(A),A3)),zero_zero(A)) ) ).

% abs_not_less_zero
tff(fact_2459_abs__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] : aa(A,$o,ord_less_eq(A,aa(A,A,abs_abs(A),aa(A,A,plus_plus(A,A3),B3))),aa(A,A,plus_plus(A,aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B3))) ) ).

% abs_triangle_ineq
tff(fact_2460_abs__mult__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,C2: A,B3: A,D2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,abs_abs(A),A3)),C2)
         => ( aa(A,$o,ord_less(A,aa(A,A,abs_abs(A),B3)),D2)
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B3))),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D2)) ) ) ) ).

% abs_mult_less
tff(fact_2461_abs__triangle__ineq2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] : aa(A,$o,ord_less_eq(A,aa(A,A,minus_minus(A,aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B3))),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,A3),B3))) ) ).

% abs_triangle_ineq2
tff(fact_2462_abs__triangle__ineq3,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] : aa(A,$o,ord_less_eq(A,aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B3)))),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,A3),B3))) ) ).

% abs_triangle_ineq3
tff(fact_2463_abs__triangle__ineq2__sym,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] : aa(A,$o,ord_less_eq(A,aa(A,A,minus_minus(A,aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B3))),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,B3),A3))) ) ).

% abs_triangle_ineq2_sym
tff(fact_2464_nonzero__abs__divide,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B3)) ) ) ) ).

% nonzero_abs_divide
tff(fact_2465_abs__leI,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),A3)),B3)
           => aa(A,$o,ord_less_eq(A,aa(A,A,abs_abs(A),A3)),B3) ) ) ) ).

% abs_leI
tff(fact_2466_abs__le__D2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,abs_abs(A),A3)),B3)
         => aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),A3)),B3) ) ) ).

% abs_le_D2
tff(fact_2467_abs__le__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,abs_abs(A),A3)),B3)
        <=> ( aa(A,$o,ord_less_eq(A,A3),B3)
            & aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),A3)),B3) ) ) ) ).

% abs_le_iff
tff(fact_2468_abs__ge__minus__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),A3)),aa(A,A,abs_abs(A),A3)) ) ).

% abs_ge_minus_self
tff(fact_2469_abs__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,abs_abs(A),A3)),B3)
        <=> ( aa(A,$o,ord_less(A,A3),B3)
            & aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),A3)),B3) ) ) ) ).

% abs_less_iff
tff(fact_2470_dense__eq0__I,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs(A)
        & dense_linorder(A) )
     => ! [X: A] :
          ( ! [E: A] :
              ( aa(A,$o,ord_less(A,zero_zero(A)),E)
             => aa(A,$o,ord_less_eq(A,aa(A,A,abs_abs(A),X)),E) )
         => ( X = zero_zero(A) ) ) ) ).

% dense_eq0_I
tff(fact_2471_abs__eq__mult,axiom,
    ! [A: $tType] :
      ( ordered_ring_abs(A)
     => ! [A3: A,B3: A] :
          ( ( ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
              | aa(A,$o,ord_less_eq(A,A3),zero_zero(A)) )
            & ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B3)
              | aa(A,$o,ord_less_eq(A,B3),zero_zero(A)) ) )
         => ( aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B3)) ) ) ) ).

% abs_eq_mult
tff(fact_2472_abs__mult__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),X)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),Y)),X) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),Y),X)) ) ) ) ).

% abs_mult_pos
tff(fact_2473_abs__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,X: A] :
          ( aa(A,$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),X)),Y) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)) ) ) ) ).

% abs_div_pos
tff(fact_2474_zero__le__power__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,Na: nat] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,power_power(A,aa(A,A,abs_abs(A),A3)),Na)) ) ).

% zero_le_power_abs
tff(fact_2475_abs__minus__le__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(A,A,abs_abs(A),A3))),zero_zero(A)) ) ).

% abs_minus_le_zero
tff(fact_2476_eq__abs__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = aa(A,A,abs_abs(A),B3) )
        <=> ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
            & ( ( B3 = A3 )
              | ( B3 = aa(A,A,uminus_uminus(A),A3) ) ) ) ) ) ).

% eq_abs_iff'
tff(fact_2477_abs__eq__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,abs_abs(A),A3) = B3 )
        <=> ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B3)
            & ( ( A3 = B3 )
              | ( A3 = aa(A,A,uminus_uminus(A),B3) ) ) ) ) ) ).

% abs_eq_iff'
tff(fact_2478_abs__if,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [A3: A] :
          aa(A,A,abs_abs(A),A3) = $ite(aa(A,$o,ord_less(A,A3),zero_zero(A)),aa(A,A,uminus_uminus(A),A3),A3) ) ).

% abs_if
tff(fact_2479_abs__if__raw,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [X2: A] :
          aa(A,A,abs_abs(A),X2) = $ite(aa(A,$o,ord_less(A,X2),zero_zero(A)),aa(A,A,uminus_uminus(A),X2),X2) ) ).

% abs_if_raw
tff(fact_2480_abs__of__neg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,A3),zero_zero(A))
         => ( aa(A,A,abs_abs(A),A3) = aa(A,A,uminus_uminus(A),A3) ) ) ) ).

% abs_of_neg
tff(fact_2481_abs__triangle__ineq4,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] : aa(A,$o,ord_less_eq(A,aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,A3),B3))),aa(A,A,plus_plus(A,aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B3))) ) ).

% abs_triangle_ineq4
tff(fact_2482_abs__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A,C2: A,D2: A] : aa(A,$o,ord_less_eq(A,aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,aa(A,A,plus_plus(A,A3),B3)),aa(A,A,plus_plus(A,C2),D2)))),aa(A,A,plus_plus(A,aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,A3),C2))),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,B3),D2)))) ) ).

% abs_diff_triangle_ineq
tff(fact_2483_abs__diff__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,A3: A,R3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,X),A3))),R3)
        <=> ( aa(A,$o,ord_less_eq(A,aa(A,A,minus_minus(A,A3),R3)),X)
            & aa(A,$o,ord_less_eq(A,X),aa(A,A,plus_plus(A,A3),R3)) ) ) ) ).

% abs_diff_le_iff
tff(fact_2484_abs__diff__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,A3: A,R3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,X),A3))),R3)
        <=> ( aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,A3),R3)),X)
            & aa(A,$o,ord_less(A,X),aa(A,A,plus_plus(A,A3),R3)) ) ) ) ).

% abs_diff_less_iff
tff(fact_2485_abs__real__def,axiom,
    ! [A3: real] :
      aa(real,real,abs_abs(real),A3) = $ite(aa(real,$o,ord_less(real,A3),zero_zero(real)),aa(real,real,uminus_uminus(real),A3),A3) ).

% abs_real_def
tff(fact_2486_lemma__interval__lt,axiom,
    ! [A3: real,X: real,B3: real] :
      ( aa(real,$o,ord_less(real,A3),X)
     => ( aa(real,$o,ord_less(real,X),B3)
       => ? [D6: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),D6)
            & ! [Y4: real] :
                ( aa(real,$o,ord_less(real,aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,X),Y4))),D6)
               => ( aa(real,$o,ord_less(real,A3),Y4)
                  & aa(real,$o,ord_less(real,Y4),B3) ) ) ) ) ) ).

% lemma_interval_lt
tff(fact_2487_abs__add__one__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] : aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,plus_plus(A,one_one(A)),aa(A,A,abs_abs(A),X))) ) ).

% abs_add_one_gt_zero
tff(fact_2488_of__int__leD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: int,X: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),Na))),X)
         => ( ( Na = zero_zero(int) )
            | aa(A,$o,ord_less_eq(A,one_one(A)),X) ) ) ) ).

% of_int_leD
tff(fact_2489_of__int__lessD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: int,X: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),Na))),X)
         => ( ( Na = zero_zero(int) )
            | aa(A,$o,ord_less(A,one_one(A)),X) ) ) ) ).

% of_int_lessD
tff(fact_2490_lemma__interval,axiom,
    ! [A3: real,X: real,B3: real] :
      ( aa(real,$o,ord_less(real,A3),X)
     => ( aa(real,$o,ord_less(real,X),B3)
       => ? [D6: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),D6)
            & ! [Y4: real] :
                ( aa(real,$o,ord_less(real,aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,X),Y4))),D6)
               => ( aa(real,$o,ord_less_eq(real,A3),Y4)
                  & aa(real,$o,ord_less_eq(real,Y4),B3) ) ) ) ) ) ).

% lemma_interval
tff(fact_2491_round__diff__minimal,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: A,M: int] : aa(A,$o,ord_less_eq(A,aa(A,A,abs_abs(A),aa(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,minus_minus(A,Z),aa(int,A,ring_1_of_int(A),M)))) ) ).

% round_diff_minimal
tff(fact_2492_abs__le__square__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,abs_abs(A),X)),aa(A,A,abs_abs(A),Y))
        <=> aa(A,$o,ord_less_eq(A,aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% abs_le_square_iff
tff(fact_2493_abs__square__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( ( aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(A) )
        <=> ( aa(A,A,abs_abs(A),X) = one_one(A) ) ) ) ).

% abs_square_eq_1
tff(fact_2494_power__even__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat,A3: A] :
          ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
         => ( aa(nat,A,power_power(A,aa(A,A,abs_abs(A),A3)),Na) = aa(nat,A,power_power(A,A3),Na) ) ) ) ).

% power_even_abs
tff(fact_2495_power2__le__iff__abs__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Y)
         => ( aa(A,$o,ord_less_eq(A,aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
          <=> aa(A,$o,ord_less_eq(A,aa(A,A,abs_abs(A),X)),Y) ) ) ) ).

% power2_le_iff_abs_le
tff(fact_2496_abs__square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A))
        <=> aa(A,$o,ord_less_eq(A,aa(A,A,abs_abs(A),X)),one_one(A)) ) ) ).

% abs_square_le_1
tff(fact_2497_abs__square__less__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( aa(A,$o,ord_less(A,aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A))
        <=> aa(A,$o,ord_less(A,aa(A,A,abs_abs(A),X)),one_one(A)) ) ) ).

% abs_square_less_1
tff(fact_2498_power__mono__even,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat,A3: A,B3: A] :
          ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
         => ( aa(A,$o,ord_less_eq(A,aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B3))
           => aa(A,$o,ord_less_eq(A,aa(nat,A,power_power(A,A3),Na)),aa(nat,A,power_power(A,B3),Na)) ) ) ) ).

% power_mono_even
tff(fact_2499_even__set__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),bit_se5668285175392031749et_bit(A,M,A3))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
            & ( M != zero_zero(nat) ) ) ) ) ).

% even_set_bit_iff
tff(fact_2500_even__unset__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2638667681897837118et_bit(A,M,A3))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
            | ( M = zero_zero(nat) ) ) ) ) ).

% even_unset_bit_iff
tff(fact_2501_of__int__round__abs__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(A,$o,ord_less_eq(A,aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),archimedean_round(A,X))),X))),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% of_int_round_abs_le
tff(fact_2502_round__unique_H,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Na: int] :
          ( aa(A,$o,ord_less(A,aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,X),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,X) = Na ) ) ) ).

% round_unique'
tff(fact_2503_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less_eq(real,X),one_one(real))
       => aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,aa(real,real,ln_ln(real),aa(real,real,plus_plus(real,one_one(real)),X))),X))),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
tff(fact_2504_mult__less__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,X: A,Y: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Z)
         => ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))
          <=> aa(A,$o,ord_less(A,X),Y) ) ) ) ).

% mult_less_iff1
tff(fact_2505_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),X)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))
     => aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,aa(real,real,ln_ln(real),aa(real,real,plus_plus(real,one_one(real)),X))),X))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% abs_ln_one_plus_x_minus_x_bound
tff(fact_2506_abs__sqrt__wlog,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P: fun(A,fun(A,$o)),X: A] :
          ( ! [X4: A] :
              ( aa(A,$o,ord_less_eq(A,zero_zero(A)),X4)
             => aa(A,$o,aa(A,fun(A,$o),P,X4),aa(nat,A,power_power(A,X4),aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
         => aa(A,$o,aa(A,fun(A,$o),P,aa(A,A,abs_abs(A),X)),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% abs_sqrt_wlog
tff(fact_2507_monoseq__arctan__series,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),X)),one_one(real))
     => topological_monoseq(real,aTP_Lamp_av(real,fun(nat,real),X)) ) ).

% monoseq_arctan_series
tff(fact_2508_summable__arctan__series,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),X)),one_one(real))
     => summable(real,aTP_Lamp_aw(real,fun(nat,real),X)) ) ).

% summable_arctan_series
tff(fact_2509_round__altdef,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          archimedean_round(A,X) = $ite(aa(A,$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,X)),archimedean_ceiling(A,X),archim6421214686448440834_floor(A,X)) ) ).

% round_altdef
tff(fact_2510_arctan__double,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,abs_abs(real),X)),one_one(real))
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,arctan,X)) = aa(real,real,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))),X)),aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ).

% arctan_double
tff(fact_2511_pochhammer__double,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Z: A,Na: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Z),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)))),comm_s3205402744901411588hammer(A,Z,Na))),comm_s3205402744901411588hammer(A,aa(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)))),Na)) ) ).

% pochhammer_double
tff(fact_2512_frac__frac,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_frac(A,archimedean_frac(A,X)) = archimedean_frac(A,X) ) ).

% frac_frac
tff(fact_2513_pochhammer__1,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] : comm_s3205402744901411588hammer(A,A3,one_one(nat)) = A3 ) ).

% pochhammer_1
tff(fact_2514_arctan__zero__zero,axiom,
    aa(real,real,arctan,zero_zero(real)) = zero_zero(real) ).

% arctan_zero_zero
tff(fact_2515_arctan__eq__zero__iff,axiom,
    ! [X: real] :
      ( ( aa(real,real,arctan,X) = zero_zero(real) )
    <=> ( X = zero_zero(real) ) ) ).

% arctan_eq_zero_iff
tff(fact_2516_pochhammer__0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] : comm_s3205402744901411588hammer(A,A3,zero_zero(nat)) = one_one(A) ) ).

% pochhammer_0
tff(fact_2517_arctan__less__zero__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,arctan,X)),zero_zero(real))
    <=> aa(real,$o,ord_less(real,X),zero_zero(real)) ) ).

% arctan_less_zero_iff
tff(fact_2518_zero__less__arctan__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,arctan,X))
    <=> aa(real,$o,ord_less(real,zero_zero(real)),X) ) ).

% zero_less_arctan_iff
tff(fact_2519_arctan__le__zero__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,arctan,X)),zero_zero(real))
    <=> aa(real,$o,ord_less_eq(real,X),zero_zero(real)) ) ).

% arctan_le_zero_iff
tff(fact_2520_zero__le__arctan__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,arctan,X))
    <=> aa(real,$o,ord_less_eq(real,zero_zero(real)),X) ) ).

% zero_le_arctan_iff
tff(fact_2521_zdvd1__eq,axiom,
    ! [X: int] :
      ( aa(int,$o,dvd_dvd(int,X),one_one(int))
    <=> ( aa(int,int,abs_abs(int),X) = one_one(int) ) ) ).

% zdvd1_eq
tff(fact_2522_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_2523_zabs__less__one__iff,axiom,
    ! [Z: int] :
      ( aa(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_2524_pochhammer__of__nat,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [X: nat,Na: nat] : comm_s3205402744901411588hammer(A,aa(nat,A,semiring_1_of_nat(A),X),Na) = aa(nat,A,semiring_1_of_nat(A),comm_s3205402744901411588hammer(nat,X,Na)) ) ).

% pochhammer_of_nat
tff(fact_2525_zdvd__antisym__abs,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,dvd_dvd(int,A3),B3)
     => ( aa(int,$o,dvd_dvd(int,B3),A3)
       => ( aa(int,int,abs_abs(int),A3) = aa(int,int,abs_abs(int),B3) ) ) ) ).

% zdvd_antisym_abs
tff(fact_2526_arctan__less__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,arctan,X)),aa(real,real,arctan,Y))
    <=> aa(real,$o,ord_less(real,X),Y) ) ).

% arctan_less_iff
tff(fact_2527_arctan__monotone,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less(real,X),Y)
     => aa(real,$o,ord_less(real,aa(real,real,arctan,X)),aa(real,real,arctan,Y)) ) ).

% arctan_monotone
tff(fact_2528_abs__zmult__eq__1,axiom,
    ! [M: int,Na: int] :
      ( ( aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),times_times(int),M),Na)) = one_one(int) )
     => ( aa(int,int,abs_abs(int),M) = one_one(int) ) ) ).

% abs_zmult_eq_1
tff(fact_2529_pochhammer__pos,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Na: nat] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),X)
         => aa(A,$o,ord_less(A,zero_zero(A)),comm_s3205402744901411588hammer(A,X,Na)) ) ) ).

% pochhammer_pos
tff(fact_2530_pochhammer__eq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,Na: nat,M: nat] :
          ( ( comm_s3205402744901411588hammer(A,A3,Na) = zero_zero(A) )
         => ( aa(nat,$o,ord_less_eq(nat,Na),M)
           => ( comm_s3205402744901411588hammer(A,A3,M) = zero_zero(A) ) ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_2531_pochhammer__neq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,M: nat,Na: nat] :
          ( ( comm_s3205402744901411588hammer(A,A3,M) != zero_zero(A) )
         => ( aa(nat,$o,ord_less_eq(nat,Na),M)
           => ( comm_s3205402744901411588hammer(A,A3,Na) != zero_zero(A) ) ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_2532_frac__ge__0,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(A,$o,ord_less_eq(A,zero_zero(A)),archimedean_frac(A,X)) ) ).

% frac_ge_0
tff(fact_2533_frac__lt__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(A,$o,ord_less(A,archimedean_frac(A,X)),one_one(A)) ) ).

% frac_lt_1
tff(fact_2534_frac__1__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_frac(A,aa(A,A,plus_plus(A,X),one_one(A))) = archimedean_frac(A,X) ) ).

% frac_1_eq
tff(fact_2535_zabs__def,axiom,
    ! [I: int] :
      aa(int,int,abs_abs(int),I) = $ite(aa(int,$o,ord_less(int,I),zero_zero(int)),aa(int,int,uminus_uminus(int),I),I) ).

% zabs_def
tff(fact_2536_pochhammer__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Na: nat] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),X)
         => aa(A,$o,ord_less_eq(A,zero_zero(A)),comm_s3205402744901411588hammer(A,X,Na)) ) ) ).

% pochhammer_nonneg
tff(fact_2537_dvd__imp__le__int,axiom,
    ! [I: int,D2: int] :
      ( ( I != zero_zero(int) )
     => ( aa(int,$o,dvd_dvd(int,D2),I)
       => aa(int,$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_2538_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_2539_zdvd__mult__cancel1,axiom,
    ! [M: int,Na: int] :
      ( ( M != zero_zero(int) )
     => ( aa(int,$o,dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),M),Na)),M)
      <=> ( aa(int,int,abs_abs(int),Na) = one_one(int) ) ) ) ).

% zdvd_mult_cancel1
tff(fact_2540_frac__def,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_frac(A,X) = aa(A,A,minus_minus(A,X),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X))) ) ).

% frac_def
tff(fact_2541_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Na: nat,K: nat] :
          ( aa(nat,$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_2542_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,ord_less(nat,Na),K) ) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_2543_pochhammer__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,Na: nat] :
          ( ( comm_s3205402744901411588hammer(A,A3,Na) = zero_zero(A) )
        <=> ? [K3: nat] :
              ( aa(nat,$o,ord_less(nat,K3),Na)
              & ( A3 = aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),K3)) ) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_2544_pochhammer__product_H,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Z: A,Na: nat,M: nat] : comm_s3205402744901411588hammer(A,Z,aa(nat,nat,plus_plus(nat,Na),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z,Na)),comm_s3205402744901411588hammer(A,aa(A,A,plus_plus(A,Z),aa(nat,A,semiring_1_of_nat(A),Na)),M)) ) ).

% pochhammer_product'
tff(fact_2545_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [K: nat,Na: nat] :
          ( aa(nat,$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_2546_even__add__abs__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,plus_plus(int,K),aa(int,int,abs_abs(int),L)))
    <=> aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,plus_plus(int,K),L)) ) ).

% even_add_abs_iff
tff(fact_2547_even__abs__add__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,plus_plus(int,aa(int,int,abs_abs(int),K)),L))
    <=> aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,plus_plus(int,K),L)) ) ).

% even_abs_add_iff
tff(fact_2548_frac__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( archimedean_frac(A,X) = X )
        <=> ( aa(A,$o,ord_less_eq(A,zero_zero(A)),X)
            & aa(A,$o,ord_less(A,X),one_one(A)) ) ) ) ).

% frac_eq
tff(fact_2549_frac__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          archimedean_frac(A,aa(A,A,plus_plus(A,X),Y)) = $ite(aa(A,$o,ord_less(A,aa(A,A,plus_plus(A,archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)),aa(A,A,plus_plus(A,archimedean_frac(A,X)),archimedean_frac(A,Y)),aa(A,A,minus_minus(A,aa(A,A,plus_plus(A,archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A))) ) ).

% frac_add
tff(fact_2550_pochhammer__product,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [M: nat,Na: nat,Z: A] :
          ( aa(nat,$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,plus_plus(A,Z),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,nat,minus_minus(nat,Na),M))) ) ) ) ).

% pochhammer_product
tff(fact_2551_monoseq__realpow,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less_eq(real,X),one_one(real))
       => topological_monoseq(real,power_power(real,X)) ) ) ).

% monoseq_realpow
tff(fact_2552_incr__lemma,axiom,
    ! [D2: int,Z: int,X: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D2)
     => aa(int,$o,ord_less(int,Z),aa(int,int,plus_plus(int,X),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,plus_plus(int,aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,X),Z))),one_one(int))),D2))) ) ).

% incr_lemma
tff(fact_2553_decr__lemma,axiom,
    ! [D2: int,X: int,Z: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D2)
     => aa(int,$o,ord_less(int,aa(int,int,minus_minus(int,X),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,plus_plus(int,aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,X),Z))),one_one(int))),D2))),Z) ) ).

% decr_lemma
tff(fact_2554_pochhammer__absorb__comp,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [R3: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,R3),aa(nat,A,semiring_1_of_nat(A),K))),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),R3),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),R3),comm_s3205402744901411588hammer(A,aa(A,A,plus_plus(A,aa(A,A,uminus_uminus(A),R3)),one_one(A)),K)) ) ).

% pochhammer_absorb_comp
tff(fact_2555_pochhammer__minus,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [B3: A,K: nat] : comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),B3),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),K)),comm_s3205402744901411588hammer(A,aa(A,A,plus_plus(A,aa(A,A,minus_minus(A,B3),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K)) ) ).

% pochhammer_minus
tff(fact_2556_pochhammer__minus_H,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [B3: A,K: nat] : comm_s3205402744901411588hammer(A,aa(A,A,plus_plus(A,aa(A,A,minus_minus(A,B3),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),K)),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),B3),K)) ) ).

% pochhammer_minus'
tff(fact_2557_floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          archim6421214686448440834_floor(A,aa(A,A,plus_plus(A,X),Y)) = $ite(aa(A,$o,ord_less(A,aa(A,A,plus_plus(A,archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)),aa(int,int,plus_plus(int,archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y)),aa(int,int,plus_plus(int,aa(int,int,plus_plus(int,archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y))),one_one(int))) ) ).

% floor_add
tff(fact_2558_nat0__intermed__int__val,axiom,
    ! [Na: nat,F2: fun(nat,int),K: int] :
      ( ! [I3: nat] :
          ( aa(nat,$o,ord_less(nat,I3),Na)
         => aa(int,$o,ord_less_eq(int,aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,aa(nat,int,F2,aa(nat,nat,plus_plus(nat,I3),one_one(nat)))),aa(nat,int,F2,I3)))),one_one(int)) )
     => ( aa(int,$o,ord_less_eq(int,aa(nat,int,F2,zero_zero(nat))),K)
       => ( aa(int,$o,ord_less_eq(int,K),aa(nat,int,F2,Na))
         => ? [I3: nat] :
              ( aa(nat,$o,ord_less_eq(nat,I3),Na)
              & ( aa(nat,int,F2,I3) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_2559_arctan__add,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),X)),one_one(real))
     => ( aa(real,$o,ord_less(real,aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( aa(real,real,plus_plus(real,aa(real,real,arctan,X)),aa(real,real,arctan,Y)) = aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,plus_plus(real,X),Y)),aa(real,real,minus_minus(real,one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),X),Y)))) ) ) ) ).

% arctan_add
tff(fact_2560_summable__divide__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_ax(fun(nat,A),fun(A,fun(nat,A)),F2),C2))
        <=> ( ( C2 = zero_zero(A) )
            | summable(A,F2) ) ) ) ).

% summable_divide_iff
tff(fact_2561_summable__cmult__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,F2: fun(nat,A)] :
          ( summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ay(A,fun(fun(nat,A),fun(nat,A)),C2),F2))
        <=> ( ( C2 = zero_zero(A) )
            | summable(A,F2) ) ) ) ).

% summable_cmult_iff
tff(fact_2562_summable__power__series,axiom,
    ! [F2: fun(nat,real),Z: real] :
      ( ! [I3: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,F2,I3)),one_one(real))
     => ( ! [I3: nat] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(nat,real,F2,I3))
       => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Z)
         => ( aa(real,$o,ord_less(real,Z),one_one(real))
           => summable(real,aa(real,fun(nat,real),aTP_Lamp_az(fun(nat,real),fun(real,fun(nat,real)),F2),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_2563_summable__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),K: nat] :
          ( summable(A,aa(nat,fun(nat,A),aTP_Lamp_ba(fun(nat,A),fun(nat,fun(nat,A)),F2),K))
        <=> summable(A,F2) ) ) ).

% summable_iff_shift
tff(fact_2564_summable__single,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [I: nat,F2: fun(nat,A)] : summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bb(nat,fun(fun(nat,A),fun(nat,A)),I),F2)) ) ).

% summable_single
tff(fact_2565_summable__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => summable(A,aTP_Lamp_bc(nat,A)) ) ).

% summable_zero
tff(fact_2566_summable__complex__of__real,axiom,
    ! [F2: fun(nat,real)] :
      ( summable(complex,aTP_Lamp_bd(fun(nat,real),fun(nat,complex),F2))
    <=> summable(real,F2) ) ).

% summable_complex_of_real
tff(fact_2567_summable__const__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [C2: A] :
          ( summable(A,aTP_Lamp_be(A,fun(nat,A),C2))
        <=> ( C2 = zero_zero(A) ) ) ) ).

% summable_const_iff
tff(fact_2568_summable__add,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F2: fun(nat,A),G: fun(nat,A)] :
          ( summable(A,F2)
         => ( summable(A,G)
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bf(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)) ) ) ) ).

% summable_add
tff(fact_2569_summable__mult,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,F2)
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_bg(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ).

% summable_mult
tff(fact_2570_summable__mult2,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,F2)
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_bh(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ).

% summable_mult2
tff(fact_2571_summable__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),G: fun(nat,A)] :
          ( summable(A,F2)
         => ( summable(A,G)
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bi(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)) ) ) ) ).

% summable_diff
tff(fact_2572_summable__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,F2)
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_ax(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ).

% summable_divide
tff(fact_2573_summable__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => summable(A,aTP_Lamp_bj(fun(nat,A),fun(nat,A),F2)) ) ) ).

% summable_minus
tff(fact_2574_summable__minus__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,aTP_Lamp_bj(fun(nat,A),fun(nat,A),F2))
        <=> summable(A,F2) ) ) ).

% summable_minus_iff
tff(fact_2575_summable__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),K: nat] :
          ( summable(A,F2)
         => summable(A,aa(nat,fun(nat,A),aTP_Lamp_ba(fun(nat,A),fun(nat,fun(nat,A)),F2),K)) ) ) ).

% summable_ignore_initial_segment
tff(fact_2576_summable__rabs__cancel,axiom,
    ! [F2: fun(nat,real)] :
      ( summable(real,aTP_Lamp_bk(fun(nat,real),fun(nat,real),F2))
     => summable(real,F2) ) ).

% summable_rabs_cancel
tff(fact_2577_summable__mult__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,F2: fun(nat,A)] :
          ( summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ay(A,fun(fun(nat,A),fun(nat,A)),C2),F2))
         => ( ( C2 != zero_zero(A) )
           => summable(A,F2) ) ) ) ).

% summable_mult_D
tff(fact_2578_summable__zero__power,axiom,
    ! [A: $tType] :
      ( ( comm_ring_1(A)
        & topolo4958980785337419405_space(A) )
     => summable(A,power_power(A,zero_zero(A))) ) ).

% summable_zero_power
tff(fact_2579_summable__of__real,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => ! [X5: fun(nat,real)] :
          ( summable(real,X5)
         => summable(A,aTP_Lamp_bl(fun(nat,real),fun(nat,A),X5)) ) ) ).

% summable_of_real
tff(fact_2580_summable__0__powser,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,A)] : summable(A,aTP_Lamp_bm(fun(nat,A),fun(nat,A),F2)) ) ).

% summable_0_powser
tff(fact_2581_summable__zero__power_H,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A)] : summable(A,aTP_Lamp_bn(fun(nat,A),fun(nat,A),F2)) ) ).

% summable_zero_power'
tff(fact_2582_summable__powser__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,A),M: nat,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_bo(fun(nat,A),fun(nat,fun(A,fun(nat,A))),F2),M),Z))
        <=> summable(A,aa(A,fun(nat,A),aTP_Lamp_bp(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) ) ) ).

% summable_powser_ignore_initial_segment
tff(fact_2583_summable__rabs__comparison__test,axiom,
    ! [F2: fun(nat,real),G: fun(nat,real)] :
      ( ? [N7: nat] :
        ! [N: nat] :
          ( aa(nat,$o,ord_less_eq(nat,N7),N)
         => aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),aa(nat,real,F2,N))),aa(nat,real,G,N)) )
     => ( summable(real,G)
       => summable(real,aTP_Lamp_bk(fun(nat,real),fun(nat,real),F2)) ) ) ).

% summable_rabs_comparison_test
tff(fact_2584_arctan__series,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),X)),one_one(real))
     => ( aa(real,real,arctan,X) = suminf(real,aTP_Lamp_aw(real,fun(nat,real),X)) ) ) ).

% arctan_series
tff(fact_2585_pochhammer__code,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,Na: nat] :
          comm_s3205402744901411588hammer(A,A3,Na) = $ite(Na = zero_zero(nat),one_one(A),set_fo6178422350223883121st_nat(A,aTP_Lamp_bq(A,fun(nat,fun(A,A)),A3),zero_zero(nat),aa(nat,nat,minus_minus(nat,Na),one_one(nat)),one_one(A))) ) ).

% pochhammer_code
tff(fact_2586_central__binomial__lower__bound,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => aa(real,$o,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(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),aa(nat,nat,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_2587_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,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,plus_plus(A,l),one_one(A))) ) ) ) ) ).

% of_int_code_if
tff(fact_2588_VEBT__internal_Obit__concat__def,axiom,
    ! [H: nat,L: nat,D2: nat] : vEBT_VEBT_bit_concat(H,L,D2) = aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),D2))),L) ).

% VEBT_internal.bit_concat_def
tff(fact_2589_set__decode__0,axiom,
    ! [X: nat] :
      ( member(nat,zero_zero(nat),nat_set_decode(X))
    <=> ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),X) ) ).

% set_decode_0
tff(fact_2590_bits__mod__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),zero_zero(A)),A3) = zero_zero(A) ) ).

% bits_mod_0
tff(fact_2591_mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),zero_zero(A)),A3) = zero_zero(A) ) ).

% mod_0
tff(fact_2592_mod__by__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),zero_zero(A)) = A3 ) ).

% mod_by_0
tff(fact_2593_mod__self,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),A3) = zero_zero(A) ) ).

% mod_self
tff(fact_2594_suminf__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ( suminf(A,aTP_Lamp_br(nat,A)) = zero_zero(A) ) ) ).

% suminf_zero
tff(fact_2595_bits__mod__by__1,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),one_one(A)) = zero_zero(A) ) ).

% bits_mod_by_1
tff(fact_2596_mod__by__1,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),one_one(A)) = zero_zero(A) ) ).

% mod_by_1
tff(fact_2597_mod__mult__self2__is__0,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),B3) = zero_zero(A) ) ).

% mod_mult_self2_is_0
tff(fact_2598_mod__mult__self1__is__0,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B3: A,A3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3)),B3) = zero_zero(A) ) ).

% mod_mult_self1_is_0
tff(fact_2599_bits__mod__div__trivial,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3)),B3) = zero_zero(A) ) ).

% bits_mod_div_trivial
tff(fact_2600_mod__div__trivial,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3)),B3) = zero_zero(A) ) ).

% mod_div_trivial
tff(fact_2601_dvd__imp__mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B3),A3) = zero_zero(A) ) ) ) ).

% dvd_imp_mod_0
tff(fact_2602_set__decode__zero,axiom,
    nat_set_decode(zero_zero(nat)) = bot_bot(set(nat)) ).

% set_decode_zero
tff(fact_2603_mod__minus1__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(A,A,uminus_uminus(A),one_one(A))) = zero_zero(A) ) ).

% mod_minus1_right
tff(fact_2604_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
     => ( aa(int,$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_2605_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,ord_less_eq(int,K),zero_zero(int))
     => ( aa(int,$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_2606_bits__one__mod__two__eq__one,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ).

% bits_one_mod_two_eq_one
tff(fact_2607_one__mod__two__eq__one,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ).

% one_mod_two_eq_one
tff(fact_2608_even__mod__2__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))))
        <=> aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3) ) ) ).

% even_mod_2_iff
tff(fact_2609_zmod__numeral__Bit0,axiom,
    ! [V2: num,W: num] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(num,int,numeral_numeral(int),bit0(V2))),aa(num,int,numeral_numeral(int),bit0(W))) = 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),V2)),aa(num,int,numeral_numeral(int),W))) ).

% zmod_numeral_Bit0
tff(fact_2610_powser__zero,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [F2: fun(nat,A)] : suminf(A,aTP_Lamp_bs(fun(nat,A),fun(nat,A),F2)) = aa(nat,A,F2,zero_zero(nat)) ) ).

% powser_zero
tff(fact_2611_not__mod__2__eq__1__eq__0,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) != one_one(A) )
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_2612_not__mod__2__eq__0__eq__1,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) != zero_zero(A) )
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_2613_bits__minus__1__mod__2__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ).

% bits_minus_1_mod_2_eq
tff(fact_2614_minus__1__mod__2__eq,axiom,
    ! [A: $tType] :
      ( euclid8789492081693882211th_nat(A)
     => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ).

% minus_1_mod_2_eq
tff(fact_2615_one__mod__exp__eq__one,axiom,
    ! [Na: nat] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))) = one_one(int) ).

% one_mod_exp_eq_one
tff(fact_2616_even__succ__mod__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
         => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
           => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,plus_plus(A,one_one(A)),A3)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa(A,A,plus_plus(A,one_one(A)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na))) ) ) ) ) ).

% even_succ_mod_exp
tff(fact_2617_of__nat__mod,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,Na: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Na)) ) ).

% of_nat_mod
tff(fact_2618_power__mod,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,B3: A,Na: nat] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3)),Na)),B3) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(nat,A,power_power(A,A3),Na)),B3) ) ).

% power_mod
tff(fact_2619_dvd__mod__imp__dvd,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,C2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3))
         => ( aa(A,$o,dvd_dvd(A,C2),B3)
           => aa(A,$o,dvd_dvd(A,C2),A3) ) ) ) ).

% dvd_mod_imp_dvd
tff(fact_2620_dvd__mod__iff,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,dvd_dvd(A,C2),B3)
         => ( aa(A,$o,dvd_dvd(A,C2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3))
          <=> aa(A,$o,dvd_dvd(A,C2),A3) ) ) ) ).

% dvd_mod_iff
tff(fact_2621_suminf__of__real,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => ! [X5: fun(nat,real)] :
          ( summable(real,X5)
         => ( aa(real,A,real_Vector_of_real(A),suminf(real,X5)) = suminf(A,aTP_Lamp_bl(fun(nat,real),fun(nat,A),X5)) ) ) ) ).

% suminf_of_real
tff(fact_2622_mod__plus__cong,axiom,
    ! [B3: int,B6: int,X: int,X6: int,Y: int,Y5: int,Z4: int] :
      ( ( B3 = B6 )
     => ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),X),B6) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),X6),B6) )
       => ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Y),B6) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Y5),B6) )
         => ( ( aa(int,int,plus_plus(int,X6),Y5) = Z4 )
           => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,plus_plus(int,X),Y)),B3) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Z4),B6) ) ) ) ) ) ).

% mod_plus_cong
tff(fact_2623_Word_Omod__minus__cong,axiom,
    ! [B3: int,B6: int,X: int,X6: int,Y: int,Y5: int,Z4: int] :
      ( ( B3 = B6 )
     => ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),X),B6) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),X6),B6) )
       => ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Y),B6) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Y5),B6) )
         => ( ( aa(int,int,minus_minus(int,X6),Y5) = Z4 )
           => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,minus_minus(int,X),Y)),B3) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Z4),B6) ) ) ) ) ) ).

% Word.mod_minus_cong
tff(fact_2624_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3)),A3) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_2625_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),B3)
         => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3)),B3) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_2626_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_2627_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_2628_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3) = A3 )
        <=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) = zero_zero(A) ) ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_2629_mod__eq__0__iff__dvd,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3) = zero_zero(A) )
        <=> aa(A,$o,dvd_dvd(A,B3),A3) ) ) ).

% mod_eq_0_iff_dvd
tff(fact_2630_dvd__eq__mod__eq__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,A3),B3)
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B3),A3) = zero_zero(A) ) ) ) ).

% dvd_eq_mod_eq_0
tff(fact_2631_mod__0__imp__dvd,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3) = zero_zero(A) )
         => aa(A,$o,dvd_dvd(A,B3),A3) ) ) ).

% mod_0_imp_dvd
tff(fact_2632_dvd__minus__mod,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [B3: A,A3: A] : aa(A,$o,dvd_dvd(A,B3),aa(A,A,minus_minus(A,A3),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3))) ) ).

% dvd_minus_mod
tff(fact_2633_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),M)
     => aa(int,$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_2634_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,ord_less(int,L),zero_zero(int))
     => aa(int,$o,ord_less(int,L),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L)) ) ).

% neg_mod_bound
tff(fact_2635_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),L)
     => aa(int,$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_2636_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_2637_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_2638_zmod__eq__0D,axiom,
    ! [M: int,D2: int] :
      ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),M),D2) = zero_zero(int) )
     => ? [Q6: int] : M = aa(int,int,aa(int,fun(int,int),times_times(int),D2),Q6) ) ).

% zmod_eq_0D
tff(fact_2639_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) )
    <=> ? [Q5: int] : M = aa(int,int,aa(int,fun(int,int),times_times(int),D2),Q5) ) ).

% zmod_eq_0_iff
tff(fact_2640_suminf__add,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F2: fun(nat,A),G: fun(nat,A)] :
          ( summable(A,F2)
         => ( summable(A,G)
           => ( aa(A,A,plus_plus(A,suminf(A,F2)),suminf(A,G)) = suminf(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bf(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)) ) ) ) ) ).

% suminf_add
tff(fact_2641_suminf__mult2,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,F2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,F2)),C2) = suminf(A,aa(A,fun(nat,A),aTP_Lamp_bh(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ) ).

% suminf_mult2
tff(fact_2642_suminf__mult,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,F2)
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_bg(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),suminf(A,F2)) ) ) ) ).

% suminf_mult
tff(fact_2643_suminf__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),G: fun(nat,A)] :
          ( summable(A,F2)
         => ( summable(A,G)
           => ( aa(A,A,minus_minus(A,suminf(A,F2)),suminf(A,G)) = suminf(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bi(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)) ) ) ) ) ).

% suminf_diff
tff(fact_2644_suminf__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,F2)
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_ax(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),suminf(A,F2)),C2) ) ) ) ).

% suminf_divide
tff(fact_2645_suminf__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( suminf(A,aTP_Lamp_bj(fun(nat,A),fun(nat,A),F2)) = aa(A,A,uminus_uminus(A),suminf(A,F2)) ) ) ) ).

% suminf_minus
tff(fact_2646_suminf__nonneg,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N: nat] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,F2,N))
           => aa(A,$o,ord_less_eq(A,zero_zero(A)),suminf(A,F2)) ) ) ) ).

% suminf_nonneg
tff(fact_2647_suminf__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N: nat] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,F2,N))
           => ( ( suminf(A,F2) = zero_zero(A) )
            <=> ! [N3: nat] : aa(nat,A,F2,N3) = zero_zero(A) ) ) ) ) ).

% suminf_eq_zero_iff
tff(fact_2648_suminf__pos,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N: nat] : aa(A,$o,ord_less(A,zero_zero(A)),aa(nat,A,F2,N))
           => aa(A,$o,ord_less(A,zero_zero(A)),suminf(A,F2)) ) ) ) ).

% suminf_pos
tff(fact_2649_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),B3)
         => aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3)) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_2650_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less(A,A3),B3)
           => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3) = A3 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_2651_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_2652_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_2653_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_2654_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_2655_mult__div__mod__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [B3: A,A3: A] : aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3)) = A3 ) ).

% mult_div_mod_eq
tff(fact_2656_mod__mult__div__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B3: A] : aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3))) = A3 ) ).

% mod_mult_div_eq
tff(fact_2657_mod__div__mult__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B3: A] : aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),B3)) = A3 ) ).

% mod_div_mult_eq
tff(fact_2658_div__mult__mod__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B3: A] : aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),B3)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3)) = A3 ) ).

% div_mult_mod_eq
tff(fact_2659_mod__div__decomp,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B3: A] : A3 = aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),B3)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3)) ) ).

% mod_div_decomp
tff(fact_2660_cancel__div__mod__rules_I1_J,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,plus_plus(A,aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),B3)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3))),C2) = aa(A,A,plus_plus(A,A3),C2) ) ).

% cancel_div_mod_rules(1)
tff(fact_2661_cancel__div__mod__rules_I2_J,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [B3: A,A3: A,C2: A] : aa(A,A,plus_plus(A,aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3))),C2) = aa(A,A,plus_plus(A,A3),C2) ) ).

% cancel_div_mod_rules(2)
tff(fact_2662_minus__mult__div__eq__mod,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B3: A] : aa(A,A,minus_minus(A,A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3))) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3) ) ).

% minus_mult_div_eq_mod
tff(fact_2663_minus__mod__eq__mult__div,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B3: A] : aa(A,A,minus_minus(A,A3),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),B3),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)) ) ).

% minus_mod_eq_mult_div
tff(fact_2664_minus__mod__eq__div__mult,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B3: A] : aa(A,A,minus_minus(A,A3),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),B3) ) ).

% minus_mod_eq_div_mult
tff(fact_2665_minus__div__mult__eq__mod,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B3: A] : aa(A,A,minus_minus(A,A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),B3)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3) ) ).

% minus_div_mult_eq_mod
tff(fact_2666_unit__imp__mod__eq__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,dvd_dvd(A,B3),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3) = zero_zero(A) ) ) ) ).

% unit_imp_mod_eq_0
tff(fact_2667_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,$o,ord_less_eq(int,zero_zero(int)),I)
          & aa(int,$o,ord_less(int,I),K) )
        | ( aa(int,$o,ord_less_eq(int,I),zero_zero(int))
          & aa(int,$o,ord_less(int,K),I) ) ) ) ).

% zmod_trivial_iff
tff(fact_2668_pos__mod__conj,axiom,
    ! [B3: int,A3: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),B3)
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B3))
        & aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B3)),B3) ) ) ).

% pos_mod_conj
tff(fact_2669_neg__mod__conj,axiom,
    ! [B3: int,A3: int] :
      ( aa(int,$o,ord_less(int,B3),zero_zero(int))
     => ( aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B3)),zero_zero(int))
        & aa(int,$o,ord_less(int,B3),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B3)) ) ) ).

% neg_mod_conj
tff(fact_2670_int__mod__ge,axiom,
    ! [A3: int,Na: int] :
      ( aa(int,$o,ord_less(int,A3),Na)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),Na)
       => aa(int,$o,ord_less_eq(int,A3),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),Na)) ) ) ).

% int_mod_ge
tff(fact_2671_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),L)
     => aa(int,$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_2672_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,ord_less(int,L),zero_zero(int))
     => aa(int,$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_2673_int__mod__lem,axiom,
    ! [Na: int,B3: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),Na)
     => ( ( aa(int,$o,ord_less_eq(int,zero_zero(int)),B3)
          & aa(int,$o,ord_less(int,B3),Na) )
      <=> ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B3),Na) = B3 ) ) ) ).

% int_mod_lem
tff(fact_2674_int__mod__eq,axiom,
    ! [B3: int,Na: int,A3: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),B3)
     => ( aa(int,$o,ord_less(int,B3),Na)
       => ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),Na) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B3),Na) )
         => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),Na) = B3 ) ) ) ) ).

% int_mod_eq
tff(fact_2675_int__mod__le_H,axiom,
    ! [B3: int,Na: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,minus_minus(int,B3),Na))
     => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B3),Na)),aa(int,int,minus_minus(int,B3),Na)) ) ).

% int_mod_le'
tff(fact_2676_zmod__zminus2__eq__if,axiom,
    ! [A3: int,B3: int] :
      aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),aa(int,int,uminus_uminus(int),B3)) = $ite(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B3) = zero_zero(int),zero_zero(int),aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B3)),B3)) ).

% zmod_zminus2_eq_if
tff(fact_2677_zmod__zminus1__eq__if,axiom,
    ! [A3: int,B3: int] :
      aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),A3)),B3) = $ite(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B3) = zero_zero(int),zero_zero(int),aa(int,int,minus_minus(int,B3),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B3))) ).

% zmod_zminus1_eq_if
tff(fact_2678_nonneg__mod__div,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),A3)
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),B3)
       => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B3))
          & aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3)) ) ) ) ).

% nonneg_mod_div
tff(fact_2679_zdiv__mono__strict,axiom,
    ! [A2: int,B2: int,Na: int] :
      ( aa(int,$o,ord_less(int,A2),B2)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),Na)
       => ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),Na) = zero_zero(int) )
         => ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B2),Na) = zero_zero(int) )
           => aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),Na)),aa(int,int,aa(int,fun(int,int),divide_divide(int),B2),Na)) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_2680_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => aa(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_2681_div__mod__decomp__int,axiom,
    ! [A2: int,Na: int] : A2 = aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),Na)),Na)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),Na)) ).

% div_mod_decomp_int
tff(fact_2682_summable__rabs,axiom,
    ! [F2: fun(nat,real)] :
      ( summable(real,aTP_Lamp_bk(fun(nat,real),fun(nat,real),F2))
     => aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),suminf(real,F2))),suminf(real,aTP_Lamp_bk(fun(nat,real),fun(nat,real),F2))) ) ).

% summable_rabs
tff(fact_2683_suminf__pos2,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),I: nat] :
          ( summable(A,F2)
         => ( ! [N: nat] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,F2,N))
           => ( aa(A,$o,ord_less(A,zero_zero(A)),aa(nat,A,F2,I))
             => aa(A,$o,ord_less(A,zero_zero(A)),suminf(A,F2)) ) ) ) ) ).

% suminf_pos2
tff(fact_2684_suminf__pos__iff,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N: nat] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,F2,N))
           => ( aa(A,$o,ord_less(A,zero_zero(A)),suminf(A,F2))
            <=> ? [I4: nat] : aa(A,$o,ord_less(A,zero_zero(A)),aa(nat,A,F2,I4)) ) ) ) ) ).

% suminf_pos_iff
tff(fact_2685_mod__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A,M: nat,Na: nat] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Na))) = aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,semiring_1_of_nat(A),M))),aa(nat,A,semiring_1_of_nat(A),Na)))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(nat,A,semiring_1_of_nat(A),M))) ) ).

% mod_mult2_eq'
tff(fact_2686_pos__mod__bound2,axiom,
    ! [A3: int] : aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% pos_mod_bound2
tff(fact_2687_int__mod__ge_H,axiom,
    ! [B3: int,Na: int] :
      ( aa(int,$o,ord_less(int,B3),zero_zero(int))
     => ( aa(int,$o,ord_less(int,zero_zero(int)),Na)
       => aa(int,$o,ord_less_eq(int,aa(int,int,plus_plus(int,B3),Na)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B3),Na)) ) ) ).

% int_mod_ge'
tff(fact_2688_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),K)
     => ( aa(int,$o,ord_less_eq(int,aa(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,plus_plus(int,K),L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_2689_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),L)
     => ( aa(int,$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,minus_minus(int,K),L)),L) ) ) ) ).

% mod_pos_geq
tff(fact_2690_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa(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,dvd_dvd(int,L),K)
        | ( ( L = zero_zero(int) )
          & aa(int,$o,ord_less_eq(int,zero_zero(int)),K) )
        | aa(int,$o,ord_less(int,zero_zero(int)),L) ) ) ).

% mod_int_pos_iff
tff(fact_2691_real__of__int__div__aux,axiom,
    ! [X: int,D2: int] : aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),X)),aa(int,real,ring_1_of_int(real),D2)) = aa(real,real,plus_plus(real,aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),divide_divide(int),X),D2))),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),modulo_modulo(int),X),D2))),aa(int,real,ring_1_of_int(real),D2))) ).

% real_of_int_div_aux
tff(fact_2692_binomial__antimono,axiom,
    ! [K: nat,K5: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),K5)
     => ( aa(nat,$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,$o,ord_less_eq(nat,K5),Na)
         => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,binomial(Na),K5)),aa(nat,nat,binomial(Na),K)) ) ) ) ).

% binomial_antimono
tff(fact_2693_binomial__maximum,axiom,
    ! [Na: nat,K: nat] : aa(nat,$o,ord_less_eq(nat,aa(nat,nat,binomial(Na),K)),aa(nat,nat,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_2694_binomial__maximum_H,axiom,
    ! [Na: nat,K: nat] : aa(nat,$o,ord_less_eq(nat,aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),K)),aa(nat,nat,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_2695_binomial__mono,axiom,
    ! [K: nat,K5: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),K5)
     => ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K5)),Na)
       => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,binomial(Na),K)),aa(nat,nat,binomial(Na),K5)) ) ) ).

% binomial_mono
tff(fact_2696_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
         => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) = aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),C2))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3)) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_2697_even__iff__mod__2__eq__zero,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_2698_odd__iff__mod__2__eq__one,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_2699_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [A: $tType,X: fun(nat,fun(A,A)),Xa2: nat,Xb: nat,Xc: A,Y: A] :
      ( ( set_fo6178422350223883121st_nat(A,X,Xa2,Xb,Xc) = Y )
     => ( Y = $ite(aa(nat,$o,ord_less(nat,Xb),Xa2),Xc,set_fo6178422350223883121st_nat(A,X,aa(nat,nat,plus_plus(nat,Xa2),one_one(nat)),Xb,aa(A,A,aa(nat,fun(A,A),X,Xa2),Xc))) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_2700_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [A: $tType,F2: fun(nat,fun(A,A)),A3: nat,B3: nat,Acc: A] :
      set_fo6178422350223883121st_nat(A,F2,A3,B3,Acc) = $ite(aa(nat,$o,ord_less(nat,B3),A3),Acc,set_fo6178422350223883121st_nat(A,F2,aa(nat,nat,plus_plus(nat,A3),one_one(nat)),B3,aa(A,A,aa(nat,fun(A,A),F2,A3),Acc))) ).

% fold_atLeastAtMost_nat.simps
tff(fact_2701_pos__mod__sign2,axiom,
    ! [A3: int] : aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),aa(num,int,numeral_numeral(int),bit0(one2)))) ).

% pos_mod_sign2
tff(fact_2702_nmod2,axiom,
    ! [Na: int] :
      ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Na),aa(num,int,numeral_numeral(int),bit0(one2))) = zero_zero(int) )
      | ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Na),aa(num,int,numeral_numeral(int),bit0(one2))) = one_one(int) ) ) ).

% nmod2
tff(fact_2703_mod__2__neq__1__eq__eq__0,axiom,
    ! [K: int] :
      ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),aa(num,int,numeral_numeral(int),bit0(one2))) != one_one(int) )
    <=> ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),aa(num,int,numeral_numeral(int),bit0(one2))) = zero_zero(int) ) ) ).

% mod_2_neq_1_eq_eq_0
tff(fact_2704_mod__exp__less__eq__exp,axiom,
    ! [A3: int,Na: nat] : aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ).

% mod_exp_less_eq_exp
tff(fact_2705_mod__power__lem,axiom,
    ! [A3: int,Na: nat,M: nat] :
      ( aa(int,$o,ord_less(int,one_one(int)),A3)
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(nat,int,power_power(int,A3),Na)),aa(nat,int,power_power(int,A3),M)) = $ite(aa(nat,$o,ord_less_eq(nat,M),Na),zero_zero(int),aa(nat,int,power_power(int,A3),Na)) ) ) ).

% mod_power_lem
tff(fact_2706_int__mod__pos__eq,axiom,
    ! [A3: int,B3: int,Q3: int,R3: int] :
      ( ( A3 = aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q3)),R3) )
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),R3)
       => ( aa(int,$o,ord_less(int,R3),B3)
         => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B3) = R3 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_2707_int__mod__neg__eq,axiom,
    ! [A3: int,B3: int,Q3: int,R3: int] :
      ( ( A3 = aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q3)),R3) )
     => ( aa(int,$o,ord_less_eq(int,R3),zero_zero(int))
       => ( aa(int,$o,ord_less(int,B3),R3)
         => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B3) = R3 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_2708_split__zmod,axiom,
    ! [P: fun(int,$o),Na: int,K: int] :
      ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Na),K))
    <=> ( ( ( K = zero_zero(int) )
         => aa(int,$o,P,Na) )
        & ( aa(int,$o,ord_less(int,zero_zero(int)),K)
         => ! [I4: int,J3: int] :
              ( ( aa(int,$o,ord_less_eq(int,zero_zero(int)),J3)
                & aa(int,$o,ord_less(int,J3),K)
                & ( Na = aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
             => aa(int,$o,P,J3) ) )
        & ( aa(int,$o,ord_less(int,K),zero_zero(int))
         => ! [I4: int,J3: int] :
              ( ( aa(int,$o,ord_less(int,K),J3)
                & aa(int,$o,ord_less_eq(int,J3),zero_zero(int))
                & ( Na = aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
             => aa(int,$o,P,J3) ) ) ) ) ).

% split_zmod
tff(fact_2709_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( aa(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,minus_minus(int,aa(int,int,minus_minus(int,L),one_one(int))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,minus_minus(int,K),one_one(int))),L)) ) ) ).

% minus_mod_int_eq
tff(fact_2710_zmod__minus1,axiom,
    ! [B3: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),B3)
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),one_one(int))),B3) = aa(int,int,minus_minus(int,B3),one_one(int)) ) ) ).

% zmod_minus1
tff(fact_2711_mod__sub__if__z,axiom,
    ! [X: int,Z: int,Y: int] :
      ( aa(int,$o,ord_less(int,X),Z)
     => ( aa(int,$o,ord_less(int,Y),Z)
       => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Y)
         => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),X)
           => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z)
             => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,minus_minus(int,X),Y)),Z) = $ite(aa(int,$o,ord_less_eq(int,Y),X),aa(int,int,minus_minus(int,X),Y),aa(int,int,plus_plus(int,aa(int,int,minus_minus(int,X),Y)),Z)) ) ) ) ) ) ) ).

% mod_sub_if_z
tff(fact_2712_mod__add__if__z,axiom,
    ! [X: int,Z: int,Y: int] :
      ( aa(int,$o,ord_less(int,X),Z)
     => ( aa(int,$o,ord_less(int,Y),Z)
       => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Y)
         => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),X)
           => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z)
             => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,plus_plus(int,X),Y)),Z) = $ite(aa(int,$o,ord_less(int,aa(int,int,plus_plus(int,X),Y)),Z),aa(int,int,plus_plus(int,X),Y),aa(int,int,minus_minus(int,aa(int,int,plus_plus(int,X),Y)),Z)) ) ) ) ) ) ) ).

% mod_add_if_z
tff(fact_2713_zmod__zmult2__eq,axiom,
    ! [C2: int,A3: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),C2)
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),aa(int,int,aa(int,fun(int,int),times_times(int),B3),C2)) = aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),times_times(int),B3),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3)),C2))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B3)) ) ) ).

% zmod_zmult2_eq
tff(fact_2714_zdiv__zminus2__eq__if,axiom,
    ! [B3: int,A3: int] :
      ( ( B3 != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),aa(int,int,uminus_uminus(int),B3)) = $ite(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B3) = zero_zero(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3)),aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3))),one_one(int))) ) ) ).

% zdiv_zminus2_eq_if
tff(fact_2715_zdiv__zminus1__eq__if,axiom,
    ! [B3: int,A3: int] :
      ( ( B3 != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),A3)),B3) = $ite(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B3) = zero_zero(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3)),aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3))),one_one(int))) ) ) ).

% zdiv_zminus1_eq_if
tff(fact_2716_binomial__strict__antimono,axiom,
    ! [K: nat,K5: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,K),K5)
     => ( aa(nat,$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,$o,ord_less_eq(nat,K5),Na)
         => aa(nat,$o,ord_less(nat,aa(nat,nat,binomial(Na),K5)),aa(nat,nat,binomial(Na),K)) ) ) ) ).

% binomial_strict_antimono
tff(fact_2717_binomial__strict__mono,axiom,
    ! [K: nat,K5: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,K),K5)
     => ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K5)),Na)
       => aa(nat,$o,ord_less(nat,aa(nat,nat,binomial(Na),K)),aa(nat,nat,binomial(Na),K5)) ) ) ).

% binomial_strict_mono
tff(fact_2718_divmod__digit__0_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),B3)
         => ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3))),B3)
           => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3) ) ) ) ) ).

% divmod_digit_0(2)
tff(fact_2719_bits__stable__imp__add__self,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) = A3 )
         => ( aa(A,A,plus_plus(A,A3),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))) = zero_zero(A) ) ) ) ).

% bits_stable_imp_add_self
tff(fact_2720_parity__cases,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
           => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) != zero_zero(A) ) )
         => ~ ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
             => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) != one_one(A) ) ) ) ) ).

% parity_cases
tff(fact_2721_mod2__eq__if,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) = $ite(aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3),zero_zero(A),one_one(A)) ) ).

% mod2_eq_if
tff(fact_2722_div__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,Na: nat,M: nat] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na))),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,plus_plus(nat,Na),M)))),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) ) ).

% div_exp_mod_exp_eq
tff(fact_2723_axxmod2,axiom,
    ! [X: int] :
      ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,plus_plus(int,aa(int,int,plus_plus(int,one_one(int)),X)),X)),aa(num,int,numeral_numeral(int),bit0(one2))) = one_one(int) )
      & ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,plus_plus(int,aa(int,int,plus_plus(int,zero_zero(int)),X)),X)),aa(num,int,numeral_numeral(int),bit0(one2))) = zero_zero(int) ) ) ).

% axxmod2
tff(fact_2724_z1pmod2,axiom,
    ! [B3: int] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B3)),one_one(int))),aa(num,int,numeral_numeral(int),bit0(one2))) = one_one(int) ).

% z1pmod2
tff(fact_2725_verit__le__mono__div__int,axiom,
    ! [A2: int,B2: int,Na: int] :
      ( aa(int,$o,ord_less(int,A2),B2)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),Na)
       => aa(int,$o,
            ord_less_eq(int,
              aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),Na)),
                $ite(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B2),Na) = zero_zero(int),one_one(int),zero_zero(int)))),
            aa(int,int,aa(int,fun(int,int),divide_divide(int),B2),Na)) ) ) ).

% verit_le_mono_div_int
tff(fact_2726_split__neg__lemma,axiom,
    ! [K: int,P: fun(int,fun(int,$o)),Na: int] :
      ( aa(int,$o,ord_less(int,K),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),P,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))
      <=> ! [I4: int,J3: int] :
            ( ( aa(int,$o,ord_less(int,K),J3)
              & aa(int,$o,ord_less_eq(int,J3),zero_zero(int))
              & ( Na = aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
           => aa(int,$o,aa(int,fun(int,$o),P,I4),J3) ) ) ) ).

% split_neg_lemma
tff(fact_2727_split__pos__lemma,axiom,
    ! [K: int,P: fun(int,fun(int,$o)),Na: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),P,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))
      <=> ! [I4: int,J3: int] :
            ( ( aa(int,$o,ord_less_eq(int,zero_zero(int)),J3)
              & aa(int,$o,ord_less(int,J3),K)
              & ( Na = aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
           => aa(int,$o,aa(int,fun(int,$o),P,I4),J3) ) ) ) ).

% split_pos_lemma
tff(fact_2728_divmod__digit__0_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),B3)
         => ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3))),B3)
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) ) ) ) ) ).

% divmod_digit_0(1)
tff(fact_2729_mult__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Na: nat,A3: A] :
          ( aa(nat,$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),A3),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M))),aa(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),A3),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,minus_minus(nat,Na),M)))),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M)) ) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_2730_p1mod22k_H,axiom,
    ! [B3: int,Na: nat] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(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))),B3))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))) = aa(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),B3),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)))) ).

% p1mod22k'
tff(fact_2731_p1mod22k,axiom,
    ! [B3: int,Na: nat] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B3)),one_one(int))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))) = aa(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),B3),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)))),one_one(int)) ).

% p1mod22k
tff(fact_2732_m1mod2k,axiom,
    ! [Na: nat] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) = aa(int,int,minus_minus(int,aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)),one_one(int)) ).

% m1mod2k
tff(fact_2733_mod__double__modulus,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: A,X: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),M)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),X)
           => ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),X),M) )
              | ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)) = aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),modulo_modulo(A),X),M)),M) ) ) ) ) ) ).

% mod_double_modulus
tff(fact_2734_divmod__digit__1_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),B3)
           => ( aa(A,$o,ord_less_eq(A,B3),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3)))
             => ( aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3))),B3) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3) ) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_2735_pos__zmod__mult__2,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),A3)
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(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))),B3))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A3)) = aa(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),B3),A3))) ) ) ).

% pos_zmod_mult_2
tff(fact_2736_eme1p,axiom,
    ! [Na: int,D2: int] :
      ( aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)
     => ( aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),D2)
       => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),D2)
         => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,plus_plus(int,one_one(int)),Na)),D2) = aa(int,int,plus_plus(int,one_one(int)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Na),D2)) ) ) ) ) ).

% eme1p
tff(fact_2737_emep1,axiom,
    ! [Na: int,D2: int] :
      ( aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)
     => ( aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),D2)
       => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),D2)
         => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,plus_plus(int,Na),one_one(int))),D2) = aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Na),D2)),one_one(int)) ) ) ) ) ).

% emep1
tff(fact_2738_sb__dec__lem_H,axiom,
    ! [K: nat,A3: int] :
      ( aa(int,$o,ord_less_eq(int,aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),K)),A3)
     => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,plus_plus(int,A3),aa(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,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),K)))),aa(int,int,plus_plus(int,aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),K))),A3)) ) ).

% sb_dec_lem'
tff(fact_2739_m1mod22k,axiom,
    ! [Na: nat] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))) = aa(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(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))),one_one(int)) ).

% m1mod22k
tff(fact_2740_neg__zmod__mult__2,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,A3),zero_zero(int))
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(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))),B3))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A3)) = aa(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,plus_plus(int,B3),one_one(int))),A3))),one_one(int)) ) ) ).

% neg_zmod_mult_2
tff(fact_2741_sb__dec__lem,axiom,
    ! [K: nat,A3: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,plus_plus(int,aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),K))),A3))
     => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,plus_plus(int,A3),aa(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,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),K)))),aa(int,int,plus_plus(int,aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),K))),A3)) ) ).

% sb_dec_lem
tff(fact_2742_divmod__digit__1_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),B3)
           => ( aa(A,$o,ord_less_eq(A,B3),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3)))
             => ( aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3)))),one_one(A)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) ) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_2743_set__decode__plus__power__2,axiom,
    ! [Na: nat,Z: nat] :
      ( ~ member(nat,Na,nat_set_decode(Z))
     => ( nat_set_decode(aa(nat,nat,plus_plus(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),Z)) = aa(set(nat),set(nat),insert(nat,Na),nat_set_decode(Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_2744_set__decode__def,axiom,
    ! [X: nat] : nat_set_decode(X) = collect(nat,aTP_Lamp_bt(nat,fun(nat,$o),X)) ).

% set_decode_def
tff(fact_2745_zero__less__binomial__iff,axiom,
    ! [Na: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(nat,nat,binomial(Na),K))
    <=> aa(nat,$o,ord_less_eq(nat,K),Na) ) ).

% zero_less_binomial_iff
tff(fact_2746_choose__two,axiom,
    ! [Na: nat] : aa(nat,nat,binomial(Na),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(nat,nat,minus_minus(nat,Na),one_one(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% choose_two
tff(fact_2747_binomial__n__0,axiom,
    ! [Na: nat] : aa(nat,nat,binomial(Na),zero_zero(nat)) = one_one(nat) ).

% binomial_n_0
tff(fact_2748_binomial__eq__0__iff,axiom,
    ! [Na: nat,K: nat] :
      ( ( aa(nat,nat,binomial(Na),K) = zero_zero(nat) )
    <=> aa(nat,$o,ord_less(nat,Na),K) ) ).

% binomial_eq_0_iff
tff(fact_2749_times__binomial__minus1__eq,axiom,
    ! [K: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,binomial(Na),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,Na),one_one(nat))),aa(nat,nat,minus_minus(nat,K),one_one(nat)))) ) ) ).

% times_binomial_minus1_eq
tff(fact_2750_choose__reduce__nat,axiom,
    ! [Na: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
       => ( aa(nat,nat,binomial(Na),K) = aa(nat,nat,plus_plus(nat,aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,Na),one_one(nat))),aa(nat,nat,minus_minus(nat,K),one_one(nat)))),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,Na),one_one(nat))),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_2751_mod__word__self,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),W),W) = zero_zero(word(A)) ) ).

% mod_word_self
tff(fact_2752_nat__mod__eq_H,axiom,
    ! [A3: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,A3),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A3),Na) = A3 ) ) ).

% nat_mod_eq'
tff(fact_2753_mod__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa(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_2754_binomial__n__n,axiom,
    ! [Na: nat] : aa(nat,nat,binomial(Na),Na) = one_one(nat) ).

% binomial_n_n
tff(fact_2755_word__mod__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),A3))),aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),B3)))) ) ).

% word_mod_no
tff(fact_2756_add__self__mod__2,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,plus_plus(nat,M),M)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = zero_zero(nat) ).

% add_self_mod_2
tff(fact_2757_mod2__gr__0,axiom,
    ! [M: nat] :
      ( aa(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_2758_unat__mod,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: 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)),modulo_modulo(word(A)),X),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) ).

% unat_mod
tff(fact_2759_unat__mod__distrib,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V2: word(A),W: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),V2),W)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),V2)),aa(word(A),nat,semiring_1_unsigned(A,nat),W)) ) ).

% unat_mod_distrib
tff(fact_2760_word__arith__nat__mod,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),A3),B3) = aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A3)),aa(word(A),nat,semiring_1_unsigned(A,nat),B3))) ) ).

% word_arith_nat_mod
tff(fact_2761_nat__mod__eq,axiom,
    ! [B3: nat,Na: nat,A3: nat] :
      ( aa(nat,$o,ord_less(nat,B3),Na)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A3),Na) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),B3),Na) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A3),Na) = B3 ) ) ) ).

% nat_mod_eq
tff(fact_2762_word__mod__by__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),K),zero_zero(word(A))) = K ) ).

% word_mod_by_0
tff(fact_2763_mod__word__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),V2: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),W),V2)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),W),V2) = W ) ) ) ).

% mod_word_less
tff(fact_2764_gcd__nat__induct,axiom,
    ! [P: fun(nat,fun(nat,$o)),M: nat,Na: nat] :
      ( ! [M4: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,M4),zero_zero(nat))
     => ( ! [M4: nat,N: nat] :
            ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),P,N),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M4),N))
             => aa(nat,$o,aa(nat,fun(nat,$o),P,M4),N) ) )
       => aa(nat,$o,aa(nat,fun(nat,$o),P,M),Na) ) ) ).

% gcd_nat_induct
tff(fact_2765_nat__mod__lem,axiom,
    ! [Na: nat,B3: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(nat,$o,ord_less(nat,B3),Na)
      <=> ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),B3),Na) = B3 ) ) ) ).

% nat_mod_lem
tff(fact_2766_mod__less__divisor,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na)),Na) ) ).

% mod_less_divisor
tff(fact_2767_word__rot__lem,axiom,
    ! [L: nat,K: nat,D2: nat,Na: nat] :
      ( ( aa(nat,nat,plus_plus(nat,L),K) = aa(nat,nat,plus_plus(nat,D2),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),K),L)) )
     => ( aa(nat,$o,ord_less(nat,Na),L)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,plus_plus(nat,D2),Na)),L) = Na ) ) ) ).

% word_rot_lem
tff(fact_2768_nat__minus__mod,axiom,
    ! [Na: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(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_2769_mod__nat__sub,axiom,
    ! [X: nat,Z: nat,Y: nat] :
      ( aa(nat,$o,ord_less(nat,X),Z)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,minus_minus(nat,X),Y)),Z) = aa(nat,nat,minus_minus(nat,X),Y) ) ) ).

% mod_nat_sub
tff(fact_2770_mod__if,axiom,
    ! [M: nat,Na: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na) = $ite(aa(nat,$o,ord_less(nat,M),Na),M,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,minus_minus(nat,M),Na)),Na)) ).

% mod_if
tff(fact_2771_mod__geq,axiom,
    ! [M: nat,Na: nat] :
      ( ~ aa(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,minus_minus(nat,M),Na)),Na) ) ) ).

% mod_geq
tff(fact_2772_mod__eq__0D,axiom,
    ! [M: nat,D2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),D2) = zero_zero(nat) )
     => ? [Q6: nat] : M = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),D2),Q6) ) ).

% mod_eq_0D
tff(fact_2773_msrevs_I2_J,axiom,
    ! [K: nat,Na: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(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),M),Na) ).

% msrevs(2)
tff(fact_2774_word__mod__less__divisor,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A),M: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),Na)
         => aa(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_2775_uint__mod,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: 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)),modulo_modulo(word(A)),X),Y)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ).

% uint_mod
tff(fact_2776_uint__mod__distrib,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V2: word(A),W: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),V2),W)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(word(A),int,semiring_1_unsigned(A,int),V2)),aa(word(A),int,semiring_1_unsigned(A,int),W)) ) ).

% uint_mod_distrib
tff(fact_2777_zmod__int,axiom,
    ! [A3: nat,B3: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A3),B3)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B3)) ).

% zmod_int
tff(fact_2778_mod__le__divisor,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na)),Na) ) ).

% mod_le_divisor
tff(fact_2779_div__less__mono,axiom,
    ! [A2: nat,B2: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,A2),B2)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
       => ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A2),Na) = zero_zero(nat) )
         => ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),B2),Na) = zero_zero(nat) )
           => aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A2),Na)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),B2),Na)) ) ) ) ) ).

% div_less_mono
tff(fact_2780_mod__nat__add,axiom,
    ! [X: nat,Z: nat,Y: nat] :
      ( aa(nat,$o,ord_less(nat,X),Z)
     => ( aa(nat,$o,ord_less(nat,Y),Z)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,plus_plus(nat,X),Y)),Z) = $ite(aa(nat,$o,ord_less(nat,aa(nat,nat,plus_plus(nat,X),Y)),Z),aa(nat,nat,plus_plus(nat,X),Y),aa(nat,nat,minus_minus(nat,aa(nat,nat,plus_plus(nat,X),Y)),Z)) ) ) ) ).

% mod_nat_add
tff(fact_2781_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na))
    <=> ~ aa(nat,$o,dvd_dvd(nat,Na),M) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_2782_div__mod__decomp,axiom,
    ! [A2: nat,Na: nat] : A2 = aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A2),Na)),Na)),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A2),Na)) ).

% div_mod_decomp
tff(fact_2783_word__mod__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),A3),B3) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(word(A),int,semiring_1_unsigned(A,int),A3)),aa(word(A),int,semiring_1_unsigned(A,int),B3))) ) ).

% word_mod_def
tff(fact_2784_word__mod__div__equality,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A),B3: word(A)] : aa(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(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Na),B3)),B3)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),Na),B3)) = Na ) ).

% word_mod_div_equality
tff(fact_2785_even__even__mod__4__iff,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
    <=> aa(nat,$o,dvd_dvd(nat,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(bit0(one2))))) ) ).

% even_even_mod_4_iff
tff(fact_2786_field__char__0__class_Oof__nat__div,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat,Na: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na)))),aa(nat,A,semiring_1_of_nat(A),Na)) ) ).

% field_char_0_class.of_nat_div
tff(fact_2787_mod__lemma,axiom,
    ! [C2: nat,R3: nat,B3: nat,Q3: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),C2)
     => ( aa(nat,$o,ord_less(nat,R3),B3)
       => aa(nat,$o,ord_less(nat,aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B3),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Q3),C2))),R3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B3),C2)) ) ) ).

% mod_lemma
tff(fact_2788_split__mod,axiom,
    ! [P: fun(nat,$o),M: nat,Na: nat] :
      ( aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na))
    <=> ( ( ( Na = zero_zero(nat) )
         => aa(nat,$o,P,M) )
        & ( ( Na != zero_zero(nat) )
         => ! [I4: nat,J3: nat] :
              ( aa(nat,$o,ord_less(nat,J3),Na)
             => ( ( M = aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),I4)),J3) )
               => aa(nat,$o,P,J3) ) ) ) ) ) ).

% split_mod
tff(fact_2789_diff__mod__le,axiom,
    ! [A3: nat,D2: nat,B3: nat] :
      ( aa(nat,$o,ord_less(nat,A3),D2)
     => ( aa(nat,$o,dvd_dvd(nat,B3),D2)
       => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,A3),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A3),B3))),aa(nat,nat,minus_minus(nat,D2),B3)) ) ) ).

% diff_mod_le
tff(fact_2790_mod__nat__eqI,axiom,
    ! [R3: nat,Na: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,R3),Na)
     => ( aa(nat,$o,ord_less_eq(nat,R3),M)
       => ( aa(nat,$o,dvd_dvd(nat,Na),aa(nat,nat,minus_minus(nat,M),R3))
         => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na) = R3 ) ) ) ) ).

% mod_nat_eqI
tff(fact_2791_uno__simps_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Z: word(A),M: nat] :
          ( aa(nat,$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_2792_real__of__nat__div__aux,axiom,
    ! [X: nat,D2: nat] : aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),X)),aa(nat,real,semiring_1_of_nat(real),D2)) = aa(real,real,plus_plus(real,aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),D2))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),D2))),aa(nat,real,semiring_1_of_nat(real),D2))) ).

% real_of_nat_div_aux
tff(fact_2793_power__mod__div,axiom,
    ! [X: nat,Na: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,Na),M))) ).

% power_mod_div
tff(fact_2794_verit__le__mono__div,axiom,
    ! [A2: nat,B2: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,A2),B2)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
       => aa(nat,$o,
            ord_less_eq(nat,
              aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A2),Na)),
                $ite(aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),B2),Na) = zero_zero(nat),one_one(nat),zero_zero(nat)))),
            aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),B2),Na)) ) ) ).

% verit_le_mono_div
tff(fact_2795_choose__one,axiom,
    ! [Na: nat] : aa(nat,nat,binomial(Na),one_one(nat)) = Na ).

% choose_one
tff(fact_2796_binomial__eq__0,axiom,
    ! [Na: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,Na),K)
     => ( aa(nat,nat,binomial(Na),K) = zero_zero(nat) ) ) ).

% binomial_eq_0
tff(fact_2797_binomial__le__pow,axiom,
    ! [R3: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,R3),Na)
     => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,binomial(Na),R3)),aa(nat,nat,power_power(nat,Na),R3)) ) ).

% binomial_le_pow
tff(fact_2798_zero__less__binomial,axiom,
    ! [K: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),Na)
     => aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(nat,nat,binomial(Na),K)) ) ).

% zero_less_binomial
tff(fact_2799_binomial__absorb__comp,axiom,
    ! [Na: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,Na),K)),aa(nat,nat,binomial(Na),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,Na),one_one(nat))),K)) ).

% binomial_absorb_comp
tff(fact_2800_binomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,ord_less_eq(nat,K),Na)
         => aa(A,$o,ord_less_eq(A,aa(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),aa(nat,nat,binomial(Na),K))) ) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_2801_binomial__le__pow2,axiom,
    ! [Na: nat,K: nat] : aa(nat,$o,ord_less_eq(nat,aa(nat,nat,binomial(Na),K)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ).

% binomial_le_pow2
tff(fact_2802_ln__series,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less(real,X),aa(num,real,numeral_numeral(real),bit0(one2)))
       => ( aa(real,real,ln_ln(real),X) = suminf(real,aTP_Lamp_bu(real,fun(nat,real),X)) ) ) ) ).

% ln_series
tff(fact_2803_pi__series,axiom,
    aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))) = suminf(real,aTP_Lamp_bv(nat,real)) ).

% pi_series
tff(fact_2804_signed__take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,A3: A] :
          bit_ri4674362597316999326ke_bit(A,Na,A3) = $ite(Na = zero_zero(nat),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_ri4674362597316999326ke_bit(A,aa(nat,nat,minus_minus(nat,Na),one_one(nat)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))))))) ) ).

% signed_take_bit_rec
tff(fact_2805_binomial__code,axiom,
    ! [Na: nat,K: nat] :
      aa(nat,nat,binomial(Na),K) = $ite(
        aa(nat,$o,ord_less(nat,Na),K),
        zero_zero(nat),
        $ite(aa(nat,$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)),aa(nat,nat,binomial(Na),aa(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,plus_plus(nat,aa(nat,nat,minus_minus(nat,Na),K)),one_one(nat)),Na,one_one(nat))),semiring_char_0_fact(nat,K))) ) ).

% binomial_code
tff(fact_2806_uint32_Osize__eq,axiom,
    ! [P2: uint32] : size_size(uint32,P2) = aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(one2)))))) ).

% uint32.size_eq
tff(fact_2807_add__scale__eq__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [R3: A,A3: A,B3: A,C2: A,D2: A] :
          ( ( R3 != zero_zero(A) )
         => ( ( ( A3 = B3 )
              & ( C2 != D2 ) )
           => ( aa(A,A,plus_plus(A,A3),aa(A,A,aa(A,fun(A,A),times_times(A),R3),C2)) != aa(A,A,plus_plus(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),R3),D2)) ) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2808_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_2809_nat_Oinject,axiom,
    ! [X22: nat,Y2: nat] :
      ( ( aa(nat,nat,suc,X22) = aa(nat,nat,suc,Y2) )
    <=> ( X22 = Y2 ) ) ).

% nat.inject
tff(fact_2810_Suc__less__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,M)),aa(nat,nat,suc,Na))
    <=> aa(nat,$o,ord_less(nat,M),Na) ) ).

% Suc_less_eq
tff(fact_2811_Suc__mono,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,M),Na)
     => aa(nat,$o,ord_less(nat,aa(nat,nat,suc,M)),aa(nat,nat,suc,Na)) ) ).

% Suc_mono
tff(fact_2812_lessI,axiom,
    ! [Na: nat] : aa(nat,$o,ord_less(nat,Na),aa(nat,nat,suc,Na)) ).

% lessI
tff(fact_2813_add__Suc__right,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,plus_plus(nat,M),aa(nat,nat,suc,Na)) = aa(nat,nat,suc,aa(nat,nat,plus_plus(nat,M),Na)) ).

% add_Suc_right
tff(fact_2814_Suc__le__mono,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,Na)),aa(nat,nat,suc,M))
    <=> aa(nat,$o,ord_less_eq(nat,Na),M) ) ).

% Suc_le_mono
tff(fact_2815_diff__Suc__Suc,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,M)),aa(nat,nat,suc,Na)) = aa(nat,nat,minus_minus(nat,M),Na) ).

% diff_Suc_Suc
tff(fact_2816_Suc__diff__diff,axiom,
    ! [M: nat,Na: nat,K: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,M)),Na)),aa(nat,nat,suc,K)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,M),Na)),K) ).

% Suc_diff_diff
tff(fact_2817_signed__take__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat] : bit_ri4674362597316999326ke_bit(A,Na,zero_zero(A)) = zero_zero(A) ) ).

% signed_take_bit_of_0
tff(fact_2818_power__0__Suc,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Na: nat] : aa(nat,A,power_power(A,zero_zero(A)),aa(nat,nat,suc,Na)) = zero_zero(A) ) ).

% power_0_Suc
tff(fact_2819_power__Suc0__right,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A] : aa(nat,A,power_power(A,A3),aa(nat,nat,suc,zero_zero(nat))) = A3 ) ).

% power_Suc0_right
tff(fact_2820_zero__less__Suc,axiom,
    ! [Na: nat] : aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(nat,nat,suc,Na)) ).

% zero_less_Suc
tff(fact_2821_less__Suc0,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,ord_less(nat,Na),aa(nat,nat,suc,zero_zero(nat)))
    <=> ( Na = zero_zero(nat) ) ) ).

% less_Suc0
tff(fact_2822_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_2823_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_2824_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_2825_power__Suc__0,axiom,
    ! [Na: nat] : aa(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_2826_nat__power__eq__Suc__0__iff,axiom,
    ! [X: nat,M: nat] :
      ( ( aa(nat,nat,power_power(nat,X),M) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( M = zero_zero(nat) )
        | ( X = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_2827_diff__Suc__1,axiom,
    ! [Na: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,Na)),one_one(nat)) = Na ).

% diff_Suc_1
tff(fact_2828_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,plus_plus(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na)) ).

% mult_Suc_right
tff(fact_2829_dvd__1__left,axiom,
    ! [K: nat] : aa(nat,$o,dvd_dvd(nat,aa(nat,nat,suc,zero_zero(nat))),K) ).

% dvd_1_left
tff(fact_2830_dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( aa(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_2831_signed__take__bit__Suc__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat] : bit_ri4674362597316999326ke_bit(A,aa(nat,nat,suc,Na),one_one(A)) = one_one(A) ) ).

% signed_take_bit_Suc_1
tff(fact_2832_signed__take__bit__numeral__of__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: num] : bit_ri4674362597316999326ke_bit(A,aa(num,nat,numeral_numeral(nat),K),one_one(A)) = one_one(A) ) ).

% signed_take_bit_numeral_of_1
tff(fact_2833_signed__take__bit__of__minus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat] : bit_ri4674362597316999326ke_bit(A,Na,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% signed_take_bit_of_minus_1
tff(fact_2834_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_2835_fact__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,zero_zero(nat)) = one_one(A) ) ) ).

% fact_0
tff(fact_2836_fact__1,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,one_one(nat)) = one_one(A) ) ) ).

% fact_1
tff(fact_2837_binomial__1,axiom,
    ! [Na: nat] : aa(nat,nat,binomial(Na),aa(nat,nat,suc,zero_zero(nat))) = Na ).

% binomial_1
tff(fact_2838_binomial__0__Suc,axiom,
    ! [K: nat] : aa(nat,nat,binomial(zero_zero(nat)),aa(nat,nat,suc,K)) = zero_zero(nat) ).

% binomial_0_Suc
tff(fact_2839_pochhammer__Suc0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,zero_zero(nat))) = A3 ) ).

% pochhammer_Suc0
tff(fact_2840_of__nat__fact,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Na: nat] : aa(nat,A,semiring_1_of_nat(A),semiring_char_0_fact(nat,Na)) = semiring_char_0_fact(A,Na) ) ).

% of_nat_fact
tff(fact_2841_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,plus_plus(A,one_one(A)),aa(nat,A,semiring_1_of_nat(A),M)) ) ).

% of_nat_Suc
tff(fact_2842_Suc__pred,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat)))) = Na ) ) ).

% Suc_pred
tff(fact_2843_one__le__mult__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa(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,ord_less_eq(nat,aa(nat,nat,suc,zero_zero(nat))),M)
        & aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,zero_zero(nat))),Na) ) ) ).

% one_le_mult_iff
tff(fact_2844_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),J)
     => ( aa(nat,nat,minus_minus(nat,I),aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,J),K))) = aa(nat,nat,minus_minus(nat,aa(nat,nat,plus_plus(nat,I),K)),aa(nat,nat,suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_2845_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),J)
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,J),K))),I) = aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,J)),aa(nat,nat,plus_plus(nat,K),I)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_2846_Suc__diff,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),Na)
     => ( aa(nat,$o,ord_less_eq(nat,one_one(nat)),M)
       => ( aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,Na),M)) = aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,minus_minus(nat,M),one_one(nat))) ) ) ) ).

% Suc_diff
tff(fact_2847_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_2848_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_2849_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,plus_plus(num,Na),one2)) ).

% Suc_numeral
tff(fact_2850_negative__zless,axiom,
    ! [Na: nat,M: nat] : aa(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_2851_add__2__eq__Suc_H,axiom,
    ! [Na: nat] : aa(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_2852_add__2__eq__Suc,axiom,
    ! [Na: nat] : aa(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_2853_Suc__1,axiom,
    aa(nat,nat,suc,one_one(nat)) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% Suc_1
tff(fact_2854_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_2855_even__Suc__Suc__iff,axiom,
    ! [Na: nat] :
      ( aa(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,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) ) ).

% even_Suc_Suc_iff
tff(fact_2856_even__Suc,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,Na))
    <=> ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) ) ).

% even_Suc
tff(fact_2857_signed__take__bit__Suc__bit0,axiom,
    ! [Na: nat,K: num] : 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),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_2858_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_2859_Suc__diff__1,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,Na),one_one(nat))) = Na ) ) ).

% Suc_diff_1
tff(fact_2860_fact__2,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(num,A,numeral_numeral(A),bit0(one2)) ) ) ).

% fact_2
tff(fact_2861_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_2862_Suc__unat__minus__one,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( ( X != zero_zero(word(A)) )
         => ( aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),minus_minus(word(A),X),one_one(word(A))))) = aa(word(A),nat,semiring_1_unsigned(A,nat),X) ) ) ) ).

% Suc_unat_minus_one
tff(fact_2863_shiftl__Suc__0,axiom,
    ! [Na: nat] : bit_Sh4282982442137083160shiftl(nat,aa(nat,nat,suc,zero_zero(nat)),Na) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) ).

% shiftl_Suc_0
tff(fact_2864_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_2865_even__Suc__div__two,axiom,
    ! [Na: nat] :
      ( aa(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_2866_odd__Suc__div__two,axiom,
    ! [Na: nat] :
      ( ~ aa(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_2867_signed__take__bit__Suc__minus__bit0,axiom,
    ! [Na: nat,K: num] : 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),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_2868_shiftl__of__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,Na: nat] : bit_Sh4282982442137083160shiftl(A,A3,aa(nat,nat,suc,Na)) = bit_Sh4282982442137083160shiftl(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))),Na) ) ).

% shiftl_of_Suc
tff(fact_2869_set__decode__Suc,axiom,
    ! [Na: nat,X: nat] :
      ( member(nat,aa(nat,nat,suc,Na),nat_set_decode(X))
    <=> member(nat,Na,nat_set_decode(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% set_decode_Suc
tff(fact_2870_odd__Suc__minus__one,axiom,
    ! [Na: nat] :
      ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => ( aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat)))) = Na ) ) ).

% odd_Suc_minus_one
tff(fact_2871_signed__take__bit__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A] : bit_ri4674362597316999326ke_bit(A,zero_zero(nat),A3) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% signed_take_bit_0
tff(fact_2872_fact__ge__Suc__0__nat,axiom,
    ! [Na: nat] : aa(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_2873_n__not__Suc__n,axiom,
    ! [Na: nat] : Na != aa(nat,nat,suc,Na) ).

% n_not_Suc_n
tff(fact_2874_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa(nat,nat,suc,X) = aa(nat,nat,suc,Y) )
     => ( X = Y ) ) ).

% Suc_inject
tff(fact_2875_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_2876_pi__neq__zero,axiom,
    pi != zero_zero(real) ).

% pi_neq_zero
tff(fact_2877_list__decode_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero(nat) )
     => ~ ! [N: nat] : X != aa(nat,nat,suc,N) ) ).

% list_decode.cases
tff(fact_2878_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,
    ! [X: nat] :
      ( ( X != zero_zero(nat) )
     => ( ( X != aa(nat,nat,suc,zero_zero(nat)) )
       => ~ ! [Va2: nat] : X != aa(nat,nat,suc,aa(nat,nat,suc,Va2)) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.cases
tff(fact_2879_exists__least__lemma,axiom,
    ! [P: fun(nat,$o)] :
      ( ~ aa(nat,$o,P,zero_zero(nat))
     => ( ? [X_13: nat] : aa(nat,$o,P,X_13)
       => ? [N: nat] :
            ( ~ aa(nat,$o,P,N)
            & aa(nat,$o,P,aa(nat,nat,suc,N)) ) ) ) ).

% exists_least_lemma
tff(fact_2880_not0__implies__Suc,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero(nat) )
     => ? [M4: nat] : Na = aa(nat,nat,suc,M4) ) ).

% not0_implies_Suc
tff(fact_2881_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero(nat) != aa(nat,nat,suc,M) ).

% Zero_not_Suc
tff(fact_2882_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero(nat) != aa(nat,nat,suc,M) ).

% Zero_neq_Suc
tff(fact_2883_Suc__neq__Zero,axiom,
    ! [M: nat] : aa(nat,nat,suc,M) != zero_zero(nat) ).

% Suc_neq_Zero
tff(fact_2884_zero__induct,axiom,
    ! [P: fun(nat,$o),K: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [N: nat] :
            ( aa(nat,$o,P,aa(nat,nat,suc,N))
           => aa(nat,$o,P,N) )
       => aa(nat,$o,P,zero_zero(nat)) ) ) ).

% zero_induct
tff(fact_2885_diff__induct,axiom,
    ! [P: fun(nat,fun(nat,$o)),M: nat,Na: nat] :
      ( ! [X4: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,X4),zero_zero(nat))
     => ( ! [Y3: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,zero_zero(nat)),aa(nat,nat,suc,Y3))
       => ( ! [X4: nat,Y3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),P,X4),Y3)
             => aa(nat,$o,aa(nat,fun(nat,$o),P,aa(nat,nat,suc,X4)),aa(nat,nat,suc,Y3)) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P,M),Na) ) ) ) ).

% diff_induct
tff(fact_2886_nat__induct,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ! [N: nat] :
            ( aa(nat,$o,P,N)
           => aa(nat,$o,P,aa(nat,nat,suc,N)) )
       => aa(nat,$o,P,Na) ) ) ).

% nat_induct
tff(fact_2887_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero(nat) )
     => ~ ! [Nat3: nat] : Y != aa(nat,nat,suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_2888_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat = aa(nat,nat,suc,X22) )
     => ( Nat != zero_zero(nat) ) ) ).

% nat.discI
tff(fact_2889_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero(nat) != aa(nat,nat,suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_2890_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa(nat,nat,suc,Nat2) != zero_zero(nat) ).

% old.nat.distinct(2)
tff(fact_2891_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] : zero_zero(nat) != aa(nat,nat,suc,X22) ).

% nat.distinct(1)
tff(fact_2892_not__less__less__Suc__eq,axiom,
    ! [Na: nat,M: nat] :
      ( ~ aa(nat,$o,ord_less(nat,Na),M)
     => ( aa(nat,$o,ord_less(nat,Na),aa(nat,nat,suc,M))
      <=> ( Na = M ) ) ) ).

% not_less_less_Suc_eq
tff(fact_2893_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,ord_less(nat,I),J)
     => ( ! [I3: nat] :
            ( ( J = aa(nat,nat,suc,I3) )
           => aa(nat,$o,P,I3) )
       => ( ! [I3: nat] :
              ( aa(nat,$o,ord_less(nat,I3),J)
             => ( aa(nat,$o,P,aa(nat,nat,suc,I3))
               => aa(nat,$o,P,I3) ) )
         => aa(nat,$o,P,I) ) ) ) ).

% strict_inc_induct
tff(fact_2894_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,fun(nat,$o))] :
      ( aa(nat,$o,ord_less(nat,I),J)
     => ( ! [I3: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,I3),aa(nat,nat,suc,I3))
       => ( ! [I3: nat,J2: nat,K2: nat] :
              ( aa(nat,$o,ord_less(nat,I3),J2)
             => ( aa(nat,$o,ord_less(nat,J2),K2)
               => ( aa(nat,$o,aa(nat,fun(nat,$o),P,I3),J2)
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),P,J2),K2)
                   => aa(nat,$o,aa(nat,fun(nat,$o),P,I3),K2) ) ) ) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P,I),J) ) ) ) ).

% less_Suc_induct
tff(fact_2895_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,I),J)
     => ( aa(nat,$o,ord_less(nat,J),K)
       => aa(nat,$o,ord_less(nat,aa(nat,nat,suc,I)),K) ) ) ).

% less_trans_Suc
tff(fact_2896_Suc__less__SucD,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,M)),aa(nat,nat,suc,Na))
     => aa(nat,$o,ord_less(nat,M),Na) ) ).

% Suc_less_SucD
tff(fact_2897_less__antisym,axiom,
    ! [Na: nat,M: nat] :
      ( ~ aa(nat,$o,ord_less(nat,Na),M)
     => ( aa(nat,$o,ord_less(nat,Na),aa(nat,nat,suc,M))
       => ( M = Na ) ) ) ).

% less_antisym
tff(fact_2898_Suc__less__eq2,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,Na)),M)
    <=> ? [M5: nat] :
          ( ( M = aa(nat,nat,suc,M5) )
          & aa(nat,$o,ord_less(nat,Na),M5) ) ) ).

% Suc_less_eq2
tff(fact_2899_Nat_OAll__less__Suc,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( ! [I4: nat] :
          ( aa(nat,$o,ord_less(nat,I4),aa(nat,nat,suc,Na))
         => aa(nat,$o,P,I4) )
    <=> ( aa(nat,$o,P,Na)
        & ! [I4: nat] :
            ( aa(nat,$o,ord_less(nat,I4),Na)
           => aa(nat,$o,P,I4) ) ) ) ).

% Nat.All_less_Suc
tff(fact_2900_not__less__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ~ aa(nat,$o,ord_less(nat,M),Na)
    <=> aa(nat,$o,ord_less(nat,Na),aa(nat,nat,suc,M)) ) ).

% not_less_eq
tff(fact_2901_less__Suc__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,M),aa(nat,nat,suc,Na))
    <=> ( aa(nat,$o,ord_less(nat,M),Na)
        | ( M = Na ) ) ) ).

% less_Suc_eq
tff(fact_2902_Ex__less__Suc,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( ? [I4: nat] :
          ( aa(nat,$o,ord_less(nat,I4),aa(nat,nat,suc,Na))
          & aa(nat,$o,P,I4) )
    <=> ( aa(nat,$o,P,Na)
        | ? [I4: nat] :
            ( aa(nat,$o,ord_less(nat,I4),Na)
            & aa(nat,$o,P,I4) ) ) ) ).

% Ex_less_Suc
tff(fact_2903_less__SucI,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,M),Na)
     => aa(nat,$o,ord_less(nat,M),aa(nat,nat,suc,Na)) ) ).

% less_SucI
tff(fact_2904_less__SucE,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,M),aa(nat,nat,suc,Na))
     => ( ~ aa(nat,$o,ord_less(nat,M),Na)
       => ( M = Na ) ) ) ).

% less_SucE
tff(fact_2905_Suc__lessI,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,M),Na)
     => ( ( aa(nat,nat,suc,M) != Na )
       => aa(nat,$o,ord_less(nat,aa(nat,nat,suc,M)),Na) ) ) ).

% Suc_lessI
tff(fact_2906_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,I)),K)
     => ~ ! [J2: nat] :
            ( aa(nat,$o,ord_less(nat,I),J2)
           => ( K != aa(nat,nat,suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_2907_Suc__lessD,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,M)),Na)
     => aa(nat,$o,ord_less(nat,M),Na) ) ).

% Suc_lessD
tff(fact_2908_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,I),K)
     => ( ( K != aa(nat,nat,suc,I) )
       => ~ ! [J2: nat] :
              ( aa(nat,$o,ord_less(nat,I),J2)
             => ( K != aa(nat,nat,suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_2909_nat__arith_Osuc1,axiom,
    ! [A2: nat,K: nat,A3: nat] :
      ( ( A2 = aa(nat,nat,plus_plus(nat,K),A3) )
     => ( aa(nat,nat,suc,A2) = aa(nat,nat,plus_plus(nat,K),aa(nat,nat,suc,A3)) ) ) ).

% nat_arith.suc1
tff(fact_2910_add__Suc,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,plus_plus(nat,aa(nat,nat,suc,M)),Na) = aa(nat,nat,suc,aa(nat,nat,plus_plus(nat,M),Na)) ).

% add_Suc
tff(fact_2911_add__Suc__shift,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,plus_plus(nat,aa(nat,nat,suc,M)),Na) = aa(nat,nat,plus_plus(nat,M),aa(nat,nat,suc,Na)) ).

% add_Suc_shift
tff(fact_2912_Suc__leD,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,M)),Na)
     => aa(nat,$o,ord_less_eq(nat,M),Na) ) ).

% Suc_leD
tff(fact_2913_le__SucE,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),aa(nat,nat,suc,Na))
     => ( ~ aa(nat,$o,ord_less_eq(nat,M),Na)
       => ( M = aa(nat,nat,suc,Na) ) ) ) ).

% le_SucE
tff(fact_2914_le__SucI,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),Na)
     => aa(nat,$o,ord_less_eq(nat,M),aa(nat,nat,suc,Na)) ) ).

% le_SucI
tff(fact_2915_Suc__le__D,axiom,
    ! [Na: nat,M6: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,Na)),M6)
     => ? [M4: nat] : M6 = aa(nat,nat,suc,M4) ) ).

% Suc_le_D
tff(fact_2916_le__Suc__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),aa(nat,nat,suc,Na))
    <=> ( aa(nat,$o,ord_less_eq(nat,M),Na)
        | ( M = aa(nat,nat,suc,Na) ) ) ) ).

% le_Suc_eq
tff(fact_2917_Suc__n__not__le__n,axiom,
    ! [Na: nat] : ~ aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,Na)),Na) ).

% Suc_n_not_le_n
tff(fact_2918_not__less__eq__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ~ aa(nat,$o,ord_less_eq(nat,M),Na)
    <=> aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,Na)),M) ) ).

% not_less_eq_eq
tff(fact_2919_full__nat__induct,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( ! [N: nat] :
          ( ! [M2: nat] :
              ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,M2)),N)
             => aa(nat,$o,P,M2) )
         => aa(nat,$o,P,N) )
     => aa(nat,$o,P,Na) ) ).

% full_nat_induct
tff(fact_2920_nat__induct__at__least,axiom,
    ! [M: nat,Na: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,ord_less_eq(nat,M),Na)
     => ( aa(nat,$o,P,M)
       => ( ! [N: nat] :
              ( aa(nat,$o,ord_less_eq(nat,M),N)
             => ( aa(nat,$o,P,N)
               => aa(nat,$o,P,aa(nat,nat,suc,N)) ) )
         => aa(nat,$o,P,Na) ) ) ) ).

% nat_induct_at_least
tff(fact_2921_transitive__stepwise__le,axiom,
    ! [M: nat,Na: nat,R: fun(nat,fun(nat,$o))] :
      ( aa(nat,$o,ord_less_eq(nat,M),Na)
     => ( ! [X4: nat] : aa(nat,$o,aa(nat,fun(nat,$o),R,X4),X4)
       => ( ! [X4: nat,Y3: nat,Z2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),R,X4),Y3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),R,Y3),Z2)
               => aa(nat,$o,aa(nat,fun(nat,$o),R,X4),Z2) ) )
         => ( ! [N: nat] : aa(nat,$o,aa(nat,fun(nat,$o),R,N),aa(nat,nat,suc,N))
           => aa(nat,$o,aa(nat,fun(nat,$o),R,M),Na) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_2922_zero__induct__lemma,axiom,
    ! [P: fun(nat,$o),K: nat,I: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [N: nat] :
            ( aa(nat,$o,P,aa(nat,nat,suc,N))
           => aa(nat,$o,P,N) )
       => aa(nat,$o,P,aa(nat,nat,minus_minus(nat,K),I)) ) ) ).

% zero_induct_lemma
tff(fact_2923_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_2924_fact__diff__Suc,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,Na),aa(nat,nat,suc,M))
     => ( semiring_char_0_fact(nat,aa(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,minus_minus(nat,aa(nat,nat,suc,M)),Na)),semiring_char_0_fact(nat,aa(nat,nat,minus_minus(nat,M),Na))) ) ) ).

% fact_diff_Suc
tff(fact_2925_summable__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,aTP_Lamp_bw(fun(nat,A),fun(nat,A),F2))
        <=> summable(A,F2) ) ) ).

% summable_Suc_iff
tff(fact_2926_fact__less__mono__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
     => ( aa(nat,$o,ord_less(nat,M),Na)
       => aa(nat,$o,ord_less(nat,semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,Na)) ) ) ).

% fact_less_mono_nat
tff(fact_2927_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_2928_fact__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat] : aa(A,$o,ord_less_eq(A,zero_zero(A)),semiring_char_0_fact(A,Na)) ) ).

% fact_ge_zero
tff(fact_2929_fact__not__neg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat] : ~ aa(A,$o,ord_less(A,semiring_char_0_fact(A,Na)),zero_zero(A)) ) ).

% fact_not_neg
tff(fact_2930_fact__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat] : aa(A,$o,ord_less(A,zero_zero(A)),semiring_char_0_fact(A,Na)) ) ).

% fact_gt_zero
tff(fact_2931_fact__ge__1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat] : aa(A,$o,ord_less_eq(A,one_one(A)),semiring_char_0_fact(A,Na)) ) ).

% fact_ge_1
tff(fact_2932_pi__gt__zero,axiom,
    aa(real,$o,ord_less(real,zero_zero(real)),pi) ).

% pi_gt_zero
tff(fact_2933_pi__not__less__zero,axiom,
    ~ aa(real,$o,ord_less(real,pi),zero_zero(real)) ).

% pi_not_less_zero
tff(fact_2934_pi__ge__zero,axiom,
    aa(real,$o,ord_less_eq(real,zero_zero(real)),pi) ).

% pi_ge_zero
tff(fact_2935_lift__Suc__mono__less__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),Na: nat,M: nat] :
          ( ! [N: nat] : aa(A,$o,ord_less(A,aa(nat,A,F2,N)),aa(nat,A,F2,aa(nat,nat,suc,N)))
         => ( aa(A,$o,ord_less(A,aa(nat,A,F2,Na)),aa(nat,A,F2,M))
          <=> aa(nat,$o,ord_less(nat,Na),M) ) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_2936_lift__Suc__mono__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),Na: nat,N6: nat] :
          ( ! [N: nat] : aa(A,$o,ord_less(A,aa(nat,A,F2,N)),aa(nat,A,F2,aa(nat,nat,suc,N)))
         => ( aa(nat,$o,ord_less(nat,Na),N6)
           => aa(A,$o,ord_less(A,aa(nat,A,F2,Na)),aa(nat,A,F2,N6)) ) ) ) ).

% lift_Suc_mono_less
tff(fact_2937_power__Suc2,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,Na: nat] : aa(nat,A,power_power(A,A3),aa(nat,nat,suc,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,A3),Na)),A3) ) ).

% power_Suc2
tff(fact_2938_power__Suc,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [A3: A,Na: nat] : aa(nat,A,power_power(A,A3),aa(nat,nat,suc,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,power_power(A,A3),Na)) ) ).

% power_Suc
tff(fact_2939_lift__Suc__antimono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),Na: nat,N6: nat] :
          ( ! [N: nat] : aa(A,$o,ord_less_eq(A,aa(nat,A,F2,aa(nat,nat,suc,N))),aa(nat,A,F2,N))
         => ( aa(nat,$o,ord_less_eq(nat,Na),N6)
           => aa(A,$o,ord_less_eq(A,aa(nat,A,F2,N6)),aa(nat,A,F2,Na)) ) ) ) ).

% lift_Suc_antimono_le
tff(fact_2940_lift__Suc__mono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),Na: nat,N6: nat] :
          ( ! [N: nat] : aa(A,$o,ord_less_eq(A,aa(nat,A,F2,N)),aa(nat,A,F2,aa(nat,nat,suc,N)))
         => ( aa(nat,$o,ord_less_eq(nat,Na),N6)
           => aa(A,$o,ord_less_eq(A,aa(nat,A,F2,Na)),aa(nat,A,F2,N6)) ) ) ) ).

% lift_Suc_mono_le
tff(fact_2941_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_2942_Comparator__Generator_OAll__less__Suc,axiom,
    ! [X: nat,P: fun(nat,$o)] :
      ( ! [I4: nat] :
          ( aa(nat,$o,ord_less(nat,I4),aa(nat,nat,suc,X))
         => aa(nat,$o,P,I4) )
    <=> ( aa(nat,$o,P,zero_zero(nat))
        & ! [I4: nat] :
            ( aa(nat,$o,ord_less(nat,I4),X)
           => aa(nat,$o,P,aa(nat,nat,suc,I4)) ) ) ) ).

% Comparator_Generator.All_less_Suc
tff(fact_2943_forall__finite_I2_J,axiom,
    ! [P: fun(nat,$o)] :
      ( ! [I4: nat] :
          ( aa(nat,$o,ord_less(nat,I4),aa(nat,nat,suc,zero_zero(nat)))
         => aa(nat,$o,P,I4) )
    <=> aa(nat,$o,P,zero_zero(nat)) ) ).

% forall_finite(2)
tff(fact_2944_forall__finite_I3_J,axiom,
    ! [X: nat,P: fun(nat,$o)] :
      ( ! [I4: nat] :
          ( aa(nat,$o,ord_less(nat,I4),aa(nat,nat,suc,aa(nat,nat,suc,X)))
         => aa(nat,$o,P,I4) )
    <=> ( aa(nat,$o,P,zero_zero(nat))
        & ! [I4: nat] :
            ( aa(nat,$o,ord_less(nat,I4),aa(nat,nat,suc,X))
           => aa(nat,$o,P,aa(nat,nat,suc,I4)) ) ) ) ).

% forall_finite(3)
tff(fact_2945_less__Suc__eq__0__disj,axiom,
    ! [M: nat,Na: nat] :
      ( aa(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,ord_less(nat,J3),Na) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_2946_gr0__implies__Suc,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ? [M4: nat] : Na = aa(nat,nat,suc,M4) ) ).

% gr0_implies_Suc
tff(fact_2947_All__less__Suc2,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( ! [I4: nat] :
          ( aa(nat,$o,ord_less(nat,I4),aa(nat,nat,suc,Na))
         => aa(nat,$o,P,I4) )
    <=> ( aa(nat,$o,P,zero_zero(nat))
        & ! [I4: nat] :
            ( aa(nat,$o,ord_less(nat,I4),Na)
           => aa(nat,$o,P,aa(nat,nat,suc,I4)) ) ) ) ).

% All_less_Suc2
tff(fact_2948_gr0__conv__Suc,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
    <=> ? [M3: nat] : Na = aa(nat,nat,suc,M3) ) ).

% gr0_conv_Suc
tff(fact_2949_Ex__less__Suc2,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( ? [I4: nat] :
          ( aa(nat,$o,ord_less(nat,I4),aa(nat,nat,suc,Na))
          & aa(nat,$o,P,I4) )
    <=> ( aa(nat,$o,P,zero_zero(nat))
        | ? [I4: nat] :
            ( aa(nat,$o,ord_less(nat,I4),Na)
            & aa(nat,$o,P,aa(nat,nat,suc,I4)) ) ) ) ).

% Ex_less_Suc2
tff(fact_2950_one__is__add,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,suc,zero_zero(nat)) = aa(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_2951_add__is__1,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(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_2952_nat__compl__induct,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ! [N: nat] :
            ( ! [Nn: nat] :
                ( aa(nat,$o,ord_less_eq(nat,Nn),N)
               => aa(nat,$o,P,Nn) )
           => aa(nat,$o,P,aa(nat,nat,suc,N)) )
       => aa(nat,$o,P,Na) ) ) ).

% nat_compl_induct
tff(fact_2953_nat__compl__induct_H,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ! [N: nat] :
            ( ! [Nn: nat] :
                ( aa(nat,$o,ord_less_eq(nat,Nn),N)
               => aa(nat,$o,P,Nn) )
           => aa(nat,$o,P,aa(nat,nat,suc,N)) )
       => aa(nat,$o,P,Na) ) ) ).

% nat_compl_induct'
tff(fact_2954_One__nat__def,axiom,
    one_one(nat) = aa(nat,nat,suc,zero_zero(nat)) ).

% One_nat_def
tff(fact_2955_less__natE,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,M),Na)
     => ~ ! [Q6: nat] : Na != aa(nat,nat,suc,aa(nat,nat,plus_plus(nat,M),Q6)) ) ).

% less_natE
tff(fact_2956_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : aa(nat,$o,ord_less(nat,I),aa(nat,nat,suc,aa(nat,nat,plus_plus(nat,I),M))) ).

% less_add_Suc1
tff(fact_2957_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : aa(nat,$o,ord_less(nat,I),aa(nat,nat,suc,aa(nat,nat,plus_plus(nat,M),I))) ).

% less_add_Suc2
tff(fact_2958_less__iff__Suc__add,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,M),Na)
    <=> ? [K3: nat] : Na = aa(nat,nat,suc,aa(nat,nat,plus_plus(nat,M),K3)) ) ).

% less_iff_Suc_add
tff(fact_2959_less__imp__Suc__add,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,M),Na)
     => ? [K2: nat] : Na = aa(nat,nat,suc,aa(nat,nat,plus_plus(nat,M),K2)) ) ).

% less_imp_Suc_add
tff(fact_2960_Suc__leI,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,M),Na)
     => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,M)),Na) ) ).

% Suc_leI
tff(fact_2961_Suc__le__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,M)),Na)
    <=> aa(nat,$o,ord_less(nat,M),Na) ) ).

% Suc_le_eq
tff(fact_2962_dec__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => ( aa(nat,$o,P,I)
       => ( ! [N: nat] :
              ( aa(nat,$o,ord_less_eq(nat,I),N)
             => ( aa(nat,$o,ord_less(nat,N),J)
               => ( aa(nat,$o,P,N)
                 => aa(nat,$o,P,aa(nat,nat,suc,N)) ) ) )
         => aa(nat,$o,P,J) ) ) ) ).

% dec_induct
tff(fact_2963_inc__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => ( aa(nat,$o,P,J)
       => ( ! [N: nat] :
              ( aa(nat,$o,ord_less_eq(nat,I),N)
             => ( aa(nat,$o,ord_less(nat,N),J)
               => ( aa(nat,$o,P,aa(nat,nat,suc,N))
                 => aa(nat,$o,P,N) ) ) )
         => aa(nat,$o,P,I) ) ) ) ).

% inc_induct
tff(fact_2964_Suc__le__lessD,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,M)),Na)
     => aa(nat,$o,ord_less(nat,M),Na) ) ).

% Suc_le_lessD
tff(fact_2965_le__less__Suc__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),Na)
     => ( aa(nat,$o,ord_less(nat,Na),aa(nat,nat,suc,M))
      <=> ( Na = M ) ) ) ).

% le_less_Suc_eq
tff(fact_2966_less__Suc__eq__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,M),aa(nat,nat,suc,Na))
    <=> aa(nat,$o,ord_less_eq(nat,M),Na) ) ).

% less_Suc_eq_le
tff(fact_2967_less__eq__Suc__le,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,Na),M)
    <=> aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,Na)),M) ) ).

% less_eq_Suc_le
tff(fact_2968_le__imp__less__Suc,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),Na)
     => aa(nat,$o,ord_less(nat,M),aa(nat,nat,suc,Na)) ) ).

% le_imp_less_Suc
tff(fact_2969_nat__in__between__eq_I1_J,axiom,
    ! [A3: nat,B3: nat] :
      ( ( aa(nat,$o,ord_less(nat,A3),B3)
        & aa(nat,$o,ord_less_eq(nat,B3),aa(nat,nat,suc,A3)) )
    <=> ( B3 = aa(nat,nat,suc,A3) ) ) ).

% nat_in_between_eq(1)
tff(fact_2970_nat__in__between__eq_I2_J,axiom,
    ! [A3: nat,B3: nat] :
      ( ( aa(nat,$o,ord_less_eq(nat,A3),B3)
        & aa(nat,$o,ord_less(nat,B3),aa(nat,nat,suc,A3)) )
    <=> ( B3 = A3 ) ) ).

% nat_in_between_eq(2)
tff(fact_2971_Suc__to__right,axiom,
    ! [Na: nat,M: nat] :
      ( ( aa(nat,nat,suc,Na) = M )
     => ( Na = aa(nat,nat,minus_minus(nat,M),aa(nat,nat,suc,zero_zero(nat))) ) ) ).

% Suc_to_right
tff(fact_2972_diff__less__Suc,axiom,
    ! [M: nat,Na: nat] : aa(nat,$o,ord_less(nat,aa(nat,nat,minus_minus(nat,M),Na)),aa(nat,nat,suc,M)) ).

% diff_less_Suc
tff(fact_2973_Suc__diff__Suc,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,Na),M)
     => ( aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,M),aa(nat,nat,suc,Na))) = aa(nat,nat,minus_minus(nat,M),Na) ) ) ).

% Suc_diff_Suc
tff(fact_2974_Suc__eq__plus1__left,axiom,
    ! [Na: nat] : aa(nat,nat,suc,Na) = aa(nat,nat,plus_plus(nat,one_one(nat)),Na) ).

% Suc_eq_plus1_left
tff(fact_2975_plus__1__eq__Suc,axiom,
    plus_plus(nat,one_one(nat)) = suc ).

% plus_1_eq_Suc
tff(fact_2976_Suc__eq__plus1,axiom,
    ! [Na: nat] : aa(nat,nat,suc,Na) = aa(nat,nat,plus_plus(nat,Na),one_one(nat)) ).

% Suc_eq_plus1
tff(fact_2977_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(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,ord_less(nat,M),Na) ) ).

% Suc_mult_less_cancel1
tff(fact_2978_Suc__diff__le,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,ord_less_eq(nat,Na),M)
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,M)),Na) = aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,M),Na)) ) ) ).

% Suc_diff_le
tff(fact_2979_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,minus_minus(nat,M),aa(nat,nat,suc,Na)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,M),one_one(nat))),Na) ).

% diff_Suc_eq_diff_pred
tff(fact_2980_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,plus_plus(nat,Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na)) ).

% mult_Suc
tff(fact_2981_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(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,ord_less_eq(nat,M),Na) ) ).

% Suc_mult_le_cancel1
tff(fact_2982_pochhammer__fact,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & comm_semiring_1(A) )
     => ! [Na: nat] : semiring_char_0_fact(A,Na) = comm_s3205402744901411588hammer(A,one_one(A),Na) ) ).

% pochhammer_fact
tff(fact_2983_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_2984_mod__induct,axiom,
    ! [P: fun(nat,$o),Na: nat,P2: nat,M: nat] :
      ( aa(nat,$o,P,Na)
     => ( aa(nat,$o,ord_less(nat,Na),P2)
       => ( aa(nat,$o,ord_less(nat,M),P2)
         => ( ! [N: nat] :
                ( aa(nat,$o,ord_less(nat,N),P2)
               => ( aa(nat,$o,P,N)
                 => aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,N)),P2)) ) )
           => aa(nat,$o,P,M) ) ) ) ) ).

% mod_induct
tff(fact_2985_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_2986_int__of__nat__induct,axiom,
    ! [P: fun(int,$o),Z: int] :
      ( ! [N: nat] : aa(int,$o,P,aa(nat,int,semiring_1_of_nat(int),N))
     => ( ! [N: nat] : aa(int,$o,P,aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N))))
       => aa(int,$o,P,Z) ) ) ).

% int_of_nat_induct
tff(fact_2987_dvd__fact,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,one_one(nat)),M)
     => ( aa(nat,$o,ord_less_eq(nat,M),Na)
       => aa(nat,$o,dvd_dvd(nat,M),semiring_char_0_fact(nat,Na)) ) ) ).

% dvd_fact
tff(fact_2988_signed__take__bit__int__less__eq,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,ord_less_eq(int,aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)),K)
     => aa(int,$o,ord_less_eq(int,bit_ri4674362597316999326ke_bit(int,Na,K)),aa(int,int,minus_minus(int,K),aa(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_2989_fact__less__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
         => ( aa(nat,$o,ord_less(nat,M),Na)
           => aa(A,$o,ord_less(A,semiring_char_0_fact(A,M)),semiring_char_0_fact(A,Na)) ) ) ) ).

% fact_less_mono
tff(fact_2990_fact__mod,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom(A)
        & semidom_modulo(A) )
     => ! [M: nat,Na: nat] :
          ( aa(nat,$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_2991_fact__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat] : aa(A,$o,ord_less_eq(A,semiring_char_0_fact(A,Na)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,power_power(nat,Na),Na))) ) ).

% fact_le_power
tff(fact_2992_power__inject__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Na: nat,B3: A] :
          ( ( aa(nat,A,power_power(A,A3),aa(nat,nat,suc,Na)) = aa(nat,A,power_power(A,B3),aa(nat,nat,suc,Na)) )
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
           => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B3)
             => ( A3 = B3 ) ) ) ) ) ).

% power_inject_base
tff(fact_2993_power__le__imp__le__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Na: nat,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,power_power(A,A3),aa(nat,nat,suc,Na))),aa(nat,A,power_power(A,B3),aa(nat,nat,suc,Na)))
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B3)
           => aa(A,$o,ord_less_eq(A,A3),B3) ) ) ) ).

% power_le_imp_le_base
tff(fact_2994_power__gt1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,ord_less(A,one_one(A)),A3)
         => aa(A,$o,ord_less(A,one_one(A)),aa(nat,A,power_power(A,A3),aa(nat,nat,suc,Na))) ) ) ).

% power_gt1
tff(fact_2995_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_2996_ex__least__nat__less,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( aa(nat,$o,P,Na)
     => ( ~ aa(nat,$o,P,zero_zero(nat))
       => ? [K2: nat] :
            ( aa(nat,$o,ord_less(nat,K2),Na)
            & ! [I2: nat] :
                ( aa(nat,$o,ord_less_eq(nat,I2),K2)
               => ~ aa(nat,$o,P,I2) )
            & aa(nat,$o,P,aa(nat,nat,suc,K2)) ) ) ) ).

% ex_least_nat_less
tff(fact_2997_nat__induct__non__zero,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(nat,$o,P,one_one(nat))
       => ( ! [N: nat] :
              ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
             => ( aa(nat,$o,P,N)
               => aa(nat,$o,P,aa(nat,nat,suc,N)) ) )
         => aa(nat,$o,P,Na) ) ) ) ).

% nat_induct_non_zero
tff(fact_2998_diff__Suc__less,axiom,
    ! [Na: nat,I: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => aa(nat,$o,ord_less(nat,aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,I))),Na) ) ).

% diff_Suc_less
tff(fact_2999_one__less__mult,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,zero_zero(nat))),Na)
     => ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,zero_zero(nat))),M)
       => aa(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_3000_n__less__m__mult__n,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,zero_zero(nat))),M)
       => aa(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_3001_n__less__n__mult__m,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,zero_zero(nat))),M)
       => aa(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_3002_power__gt__expt,axiom,
    ! [Na: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,zero_zero(nat))),Na)
     => aa(nat,$o,ord_less(nat,K),aa(nat,nat,power_power(nat,Na),K)) ) ).

% power_gt_expt
tff(fact_3003_nat__one__le__power,axiom,
    ! [I: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,zero_zero(nat))),I)
     => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,power_power(nat,I),Na)) ) ).

% nat_one_le_power
tff(fact_3004_realpow__pos__nth2,axiom,
    ! [A3: real,Na: nat] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
     => ? [R2: real] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),R2)
          & ( aa(nat,real,power_power(real,R2),aa(nat,nat,suc,Na)) = A3 ) ) ) ).

% realpow_pos_nth2
tff(fact_3005_unat__eq__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( ( aa(word(A),nat,semiring_1_unsigned(A,nat),X) = aa(nat,nat,suc,zero_zero(nat)) )
        <=> ( X = one_one(word(A)) ) ) ) ).

% unat_eq_1
tff(fact_3006_int__Suc,axiom,
    ! [Na: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,Na)) = aa(int,int,plus_plus(int,aa(nat,int,semiring_1_of_nat(int),Na)),one_one(int)) ).

% int_Suc
tff(fact_3007_int__ops_I4_J,axiom,
    ! [A3: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,A3)) = aa(int,int,plus_plus(int,aa(nat,int,semiring_1_of_nat(int),A3)),one_one(int)) ).

% int_ops(4)
tff(fact_3008_zless__iff__Suc__zadd,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,ord_less(int,W),Z)
    <=> ? [N3: nat] : Z = aa(int,int,plus_plus(int,W),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N3))) ) ).

% zless_iff_Suc_zadd
tff(fact_3009_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_3010_powser__split__head_I3_J,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_bx(fun(nat,A),fun(A,fun(nat,A)),F2),Z))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_by(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) ) ) ).

% powser_split_head(3)
tff(fact_3011_summable__powser__split__head,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_bz(fun(nat,A),fun(A,fun(nat,A)),F2),Z))
        <=> summable(A,aa(A,fun(nat,A),aTP_Lamp_bp(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) ) ) ).

% summable_powser_split_head
tff(fact_3012_signed__take__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,A3: A] : bit_ri4674362597316999326ke_bit(A,aa(nat,nat,suc,Na),A3) = aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_ri4674362597316999326ke_bit(A,Na,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% signed_take_bit_Suc
tff(fact_3013_fact__div__fact__le__pow,axiom,
    ! [R3: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,R3),Na)
     => aa(nat,$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,minus_minus(nat,Na),R3)))),aa(nat,nat,power_power(nat,Na),R3)) ) ).

% fact_div_fact_le_pow
tff(fact_3014_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,ord_less(int,K),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)))
     => aa(int,$o,ord_less_eq(int,aa(int,int,plus_plus(int,K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,Na)))),bit_ri4674362597316999326ke_bit(int,Na,K)) ) ).

% signed_take_bit_int_greater_eq
tff(fact_3015_pi__less__4,axiom,
    aa(real,$o,ord_less(real,pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))) ).

% pi_less_4
tff(fact_3016_pi__ge__two,axiom,
    aa(real,$o,ord_less_eq(real,aa(num,real,numeral_numeral(real),bit0(one2))),pi) ).

% pi_ge_two
tff(fact_3017_power__Suc__le__self,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less_eq(A,A3),one_one(A))
           => aa(A,$o,ord_less_eq(A,aa(nat,A,power_power(A,A3),aa(nat,nat,suc,Na))),A3) ) ) ) ).

% power_Suc_le_self
tff(fact_3018_pi__half__neq__two,axiom,
    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_neq_two
tff(fact_3019_power__Suc__less__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less(A,A3),one_one(A))
           => aa(A,$o,ord_less(A,aa(nat,A,power_power(A,A3),aa(nat,nat,suc,Na))),one_one(A)) ) ) ) ).

% power_Suc_less_one
tff(fact_3020_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_3021_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_3022_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_3023_nz__le__conv__less,axiom,
    ! [K: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
     => ( aa(nat,$o,ord_less_eq(nat,K),M)
       => aa(nat,$o,ord_less(nat,aa(nat,nat,minus_minus(nat,K),aa(nat,nat,suc,zero_zero(nat)))),M) ) ) ).

% nz_le_conv_less
tff(fact_3024_Suc__pred_H,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( Na = aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,Na),one_one(nat))) ) ) ).

% Suc_pred'
tff(fact_3025_Suc__diff__eq__diff__pred,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,M)),Na) = aa(nat,nat,minus_minus(nat,M),aa(nat,nat,minus_minus(nat,Na),one_one(nat))) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_3026_div__geq,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( ~ aa(nat,$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,minus_minus(nat,M),Na)),Na)) ) ) ) ).

% div_geq
tff(fact_3027_div__if,axiom,
    ! [M: nat,Na: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na) = $ite(
        ( aa(nat,$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,minus_minus(nat,M),Na)),Na)) ) ).

% div_if
tff(fact_3028_add__eq__if,axiom,
    ! [M: nat,Na: nat] :
      aa(nat,nat,plus_plus(nat,M),Na) = $ite(M = zero_zero(nat),Na,aa(nat,nat,suc,aa(nat,nat,plus_plus(nat,aa(nat,nat,minus_minus(nat,M),one_one(nat))),Na))) ).

% add_eq_if
tff(fact_3029_Suc__n__minus__m__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),Na)
     => ( aa(nat,$o,ord_less(nat,one_one(nat)),M)
       => ( aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,Na),M)) = aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,minus_minus(nat,M),one_one(nat))) ) ) ) ).

% Suc_n_minus_m_eq
tff(fact_3030_div__nat__eqI,axiom,
    ! [Na: nat,Q3: nat,M: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),Q3)),M)
     => ( aa(nat,$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_3031_pochhammer__rec,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,Na: nat] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),comm_s3205402744901411588hammer(A,aa(A,A,plus_plus(A,A3),one_one(A)),Na)) ) ).

% pochhammer_rec
tff(fact_3032_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,plus_plus(A,Z),aa(nat,A,semiring_1_of_nat(A),Na))),comm_s3205402744901411588hammer(A,Z,Na)) ) ).

% pochhammer_rec'
tff(fact_3033_pochhammer__Suc,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,Na: nat] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,A3,Na)),aa(A,A,plus_plus(A,A3),aa(nat,A,semiring_1_of_nat(A),Na))) ) ).

% pochhammer_Suc
tff(fact_3034_Suc__nat__number__of__add,axiom,
    ! [V2: num,Na: nat] : aa(nat,nat,suc,aa(nat,nat,plus_plus(nat,aa(num,nat,numeral_numeral(nat),V2)),Na)) = aa(nat,nat,plus_plus(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,plus_plus(num,V2),one2))),Na) ).

% Suc_nat_number_of_add
tff(fact_3035_not__zle__0__negative,axiom,
    ! [Na: nat] : ~ aa(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_3036_negD,axiom,
    ! [X: int] :
      ( aa(int,$o,ord_less(int,X),zero_zero(int))
     => ? [N: nat] : X = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N))) ) ).

% negD
tff(fact_3037_negative__zless__0,axiom,
    ! [Na: nat] : aa(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_3038_binomial__absorption,axiom,
    ! [K: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),aa(nat,nat,binomial(Na),aa(nat,nat,suc,K))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,Na),one_one(nat))),K)) ).

% binomial_absorption
tff(fact_3039_unatSuc2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A)] :
          ( ( aa(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),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_3040_unatSuc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A)] :
          ( ( aa(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),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_3041_Suc__unat__diff__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),one_one(word(A))),X)
         => ( aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),minus_minus(word(A),X),one_one(word(A))))) = aa(word(A),nat,semiring_1_unsigned(A,nat),X) ) ) ) ).

% Suc_unat_diff_1
tff(fact_3042_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),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_3043_suminf__split__head,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( suminf(A,aTP_Lamp_bw(fun(nat,A),fun(nat,A),F2)) = aa(A,A,minus_minus(A,suminf(A,F2)),aa(nat,A,F2,zero_zero(nat))) ) ) ) ).

% suminf_split_head
tff(fact_3044_signed__take__bit__int__less__exp,axiom,
    ! [Na: nat,K: int] : aa(int,$o,ord_less(int,bit_ri4674362597316999326ke_bit(int,Na,K)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ).

% signed_take_bit_int_less_exp
tff(fact_3045_num_Osize_I5_J,axiom,
    ! [X22: num] : size_size(num,bit0(X22)) = aa(nat,nat,plus_plus(nat,size_size(num,X22)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size(5)
tff(fact_3046_even__signed__take__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,A3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),bit_ri4674362597316999326ke_bit(A,M,A3))
        <=> aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3) ) ) ).

% even_signed_take_bit_iff
tff(fact_3047_pi__half__neq__zero,axiom,
    aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))) != zero_zero(real) ).

% pi_half_neq_zero
tff(fact_3048_pi__half__less__two,axiom,
    aa(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_3049_pi__half__le__two,axiom,
    aa(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_3050_nat__approx__posE,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [E2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),E2)
         => ~ ! [N: nat] : ~ aa(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)))),E2) ) ) ).

% nat_approx_posE
tff(fact_3051_less__2__cases__iff,axiom,
    ! [Na: nat] :
      ( aa(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_3052_less__2__cases,axiom,
    ! [Na: nat] :
      ( aa(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_3053_le__div__geq,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(nat,$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,minus_minus(nat,M),Na)),Na)) ) ) ) ).

% le_div_geq
tff(fact_3054_split__div_H,axiom,
    ! [P: fun(nat,$o),M: nat,Na: nat] :
      ( aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na))
    <=> ( ( ( Na = zero_zero(nat) )
          & aa(nat,$o,P,zero_zero(nat)) )
        | ? [Q5: nat] :
            ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),Q5)),M)
            & aa(nat,$o,ord_less(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(nat,nat,suc,Q5)))
            & aa(nat,$o,P,Q5) ) ) ) ).

% split_div'
tff(fact_3055_Suc__times__mod__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa(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_3056_signed__take__bit__int__less__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,ord_less(int,bit_ri4674362597316999326ke_bit(int,Na,K)),K)
    <=> aa(int,$o,ord_less_eq(int,aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)),K) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_3057_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,ord_less_eq(int,K),bit_ri4674362597316999326ke_bit(int,Na,K))
    <=> aa(int,$o,ord_less(int,K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_3058_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,ord_less_eq(int,bit_ri4674362597316999326ke_bit(int,Na,K)),K)
    <=> aa(int,$o,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(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_3059_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [Na: nat,K: int] : aa(int,$o,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))),bit_ri4674362597316999326ke_bit(int,Na,K)) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_3060_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,ord_less(int,K),bit_ri4674362597316999326ke_bit(int,Na,K))
    <=> aa(int,$o,ord_less(int,K),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_3061_fact__code,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Na: nat] : semiring_char_0_fact(A,Na) = aa(nat,A,semiring_1_of_nat(A),set_fo6178422350223883121st_nat(nat,times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2)),Na,one_one(nat))) ) ).

% fact_code
tff(fact_3062_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,minus_minus(nat,M),one_one(nat))))) ) ).

% fact_num_eq_if
tff(fact_3063_fact__reduce,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Na: nat] :
          ( aa(nat,$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,minus_minus(nat,Na),one_one(nat)))) ) ) ) ).

% fact_reduce
tff(fact_3064_pochhammer__same,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & comm_ring_1(A)
        & semiri3467727345109120633visors(A) )
     => ! [Na: nat] : comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Na)),Na) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),Na)),semiring_char_0_fact(A,Na)) ) ).

% pochhammer_same
tff(fact_3065_pi__half__gt__zero,axiom,
    aa(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_3066_pi__half__ge__zero,axiom,
    aa(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_3067_binomial__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,ord_less_eq(nat,K),Na)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,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,minus_minus(nat,Na),K)))) ) ) ) ).

% binomial_fact
tff(fact_3068_fact__binomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Na: nat] :
          ( aa(nat,$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),aa(nat,nat,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,minus_minus(nat,Na),K))) ) ) ) ).

% fact_binomial
tff(fact_3069_m2pi__less__pi,axiom,
    aa(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_3070_power__odd__eq,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,Na: nat] : aa(nat,A,power_power(A,A3),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,power_power(A,aa(nat,A,power_power(A,A3),Na)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% power_odd_eq
tff(fact_3071_Suc__n__div__2__gt__zero,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => aa(nat,$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_3072_div__2__gt__zero,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,zero_zero(nat))),Na)
     => aa(nat,$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_3073_nat__bit__induct,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ! [N: nat] :
            ( aa(nat,$o,P,N)
           => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
             => aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ) )
       => ( ! [N: nat] :
              ( aa(nat,$o,P,N)
             => aa(nat,$o,P,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) )
         => aa(nat,$o,P,Na) ) ) ) ).

% nat_bit_induct
tff(fact_3074_arctan__ubound,axiom,
    ! [Y: real] : aa(real,$o,ord_less(real,aa(real,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_3075_arctan__one,axiom,
    aa(real,real,arctan,one_one(real)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))) ).

% arctan_one
tff(fact_3076_binomial__less__binomial__Suc,axiom,
    ! [K: nat,Na: nat] :
      ( aa(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,ord_less(nat,aa(nat,nat,binomial(Na),K)),aa(nat,nat,binomial(Na),aa(nat,nat,suc,K))) ) ).

% binomial_less_binomial_Suc
tff(fact_3077_central__binomial__odd,axiom,
    ! [Na: nat] :
      ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => ( aa(nat,nat,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))))) = aa(nat,nat,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_3078_binomial__addition__formula,axiom,
    ! [Na: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(nat,nat,binomial(Na),aa(nat,nat,suc,K)) = aa(nat,nat,plus_plus(nat,aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,Na),one_one(nat))),aa(nat,nat,suc,K))),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,Na),one_one(nat))),K)) ) ) ).

% binomial_addition_formula
tff(fact_3079_nat__intermed__int__val,axiom,
    ! [M: nat,Na: nat,F2: fun(nat,int),K: int] :
      ( ! [I3: nat] :
          ( ( aa(nat,$o,ord_less_eq(nat,M),I3)
            & aa(nat,$o,ord_less(nat,I3),Na) )
         => aa(int,$o,ord_less_eq(int,aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,aa(nat,int,F2,aa(nat,nat,suc,I3))),aa(nat,int,F2,I3)))),one_one(int)) )
     => ( aa(nat,$o,ord_less_eq(nat,M),Na)
       => ( aa(int,$o,ord_less_eq(int,aa(nat,int,F2,M)),K)
         => ( aa(int,$o,ord_less_eq(int,K),aa(nat,int,F2,Na))
           => ? [I3: nat] :
                ( aa(nat,$o,ord_less_eq(nat,M),I3)
                & aa(nat,$o,ord_less_eq(nat,I3),Na)
                & ( aa(nat,int,F2,I3) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_3080_add__0__iff,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [B3: A,A3: A] :
          ( ( B3 = aa(A,A,plus_plus(A,B3),A3) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% add_0_iff
tff(fact_3081_signed__take__bit__int__eq__self,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))),K)
     => ( aa(int,$o,ord_less(int,K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))
       => ( bit_ri4674362597316999326ke_bit(int,Na,K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_3082_signed__take__bit__int__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( ( bit_ri4674362597316999326ke_bit(int,Na,K) = K )
    <=> ( aa(int,$o,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))),K)
        & aa(int,$o,ord_less(int,K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_3083_powser__split__head_I1_J,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_bx(fun(nat,A),fun(A,fun(nat,A)),F2),Z))
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_bx(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) = aa(A,A,plus_plus(A,aa(nat,A,F2,zero_zero(nat))),aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,aa(A,fun(nat,A),aTP_Lamp_by(fun(nat,A),fun(A,fun(nat,A)),F2),Z))),Z)) ) ) ) ).

% powser_split_head(1)
tff(fact_3084_powser__split__head_I2_J,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_bx(fun(nat,A),fun(A,fun(nat,A)),F2),Z))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,aa(A,fun(nat,A),aTP_Lamp_by(fun(nat,A),fun(A,fun(nat,A)),F2),Z))),Z) = aa(A,A,minus_minus(A,suminf(A,aa(A,fun(nat,A),aTP_Lamp_bx(fun(nat,A),fun(A,fun(nat,A)),F2),Z))),aa(nat,A,F2,zero_zero(nat))) ) ) ) ).

% powser_split_head(2)
tff(fact_3085_odd__0__le__power__imp__0__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,power_power(A,A3),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))))
         => aa(A,$o,ord_less_eq(A,zero_zero(A)),A3) ) ) ).

% odd_0_le_power_imp_0_le
tff(fact_3086_odd__power__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,ord_less(A,A3),zero_zero(A))
         => aa(A,$o,ord_less(A,aa(nat,A,power_power(A,A3),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)))),zero_zero(A)) ) ) ).

% odd_power_less_zero
tff(fact_3087_minus__pi__half__less__zero,axiom,
    aa(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_3088_power__minus1__odd,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Na: nat] : aa(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_3089_arctan__lbound,axiom,
    ! [Y: real] : aa(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,arctan,Y)) ).

% arctan_lbound
tff(fact_3090_arctan__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arctan,Y))
      & aa(real,$o,ord_less(real,aa(real,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_3091_int__power__div__base,axiom,
    ! [M: nat,K: int] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),K)
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,power_power(int,K),M)),K) = aa(nat,int,power_power(int,K),aa(nat,nat,minus_minus(nat,M),aa(nat,nat,suc,zero_zero(nat)))) ) ) ) ).

% int_power_div_base
tff(fact_3092_nat__ivt__aux,axiom,
    ! [Na: nat,F2: fun(nat,int),K: int] :
      ( ! [I3: nat] :
          ( aa(nat,$o,ord_less(nat,I3),Na)
         => aa(int,$o,ord_less_eq(int,aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,aa(nat,int,F2,aa(nat,nat,suc,I3))),aa(nat,int,F2,I3)))),one_one(int)) )
     => ( aa(int,$o,ord_less_eq(int,aa(nat,int,F2,zero_zero(nat))),K)
       => ( aa(int,$o,ord_less_eq(int,K),aa(nat,int,F2,Na))
         => ? [I3: nat] :
              ( aa(nat,$o,ord_less_eq(nat,I3),Na)
              & ( aa(nat,int,F2,I3) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_3093_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,$o,ord_less_eq(nat,M),Na)
        & aa(nat,$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_3094_unset__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A3: A] : bit_se2638667681897837118et_bit(A,aa(nat,nat,suc,Na),A3) = aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2638667681897837118et_bit(A,Na,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% unset_bit_Suc
tff(fact_3095_set__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A3: A] : bit_se5668285175392031749et_bit(A,aa(nat,nat,suc,Na),A3) = aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se5668285175392031749et_bit(A,Na,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% set_bit_Suc
tff(fact_3096_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,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,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_3097_sb__inc__lem,axiom,
    ! [A3: int,K: nat] :
      ( aa(int,$o,ord_less(int,aa(int,int,plus_plus(int,A3),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),K))),zero_zero(int))
     => aa(int,$o,ord_less_eq(int,aa(int,int,plus_plus(int,aa(int,int,plus_plus(int,A3),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),K))),aa(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,plus_plus(int,A3),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),K))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,K)))) ) ).

% sb_inc_lem
tff(fact_3098_sb__inc__lem_H,axiom,
    ! [A3: int,K: nat] :
      ( aa(int,$o,ord_less(int,A3),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),K)))
     => aa(int,$o,ord_less_eq(int,aa(int,int,plus_plus(int,aa(int,int,plus_plus(int,A3),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),K))),aa(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,plus_plus(int,A3),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),K))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,K)))) ) ).

% sb_inc_lem'
tff(fact_3099_fact__double,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Na: nat] : semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) = 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,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))),Na))),semiring_char_0_fact(A,Na)) ) ).

% fact_double
tff(fact_3100_lemma__termdiff3,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [H: A,Z: A,K4: real,Na: nat] :
          ( ( H != zero_zero(A) )
         => ( aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,Z)),K4)
           => ( aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,plus_plus(A,Z),H))),K4)
             => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(A,A,plus_plus(A,Z),H)),Na)),aa(nat,A,power_power(A,Z),Na))),H)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),aa(nat,A,power_power(A,Z),aa(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,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat)))))),aa(nat,real,power_power(real,K4),aa(nat,nat,minus_minus(nat,Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),real_V7770717601297561774m_norm(A,H))) ) ) ) ) ).

% lemma_termdiff3
tff(fact_3101_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,power_power(real,aa(real,real,uminus_uminus(real),one_one(real))),Na) ).

% sin_cos_npi
tff(fact_3102_Suc__if__eq,axiom,
    ! [A: $tType,F2: fun(nat,A),H: fun(nat,A),G: A,Na: nat] :
      ( ! [N: nat] : aa(nat,A,F2,aa(nat,nat,suc,N)) = aa(nat,A,H,N)
     => ( ( aa(nat,A,F2,zero_zero(nat)) = G )
       => ( aa(nat,A,F2,Na) = $ite(Na = zero_zero(nat),G,aa(nat,A,H,aa(nat,nat,minus_minus(nat,Na),one_one(nat)))) ) ) ) ).

% Suc_if_eq
tff(fact_3103_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_3104_norm__eq__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] :
          ( ( real_V7770717601297561774m_norm(A,X) = zero_zero(real) )
        <=> ( X = zero_zero(A) ) ) ) ).

% norm_eq_zero
tff(fact_3105_norm__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ( real_V7770717601297561774m_norm(A,zero_zero(A)) = zero_zero(real) ) ) ).

% norm_zero
tff(fact_3106_norm__one,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ( real_V7770717601297561774m_norm(A,one_one(A)) = one_one(real) ) ) ).

% norm_one
tff(fact_3107_norm__numeral,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [W: num] : real_V7770717601297561774m_norm(A,aa(num,A,numeral_numeral(A),W)) = aa(num,real,numeral_numeral(real),W) ) ).

% norm_numeral
tff(fact_3108_norm__of__nat,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Na: nat] : real_V7770717601297561774m_norm(A,aa(nat,A,semiring_1_of_nat(A),Na)) = aa(nat,real,semiring_1_of_nat(real),Na) ) ).

% norm_of_nat
tff(fact_3109_sin__pi,axiom,
    sin(real,pi) = zero_zero(real) ).

% sin_pi
tff(fact_3110_zero__less__norm__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),real_V7770717601297561774m_norm(A,X))
        <=> ( X != zero_zero(A) ) ) ) ).

% zero_less_norm_iff
tff(fact_3111_norm__le__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] :
          ( aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,X)),zero_zero(real))
        <=> ( X = zero_zero(A) ) ) ) ).

% norm_le_zero_iff
tff(fact_3112_norm__neg__numeral,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [W: num] : real_V7770717601297561774m_norm(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) = aa(num,real,numeral_numeral(real),W) ) ).

% norm_neg_numeral
tff(fact_3113_sin__of__real__pi,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( sin(A,aa(real,A,real_Vector_of_real(A),pi)) = zero_zero(A) ) ) ).

% sin_of_real_pi
tff(fact_3114_norm__mult__numeral1,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [W: num,A3: A] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),A3)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),W)),real_V7770717601297561774m_norm(A,A3)) ) ).

% norm_mult_numeral1
tff(fact_3115_norm__mult__numeral2,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A3: A,W: num] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W))) = aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,A3)),aa(num,real,numeral_numeral(real),W)) ) ).

% norm_mult_numeral2
tff(fact_3116_norm__divide__numeral,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A3: A,W: num] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),W))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(A,A3)),aa(num,real,numeral_numeral(real),W)) ) ).

% norm_divide_numeral
tff(fact_3117_sin__npi,axiom,
    ! [Na: nat] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),pi)) = zero_zero(real) ).

% sin_npi
tff(fact_3118_sin__npi2,axiom,
    ! [Na: nat] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(nat,real,semiring_1_of_nat(real),Na))) = zero_zero(real) ).

% sin_npi2
tff(fact_3119_sin__npi__int,axiom,
    ! [Na: int] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(int,real,ring_1_of_int(real),Na))) = zero_zero(real) ).

% sin_npi_int
tff(fact_3120_summable__geometric__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( summable(A,power_power(A,C2))
        <=> aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,C2)),one_one(real)) ) ) ).

% summable_geometric_iff
tff(fact_3121_sin__two__pi,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)) = zero_zero(real) ).

% sin_two_pi
tff(fact_3122_sin__pi__half,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) = one_one(real) ).

% sin_pi_half
tff(fact_3123_norm__of__real__add1,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: real] : real_V7770717601297561774m_norm(A,aa(A,A,plus_plus(A,aa(real,A,real_Vector_of_real(A),X)),one_one(A))) = aa(real,real,abs_abs(real),aa(real,real,plus_plus(real,X),one_one(real))) ) ).

% norm_of_real_add1
tff(fact_3124_norm__of__real__addn,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: real,B3: num] : real_V7770717601297561774m_norm(A,aa(A,A,plus_plus(A,aa(real,A,real_Vector_of_real(A),X)),aa(num,A,numeral_numeral(A),B3))) = aa(real,real,abs_abs(real),aa(real,real,plus_plus(real,X),aa(num,real,numeral_numeral(real),B3))) ) ).

% norm_of_real_addn
tff(fact_3125_sin__periodic,axiom,
    ! [X: real] : sin(real,aa(real,real,plus_plus(real,X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))) = sin(real,X) ).

% sin_periodic
tff(fact_3126_sin__of__real__pi__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V7773925162809079976_field(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( sin(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(real,A,real_Vector_of_real(A),pi)),aa(num,A,numeral_numeral(A),bit0(one2)))) = one_one(A) ) ) ).

% sin_of_real_pi_half
tff(fact_3127_sin__2npi,axiom,
    ! [Na: nat] : sin(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(nat,real,semiring_1_of_nat(real),Na))),pi)) = zero_zero(real) ).

% sin_2npi
tff(fact_3128_sin__2pi__minus,axiom,
    ! [X: real] : sin(real,aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)),X)) = aa(real,real,uminus_uminus(real),sin(real,X)) ).

% sin_2pi_minus
tff(fact_3129_sin__int__2pin,axiom,
    ! [Na: int] : sin(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))),pi)),aa(int,real,ring_1_of_int(real),Na))) = zero_zero(real) ).

% sin_int_2pin
tff(fact_3130_sin__x__le__x,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => aa(real,$o,ord_less_eq(real,sin(real,X)),X) ) ).

% sin_x_le_x
tff(fact_3131_sin__le__one,axiom,
    ! [X: real] : aa(real,$o,ord_less_eq(real,sin(real,X)),one_one(real)) ).

% sin_le_one
tff(fact_3132_norm__not__less__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] : ~ aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,X)),zero_zero(real)) ) ).

% norm_not_less_zero
tff(fact_3133_norm__ge__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] : aa(real,$o,ord_less_eq(real,zero_zero(real)),real_V7770717601297561774m_norm(A,X)) ) ).

% norm_ge_zero
tff(fact_3134_norm__power,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: A,Na: nat] : real_V7770717601297561774m_norm(A,aa(nat,A,power_power(A,X),Na)) = aa(nat,real,power_power(real,real_V7770717601297561774m_norm(A,X)),Na) ) ).

% norm_power
tff(fact_3135_summable__norm__cancel,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_ca(fun(nat,A),fun(nat,real),F2))
         => summable(A,F2) ) ) ).

% summable_norm_cancel
tff(fact_3136_power__eq__imp__eq__norm,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [W: A,Na: nat,Z: A] :
          ( ( aa(nat,A,power_power(A,W),Na) = aa(nat,A,power_power(A,Z),Na) )
         => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
           => ( real_V7770717601297561774m_norm(A,W) = real_V7770717601297561774m_norm(A,Z) ) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_3137_nonzero__norm__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [B3: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(A,A3)),real_V7770717601297561774m_norm(A,B3)) ) ) ) ).

% nonzero_norm_divide
tff(fact_3138_sin__gt__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less(real,X),pi)
       => aa(real,$o,ord_less(real,zero_zero(real)),sin(real,X)) ) ) ).

% sin_gt_zero
tff(fact_3139_sin__x__ge__neg__x,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),X)),sin(real,X)) ) ).

% sin_x_ge_neg_x
tff(fact_3140_sin__ge__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less_eq(real,X),pi)
       => aa(real,$o,ord_less_eq(real,zero_zero(real)),sin(real,X)) ) ) ).

% sin_ge_zero
tff(fact_3141_sin__ge__minus__one,axiom,
    ! [X: real] : aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),sin(real,X)) ).

% sin_ge_minus_one
tff(fact_3142_norm__mult__less,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [X: A,R3: real,Y: A,Sa: real] :
          ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,X)),R3)
         => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Y)),Sa)
           => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Y))),aa(real,real,aa(real,fun(real,real),times_times(real),R3),Sa)) ) ) ) ).

% norm_mult_less
tff(fact_3143_abs__sin__le__one,axiom,
    ! [X: real] : aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),sin(real,X))),one_one(real)) ).

% abs_sin_le_one
tff(fact_3144_norm__triangle__lt,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y: A,E2: real] :
          ( aa(real,$o,ord_less(real,aa(real,real,plus_plus(real,real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,Y))),E2)
         => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,plus_plus(A,X),Y))),E2) ) ) ).

% norm_triangle_lt
tff(fact_3145_norm__add__less,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,R3: real,Y: A,Sa: real] :
          ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,X)),R3)
         => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Y)),Sa)
           => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,plus_plus(A,X),Y))),aa(real,real,plus_plus(real,R3),Sa)) ) ) ) ).

% norm_add_less
tff(fact_3146_norm__diff__triangle__less,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y: A,E1: real,Z: A,E22: real] :
          ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X),Y))),E1)
         => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Y),Z))),E22)
           => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X),Z))),aa(real,real,plus_plus(real,E1),E22)) ) ) ) ).

% norm_diff_triangle_less
tff(fact_3147_norm__power__ineq,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [X: A,Na: nat] : aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(nat,A,power_power(A,X),Na))),aa(nat,real,power_power(real,real_V7770717601297561774m_norm(A,X)),Na)) ) ).

% norm_power_ineq
tff(fact_3148_summable__norm__comparison__test,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( ? [N7: nat] :
            ! [N: nat] :
              ( aa(nat,$o,ord_less_eq(nat,N7),N)
             => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(nat,A,F2,N))),aa(nat,real,G,N)) )
         => ( summable(real,G)
           => summable(real,aTP_Lamp_cb(fun(nat,A),fun(nat,real),F2)) ) ) ) ).

% summable_norm_comparison_test
tff(fact_3149_power__eq__1__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [W: A,Na: nat] :
          ( ( aa(nat,A,power_power(A,W),Na) = one_one(A) )
         => ( ( real_V7770717601297561774m_norm(A,W) = one_one(real) )
            | ( Na = zero_zero(nat) ) ) ) ) ).

% power_eq_1_iff
tff(fact_3150_sin__eq__0__pi,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),pi)),X)
     => ( aa(real,$o,ord_less(real,X),pi)
       => ( ( sin(real,X) = zero_zero(real) )
         => ( X = zero_zero(real) ) ) ) ) ).

% sin_eq_0_pi
tff(fact_3151_norm__less__p1,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [X: A] : aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,aa(A,A,plus_plus(A,aa(real,A,real_Vector_of_real(A),real_V7770717601297561774m_norm(A,X))),one_one(A)))) ) ).

% norm_less_p1
tff(fact_3152_sin__zero__pi__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,abs_abs(real),X)),pi)
     => ( ( sin(real,X) = zero_zero(real) )
      <=> ( X = zero_zero(real) ) ) ) ).

% sin_zero_pi_iff
tff(fact_3153_sin__zero__iff__int2,axiom,
    ! [X: real] :
      ( ( sin(real,X) = zero_zero(real) )
    <=> ? [I4: int] : X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),I4)),pi) ) ).

% sin_zero_iff_int2
tff(fact_3154_powser__inside,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(nat,A),X: A,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_bx(fun(nat,A),fun(A,fun(nat,A)),F2),X))
         => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Z)),real_V7770717601297561774m_norm(A,X))
           => summable(A,aa(A,fun(nat,A),aTP_Lamp_bx(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) ) ) ) ).

% powser_inside
tff(fact_3155_complete__algebra__summable__geometric,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] :
          ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,X)),one_one(real))
         => summable(A,power_power(A,X)) ) ) ).

% complete_algebra_summable_geometric
tff(fact_3156_summable__geometric,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,C2)),one_one(real))
         => summable(A,power_power(A,C2)) ) ) ).

% summable_geometric
tff(fact_3157_sin__gt__zero__02,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less(real,X),aa(num,real,numeral_numeral(real),bit0(one2)))
       => aa(real,$o,ord_less(real,zero_zero(real)),sin(real,X)) ) ) ).

% sin_gt_zero_02
tff(fact_3158_summable__norm,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_ca(fun(nat,A),fun(nat,real),F2))
         => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,suminf(A,F2))),suminf(real,aTP_Lamp_ca(fun(nat,A),fun(nat,real),F2))) ) ) ).

% summable_norm
tff(fact_3159_lemma__NBseq__def2,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X5: fun(A,B)] :
          ( ? [K6: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),K6)
              & ! [N3: A] : aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(A,B,X5,N3))),K6) )
        <=> ? [N8: nat] :
            ! [N3: A] : aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(B,aa(A,B,X5,N3))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N8))) ) ) ).

% lemma_NBseq_def2
tff(fact_3160_lemma__NBseq__def,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X5: fun(A,B)] :
          ( ? [K6: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),K6)
              & ! [N3: A] : aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(A,B,X5,N3))),K6) )
        <=> ? [N8: nat] :
            ! [N3: A] : aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(A,B,X5,N3))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N8))) ) ) ).

% lemma_NBseq_def
tff(fact_3161_powser__insidea,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,A),X: A,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_bp(fun(nat,A),fun(A,fun(nat,A)),F2),X))
         => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Z)),real_V7770717601297561774m_norm(A,X))
           => summable(real,aa(A,fun(nat,real),aTP_Lamp_cc(fun(nat,A),fun(A,fun(nat,real)),F2),Z)) ) ) ) ).

% powser_insidea
tff(fact_3162_square__fact__le__2__fact,axiom,
    ! [Na: nat] : aa(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_3163_square__norm__one,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: A] :
          ( ( aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(A) )
         => ( real_V7770717601297561774m_norm(A,X) = one_one(real) ) ) ) ).

% square_norm_one
tff(fact_3164_sin__pi__divide__n__ge__0,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero(nat) )
     => aa(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_3165_norm__power__diff,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A,W: A,M: nat] :
          ( aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,Z)),one_one(real))
         => ( aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,W)),one_one(real))
           => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,power_power(A,Z),M)),aa(nat,A,power_power(A,W),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,minus_minus(A,Z),W)))) ) ) ) ).

% norm_power_diff
tff(fact_3166_suminf__exist__split,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R3: real,F2: fun(nat,A)] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),R3)
         => ( summable(A,F2)
           => ? [N9: nat] :
              ! [N10: nat] :
                ( aa(nat,$o,ord_less_eq(nat,N9),N10)
               => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,suminf(A,aa(nat,fun(nat,A),aTP_Lamp_ba(fun(nat,A),fun(nat,fun(nat,A)),F2),N10)))),R3) ) ) ) ) ).

% suminf_exist_split
tff(fact_3167_Abel__lemma,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R3: real,R0: real,A3: fun(nat,A),M7: real] :
          ( aa(real,$o,ord_less_eq(real,zero_zero(real)),R3)
         => ( aa(real,$o,ord_less(real,R3),R0)
           => ( ! [N: nat] : aa(real,$o,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,aa(nat,A,A3,N))),aa(nat,real,power_power(real,R0),N))),M7)
             => summable(real,aa(fun(nat,A),fun(nat,real),aTP_Lamp_cd(real,fun(fun(nat,A),fun(nat,real)),R3),A3)) ) ) ) ) ).

% Abel_lemma
tff(fact_3168_sin__gt__zero2,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ( aa(real,$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,ord_less(real,zero_zero(real)),sin(real,X)) ) ) ).

% sin_gt_zero2
tff(fact_3169_sin__lt__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,pi),X)
     => ( aa(real,$o,ord_less(real,X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
       => aa(real,$o,ord_less(real,sin(real,X)),zero_zero(real)) ) ) ).

% sin_lt_zero
tff(fact_3170_sin__monotone__2pi__le,axiom,
    ! [Y: real,X: real] :
      ( aa(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,ord_less_eq(real,Y),X)
       => ( aa(real,$o,ord_less_eq(real,X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,ord_less_eq(real,sin(real,Y)),sin(real,X)) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_3171_sin__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( aa(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,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))))
       => ( aa(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,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,$o,ord_less_eq(real,sin(real,X)),sin(real,Y))
            <=> aa(real,$o,ord_less_eq(real,X),Y) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_3172_sin__inj__pi,axiom,
    ! [X: real,Y: real] :
      ( aa(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,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))))
       => ( aa(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,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,X) = sin(real,Y) )
             => ( X = Y ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_3173_summable__ratio__test,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [C2: real,N2: nat,F2: fun(nat,A)] :
          ( aa(real,$o,ord_less(real,C2),one_one(real))
         => ( ! [N: nat] :
                ( aa(nat,$o,ord_less_eq(nat,N2),N)
               => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(nat,A,F2,aa(nat,nat,suc,N)))),aa(real,real,aa(real,fun(real,real),times_times(real),C2),real_V7770717601297561774m_norm(A,aa(nat,A,F2,N)))) )
           => summable(A,F2) ) ) ) ).

% summable_ratio_test
tff(fact_3174_suminf__geometric,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,C2)),one_one(real))
         => ( suminf(A,power_power(A,C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(A,A,minus_minus(A,one_one(A)),C2)) ) ) ) ).

% suminf_geometric
tff(fact_3175_sin__le__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,pi),X)
     => ( aa(real,$o,ord_less(real,X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
       => aa(real,$o,ord_less_eq(real,sin(real,X)),zero_zero(real)) ) ) ).

% sin_le_zero
tff(fact_3176_sin__less__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),X)
     => ( aa(real,$o,ord_less(real,X),zero_zero(real))
       => aa(real,$o,ord_less(real,sin(real,X)),zero_zero(real)) ) ) ).

% sin_less_zero
tff(fact_3177_sin__monotone__2pi,axiom,
    ! [Y: real,X: real] :
      ( aa(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,ord_less(real,Y),X)
       => ( aa(real,$o,ord_less_eq(real,X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,ord_less(real,sin(real,Y)),sin(real,X)) ) ) ) ).

% sin_monotone_2pi
tff(fact_3178_sin__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( aa(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,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))))
       => ( aa(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,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,$o,ord_less(real,sin(real,X)),sin(real,Y))
            <=> aa(real,$o,ord_less(real,X),Y) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_3179_sin__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
       => ? [X4: real] :
            ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X4)
            & aa(real,$o,ord_less_eq(real,X4),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
            & ( sin(real,X4) = Y )
            & ! [Y4: real] :
                ( ( aa(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,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 = X4 ) ) ) ) ) ).

% sin_total
tff(fact_3180_sin__pi__divide__n__gt__0,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => aa(real,$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_3181_sin__zero__iff__int,axiom,
    ! [X: real] :
      ( ( sin(real,X) = zero_zero(real) )
    <=> ? [I4: int] :
          ( aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),I4)
          & ( X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),I4)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ).

% sin_zero_iff_int
tff(fact_3182_sin__zero__lemma,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => ( ( sin(real,X) = zero_zero(real) )
       => ? [N: nat] :
            ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
            & ( X = 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_3183_sin__zero__iff,axiom,
    ! [X: real] :
      ( ( sin(real,X) = zero_zero(real) )
    <=> ( ? [N3: nat] :
            ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N3)
            & ( X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N3)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) )
        | ? [N3: nat] :
            ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N3)
            & ( X = aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N3)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))) ) ) ) ) ).

% sin_zero_iff
tff(fact_3184_sin__coeff__def,axiom,
    ! [X2: nat] :
      sin_coeff(X2) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),X2),zero_zero(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(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,minus_minus(nat,X2),aa(nat,nat,suc,zero_zero(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),semiring_char_0_fact(real,X2))) ).

% sin_coeff_def
tff(fact_3185_cos__coeff__def,axiom,
    ! [X2: nat] :
      cos_coeff(X2) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),X2),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(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),X2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),semiring_char_0_fact(real,X2)),zero_zero(real)) ).

% cos_coeff_def
tff(fact_3186_flip__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A3: A] : bit_se8732182000553998342ip_bit(A,aa(nat,nat,suc,Na),A3) = aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se8732182000553998342ip_bit(A,Na,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% flip_bit_Suc
tff(fact_3187_signed__take__bit__Suc__minus__bit1,axiom,
    ! [Na: nat,K: num] : 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,plus_plus(int,aa(int,int,aa(int,fun(int,int),times_times(int),bit_ri4674362597316999326ke_bit(int,Na,aa(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_3188_VEBT__internal_OT__vebt__buildupi_Oelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( vEBT_V441764108873111860ildupi(X) = Y )
     => ( ( ( X = zero_zero(nat) )
         => ( Y != aa(nat,nat,suc,zero_zero(nat)) ) )
       => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != aa(nat,nat,suc,zero_zero(nat)) ) )
         => ~ ! [N: nat] :
                ( ( X = aa(nat,nat,suc,aa(nat,nat,suc,N)) )
               => ( Y != $ite(aa(nat,$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,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,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,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,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,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,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,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,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_3189_verit__eq__simplify_I9_J,axiom,
    ! [X32: num,Y32: num] :
      ( ( bit1(X32) = bit1(Y32) )
    <=> ( X32 = Y32 ) ) ).

% verit_eq_simplify(9)
tff(fact_3190_semiring__norm_I90_J,axiom,
    ! [M: num,Na: num] :
      ( ( bit1(M) = bit1(Na) )
    <=> ( M = Na ) ) ).

% semiring_norm(90)
tff(fact_3191_semiring__norm_I88_J,axiom,
    ! [M: num,Na: num] : bit0(M) != bit1(Na) ).

% semiring_norm(88)
tff(fact_3192_semiring__norm_I89_J,axiom,
    ! [M: num,Na: num] : bit1(M) != bit0(Na) ).

% semiring_norm(89)
tff(fact_3193_semiring__norm_I84_J,axiom,
    ! [Na: num] : one2 != bit1(Na) ).

% semiring_norm(84)
tff(fact_3194_semiring__norm_I86_J,axiom,
    ! [M: num] : bit1(M) != one2 ).

% semiring_norm(86)
tff(fact_3195_flip__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),bit_se8732182000553998342ip_bit(int,Na,K))
    <=> aa(int,$o,ord_less_eq(int,zero_zero(int)),K) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_3196_flip__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,ord_less(int,bit_se8732182000553998342ip_bit(int,Na,K)),zero_zero(int))
    <=> aa(int,$o,ord_less(int,K),zero_zero(int)) ) ).

% flip_bit_negative_int_iff
tff(fact_3197_semiring__norm_I73_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,ord_less_eq(num,bit1(M)),bit1(Na))
    <=> aa(num,$o,ord_less_eq(num,M),Na) ) ).

% semiring_norm(73)
tff(fact_3198_semiring__norm_I80_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,ord_less(num,bit1(M)),bit1(Na))
    <=> aa(num,$o,ord_less(num,M),Na) ) ).

% semiring_norm(80)
tff(fact_3199_semiring__norm_I9_J,axiom,
    ! [M: num,Na: num] : aa(num,num,plus_plus(num,bit1(M)),bit0(Na)) = bit1(aa(num,num,plus_plus(num,M),Na)) ).

% semiring_norm(9)
tff(fact_3200_semiring__norm_I7_J,axiom,
    ! [M: num,Na: num] : aa(num,num,plus_plus(num,bit0(M)),bit1(Na)) = bit1(aa(num,num,plus_plus(num,M),Na)) ).

% semiring_norm(7)
tff(fact_3201_semiring__norm_I15_J,axiom,
    ! [M: num,Na: num] : aa(num,num,aa(num,fun(num,num),times_times(num),bit1(M)),bit0(Na)) = bit0(aa(num,num,aa(num,fun(num,num),times_times(num),bit1(M)),Na)) ).

% semiring_norm(15)
tff(fact_3202_semiring__norm_I14_J,axiom,
    ! [M: num,Na: num] : aa(num,num,aa(num,fun(num,num),times_times(num),bit0(M)),bit1(Na)) = bit0(aa(num,num,aa(num,fun(num,num),times_times(num),M),bit1(Na))) ).

% semiring_norm(14)
tff(fact_3203_semiring__norm_I72_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,ord_less_eq(num,bit0(M)),bit1(Na))
    <=> aa(num,$o,ord_less_eq(num,M),Na) ) ).

% semiring_norm(72)
tff(fact_3204_semiring__norm_I81_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,ord_less(num,bit1(M)),bit0(Na))
    <=> aa(num,$o,ord_less(num,M),Na) ) ).

% semiring_norm(81)
tff(fact_3205_semiring__norm_I70_J,axiom,
    ! [M: num] : ~ aa(num,$o,ord_less_eq(num,bit1(M)),one2) ).

% semiring_norm(70)
tff(fact_3206_semiring__norm_I77_J,axiom,
    ! [Na: num] : aa(num,$o,ord_less(num,one2),bit1(Na)) ).

% semiring_norm(77)
tff(fact_3207_sin__coeff__0,axiom,
    sin_coeff(zero_zero(nat)) = zero_zero(real) ).

% sin_coeff_0
tff(fact_3208_cos__coeff__0,axiom,
    cos_coeff(zero_zero(nat)) = one_one(real) ).

% cos_coeff_0
tff(fact_3209_dbl__inc__simps_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_dbl_inc(A,aa(num,A,numeral_numeral(A),K)) = aa(num,A,numeral_numeral(A),bit1(K)) ) ).

% dbl_inc_simps(5)
tff(fact_3210_zdiv__numeral__Bit1,axiom,
    ! [V2: num,W: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),bit1(V2))),aa(num,int,numeral_numeral(int),bit0(W))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),V2)),aa(num,int,numeral_numeral(int),W)) ).

% zdiv_numeral_Bit1
tff(fact_3211_semiring__norm_I3_J,axiom,
    ! [Na: num] : aa(num,num,plus_plus(num,one2),bit0(Na)) = bit1(Na) ).

% semiring_norm(3)
tff(fact_3212_semiring__norm_I4_J,axiom,
    ! [Na: num] : aa(num,num,plus_plus(num,one2),bit1(Na)) = bit0(aa(num,num,plus_plus(num,Na),one2)) ).

% semiring_norm(4)
tff(fact_3213_semiring__norm_I5_J,axiom,
    ! [M: num] : aa(num,num,plus_plus(num,bit0(M)),one2) = bit1(M) ).

% semiring_norm(5)
tff(fact_3214_semiring__norm_I8_J,axiom,
    ! [M: num] : aa(num,num,plus_plus(num,bit1(M)),one2) = bit0(aa(num,num,plus_plus(num,M),one2)) ).

% semiring_norm(8)
tff(fact_3215_semiring__norm_I10_J,axiom,
    ! [M: num,Na: num] : aa(num,num,plus_plus(num,bit1(M)),bit1(Na)) = bit0(aa(num,num,plus_plus(num,aa(num,num,plus_plus(num,M),Na)),one2)) ).

% semiring_norm(10)
tff(fact_3216_semiring__norm_I16_J,axiom,
    ! [M: num,Na: num] : aa(num,num,aa(num,fun(num,num),times_times(num),bit1(M)),bit1(Na)) = bit1(aa(num,num,plus_plus(num,aa(num,num,plus_plus(num,M),Na)),bit0(aa(num,num,aa(num,fun(num,num),times_times(num),M),Na)))) ).

% semiring_norm(16)
tff(fact_3217_semiring__norm_I79_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,ord_less(num,bit0(M)),bit1(Na))
    <=> aa(num,$o,ord_less_eq(num,M),Na) ) ).

% semiring_norm(79)
tff(fact_3218_semiring__norm_I74_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,ord_less_eq(num,bit1(M)),bit0(Na))
    <=> aa(num,$o,ord_less(num,M),Na) ) ).

% semiring_norm(74)
tff(fact_3219_dbl__inc__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl_inc(A,one_one(A)) = aa(num,A,numeral_numeral(A),bit1(one2)) ) ) ).

% dbl_inc_simps(3)
tff(fact_3220_Suc__div__eq__add3__div__numeral,axiom,
    ! [M: nat,V2: 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),V2)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,plus_plus(nat,aa(num,nat,numeral_numeral(nat),bit1(one2))),M)),aa(num,nat,numeral_numeral(nat),V2)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_3221_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,plus_plus(nat,aa(num,nat,numeral_numeral(nat),bit1(one2))),Na)) ).

% div_Suc_eq_div_add3
tff(fact_3222_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M: nat,V2: 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),V2)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,plus_plus(nat,aa(num,nat,numeral_numeral(nat),bit1(one2))),M)),aa(num,nat,numeral_numeral(nat),V2)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_3223_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,plus_plus(nat,aa(num,nat,numeral_numeral(nat),bit1(one2))),Na)) ).

% mod_Suc_eq_mod_add3
tff(fact_3224_dbl__dec__simps_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl_dec(A,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit1(one2))) ) ) ).

% dbl_dec_simps(4)
tff(fact_3225_zmod__numeral__Bit1,axiom,
    ! [V2: num,W: num] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(num,int,numeral_numeral(int),bit1(V2))),aa(num,int,numeral_numeral(int),bit0(W))) = aa(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),V2)),aa(num,int,numeral_numeral(int),W)))),one_one(int)) ).

% zmod_numeral_Bit1
tff(fact_3226_signed__take__bit__Suc__bit1,axiom,
    ! [Na: nat,K: num] : bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,Na),aa(num,int,numeral_numeral(int),bit1(K))) = aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),times_times(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_3227_sin__3over2__pi,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),bit1(one2))),aa(num,real,numeral_numeral(real),bit0(one2)))),pi)) = aa(real,real,uminus_uminus(real),one_one(real)) ).

% sin_3over2_pi
tff(fact_3228_verit__eq__simplify_I14_J,axiom,
    ! [X22: num,X32: num] : bit0(X22) != bit1(X32) ).

% verit_eq_simplify(14)
tff(fact_3229_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] : one2 != bit1(X32) ).

% verit_eq_simplify(12)
tff(fact_3230_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one2 )
     => ( ! [X23: num] : Y != bit0(X23)
       => ~ ! [X33: num] : Y != bit1(X33) ) ) ).

% num.exhaust
tff(fact_3231_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_3232_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_3233_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_3234_VEBT__internal_OT__vebt__buildupi__gq__0,axiom,
    ! [Na: nat] : aa(nat,$o,ord_less(nat,zero_zero(nat)),vEBT_V441764108873111860ildupi(Na)) ).

% VEBT_internal.T_vebt_buildupi_gq_0
tff(fact_3235_numeral__Bit1,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Na: num] : aa(num,A,numeral_numeral(A),bit1(Na)) = aa(A,A,plus_plus(A,aa(A,A,plus_plus(A,aa(num,A,numeral_numeral(A),Na)),aa(num,A,numeral_numeral(A),Na))),one_one(A)) ) ).

% numeral_Bit1
tff(fact_3236_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_3237_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_3238_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_3239_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_3240_power__minus__Bit1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: A,K: num] : aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),X)),aa(num,nat,numeral_numeral(nat),bit1(K))) = aa(A,A,uminus_uminus(A),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit1(K)))) ) ).

% power_minus_Bit1
tff(fact_3241_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_3242_VEBT__internal_OT__vebt__buildupi__univ,axiom,
    ! [U: nat,Na: nat] :
      ( ( U = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) )
     => aa(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)) ) ).

% VEBT_internal.T_vebt_buildupi_univ
tff(fact_3243_numeral__code_I3_J,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Na: num] :
          aa(num,A,numeral_numeral(A),bit1(Na)) = $let(
            m: A,
            m:= aa(num,A,numeral_numeral(A),Na),
            aa(A,A,plus_plus(A,aa(A,A,plus_plus(A,m),m)),one_one(A)) ) ) ).

% numeral_code(3)
tff(fact_3244_power__numeral__odd,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Z: A,W: num] :
          aa(nat,A,power_power(A,Z),aa(num,nat,numeral_numeral(nat),bit1(W))) = $let(
            w: A,
            w:= aa(nat,A,power_power(A,Z),aa(num,nat,numeral_numeral(nat),W)),
            aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),w)),w) ) ) ).

% power_numeral_odd
tff(fact_3245_numeral__Bit1__div__2,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Na: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),bit1(Na))),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(num,A,numeral_numeral(A),Na) ) ).

% numeral_Bit1_div_2
tff(fact_3246_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_3247_odd__numeral,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Na: num] : ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(num,A,numeral_numeral(A),bit1(Na))) ) ).

% odd_numeral
tff(fact_3248_power3__eq__cube,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A] : aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),bit1(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3)),A3) ) ).

% power3_eq_cube
tff(fact_3249_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_3250_Suc3__eq__add__3,axiom,
    ! [Na: nat] : aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Na))) = aa(nat,nat,plus_plus(nat,aa(num,nat,numeral_numeral(nat),bit1(one2))),Na) ).

% Suc3_eq_add_3
tff(fact_3251_num_Osize_I6_J,axiom,
    ! [X32: num] : size_size(num,bit1(X32)) = aa(nat,nat,plus_plus(nat,size_size(num,X32)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size(6)
tff(fact_3252_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_3253_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_3254_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,plus_plus(nat,aa(num,nat,numeral_numeral(nat),bit1(one2))),M)),Na) ).

% Suc_div_eq_add3_div
tff(fact_3255_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,plus_plus(nat,aa(num,nat,numeral_numeral(nat),bit1(one2))),M)),Na) ).

% Suc_mod_eq_add3_mod
tff(fact_3256_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_3257_sin__30,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit1(one2))))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% sin_30
tff(fact_3258_small__powers__of__2,axiom,
    ! [X: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit1(one2))),X)
     => aa(nat,$o,ord_less(nat,X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,X),one_one(nat)))) ) ).

% small_powers_of_2
tff(fact_3259_even__flip__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),bit_se8732182000553998342ip_bit(A,M,A3))
        <=> ~ ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
            <=> ( M = zero_zero(nat) ) ) ) ) ).

% even_flip_bit_iff
tff(fact_3260_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,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,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_3261_machin__Euler,axiom,
    aa(real,real,plus_plus(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit1(bit0(one2)))),aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit1(bit1(one2))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),bit1(one2))),aa(num,real,numeral_numeral(real),bit1(bit1(bit1(bit1(bit0(bit0(one2))))))))))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))) ).

% machin_Euler
tff(fact_3262_machin,axiom,
    aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))) = aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))),aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit1(bit0(one2))))))),aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit1(bit1(bit1(bit1(bit0(bit1(bit1(one2))))))))))) ).

% machin
tff(fact_3263_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,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,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,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,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,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,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,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,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,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_3264_VEBT__internal_OT__vebt__buildupi_H_Oelims,axiom,
    ! [X: nat,Y: int] :
      ( ( vEBT_V9176841429113362141ildupi(X) = Y )
     => ( ( ( X = zero_zero(nat) )
         => ( Y != one_one(int) ) )
       => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != one_one(int) ) )
         => ~ ! [N: nat] :
                ( ( X = aa(nat,nat,suc,aa(nat,nat,suc,N)) )
               => ( Y != $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N),aa(int,int,plus_plus(int,aa(num,int,numeral_numeral(int),bit1(one2))),aa(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,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,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,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,plus_plus(int,aa(num,int,numeral_numeral(int),bit1(one2))),aa(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,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,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,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_3265_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,
    ! [X: nat,Y: nat] :
      ( ( vEBT_V8646137997579335489_i_l_d(X) = Y )
     => ( ( ( X = zero_zero(nat) )
         => ( Y != aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ) )
       => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ) )
         => ~ ! [Va2: nat] :
                ( ( X = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
               => ( Y != $ite(
                      aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),
                      aa(nat,nat,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,plus_plus(nat,aa(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,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,plus_plus(nat,aa(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,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_3266_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,
    ! [X: nat,Y: nat] :
      ( ( vEBT_V8346862874174094_d_u_p(X) = Y )
     => ( ( ( X = zero_zero(nat) )
         => ( Y != aa(num,nat,numeral_numeral(nat),bit1(one2)) ) )
       => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != aa(num,nat,numeral_numeral(nat),bit1(one2)) ) )
         => ~ ! [Va2: nat] :
                ( ( X = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
               => ( Y != $ite(
                      aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),
                      aa(nat,nat,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,plus_plus(nat,aa(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,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),aa(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,plus_plus(nat,aa(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,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),aa(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_3267_VEBT__internal_OTb_Oelims,axiom,
    ! [X: nat,Y: int] :
      ( ( vEBT_VEBT_Tb(X) = Y )
     => ( ( ( X = zero_zero(nat) )
         => ( Y != aa(num,int,numeral_numeral(int),bit1(one2)) ) )
       => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != aa(num,int,numeral_numeral(int),bit1(one2)) ) )
         => ~ ! [N: nat] :
                ( ( X = aa(nat,nat,suc,aa(nat,nat,suc,N)) )
               => ( Y != $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N),aa(int,int,plus_plus(int,aa(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,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,plus_plus(int,aa(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,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_3268_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,
    ! [Va: nat] :
      vEBT_V8646137997579335489_i_l_d(aa(nat,nat,suc,aa(nat,nat,suc,Va))) = $ite(
        aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va))),
        aa(nat,nat,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,plus_plus(nat,aa(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,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,plus_plus(nat,aa(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,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_3269_VEBT__internal_Obuildup__build__time,axiom,
    ! [Na: nat] : aa(nat,$o,ord_less(nat,vEBT_V8346862874174094_d_u_p(Na)),vEBT_V8646137997579335489_i_l_d(Na)) ).

% VEBT_internal.buildup_build_time
tff(fact_3270_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_3271_VEBT__internal_OTbuildupi__buildupi_H,axiom,
    ! [Na: nat] : aa(nat,int,semiring_1_of_nat(int),vEBT_V441764108873111860ildupi(Na)) = vEBT_V9176841429113362141ildupi(Na) ).

% VEBT_internal.Tbuildupi_buildupi'
tff(fact_3272_VEBT__internal_OTb__T__vebt__buildupi_H,axiom,
    ! [Na: nat] : aa(int,$o,ord_less_eq(int,vEBT_V9176841429113362141ildupi(Na)),aa(int,int,minus_minus(int,vEBT_VEBT_Tb(Na)),aa(num,int,numeral_numeral(int),bit0(one2)))) ).

% VEBT_internal.Tb_T_vebt_buildupi'
tff(fact_3273_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_3274_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_3275_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_3276_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_3277_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_3278_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_3279_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_3280_VEBT__internal_OTb__T__vebt__buildupi,axiom,
    ! [Na: nat] : aa(int,$o,ord_less_eq(int,aa(nat,int,semiring_1_of_nat(int),vEBT_V441764108873111860ildupi(Na))),aa(int,int,minus_minus(int,vEBT_VEBT_Tb(Na)),aa(num,int,numeral_numeral(int),bit0(one2)))) ).

% VEBT_internal.Tb_T_vebt_buildupi
tff(fact_3281_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,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na),aa(int,int,plus_plus(int,aa(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,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,plus_plus(int,aa(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,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_3282_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,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na),aa(int,int,plus_plus(int,aa(num,int,numeral_numeral(int),bit1(one2))),aa(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,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,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,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,plus_plus(int,aa(num,int,numeral_numeral(int),bit1(one2))),aa(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,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,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,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_3283_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,
    ! [Va: nat] :
      vEBT_V8346862874174094_d_u_p(aa(nat,nat,suc,aa(nat,nat,suc,Va))) = $ite(
        aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va))),
        aa(nat,nat,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,plus_plus(nat,aa(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,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),aa(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,plus_plus(nat,aa(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,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),aa(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_3284_vebt__inst_Otime__vebt__buildup,axiom,
    ! [U: nat,Na: nat] :
      ( ( U = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) )
     => aa(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_inst.time_vebt_buildup
tff(fact_3285_vebt__buildupi__rule,axiom,
    ! [Na: nat] : time_htt(vEBT_VEBTi,pure_assn(aa(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,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ).

% vebt_buildupi_rule
tff(fact_3286_VEBT__internal_OTb_H_Oelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( vEBT_VEBT_Tb2(X) = Y )
     => ( ( ( X = zero_zero(nat) )
         => ( Y != aa(num,nat,numeral_numeral(nat),bit1(one2)) ) )
       => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != aa(num,nat,numeral_numeral(nat),bit1(one2)) ) )
         => ~ ! [N: nat] :
                ( ( X = aa(nat,nat,suc,aa(nat,nat,suc,N)) )
               => ( Y != $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N),aa(nat,nat,plus_plus(nat,aa(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,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,plus_plus(nat,aa(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,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_3287_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit1(K)))) = aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),times_times(int),bit_ri4674362597316999326ke_bit(int,pred_numeral(L),aa(int,int,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_numeral_minus_bit1
tff(fact_3288_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,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na),aa(nat,nat,plus_plus(nat,aa(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,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,plus_plus(nat,aa(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,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_3289_pred__numeral__simps_I1_J,axiom,
    pred_numeral(one2) = zero_zero(nat) ).

% pred_numeral_simps(1)
tff(fact_3290_eq__numeral__Suc,axiom,
    ! [K: num,Na: nat] :
      ( ( aa(num,nat,numeral_numeral(nat),K) = aa(nat,nat,suc,Na) )
    <=> ( pred_numeral(K) = Na ) ) ).

% eq_numeral_Suc
tff(fact_3291_Suc__eq__numeral,axiom,
    ! [Na: nat,K: num] :
      ( ( aa(nat,nat,suc,Na) = aa(num,nat,numeral_numeral(nat),K) )
    <=> ( Na = pred_numeral(K) ) ) ).

% Suc_eq_numeral
tff(fact_3292_pred__numeral__simps_I3_J,axiom,
    ! [K: num] : pred_numeral(bit1(K)) = aa(num,nat,numeral_numeral(nat),bit0(K)) ).

% pred_numeral_simps(3)
tff(fact_3293_less__numeral__Suc,axiom,
    ! [K: num,Na: nat] :
      ( aa(nat,$o,ord_less(nat,aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Na))
    <=> aa(nat,$o,ord_less(nat,pred_numeral(K)),Na) ) ).

% less_numeral_Suc
tff(fact_3294_less__Suc__numeral,axiom,
    ! [Na: nat,K: num] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,Na)),aa(num,nat,numeral_numeral(nat),K))
    <=> aa(nat,$o,ord_less(nat,Na),pred_numeral(K)) ) ).

% less_Suc_numeral
tff(fact_3295_le__Suc__numeral,axiom,
    ! [Na: nat,K: num] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,Na)),aa(num,nat,numeral_numeral(nat),K))
    <=> aa(nat,$o,ord_less_eq(nat,Na),pred_numeral(K)) ) ).

% le_Suc_numeral
tff(fact_3296_le__numeral__Suc,axiom,
    ! [K: num,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Na))
    <=> aa(nat,$o,ord_less_eq(nat,pred_numeral(K)),Na) ) ).

% le_numeral_Suc
tff(fact_3297_diff__numeral__Suc,axiom,
    ! [K: num,Na: nat] : aa(nat,nat,minus_minus(nat,aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Na)) = aa(nat,nat,minus_minus(nat,pred_numeral(K)),Na) ).

% diff_numeral_Suc
tff(fact_3298_diff__Suc__numeral,axiom,
    ! [Na: nat,K: num] : aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,Na)),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,minus_minus(nat,Na),pred_numeral(K)) ).

% diff_Suc_numeral
tff(fact_3299_signed__take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] : bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L),aa(num,int,numeral_numeral(int),bit0(K))) = aa(int,int,aa(int,fun(int,int),times_times(int),bit_ri4674362597316999326ke_bit(int,pred_numeral(L),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% signed_take_bit_numeral_bit0
tff(fact_3300_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),bit_ri4674362597316999326ke_bit(int,pred_numeral(L),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% signed_take_bit_numeral_minus_bit0
tff(fact_3301_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L),aa(num,int,numeral_numeral(int),bit1(K))) = aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),times_times(int),bit_ri4674362597316999326ke_bit(int,pred_numeral(L),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).

% signed_take_bit_numeral_bit1
tff(fact_3302_numeral__eq__Suc,axiom,
    ! [K: num] : aa(num,nat,numeral_numeral(nat),K) = aa(nat,nat,suc,pred_numeral(K)) ).

% numeral_eq_Suc
tff(fact_3303_pred__numeral__def,axiom,
    ! [K: num] : pred_numeral(K) = aa(nat,nat,minus_minus(nat,aa(num,nat,numeral_numeral(nat),K)),one_one(nat)) ).

% pred_numeral_def
tff(fact_3304_VEBT__internal_OTb__Tb_H,axiom,
    ! [Ta: nat] : vEBT_VEBT_Tb(Ta) = aa(nat,int,semiring_1_of_nat(int),vEBT_VEBT_Tb2(Ta)) ).

% VEBT_internal.Tb_Tb'
tff(fact_3305_fact__numeral,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [K: num] : semiring_char_0_fact(A,aa(num,nat,numeral_numeral(nat),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),K)),semiring_char_0_fact(A,pred_numeral(K))) ) ).

% fact_numeral
tff(fact_3306_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_3307_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_3308_VEBT__internal_OTb__T__vebt__buildupi_H_H,axiom,
    ! [Na: nat] : aa(nat,$o,ord_less_eq(nat,vEBT_V441764108873111860ildupi(Na)),aa(nat,nat,minus_minus(nat,vEBT_VEBT_Tb2(Na)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% VEBT_internal.Tb_T_vebt_buildupi''
tff(fact_3309_vebt__inserti__rule,axiom,
    ! [X: nat,Na: nat,Sa: set(nat),Ti: vEBT_VEBTi] :
      ( aa(nat,$o,ord_less(nat,X),aa(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),Ti),vEBT_vebt_inserti(Ti,X),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),insert(nat,X),bot_bot(set(nat))))),aa(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))))),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_3310_sincos__total__2pi,axiom,
    ! [X: real,Y: real] :
      ( ( aa(real,real,plus_plus(real,aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(real) )
     => ~ ! [T2: real] :
            ( aa(real,$o,ord_less_eq(real,zero_zero(real)),T2)
           => ( aa(real,$o,ord_less(real,T2),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
             => ( ( X = cos(real,T2) )
               => ( Y != sin(real,T2) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_3311_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_3312_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_3313_sin__paired,axiom,
    ! [X: real] : aa(real,$o,sums(real,aTP_Lamp_ce(real,fun(nat,real),X)),sin(real,X)) ).

% sin_paired
tff(fact_3314_nat__int,axiom,
    ! [Na: nat] : nat2(aa(nat,int,semiring_1_of_nat(int),Na)) = Na ).

% nat_int
tff(fact_3315_sums__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => aa(A,$o,sums(A,aTP_Lamp_bc(nat,A)),zero_zero(A)) ) ).

% sums_zero
tff(fact_3316_triangle__0,axiom,
    nat_triangle(zero_zero(nat)) = zero_zero(nat) ).

% triangle_0
tff(fact_3317_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_3318_nat__numeral,axiom,
    ! [K: num] : nat2(aa(num,int,numeral_numeral(int),K)) = aa(num,nat,numeral_numeral(nat),K) ).

% nat_numeral
tff(fact_3319_nat__1,axiom,
    nat2(one_one(int)) = aa(nat,nat,suc,zero_zero(nat)) ).

% nat_1
tff(fact_3320_nat__0__iff,axiom,
    ! [I: int] :
      ( ( nat2(I) = zero_zero(nat) )
    <=> aa(int,$o,ord_less_eq(int,I),zero_zero(int)) ) ).

% nat_0_iff
tff(fact_3321_nat__le__0,axiom,
    ! [Z: int] :
      ( aa(int,$o,ord_less_eq(int,Z),zero_zero(int))
     => ( nat2(Z) = zero_zero(nat) ) ) ).

% nat_le_0
tff(fact_3322_nat__neg__numeral,axiom,
    ! [K: num] : nat2(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))) = zero_zero(nat) ).

% nat_neg_numeral
tff(fact_3323_zless__nat__conj,axiom,
    ! [W: int,Z: int] :
      ( aa(nat,$o,ord_less(nat,nat2(W)),nat2(Z))
    <=> ( aa(int,$o,ord_less(int,zero_zero(int)),Z)
        & aa(int,$o,ord_less(int,W),Z) ) ) ).

% zless_nat_conj
tff(fact_3324_nat__zminus__int,axiom,
    ! [Na: 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_3325_int__nat__eq,axiom,
    ! [Z: int] :
      aa(nat,int,semiring_1_of_nat(int),nat2(Z)) = $ite(aa(int,$o,ord_less_eq(int,zero_zero(int)),Z),Z,zero_zero(int)) ).

% int_nat_eq
tff(fact_3326_cos__pi,axiom,
    cos(real,pi) = aa(real,real,uminus_uminus(real),one_one(real)) ).

% cos_pi
tff(fact_3327_sin__cos__squared__add3,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),cos(A,X))),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,X)),sin(A,X))) = one_one(A) ) ).

% sin_cos_squared_add3
tff(fact_3328_zero__less__nat__eq,axiom,
    ! [Z: int] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),nat2(Z))
    <=> aa(int,$o,ord_less(int,zero_zero(int)),Z) ) ).

% zero_less_nat_eq
tff(fact_3329_diff__nat__numeral,axiom,
    ! [V2: num,V3: num] : aa(nat,nat,minus_minus(nat,aa(num,nat,numeral_numeral(nat),V2)),aa(num,nat,numeral_numeral(nat),V3)) = nat2(aa(int,int,minus_minus(int,aa(num,int,numeral_numeral(int),V2)),aa(num,int,numeral_numeral(int),V3))) ).

% diff_nat_numeral
tff(fact_3330_of__nat__nat,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: int] :
          ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z)
         => ( aa(nat,A,semiring_1_of_nat(A),nat2(Z)) = aa(int,A,ring_1_of_int(A),Z) ) ) ) ).

% of_nat_nat
tff(fact_3331_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,Na: nat] :
      ( ( nat2(Y) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),X)),Na) )
    <=> ( Y = aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),Na) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_3332_numeral__power__eq__nat__cancel__iff,axiom,
    ! [X: num,Na: nat,Y: int] :
      ( ( aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),X)),Na) = nat2(Y) )
    <=> ( aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),Na) = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_3333_cos__of__real__pi,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( cos(A,aa(real,A,real_Vector_of_real(A),pi)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% cos_of_real_pi
tff(fact_3334_dvd__nat__abs__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(nat,$o,dvd_dvd(nat,Na),nat2(aa(int,int,abs_abs(int),K)))
    <=> aa(int,$o,dvd_dvd(int,aa(nat,int,semiring_1_of_nat(int),Na)),K) ) ).

% dvd_nat_abs_iff
tff(fact_3335_nat__abs__dvd__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa(nat,$o,dvd_dvd(nat,nat2(aa(int,int,abs_abs(int),K))),Na)
    <=> aa(int,$o,dvd_dvd(int,K),aa(nat,int,semiring_1_of_nat(int),Na)) ) ).

% nat_abs_dvd_iff
tff(fact_3336_powser__sums__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A3: fun(nat,A),X: A] :
          ( aa(A,$o,sums(A,aTP_Lamp_bm(fun(nat,A),fun(nat,A),A3)),X)
        <=> ( aa(nat,A,A3,zero_zero(nat)) = X ) ) ) ).

% powser_sums_zero_iff
tff(fact_3337_nat__ceiling__le__eq,axiom,
    ! [X: real,A3: nat] :
      ( aa(nat,$o,ord_less_eq(nat,nat2(archimedean_ceiling(real,X))),A3)
    <=> aa(real,$o,ord_less_eq(real,X),aa(nat,real,semiring_1_of_nat(real),A3)) ) ).

% nat_ceiling_le_eq
tff(fact_3338_one__less__nat__eq,axiom,
    ! [Z: int] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,zero_zero(nat))),nat2(Z))
    <=> aa(int,$o,ord_less(int,one_one(int)),Z) ) ).

% one_less_nat_eq
tff(fact_3339_nat__numeral__diff__1,axiom,
    ! [V2: num] : aa(nat,nat,minus_minus(nat,aa(num,nat,numeral_numeral(nat),V2)),one_one(nat)) = nat2(aa(int,int,minus_minus(int,aa(num,int,numeral_numeral(int),V2)),one_one(int))) ).

% nat_numeral_diff_1
tff(fact_3340_nat__less__numeral__power__cancel__iff,axiom,
    ! [A3: int,X: num,Na: nat] :
      ( aa(nat,$o,ord_less(nat,nat2(A3)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),X)),Na))
    <=> aa(int,$o,ord_less(int,A3),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),Na)) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_3341_numeral__power__less__nat__cancel__iff,axiom,
    ! [X: num,Na: nat,A3: int] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),X)),Na)),nat2(A3))
    <=> aa(int,$o,ord_less(int,aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),Na)),A3) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_3342_nat__le__numeral__power__cancel__iff,axiom,
    ! [A3: int,X: num,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,nat2(A3)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),X)),Na))
    <=> aa(int,$o,ord_less_eq(int,A3),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),Na)) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_3343_numeral__power__le__nat__cancel__iff,axiom,
    ! [X: num,Na: nat,A3: int] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),X)),Na)),nat2(A3))
    <=> aa(int,$o,ord_less_eq(int,aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),Na)),A3) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_3344_cos__pi__half,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) = zero_zero(real) ).

% cos_pi_half
tff(fact_3345_cos__two__pi,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)) = one_one(real) ).

% cos_two_pi
tff(fact_3346_cos__periodic,axiom,
    ! [X: real] : cos(real,aa(real,real,plus_plus(real,X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))) = cos(real,X) ).

% cos_periodic
tff(fact_3347_cos__2pi__minus,axiom,
    ! [X: real] : cos(real,aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)),X)) = cos(real,X) ).

% cos_2pi_minus
tff(fact_3348_cos__npi2,axiom,
    ! [Na: nat] : cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(nat,real,semiring_1_of_nat(real),Na))) = aa(nat,real,power_power(real,aa(real,real,uminus_uminus(real),one_one(real))),Na) ).

% cos_npi2
tff(fact_3349_cos__npi,axiom,
    ! [Na: nat] : cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),pi)) = aa(nat,real,power_power(real,aa(real,real,uminus_uminus(real),one_one(real))),Na) ).

% cos_npi
tff(fact_3350_sin__cos__squared__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,plus_plus(A,aa(nat,A,power_power(A,sin(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,cos(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(A) ) ).

% sin_cos_squared_add
tff(fact_3351_sin__cos__squared__add2,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,plus_plus(A,aa(nat,A,power_power(A,cos(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,sin(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(A) ) ).

% sin_cos_squared_add2
tff(fact_3352_cos__of__real__pi__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V7773925162809079976_field(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( cos(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(real,A,real_Vector_of_real(A),pi)),aa(num,A,numeral_numeral(A),bit0(one2)))) = zero_zero(A) ) ) ).

% cos_of_real_pi_half
tff(fact_3353_cos__2npi,axiom,
    ! [Na: nat] : cos(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(nat,real,semiring_1_of_nat(real),Na))),pi)) = one_one(real) ).

% cos_2npi
tff(fact_3354_cos__int__2pin,axiom,
    ! [Na: int] : cos(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))),pi)),aa(int,real,ring_1_of_int(real),Na))) = one_one(real) ).

% cos_int_2pin
tff(fact_3355_cos__3over2__pi,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),bit1(one2))),aa(num,real,numeral_numeral(real),bit0(one2)))),pi)) = zero_zero(real) ).

% cos_3over2_pi
tff(fact_3356_cos__npi__int,axiom,
    ! [Na: int] :
      cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(int,real,ring_1_of_int(real),Na))) = $ite(aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na),one_one(real),aa(real,real,uminus_uminus(real),one_one(real))) ).

% cos_npi_int
tff(fact_3357_sums__of__real,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => ! [X5: fun(nat,real),A3: real] :
          ( aa(real,$o,sums(real,X5),A3)
         => aa(A,$o,sums(A,aTP_Lamp_bl(fun(nat,real),fun(nat,A),X5)),aa(real,A,real_Vector_of_real(A),A3)) ) ) ).

% sums_of_real
tff(fact_3358_sums__of__real__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,real),C2: real] :
          ( aa(A,$o,sums(A,aTP_Lamp_cf(fun(nat,real),fun(nat,A),F2)),aa(real,A,real_Vector_of_real(A),C2))
        <=> aa(real,$o,sums(real,F2),C2) ) ) ).

% sums_of_real_iff
tff(fact_3359_sums__0,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A)] :
          ( ! [N: nat] : aa(nat,A,F2,N) = zero_zero(A)
         => aa(A,$o,sums(A,F2),zero_zero(A)) ) ) ).

% sums_0
tff(fact_3360_sums__single,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [I: nat,F2: fun(nat,A)] : aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bb(nat,fun(fun(nat,A),fun(nat,A)),I),F2)),aa(nat,A,F2,I)) ) ).

% sums_single
tff(fact_3361_sums__add,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F2: fun(nat,A),A3: A,G: fun(nat,A),B3: A] :
          ( aa(A,$o,sums(A,F2),A3)
         => ( aa(A,$o,sums(A,G),B3)
           => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bf(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)),aa(A,A,plus_plus(A,A3),B3)) ) ) ) ).

% sums_add
tff(fact_3362_sums__mult,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),A3: A,C2: A] :
          ( aa(A,$o,sums(A,F2),A3)
         => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_bg(fun(nat,A),fun(A,fun(nat,A)),F2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)) ) ) ).

% sums_mult
tff(fact_3363_sums__mult2,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),A3: A,C2: A] :
          ( aa(A,$o,sums(A,F2),A3)
         => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_bh(fun(nat,A),fun(A,fun(nat,A)),F2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ).

% sums_mult2
tff(fact_3364_sums__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),A3: A,G: fun(nat,A),B3: A] :
          ( aa(A,$o,sums(A,F2),A3)
         => ( aa(A,$o,sums(A,G),B3)
           => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bi(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)),aa(A,A,minus_minus(A,A3),B3)) ) ) ) ).

% sums_diff
tff(fact_3365_sums__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(nat,A),A3: A,C2: A] :
          ( aa(A,$o,sums(A,F2),A3)
         => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_ax(fun(nat,A),fun(A,fun(nat,A)),F2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)) ) ) ).

% sums_divide
tff(fact_3366_sums__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),A3: A] :
          ( aa(A,$o,sums(A,F2),A3)
         => aa(A,$o,sums(A,aTP_Lamp_bj(fun(nat,A),fun(nat,A),F2)),aa(A,A,uminus_uminus(A),A3)) ) ) ).

% sums_minus
tff(fact_3367_nat__numeral__as__int,axiom,
    ! [X2: num] : aa(num,nat,numeral_numeral(nat),X2) = nat2(aa(num,int,numeral_numeral(int),X2)) ).

% nat_numeral_as_int
tff(fact_3368_nat__zero__as__int,axiom,
    zero_zero(nat) = nat2(zero_zero(int)) ).

% nat_zero_as_int
tff(fact_3369_cos__le__one,axiom,
    ! [X: real] : aa(real,$o,ord_less_eq(real,cos(real,X)),one_one(real)) ).

% cos_le_one
tff(fact_3370_nat__mono,axiom,
    ! [X: int,Y: int] :
      ( aa(int,$o,ord_less_eq(int,X),Y)
     => aa(nat,$o,ord_less_eq(nat,nat2(X)),nat2(Y)) ) ).

% nat_mono
tff(fact_3371_nat__one__as__int,axiom,
    one_one(nat) = nat2(one_one(int)) ).

% nat_one_as_int
tff(fact_3372_eq__nat__nat__iff,axiom,
    ! [Z: int,Z4: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z)
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z4)
       => ( ( nat2(Z) = nat2(Z4) )
        <=> ( Z = Z4 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_3373_all__nat,axiom,
    ! [P: fun(nat,$o)] :
      ( ! [X_1: nat] : aa(nat,$o,P,X_1)
    <=> ! [X3: int] :
          ( aa(int,$o,ord_less_eq(int,zero_zero(int)),X3)
         => aa(nat,$o,P,nat2(X3)) ) ) ).

% all_nat
tff(fact_3374_ex__nat,axiom,
    ! [P: fun(nat,$o)] :
      ( ? [X_1: nat] : aa(nat,$o,P,X_1)
    <=> ? [X3: int] :
          ( aa(int,$o,ord_less_eq(int,zero_zero(int)),X3)
          & aa(nat,$o,P,nat2(X3)) ) ) ).

% ex_nat
tff(fact_3375_sums__mult2__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [C2: A,F2: fun(nat,A),D2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cg(A,fun(fun(nat,A),fun(nat,A)),C2),F2)),aa(A,A,aa(A,fun(A,A),times_times(A),D2),C2))
          <=> aa(A,$o,sums(A,F2),D2) ) ) ) ).

% sums_mult2_iff
tff(fact_3376_sums__mult__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [C2: A,F2: fun(nat,A),D2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ch(A,fun(fun(nat,A),fun(nat,A)),C2),F2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D2))
          <=> aa(A,$o,sums(A,F2),D2) ) ) ) ).

% sums_mult_iff
tff(fact_3377_unat__eq__nat__uint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),W) = nat2(aa(word(A),int,semiring_1_unsigned(A,int),W)) ) ).

% unat_eq_nat_uint
tff(fact_3378_cos__arctan__not__zero,axiom,
    ! [X: real] : cos(real,aa(real,real,arctan,X)) != zero_zero(real) ).

% cos_arctan_not_zero
tff(fact_3379_unset__bit__nat__def,axiom,
    ! [M: nat,Na: nat] : bit_se2638667681897837118et_bit(nat,M,Na) = nat2(bit_se2638667681897837118et_bit(int,M,aa(nat,int,semiring_1_of_nat(int),Na))) ).

% unset_bit_nat_def
tff(fact_3380_cos__one__sin__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cos(A,X) = one_one(A) )
         => ( sin(A,X) = zero_zero(A) ) ) ) ).

% cos_one_sin_zero
tff(fact_3381_sums__mult__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,F2: fun(nat,A),A3: A] :
          ( aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ay(A,fun(fun(nat,A),fun(nat,A)),C2),F2)),A3)
         => ( ( C2 != zero_zero(A) )
           => aa(A,$o,sums(A,F2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)) ) ) ) ).

% sums_mult_D
tff(fact_3382_sums__Suc__imp,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),Sa: A] :
          ( ( aa(nat,A,F2,zero_zero(nat)) = zero_zero(A) )
         => ( aa(A,$o,sums(A,aTP_Lamp_bw(fun(nat,A),fun(nat,A),F2)),Sa)
           => aa(A,$o,sums(A,F2),Sa) ) ) ) ).

% sums_Suc_imp
tff(fact_3383_sums__Suc,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F2: fun(nat,A),L: A] :
          ( aa(A,$o,sums(A,aTP_Lamp_ci(fun(nat,A),fun(nat,A),F2)),L)
         => aa(A,$o,sums(A,F2),aa(A,A,plus_plus(A,L),aa(nat,A,F2,zero_zero(nat)))) ) ) ).

% sums_Suc
tff(fact_3384_sums__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),Sa: A] :
          ( aa(A,$o,sums(A,aTP_Lamp_bw(fun(nat,A),fun(nat,A),F2)),Sa)
        <=> aa(A,$o,sums(A,F2),aa(A,A,plus_plus(A,Sa),aa(nat,A,F2,zero_zero(nat)))) ) ) ).

% sums_Suc_iff
tff(fact_3385_sums__zero__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Na: nat,F2: fun(nat,A),Sa: A] :
          ( ! [I3: nat] :
              ( aa(nat,$o,ord_less(nat,I3),Na)
             => ( aa(nat,A,F2,I3) = zero_zero(A) ) )
         => ( aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cj(nat,fun(fun(nat,A),fun(nat,A)),Na),F2)),Sa)
          <=> aa(A,$o,sums(A,F2),Sa) ) ) ) ).

% sums_zero_iff_shift
tff(fact_3386_nat__mono__iff,axiom,
    ! [Z: int,W: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),Z)
     => ( aa(nat,$o,ord_less(nat,nat2(W)),nat2(Z))
      <=> aa(int,$o,ord_less(int,W),Z) ) ) ).

% nat_mono_iff
tff(fact_3387_cos__inj__pi,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less_eq(real,X),pi)
       => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
         => ( aa(real,$o,ord_less_eq(real,Y),pi)
           => ( ( cos(real,X) = cos(real,Y) )
             => ( X = Y ) ) ) ) ) ) ).

% cos_inj_pi
tff(fact_3388_cos__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less_eq(real,X),pi)
       => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
         => ( aa(real,$o,ord_less_eq(real,Y),pi)
           => ( aa(real,$o,ord_less_eq(real,cos(real,X)),cos(real,Y))
            <=> aa(real,$o,ord_less_eq(real,Y),X) ) ) ) ) ) ).

% cos_mono_le_eq
tff(fact_3389_cos__monotone__0__pi__le,axiom,
    ! [Y: real,X: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),X)
       => ( aa(real,$o,ord_less_eq(real,X),pi)
         => aa(real,$o,ord_less_eq(real,cos(real,X)),cos(real,Y)) ) ) ) ).

% cos_monotone_0_pi_le
tff(fact_3390_cos__ge__minus__one,axiom,
    ! [X: real] : aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),cos(real,X)) ).

% cos_ge_minus_one
tff(fact_3391_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z: int] :
      ( aa(nat,$o,ord_less(nat,M),nat2(Z))
    <=> aa(int,$o,ord_less(int,aa(nat,int,semiring_1_of_nat(int),M)),Z) ) ).

% zless_nat_eq_int_zless
tff(fact_3392_of__nat__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R3: A] : aa(A,$o,ord_less_eq(A,R3),aa(nat,A,semiring_1_of_nat(A),nat2(archimedean_ceiling(A,R3)))) ) ).

% of_nat_ceiling
tff(fact_3393_nat__le__iff,axiom,
    ! [X: int,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,nat2(X)),Na)
    <=> aa(int,$o,ord_less_eq(int,X),aa(nat,int,semiring_1_of_nat(int),Na)) ) ).

% nat_le_iff
tff(fact_3394_nat__int__add,axiom,
    ! [A3: nat,B3: nat] : nat2(aa(int,int,plus_plus(int,aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B3))) = aa(nat,nat,plus_plus(nat,A3),B3) ).

% nat_int_add
tff(fact_3395_int__eq__iff,axiom,
    ! [M: nat,Z: int] :
      ( ( aa(nat,int,semiring_1_of_nat(int),M) = Z )
    <=> ( ( M = nat2(Z) )
        & aa(int,$o,ord_less_eq(int,zero_zero(int)),Z) ) ) ).

% int_eq_iff
tff(fact_3396_nat__0__le,axiom,
    ! [Z: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z)
     => ( aa(nat,int,semiring_1_of_nat(int),nat2(Z)) = Z ) ) ).

% nat_0_le
tff(fact_3397_abs__cos__le__one,axiom,
    ! [X: real] : aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),cos(real,X))),one_one(real)) ).

% abs_cos_le_one
tff(fact_3398_nat__uint__less__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Z: nat,X: word(A)] :
          ( ( nat2(aa(word(A),int,semiring_1_unsigned(A,int),Y)) = Z )
         => ( aa(word(A),$o,ord_less(word(A),X),Y)
           => aa(nat,$o,ord_less(nat,nat2(aa(word(A),int,semiring_1_unsigned(A,int),X))),Z) ) ) ) ).

% nat_uint_less_helper
tff(fact_3399_int__minus,axiom,
    ! [Na: nat,M: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,minus_minus(nat,Na),M)) = aa(nat,int,semiring_1_of_nat(int),nat2(aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),Na)),aa(nat,int,semiring_1_of_nat(int),M)))) ).

% int_minus
tff(fact_3400_nat__abs__mult__distrib,axiom,
    ! [W: int,Z: int] : nat2(aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),nat2(aa(int,int,abs_abs(int),W))),nat2(aa(int,int,abs_abs(int),Z))) ).

% nat_abs_mult_distrib
tff(fact_3401_real__nat__ceiling__ge,axiom,
    ! [X: real] : aa(real,$o,ord_less_eq(real,X),aa(nat,real,semiring_1_of_nat(real),nat2(archimedean_ceiling(real,X)))) ).

% real_nat_ceiling_ge
tff(fact_3402_nat__plus__as__int,axiom,
    ! [X2: nat,Xa: nat] : aa(nat,nat,plus_plus(nat,X2),Xa) = nat2(aa(int,int,plus_plus(int,aa(nat,int,semiring_1_of_nat(int),X2)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_plus_as_int
tff(fact_3403_nat__times__as__int,axiom,
    ! [X2: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),X2),Xa) = nat2(aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),X2)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_times_as_int
tff(fact_3404_nat__minus__as__int,axiom,
    ! [X2: nat,Xa: nat] : aa(nat,nat,minus_minus(nat,X2),Xa) = nat2(aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),X2)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_minus_as_int
tff(fact_3405_nat__div__as__int,axiom,
    ! [X2: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X2),Xa) = nat2(aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,semiring_1_of_nat(int),X2)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_div_as_int
tff(fact_3406_nat__mod__as__int,axiom,
    ! [X2: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X2),Xa) = nat2(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(nat,int,semiring_1_of_nat(int),X2)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_mod_as_int
tff(fact_3407_powser__sums__if,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [M: nat,Z: A] : aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_ck(nat,fun(A,fun(nat,A)),M),Z)),aa(nat,A,power_power(A,Z),M)) ) ).

% powser_sums_if
tff(fact_3408_powser__sums__zero,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A3: fun(nat,A)] : aa(A,$o,sums(A,aTP_Lamp_bm(fun(nat,A),fun(nat,A),A3)),aa(nat,A,A3,zero_zero(nat))) ) ).

% powser_sums_zero
tff(fact_3409_sin__zero__norm__cos__one,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( sin(A,X) = zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,cos(A,X)) = one_one(real) ) ) ) ).

% sin_zero_norm_cos_one
tff(fact_3410_cos__two__neq__zero,axiom,
    cos(real,aa(num,real,numeral_numeral(real),bit0(one2))) != zero_zero(real) ).

% cos_two_neq_zero
tff(fact_3411_of__nat__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),R3)
         => aa(A,$o,ord_less_eq(A,aa(nat,A,semiring_1_of_nat(A),nat2(archim6421214686448440834_floor(A,R3)))),R3) ) ) ).

% of_nat_floor
tff(fact_3412_cos__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less_eq(real,X),pi)
       => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
         => ( aa(real,$o,ord_less_eq(real,Y),pi)
           => ( aa(real,$o,ord_less(real,cos(real,X)),cos(real,Y))
            <=> aa(real,$o,ord_less(real,Y),X) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_3413_cos__monotone__0__pi,axiom,
    ! [Y: real,X: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
     => ( aa(real,$o,ord_less(real,Y),X)
       => ( aa(real,$o,ord_less_eq(real,X),pi)
         => aa(real,$o,ord_less(real,cos(real,X)),cos(real,Y)) ) ) ) ).

% cos_monotone_0_pi
tff(fact_3414_nat__less__eq__zless,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),W)
     => ( aa(nat,$o,ord_less(nat,nat2(W)),nat2(Z))
      <=> aa(int,$o,ord_less(int,W),Z) ) ) ).

% nat_less_eq_zless
tff(fact_3415_nat__eq__iff,axiom,
    ! [W: int,M: nat] :
      ( ( nat2(W) = M )
    <=> $ite(aa(int,$o,ord_less_eq(int,zero_zero(int)),W),W = aa(nat,int,semiring_1_of_nat(int),M),M = zero_zero(nat)) ) ).

% nat_eq_iff
tff(fact_3416_nat__eq__iff2,axiom,
    ! [M: nat,W: int] :
      ( ( M = nat2(W) )
    <=> $ite(aa(int,$o,ord_less_eq(int,zero_zero(int)),W),W = aa(nat,int,semiring_1_of_nat(int),M),M = zero_zero(nat)) ) ).

% nat_eq_iff2
tff(fact_3417_split__nat,axiom,
    ! [P: fun(nat,$o),I: int] :
      ( aa(nat,$o,P,nat2(I))
    <=> ( ! [N3: nat] :
            ( ( I = aa(nat,int,semiring_1_of_nat(int),N3) )
           => aa(nat,$o,P,N3) )
        & ( aa(int,$o,ord_less(int,I),zero_zero(int))
         => aa(nat,$o,P,zero_zero(nat)) ) ) ) ).

% split_nat
tff(fact_3418_nat__le__eq__zle,axiom,
    ! [W: int,Z: int] :
      ( ( aa(int,$o,ord_less(int,zero_zero(int)),W)
        | aa(int,$o,ord_less_eq(int,zero_zero(int)),Z) )
     => ( aa(nat,$o,ord_less_eq(nat,nat2(W)),nat2(Z))
      <=> aa(int,$o,ord_less_eq(int,W),Z) ) ) ).

% nat_le_eq_zle
tff(fact_3419_nat__add__distrib,axiom,
    ! [Z: int,Z4: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z)
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z4)
       => ( nat2(aa(int,int,plus_plus(int,Z),Z4)) = aa(nat,nat,plus_plus(nat,nat2(Z)),nat2(Z4)) ) ) ) ).

% nat_add_distrib
tff(fact_3420_cos__monotone__minus__pi__0_H,axiom,
    ! [Y: real,X: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),pi)),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),X)
       => ( aa(real,$o,ord_less_eq(real,X),zero_zero(real))
         => aa(real,$o,ord_less_eq(real,cos(real,Y)),cos(real,X)) ) ) ) ).

% cos_monotone_minus_pi_0'
tff(fact_3421_le__nat__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
     => ( aa(nat,$o,ord_less_eq(nat,Na),nat2(K))
      <=> aa(int,$o,ord_less_eq(int,aa(nat,int,semiring_1_of_nat(int),Na)),K) ) ) ).

% le_nat_iff
tff(fact_3422_Suc__as__int,axiom,
    ! [X2: nat] : aa(nat,nat,suc,X2) = nat2(aa(int,int,plus_plus(int,aa(nat,int,semiring_1_of_nat(int),X2)),one_one(int))) ).

% Suc_as_int
tff(fact_3423_le__mult__nat__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A,B3: A] : aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),nat2(archim6421214686448440834_floor(A,A3))),nat2(archim6421214686448440834_floor(A,B3)))),nat2(archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)))) ) ).

% le_mult_nat_floor
tff(fact_3424_nat__mult__distrib,axiom,
    ! [Z: int,Z4: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z)
     => ( nat2(aa(int,int,aa(int,fun(int,int),times_times(int),Z),Z4)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),nat2(Z)),nat2(Z4)) ) ) ).

% nat_mult_distrib
tff(fact_3425_nat__diff__distrib,axiom,
    ! [Z4: int,Z: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z4)
     => ( aa(int,$o,ord_less_eq(int,Z4),Z)
       => ( nat2(aa(int,int,minus_minus(int,Z),Z4)) = aa(nat,nat,minus_minus(nat,nat2(Z)),nat2(Z4)) ) ) ) ).

% nat_diff_distrib
tff(fact_3426_nat__diff__distrib_H,axiom,
    ! [X: int,Y: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),X)
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Y)
       => ( nat2(aa(int,int,minus_minus(int,X),Y)) = aa(nat,nat,minus_minus(nat,nat2(X)),nat2(Y)) ) ) ) ).

% nat_diff_distrib'
tff(fact_3427_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : aa(nat,$o,ord_less_eq(nat,nat2(aa(int,int,abs_abs(int),aa(int,int,plus_plus(int,K),L)))),aa(nat,nat,plus_plus(nat,nat2(aa(int,int,abs_abs(int),K))),nat2(aa(int,int,abs_abs(int),L)))) ).

% nat_abs_triangle_ineq
tff(fact_3428_nat__div__distrib,axiom,
    ! [X: int,Y: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),X)
     => ( nat2(aa(int,int,aa(int,fun(int,int),divide_divide(int),X),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),nat2(X)),nat2(Y)) ) ) ).

% nat_div_distrib
tff(fact_3429_nat__div__distrib_H,axiom,
    ! [Y: int,X: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Y)
     => ( nat2(aa(int,int,aa(int,fun(int,int),divide_divide(int),X),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),nat2(X)),nat2(Y)) ) ) ).

% nat_div_distrib'
tff(fact_3430_nat__power__eq,axiom,
    ! [Z: int,Na: nat] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z)
     => ( nat2(aa(nat,int,power_power(int,Z),Na)) = aa(nat,nat,power_power(nat,nat2(Z)),Na) ) ) ).

% nat_power_eq
tff(fact_3431_nat__mod__distrib,axiom,
    ! [X: int,Y: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),X)
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Y)
       => ( nat2(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),X),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),nat2(X)),nat2(Y)) ) ) ) ).

% nat_mod_distrib
tff(fact_3432_sin__zero__abs__cos__one,axiom,
    ! [X: real] :
      ( ( sin(real,X) = zero_zero(real) )
     => ( aa(real,real,abs_abs(real),cos(real,X)) = one_one(real) ) ) ).

% sin_zero_abs_cos_one
tff(fact_3433_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),nat2(aa(int,int,abs_abs(int),K))),nat2(aa(int,int,abs_abs(int),L)))) ).

% div_abs_eq_div_nat
tff(fact_3434_word__of__int__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: int] :
          ( aa(int,$o,ord_less_eq(int,zero_zero(int)),X)
         => ( aa(int,word(A),ring_1_of_int(word(A)),X) = aa(nat,word(A),semiring_1_of_nat(word(A)),nat2(X)) ) ) ) ).

% word_of_int_nat
tff(fact_3435_nat__floor__neg,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,X),zero_zero(real))
     => ( nat2(archim6421214686448440834_floor(real,X)) = zero_zero(nat) ) ) ).

% nat_floor_neg
tff(fact_3436_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),nat2(aa(int,int,abs_abs(int),K))),nat2(aa(int,int,abs_abs(int),L)))) ).

% mod_abs_eq_div_nat
tff(fact_3437_floor__eq3,axiom,
    ! [Na: nat,X: real] :
      ( aa(real,$o,ord_less(real,aa(nat,real,semiring_1_of_nat(real),Na)),X)
     => ( aa(real,$o,ord_less(real,X),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Na)))
       => ( nat2(archim6421214686448440834_floor(real,X)) = Na ) ) ) ).

% floor_eq3
tff(fact_3438_le__nat__floor,axiom,
    ! [X: nat,A3: real] :
      ( aa(real,$o,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),X)),A3)
     => aa(nat,$o,ord_less_eq(nat,X),nat2(archim6421214686448440834_floor(real,A3))) ) ).

% le_nat_floor
tff(fact_3439_sin__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : sin(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),X)) = 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))),sin(A,X))),cos(A,X)) ) ).

% sin_double
tff(fact_3440_cos__paired,axiom,
    ! [X: real] : aa(real,$o,sums(real,aTP_Lamp_cl(real,fun(nat,real),X)),cos(real,X)) ).

% cos_paired
tff(fact_3441_nat__2,axiom,
    nat2(aa(num,int,numeral_numeral(int),bit0(one2))) = aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))) ).

% nat_2
tff(fact_3442_cos__two__less__zero,axiom,
    aa(real,$o,ord_less(real,cos(real,aa(num,real,numeral_numeral(real),bit0(one2)))),zero_zero(real)) ).

% cos_two_less_zero
tff(fact_3443_cos__two__le__zero,axiom,
    aa(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_3444_cos__is__zero,axiom,
    ? [X4: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X4)
      & aa(real,$o,ord_less_eq(real,X4),aa(num,real,numeral_numeral(real),bit0(one2)))
      & ( cos(real,X4) = zero_zero(real) )
      & ! [Y4: real] :
          ( ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y4)
            & aa(real,$o,ord_less_eq(real,Y4),aa(num,real,numeral_numeral(real),bit0(one2)))
            & ( cos(real,Y4) = zero_zero(real) ) )
         => ( Y4 = X4 ) ) ) ).

% cos_is_zero
tff(fact_3445_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,X: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),pi)),Y)
     => ( aa(real,$o,ord_less(real,Y),X)
       => ( aa(real,$o,ord_less_eq(real,X),zero_zero(real))
         => aa(real,$o,ord_less(real,cos(real,Y)),cos(real,X)) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_3446_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z)
     => ( aa(nat,nat,suc,nat2(Z)) = nat2(aa(int,int,plus_plus(int,one_one(int)),Z)) ) ) ).

% Suc_nat_eq_nat_zadd1
tff(fact_3447_cos__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
       => ? [X4: real] :
            ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X4)
            & aa(real,$o,ord_less_eq(real,X4),pi)
            & ( cos(real,X4) = Y )
            & ! [Y4: real] :
                ( ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y4)
                  & aa(real,$o,ord_less_eq(real,Y4),pi)
                  & ( cos(real,Y4) = Y ) )
               => ( Y4 = X4 ) ) ) ) ) ).

% cos_total
tff(fact_3448_nat__less__iff,axiom,
    ! [W: int,M: nat] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),W)
     => ( aa(nat,$o,ord_less(nat,nat2(W)),M)
      <=> aa(int,$o,ord_less(int,W),aa(nat,int,semiring_1_of_nat(int),M)) ) ) ).

% nat_less_iff
tff(fact_3449_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z4: int] :
      ( aa(int,$o,ord_less_eq(int,Z),zero_zero(int))
     => ( nat2(aa(int,int,aa(int,fun(int,int),times_times(int),Z),Z4)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),nat2(aa(int,int,uminus_uminus(int),Z))),nat2(aa(int,int,uminus_uminus(int),Z4))) ) ) ).

% nat_mult_distrib_neg
tff(fact_3450_sincos__principal__value,axiom,
    ! [X: real] :
    ? [Y3: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),pi)),Y3)
      & aa(real,$o,ord_less_eq(real,Y3),pi)
      & ( sin(real,Y3) = sin(real,X) )
      & ( cos(real,Y3) = cos(real,X) ) ) ).

% sincos_principal_value
tff(fact_3451_nat__abs__int__diff,axiom,
    ! [A3: nat,B3: nat] :
      nat2(aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B3)))) = $ite(aa(nat,$o,ord_less_eq(nat,A3),B3),aa(nat,nat,minus_minus(nat,B3),A3),aa(nat,nat,minus_minus(nat,A3),B3)) ).

% nat_abs_int_diff
tff(fact_3452_floor__eq4,axiom,
    ! [Na: nat,X: real] :
      ( aa(real,$o,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),Na)),X)
     => ( aa(real,$o,ord_less(real,X),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Na)))
       => ( nat2(archim6421214686448440834_floor(real,X)) = Na ) ) ) ).

% floor_eq4
tff(fact_3453_diff__nat__eq__if,axiom,
    ! [Z: int,Z4: int] :
      aa(nat,nat,minus_minus(nat,nat2(Z)),nat2(Z4)) = $ite(
        aa(int,$o,ord_less(int,Z4),zero_zero(int)),
        nat2(Z),
        $let(
          d: int,
          d:= aa(int,int,minus_minus(int,Z),Z4),
          $ite(aa(int,$o,ord_less(int,d),zero_zero(int)),zero_zero(nat),nat2(d)) ) ) ).

% diff_nat_eq_if
tff(fact_3454_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,ord_less(int,K),zero_zero(int)),aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),nat2(aa(int,int,uminus_uminus(int),K)))),aa(nat,A,semiring_1_of_nat(A),nat2(K))) ) ).

% of_int_of_nat
tff(fact_3455_nat__dvd__iff,axiom,
    ! [Z: int,M: nat] :
      ( aa(nat,$o,dvd_dvd(nat,nat2(Z)),M)
    <=> $ite(aa(int,$o,ord_less_eq(int,zero_zero(int)),Z),aa(int,$o,dvd_dvd(int,Z),aa(nat,int,semiring_1_of_nat(int),M)),M = zero_zero(nat)) ) ).

% nat_dvd_iff
tff(fact_3456_sin__cos__le1,axiom,
    ! [X: real,Y: real] : aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),aa(real,real,plus_plus(real,aa(real,real,aa(real,fun(real,real),times_times(real),sin(real,X)),sin(real,Y))),aa(real,real,aa(real,fun(real,real),times_times(real),cos(real,X)),cos(real,Y))))),one_one(real)) ).

% sin_cos_le1
tff(fact_3457_cos__times__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,W)),cos(A,Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,plus_plus(A,cos(A,aa(A,A,minus_minus(A,W),Z))),cos(A,aa(A,A,plus_plus(A,W),Z)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% cos_times_cos
tff(fact_3458_cos__plus__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,plus_plus(A,cos(A,W)),cos(A,Z)) = 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))),cos(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,plus_plus(A,W),Z)),aa(num,A,numeral_numeral(A),bit0(one2)))))),cos(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,W),Z)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% cos_plus_cos
tff(fact_3459_cos__squared__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(nat,A,power_power(A,cos(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,power_power(A,sin(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% cos_squared_eq
tff(fact_3460_sin__squared__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(nat,A,power_power(A,sin(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,power_power(A,cos(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% sin_squared_eq
tff(fact_3461_geometric__sums,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,C2)),one_one(real))
         => aa(A,$o,sums(A,power_power(A,C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(A,A,minus_minus(A,one_one(A)),C2))) ) ) ).

% geometric_sums
tff(fact_3462_power__half__series,axiom,
    aa(real,$o,sums(real,aTP_Lamp_cm(nat,real)),one_one(real)) ).

% power_half_series
tff(fact_3463_cos__double__less__one,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less(real,X),aa(num,real,numeral_numeral(real),bit0(one2)))
       => aa(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))),X))),one_one(real)) ) ) ).

% cos_double_less_one
tff(fact_3464_cos__gt__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ( aa(real,$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,ord_less(real,zero_zero(real)),cos(real,X)) ) ) ).

% cos_gt_zero
tff(fact_3465_cos__60,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit1(one2)))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% cos_60
tff(fact_3466_cos__one__2pi__int,axiom,
    ! [X: real] :
      ( ( cos(real,X) = one_one(real) )
    <=> ? [X3: int] : X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),X3)),aa(num,real,numeral_numeral(real),bit0(one2)))),pi) ) ).

% cos_one_2pi_int
tff(fact_3467_cos__double__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A] : cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),W)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,power_power(A,cos(A,W)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),one_one(A)) ) ).

% cos_double_cos
tff(fact_3468_cos__treble__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit1(one2))),X)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(bit0(one2)))),aa(nat,A,power_power(A,cos(A,X)),aa(num,nat,numeral_numeral(nat),bit1(one2))))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit1(one2))),cos(A,X))) ) ).

% cos_treble_cos
tff(fact_3469_sin__times__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,W)),sin(A,Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,cos(A,aa(A,A,minus_minus(A,W),Z))),cos(A,aa(A,A,plus_plus(A,W),Z)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% sin_times_sin
tff(fact_3470_sin__times__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,W)),cos(A,Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,plus_plus(A,sin(A,aa(A,A,plus_plus(A,W),Z))),sin(A,aa(A,A,minus_minus(A,W),Z)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% sin_times_cos
tff(fact_3471_cos__times__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,W)),sin(A,Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,sin(A,aa(A,A,plus_plus(A,W),Z))),sin(A,aa(A,A,minus_minus(A,W),Z)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% cos_times_sin
tff(fact_3472_sin__plus__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,plus_plus(A,sin(A,W)),sin(A,Z)) = 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))),sin(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,plus_plus(A,W),Z)),aa(num,A,numeral_numeral(A),bit0(one2)))))),cos(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,W),Z)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% sin_plus_sin
tff(fact_3473_sin__diff__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,minus_minus(A,sin(A,W)),sin(A,Z)) = 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))),sin(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,W),Z)),aa(num,A,numeral_numeral(A),bit0(one2)))))),cos(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,plus_plus(A,W),Z)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% sin_diff_sin
tff(fact_3474_cos__diff__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,minus_minus(A,cos(A,W)),cos(A,Z)) = 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))),sin(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,plus_plus(A,W),Z)),aa(num,A,numeral_numeral(A),bit0(one2)))))),sin(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,Z),W)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% cos_diff_cos
tff(fact_3475_cos__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),X)) = aa(A,A,minus_minus(A,aa(nat,A,power_power(A,cos(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,sin(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% cos_double
tff(fact_3476_sums__if_H,axiom,
    ! [G: fun(nat,real),X: real] :
      ( aa(real,$o,sums(real,G),X)
     => aa(real,$o,sums(real,aTP_Lamp_cn(fun(nat,real),fun(nat,real),G)),X) ) ).

% sums_if'
tff(fact_3477_cos__sin__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : cos(A,X) = sin(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(real,A,real_Vector_of_real(A),pi)),aa(num,A,numeral_numeral(A),bit0(one2)))),X)) ) ).

% cos_sin_eq
tff(fact_3478_sin__cos__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : sin(A,X) = cos(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(real,A,real_Vector_of_real(A),pi)),aa(num,A,numeral_numeral(A),bit0(one2)))),X)) ) ).

% sin_cos_eq
tff(fact_3479_sums__if,axiom,
    ! [G: fun(nat,real),X: real,F2: fun(nat,real),Y: real] :
      ( aa(real,$o,sums(real,G),X)
     => ( aa(real,$o,sums(real,F2),Y)
       => aa(real,$o,sums(real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_co(fun(nat,real),fun(fun(nat,real),fun(nat,real)),G),F2)),aa(real,real,plus_plus(real,X),Y)) ) ) ).

% sums_if
tff(fact_3480_cos__gt__zero__pi,axiom,
    ! [X: real] :
      ( aa(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,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,ord_less(real,zero_zero(real)),cos(real,X)) ) ) ).

% cos_gt_zero_pi
tff(fact_3481_cos__ge__zero,axiom,
    ! [X: real] :
      ( aa(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,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))))
       => aa(real,$o,ord_less_eq(real,zero_zero(real)),cos(real,X)) ) ) ).

% cos_ge_zero
tff(fact_3482_even__nat__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
     => ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),nat2(K))
      <=> aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K) ) ) ).

% even_nat_iff
tff(fact_3483_cos__one__2pi,axiom,
    ! [X: real] :
      ( ( cos(real,X) = one_one(real) )
    <=> ( ? [X3: nat] : X = 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),X3)),aa(num,real,numeral_numeral(real),bit0(one2)))),pi)
        | ? [X3: nat] : X = aa(real,real,uminus_uminus(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),X3)),aa(num,real,numeral_numeral(real),bit0(one2)))),pi)) ) ) ).

% cos_one_2pi
tff(fact_3484_cos__double__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A] : cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),W)) = aa(A,A,minus_minus(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,power_power(A,sin(A,W)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% cos_double_sin
tff(fact_3485_minus__sin__cos__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,uminus_uminus(A),sin(A,X)) = cos(A,aa(A,A,plus_plus(A,X),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(real,A,real_Vector_of_real(A),pi)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% minus_sin_cos_eq
tff(fact_3486_sincos__total__pi,axiom,
    ! [Y: real,X: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
     => ( ( aa(real,real,plus_plus(real,aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(real) )
       => ? [T2: real] :
            ( aa(real,$o,ord_less_eq(real,zero_zero(real)),T2)
            & aa(real,$o,ord_less_eq(real,T2),pi)
            & ( X = cos(real,T2) )
            & ( Y = sin(real,T2) ) ) ) ) ).

% sincos_total_pi
tff(fact_3487_sin__expansion__lemma,axiom,
    ! [X: real,M: nat] : sin(real,aa(real,real,plus_plus(real,X),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,plus_plus(real,X),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_3488_cos__zero__iff__int,axiom,
    ! [X: real] :
      ( ( cos(real,X) = zero_zero(real) )
    <=> ? [I4: int] :
          ( ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),I4)
          & ( X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),I4)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ).

% cos_zero_iff_int
tff(fact_3489_powr__int,axiom,
    ! [X: real,I: int] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ( powr(real,X,aa(int,real,ring_1_of_int(real),I)) = $ite(aa(int,$o,ord_less_eq(int,zero_zero(int)),I),aa(nat,real,power_power(real,X),nat2(I)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,power_power(real,X),nat2(aa(int,int,uminus_uminus(int),I))))) ) ) ).

% powr_int
tff(fact_3490_cos__zero__lemma,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => ( ( cos(real,X) = zero_zero(real) )
       => ? [N: nat] :
            ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
            & ( X = 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_3491_cos__zero__iff,axiom,
    ! [X: real] :
      ( ( cos(real,X) = zero_zero(real) )
    <=> ( ? [N3: nat] :
            ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N3)
            & ( X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N3)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) )
        | ? [N3: nat] :
            ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N3)
            & ( X = aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N3)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))) ) ) ) ) ).

% cos_zero_iff
tff(fact_3492_cos__expansion__lemma,axiom,
    ! [X: real,M: nat] : cos(real,aa(real,real,plus_plus(real,X),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,plus_plus(real,X),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_3493_geometric__deriv__sums,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z: A] :
          ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Z)),one_one(real))
         => aa(A,$o,sums(A,aTP_Lamp_cp(A,fun(nat,A),Z)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,power_power(A,aa(A,A,minus_minus(A,one_one(A)),Z)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% geometric_deriv_sums
tff(fact_3494_sincos__total__pi__half,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
       => ( ( aa(real,real,plus_plus(real,aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(real) )
         => ? [T2: real] :
              ( aa(real,$o,ord_less_eq(real,zero_zero(real)),T2)
              & aa(real,$o,ord_less_eq(real,T2),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
              & ( X = cos(real,T2) )
              & ( Y = sin(real,T2) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_3495_sincos__total__2pi__le,axiom,
    ! [X: real,Y: real] :
      ( ( aa(real,real,plus_plus(real,aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(real) )
     => ? [T2: real] :
          ( aa(real,$o,ord_less_eq(real,zero_zero(real)),T2)
          & aa(real,$o,ord_less_eq(real,T2),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
          & ( X = cos(real,T2) )
          & ( Y = sin(real,T2) ) ) ) ).

% sincos_total_2pi_le
tff(fact_3496_vebt__deletei__rule,axiom,
    ! [Na: nat,Sa: set(nat),Ti: vEBT_VEBTi,X: nat] : time_htt(vEBT_VEBTi,aa(vEBT_VEBTi,assn,vEBT_Intf_vebt_assn(Na,Sa),Ti),vEBT_vebt_deletei(Ti,X),vEBT_Intf_vebt_assn(Na,aa(set(nat),set(nat),minus_minus(set(nat),Sa),aa(set(nat),set(nat),insert(nat,X),bot_bot(set(nat))))),aa(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)))))),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_deletei_rule
tff(fact_3497_diffs__equiv,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & ring_1(A) )
     => ! [C2: fun(nat,A),X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_cq(fun(nat,A),fun(A,fun(nat,A)),C2),X))
         => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_cr(fun(nat,A),fun(A,fun(nat,A)),C2),X)),suminf(A,aa(A,fun(nat,A),aTP_Lamp_cq(fun(nat,A),fun(A,fun(nat,A)),C2),X))) ) ) ).

% diffs_equiv
tff(fact_3498_tan__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( ( cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),X)) != 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))),X)) = 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),X))),aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,power_power(A,aa(A,A,tan(A),X)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ) ).

% tan_double
tff(fact_3499_vebt__memberi__rule,axiom,
    ! [Na: nat,Sa: set(nat),Ti: vEBT_VEBTi,X: nat] : time_htt($o,aa(vEBT_VEBTi,assn,vEBT_Intf_vebt_assn(Na,Sa),Ti),vEBT_vebt_memberi(Ti,X),aa(nat,fun($o,assn),aa(vEBT_VEBTi,fun(nat,fun($o,assn)),aa(set(nat),fun(vEBT_VEBTi,fun(nat,fun($o,assn))),aTP_Lamp_cs(nat,fun(set(nat),fun(vEBT_VEBTi,fun(nat,fun($o,assn)))),Na),Sa),Ti),X),aa(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)))),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_3500_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( real_V7770717601297561774m_norm(complex,Z) = one_one(real) )
     => ~ ! [T2: real] :
            ( aa(real,$o,ord_less_eq(real,zero_zero(real)),T2)
           => ( aa(real,$o,ord_less(real,T2),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
             => ( Z != complex2(cos(real,T2),sin(real,T2)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_3501_tan__pi,axiom,
    aa(real,real,tan(real),pi) = zero_zero(real) ).

% tan_pi
tff(fact_3502_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_3503_tan__npi,axiom,
    ! [Na: nat] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),pi)) = zero_zero(real) ).

% tan_npi
tff(fact_3504_tan__periodic__n,axiom,
    ! [X: real,Na: num] : aa(real,real,tan(real),aa(real,real,plus_plus(real,X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),Na)),pi))) = aa(real,real,tan(real),X) ).

% tan_periodic_n
tff(fact_3505_tan__periodic__nat,axiom,
    ! [X: real,Na: nat] : aa(real,real,tan(real),aa(real,real,plus_plus(real,X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),pi))) = aa(real,real,tan(real),X) ).

% tan_periodic_nat
tff(fact_3506_norm__cos__sin,axiom,
    ! [Ta: real] : real_V7770717601297561774m_norm(complex,complex2(cos(real,Ta),sin(real,Ta))) = one_one(real) ).

% norm_cos_sin
tff(fact_3507_tan__periodic,axiom,
    ! [X: real] : aa(real,real,tan(real),aa(real,real,plus_plus(real,X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))) = aa(real,real,tan(real),X) ).

% tan_periodic
tff(fact_3508_zero__complex_Ocode,axiom,
    zero_zero(complex) = complex2(zero_zero(real),zero_zero(real)) ).

% zero_complex.code
tff(fact_3509_Complex__eq__0,axiom,
    ! [A3: real,B3: real] :
      ( ( complex2(A3,B3) = zero_zero(complex) )
    <=> ( ( A3 = zero_zero(real) )
        & ( B3 = zero_zero(real) ) ) ) ).

% Complex_eq_0
tff(fact_3510_diffs__of__real,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [F2: fun(nat,real),X2: nat] : aa(nat,A,diffs(A,aTP_Lamp_ct(fun(nat,real),fun(nat,A),F2)),X2) = aa(real,A,real_Vector_of_real(A),aa(nat,real,diffs(real,F2),X2)) ) ).

% diffs_of_real
tff(fact_3511_one__complex_Ocode,axiom,
    one_one(complex) = complex2(one_one(real),zero_zero(real)) ).

% one_complex.code
tff(fact_3512_Complex__eq__1,axiom,
    ! [A3: real,B3: real] :
      ( ( complex2(A3,B3) = one_one(complex) )
    <=> ( ( A3 = one_one(real) )
        & ( B3 = zero_zero(real) ) ) ) ).

% Complex_eq_1
tff(fact_3513_diffs__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [C2: fun(nat,A),X2: nat] : aa(nat,A,diffs(A,aTP_Lamp_cu(fun(nat,A),fun(nat,A),C2)),X2) = aa(A,A,uminus_uminus(A),aa(nat,A,diffs(A,C2),X2)) ) ).

% diffs_minus
tff(fact_3514_Complex__eq__numeral,axiom,
    ! [A3: real,B3: real,W: num] :
      ( ( complex2(A3,B3) = aa(num,complex,numeral_numeral(complex),W) )
    <=> ( ( A3 = aa(num,real,numeral_numeral(real),W) )
        & ( B3 = zero_zero(real) ) ) ) ).

% Complex_eq_numeral
tff(fact_3515_complex__eq__cancel__iff2,axiom,
    ! [X: real,Y: real,Xa2: real] :
      ( ( complex2(X,Y) = aa(real,complex,real_Vector_of_real(complex),Xa2) )
    <=> ( ( X = Xa2 )
        & ( Y = zero_zero(real) ) ) ) ).

% complex_eq_cancel_iff2
tff(fact_3516_complex__of__real__code,axiom,
    ! [X2: real] : aa(real,complex,real_Vector_of_real(complex),X2) = complex2(X2,zero_zero(real)) ).

% complex_of_real_code
tff(fact_3517_complex__of__real__def,axiom,
    ! [R3: real] : aa(real,complex,real_Vector_of_real(complex),R3) = complex2(R3,zero_zero(real)) ).

% complex_of_real_def
tff(fact_3518_Complex__eq__neg__1,axiom,
    ! [A3: real,B3: real] :
      ( ( complex2(A3,B3) = aa(complex,complex,uminus_uminus(complex),one_one(complex)) )
    <=> ( ( A3 = aa(real,real,uminus_uminus(real),one_one(real)) )
        & ( B3 = zero_zero(real) ) ) ) ).

% Complex_eq_neg_1
tff(fact_3519_Complex__eq__neg__numeral,axiom,
    ! [A3: real,B3: real,W: num] :
      ( ( complex2(A3,B3) = aa(complex,complex,uminus_uminus(complex),aa(num,complex,numeral_numeral(complex),W)) )
    <=> ( ( A3 = aa(real,real,uminus_uminus(real),aa(num,real,numeral_numeral(real),W)) )
        & ( B3 = zero_zero(real) ) ) ) ).

% Complex_eq_neg_numeral
tff(fact_3520_diffs__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [C2: fun(nat,A),X2: nat] : aa(nat,A,diffs(A,C2),X2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,X2))),aa(nat,A,C2,aa(nat,nat,suc,X2))) ) ).

% diffs_def
tff(fact_3521_termdiff__converges__all,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),X: A] :
          ( ! [X4: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_cv(fun(nat,A),fun(A,fun(nat,A)),C2),X4))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_cw(fun(nat,A),fun(A,fun(nat,A)),C2),X)) ) ) ).

% termdiff_converges_all
tff(fact_3522_tan__45,axiom,
    aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2))))) = one_one(real) ).

% tan_45
tff(fact_3523_tan__gt__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ( aa(real,$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,ord_less(real,zero_zero(real)),aa(real,real,tan(real),X)) ) ) ).

% tan_gt_zero
tff(fact_3524_lemma__tan__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
     => ? [X4: real] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),X4)
          & aa(real,$o,ord_less(real,X4),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
          & aa(real,$o,ord_less(real,Y),aa(real,real,tan(real),X4)) ) ) ).

% lemma_tan_total
tff(fact_3525_tan__total,axiom,
    ! [Y: real] :
    ? [X4: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X4)
      & aa(real,$o,ord_less(real,X4),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),X4) = Y )
      & ! [Y4: real] :
          ( ( aa(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,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 = X4 ) ) ) ).

% tan_total
tff(fact_3526_tan__monotone,axiom,
    ! [Y: real,X: real] :
      ( aa(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,ord_less(real,Y),X)
       => ( aa(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,ord_less(real,aa(real,real,tan(real),Y)),aa(real,real,tan(real),X)) ) ) ) ).

% tan_monotone
tff(fact_3527_tan__monotone_H,axiom,
    ! [Y: real,X: real] :
      ( aa(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,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,$o,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X)
         => ( aa(real,$o,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,$o,ord_less(real,Y),X)
            <=> aa(real,$o,ord_less(real,aa(real,real,tan(real),Y)),aa(real,real,tan(real),X)) ) ) ) ) ) ).

% tan_monotone'
tff(fact_3528_tan__mono__lt__eq,axiom,
    ! [X: real,Y: real] :
      ( aa(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,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,$o,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
         => ( aa(real,$o,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,$o,ord_less(real,aa(real,real,tan(real),X)),aa(real,real,tan(real),Y))
            <=> aa(real,$o,ord_less(real,X),Y) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_3529_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X4: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X4)
      & aa(real,$o,ord_less(real,X4),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),X4) = Y ) ) ).

% lemma_tan_total1
tff(fact_3530_tan__minus__45,axiom,
    aa(real,real,tan(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)))))) = aa(real,real,uminus_uminus(real),one_one(real)) ).

% tan_minus_45
tff(fact_3531_tan__inverse,axiom,
    ! [Y: real] : aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,tan(real),Y)) = aa(real,real,tan(real),aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),Y)) ).

% tan_inverse
tff(fact_3532_vebt__heap__rules_I2_J,axiom,
    ! [Na: nat,Sa: set(nat),Ti: vEBT_VEBTi,X: nat] : hoare_hoare_triple($o,aa(vEBT_VEBTi,assn,vEBT_Intf_vebt_assn(Na,Sa),Ti),vEBT_vebt_memberi(Ti,X),aa(nat,fun($o,assn),aa(vEBT_VEBTi,fun(nat,fun($o,assn)),aa(set(nat),fun(vEBT_VEBTi,fun(nat,fun($o,assn))),aTP_Lamp_cs(nat,fun(set(nat),fun(vEBT_VEBTi,fun(nat,fun($o,assn)))),Na),Sa),Ti),X)) ).

% vebt_heap_rules(2)
tff(fact_3533_add__tan__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( aa(A,A,plus_plus(A,aa(A,A,tan(A),X)),aa(A,A,tan(A),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),sin(A,aa(A,A,plus_plus(A,X),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),cos(A,Y))) ) ) ) ) ).

% add_tan_eq
tff(fact_3534_termdiff__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,K4: real,C2: fun(nat,A)] :
          ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,X)),K4)
         => ( ! [X4: A] :
                ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,X4)),K4)
               => summable(A,aa(A,fun(nat,A),aTP_Lamp_cv(fun(nat,A),fun(A,fun(nat,A)),C2),X4)) )
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cx(A,fun(fun(nat,A),fun(nat,A)),X),C2)) ) ) ) ).

% termdiff_converges
tff(fact_3535_tan__total__pos,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
     => ? [X4: real] :
          ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X4)
          & aa(real,$o,ord_less(real,X4),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),X4) = Y ) ) ) ).

% tan_total_pos
tff(fact_3536_tan__pos__pi2__le,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => ( aa(real,$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,ord_less_eq(real,zero_zero(real)),aa(real,real,tan(real),X)) ) ) ).

% tan_pos_pi2_le
tff(fact_3537_tan__less__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),X)
     => ( aa(real,$o,ord_less(real,X),zero_zero(real))
       => aa(real,$o,ord_less(real,aa(real,real,tan(real),X)),zero_zero(real)) ) ) ).

% tan_less_zero
tff(fact_3538_tan__mono__le,axiom,
    ! [X: real,Y: real] :
      ( aa(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,ord_less_eq(real,X),Y)
       => ( aa(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,ord_less_eq(real,aa(real,real,tan(real),X)),aa(real,real,tan(real),Y)) ) ) ) ).

% tan_mono_le
tff(fact_3539_tan__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( aa(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,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,$o,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
         => ( aa(real,$o,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,$o,ord_less_eq(real,aa(real,real,tan(real),X)),aa(real,real,tan(real),Y))
            <=> aa(real,$o,ord_less_eq(real,X),Y) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_3540_tan__bound__pi2,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,abs_abs(real),X)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))))
     => aa(real,$o,ord_less(real,aa(real,real,abs_abs(real),aa(real,real,tan(real),X))),one_one(real)) ) ).

% tan_bound_pi2
tff(fact_3541_arctan,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arctan,Y))
      & aa(real,$o,ord_less(real,aa(real,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),aa(real,real,arctan,Y)) = Y ) ) ).

% arctan
tff(fact_3542_arctan__tan,axiom,
    ! [X: real] :
      ( aa(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,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,arctan,aa(real,real,tan(real),X)) = X ) ) ) ).

% arctan_tan
tff(fact_3543_arctan__unique,axiom,
    ! [X: real,Y: real] :
      ( aa(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,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 )
         => ( aa(real,real,arctan,Y) = X ) ) ) ) ).

% arctan_unique
tff(fact_3544_lemma__tan__add1,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( aa(A,A,minus_minus(A,one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),X)),aa(A,A,tan(A),Y))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),cos(A,aa(A,A,plus_plus(A,X),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),cos(A,Y))) ) ) ) ) ).

% lemma_tan_add1
tff(fact_3545_tan__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( ( cos(A,aa(A,A,minus_minus(A,X),Y)) != zero_zero(A) )
             => ( aa(A,A,tan(A),aa(A,A,minus_minus(A,X),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,tan(A),X)),aa(A,A,tan(A),Y))),aa(A,A,plus_plus(A,one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),X)),aa(A,A,tan(A),Y)))) ) ) ) ) ) ).

% tan_diff
tff(fact_3546_tan__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( ( cos(A,aa(A,A,plus_plus(A,X),Y)) != zero_zero(A) )
             => ( aa(A,A,tan(A),aa(A,A,plus_plus(A,X),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,plus_plus(A,aa(A,A,tan(A),X)),aa(A,A,tan(A),Y))),aa(A,A,minus_minus(A,one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),X)),aa(A,A,tan(A),Y)))) ) ) ) ) ) ).

% tan_add
tff(fact_3547_tan__total__pi4,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,abs_abs(real),X)),one_one(real))
     => ? [Z2: real] :
          ( aa(real,$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)))))),Z2)
          & aa(real,$o,ord_less(real,Z2),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),Z2) = X ) ) ) ).

% tan_total_pi4
tff(fact_3548_vebt__heap__rules_I8_J,axiom,
    ! [Na: nat,Sa: set(nat),Ti: vEBT_VEBTi,X: nat] : hoare_hoare_triple(vEBT_VEBTi,aa(vEBT_VEBTi,assn,vEBT_Intf_vebt_assn(Na,Sa),Ti),vEBT_vebt_deletei(Ti,X),vEBT_Intf_vebt_assn(Na,aa(set(nat),set(nat),minus_minus(set(nat),Sa),aa(set(nat),set(nat),insert(nat,X),bot_bot(set(nat)))))) ).

% vebt_heap_rules(8)
tff(fact_3549_tan__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,tan(A),X) = aa(A,A,aa(A,fun(A,A),divide_divide(A),sin(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),X))),aa(A,A,plus_plus(A,cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),X))),one_one(A))) ) ).

% tan_half
tff(fact_3550_sin__tan,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,abs_abs(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) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,tan(real),X)),aa(real,real,sqrt,aa(real,real,plus_plus(real,one_one(real)),aa(nat,real,power_power(real,aa(real,real,tan(real),X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ).

% sin_tan
tff(fact_3551_cos__tan,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,abs_abs(real),X)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
     => ( cos(real,X) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,sqrt,aa(real,real,plus_plus(real,one_one(real)),aa(nat,real,power_power(real,aa(real,real,tan(real),X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ).

% cos_tan
tff(fact_3552_cot__less__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),X)
     => ( aa(real,$o,ord_less(real,X),zero_zero(real))
       => aa(real,$o,ord_less(real,aa(real,real,cot(real),X)),zero_zero(real)) ) ) ).

% cot_less_zero
tff(fact_3553_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K2: int] : aa(real,real,arccos,cos(real,Theta)) != aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,Theta),aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),K2)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)))) ).

% arccos_cos_eq_abs_2pi
tff(fact_3554_sint__range__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( aa(int,$o,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,size_size(word(A),W)),aa(nat,nat,suc,zero_zero(nat)))))),aa(word(A),int,ring_1_signed(A,int),W))
          & aa(int,$o,ord_less(int,aa(word(A),int,ring_1_signed(A,int),W)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,size_size(word(A),W)),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).

% sint_range_size
tff(fact_3555_real__sqrt__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa(real,real,sqrt,X) = aa(real,real,sqrt,Y) )
    <=> ( X = Y ) ) ).

% real_sqrt_eq_iff
tff(fact_3556_real__sqrt__eq__zero__cancel__iff,axiom,
    ! [X: real] :
      ( ( aa(real,real,sqrt,X) = zero_zero(real) )
    <=> ( X = zero_zero(real) ) ) ).

% real_sqrt_eq_zero_cancel_iff
tff(fact_3557_real__sqrt__zero,axiom,
    aa(real,real,sqrt,zero_zero(real)) = zero_zero(real) ).

% real_sqrt_zero
tff(fact_3558_real__sqrt__less__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,sqrt,X)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,ord_less(real,X),Y) ) ).

% real_sqrt_less_iff
tff(fact_3559_real__sqrt__le__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,sqrt,X)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,ord_less_eq(real,X),Y) ) ).

% real_sqrt_le_iff
tff(fact_3560_real__sqrt__eq__1__iff,axiom,
    ! [X: real] :
      ( ( aa(real,real,sqrt,X) = one_one(real) )
    <=> ( X = one_one(real) ) ) ).

% real_sqrt_eq_1_iff
tff(fact_3561_real__sqrt__one,axiom,
    aa(real,real,sqrt,one_one(real)) = one_one(real) ).

% real_sqrt_one
tff(fact_3562_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_3563_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_3564_real__sqrt__lt__0__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,sqrt,X)),zero_zero(real))
    <=> aa(real,$o,ord_less(real,X),zero_zero(real)) ) ).

% real_sqrt_lt_0_iff
tff(fact_3565_real__sqrt__gt__0__iff,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,ord_less(real,zero_zero(real)),Y) ) ).

% real_sqrt_gt_0_iff
tff(fact_3566_real__sqrt__le__0__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,sqrt,X)),zero_zero(real))
    <=> aa(real,$o,ord_less_eq(real,X),zero_zero(real)) ) ).

% real_sqrt_le_0_iff
tff(fact_3567_real__sqrt__ge__0__iff,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,ord_less_eq(real,zero_zero(real)),Y) ) ).

% real_sqrt_ge_0_iff
tff(fact_3568_real__sqrt__lt__1__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,sqrt,X)),one_one(real))
    <=> aa(real,$o,ord_less(real,X),one_one(real)) ) ).

% real_sqrt_lt_1_iff
tff(fact_3569_real__sqrt__gt__1__iff,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,ord_less(real,one_one(real)),Y) ) ).

% real_sqrt_gt_1_iff
tff(fact_3570_real__sqrt__le__1__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,sqrt,X)),one_one(real))
    <=> aa(real,$o,ord_less_eq(real,X),one_one(real)) ) ).

% real_sqrt_le_1_iff
tff(fact_3571_real__sqrt__ge__1__iff,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,one_one(real)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,ord_less_eq(real,one_one(real)),Y) ) ).

% real_sqrt_ge_1_iff
tff(fact_3572_More__Word_Osint__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( ( aa(word(A),int,ring_1_signed(A,int),X) = zero_zero(int) )
        <=> ( X = zero_zero(word(A)) ) ) ) ).

% More_Word.sint_0
tff(fact_3573_Word_Oof__int__sint,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_1(A) )
     => ! [A3: word(B)] : aa(int,A,ring_1_of_int(A),aa(word(B),int,ring_1_signed(B,int),A3)) = aa(word(B),A,ring_1_signed(B,A),A3) ) ).

% Word.of_int_sint
tff(fact_3574_real__sqrt__abs2,axiom,
    ! [X: real] : aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),X),X)) = aa(real,real,abs_abs(real),X) ).

% real_sqrt_abs2
tff(fact_3575_real__sqrt__mult__self,axiom,
    ! [A3: real] : aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sqrt,A3)),aa(real,real,sqrt,A3)) = aa(real,real,abs_abs(real),A3) ).

% real_sqrt_mult_self
tff(fact_3576_arccos__1,axiom,
    aa(real,real,arccos,one_one(real)) = zero_zero(real) ).

% arccos_1
tff(fact_3577_cot__pi,axiom,
    aa(real,real,cot(real),pi) = zero_zero(real) ).

% cot_pi
tff(fact_3578_real__sqrt__four,axiom,
    aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(bit0(one2)))) = aa(num,real,numeral_numeral(real),bit0(one2)) ).

% real_sqrt_four
tff(fact_3579_arccos__minus__1,axiom,
    aa(real,real,arccos,aa(real,real,uminus_uminus(real),one_one(real))) = pi ).

% arccos_minus_1
tff(fact_3580_signed__minus__1,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ring_1(A)
        & type_len(B) )
     => ( aa(word(B),A,ring_1_signed(B,A),aa(word(B),word(B),uminus_uminus(word(B)),one_one(word(B)))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% signed_minus_1
tff(fact_3581_sint__minus1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( ( aa(word(A),int,ring_1_signed(A,int),X) = aa(int,int,uminus_uminus(int),one_one(int)) )
        <=> ( X = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% sint_minus1
tff(fact_3582_cos__arccos,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
       => ( cos(real,aa(real,real,arccos,Y)) = Y ) ) ) ).

% cos_arccos
tff(fact_3583_cot__npi,axiom,
    ! [Na: nat] : aa(real,real,cot(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),pi)) = zero_zero(real) ).

% cot_npi
tff(fact_3584_real__sqrt__abs,axiom,
    ! [X: real] : aa(real,real,sqrt,aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(real,real,abs_abs(real),X) ).

% real_sqrt_abs
tff(fact_3585_real__sqrt__pow2,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => ( aa(nat,real,power_power(real,aa(real,real,sqrt,X)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = X ) ) ).

% real_sqrt_pow2
tff(fact_3586_real__sqrt__pow2__iff,axiom,
    ! [X: real] :
      ( ( aa(nat,real,power_power(real,aa(real,real,sqrt,X)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = X )
    <=> aa(real,$o,ord_less_eq(real,zero_zero(real)),X) ) ).

% real_sqrt_pow2_iff
tff(fact_3587_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [X: real,Y: real,Xa2: real,Ya: real] : aa(nat,real,power_power(real,aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,plus_plus(real,aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(real,real,plus_plus(real,aa(nat,real,power_power(real,Xa2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Ya),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,plus_plus(real,aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(real,real,plus_plus(real,aa(nat,real,power_power(real,Xa2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Ya),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% real_sqrt_sum_squares_mult_squared_eq
tff(fact_3588_arccos__0,axiom,
    aa(real,real,arccos,zero_zero(real)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% arccos_0
tff(fact_3589_cot__periodic,axiom,
    ! [X: real] : aa(real,real,cot(real),aa(real,real,plus_plus(real,X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))) = aa(real,real,cot(real),X) ).

% cot_periodic
tff(fact_3590_signed__word__eqI,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_char_0(A) )
     => ! [V2: word(B),W: word(B)] :
          ( ( aa(word(B),A,ring_1_signed(B,A),V2) = aa(word(B),A,ring_1_signed(B,A),W) )
         => ( V2 = W ) ) ) ).

% signed_word_eqI
tff(fact_3591_word__eq__iff__signed,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & ring_char_0(B) )
     => ! [V2: word(A),W: word(A)] :
          ( ( V2 = W )
        <=> ( aa(word(A),B,ring_1_signed(A,B),V2) = aa(word(A),B,ring_1_signed(A,B),W) ) ) ) ).

% word_eq_iff_signed
tff(fact_3592_More__Word_Oof__int__sint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : aa(int,word(A),ring_1_of_int(word(A)),aa(word(A),int,ring_1_signed(A,int),A3)) = A3 ) ).

% More_Word.of_int_sint
tff(fact_3593_real__sqrt__minus,axiom,
    ! [X: real] : aa(real,real,sqrt,aa(real,real,uminus_uminus(real),X)) = aa(real,real,uminus_uminus(real),aa(real,real,sqrt,X)) ).

% real_sqrt_minus
tff(fact_3594_real__sqrt__power,axiom,
    ! [X: real,K: nat] : aa(real,real,sqrt,aa(nat,real,power_power(real,X),K)) = aa(nat,real,power_power(real,aa(real,real,sqrt,X)),K) ).

% real_sqrt_power
tff(fact_3595_real__sqrt__mult,axiom,
    ! [X: real,Y: real] : aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),X),Y)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sqrt,X)),aa(real,real,sqrt,Y)) ).

% real_sqrt_mult
tff(fact_3596_real__sqrt__divide,axiom,
    ! [X: real,Y: real] : aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),divide_divide(real),X),Y)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,X)),aa(real,real,sqrt,Y)) ).

% real_sqrt_divide
tff(fact_3597_real__sqrt__le__mono,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,X),Y)
     => aa(real,$o,ord_less_eq(real,aa(real,real,sqrt,X)),aa(real,real,sqrt,Y)) ) ).

% real_sqrt_le_mono
tff(fact_3598_real__sqrt__less__mono,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less(real,X),Y)
     => aa(real,$o,ord_less(real,aa(real,real,sqrt,X)),aa(real,real,sqrt,Y)) ) ).

% real_sqrt_less_mono
tff(fact_3599_real__sqrt__gt__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,sqrt,X)) ) ).

% real_sqrt_gt_zero
tff(fact_3600_real__sqrt__eq__zero__cancel,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => ( ( aa(real,real,sqrt,X) = zero_zero(real) )
       => ( X = zero_zero(real) ) ) ) ).

% real_sqrt_eq_zero_cancel
tff(fact_3601_real__sqrt__ge__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,sqrt,X)) ) ).

% real_sqrt_ge_zero
tff(fact_3602_real__sqrt__ge__one,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,one_one(real)),X)
     => aa(real,$o,ord_less_eq(real,one_one(real)),aa(real,real,sqrt,X)) ) ).

% real_sqrt_ge_one
tff(fact_3603_Word_Osint__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),int,ring_1_signed(A,int),zero_zero(word(A))) = zero_zero(int) ) ) ).

% Word.sint_0
tff(fact_3604_signed__eq__0__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_char_0(A) )
     => ! [W: word(B)] :
          ( ( aa(word(B),A,ring_1_signed(B,A),W) = zero_zero(A) )
        <=> ( W = zero_zero(word(B)) ) ) ) ).

% signed_eq_0_iff
tff(fact_3605_real__div__sqrt,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => ( aa(real,real,aa(real,fun(real,real),divide_divide(real),X),aa(real,real,sqrt,X)) = aa(real,real,sqrt,X) ) ) ).

% real_div_sqrt
tff(fact_3606_sqrt__add__le__add__sqrt,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
       => aa(real,$o,ord_less_eq(real,aa(real,real,sqrt,aa(real,real,plus_plus(real,X),Y))),aa(real,real,plus_plus(real,aa(real,real,sqrt,X)),aa(real,real,sqrt,Y))) ) ) ).

% sqrt_add_le_add_sqrt
tff(fact_3607_le__real__sqrt__sumsq,axiom,
    ! [X: real,Y: real] : aa(real,$o,ord_less_eq(real,X),aa(real,real,sqrt,aa(real,real,plus_plus(real,aa(real,real,aa(real,fun(real,real),times_times(real),X),X)),aa(real,real,aa(real,fun(real,real),times_times(real),Y),Y)))) ).

% le_real_sqrt_sumsq
tff(fact_3608_arccos__le__arccos,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,ord_less_eq(real,X),Y)
       => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
         => aa(real,$o,ord_less_eq(real,aa(real,real,arccos,Y)),aa(real,real,arccos,X)) ) ) ) ).

% arccos_le_arccos
tff(fact_3609_arccos__le__mono,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),X)),one_one(real))
     => ( aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( aa(real,$o,ord_less_eq(real,aa(real,real,arccos,X)),aa(real,real,arccos,Y))
        <=> aa(real,$o,ord_less_eq(real,Y),X) ) ) ) ).

% arccos_le_mono
tff(fact_3610_arccos__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),X)),one_one(real))
        & aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),Y)),one_one(real)) )
     => ( ( aa(real,real,arccos,X) = aa(real,real,arccos,Y) )
      <=> ( X = Y ) ) ) ).

% arccos_eq_iff
tff(fact_3611_sqrt2__less__2,axiom,
    aa(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_3612_sint__n1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),int,ring_1_signed(A,int),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ).

% sint_n1
tff(fact_3613_sin__arccos,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,ord_less_eq(real,X),one_one(real))
       => ( sin(real,aa(real,real,arccos,X)) = aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ).

% sin_arccos
tff(fact_3614_sin__arccos__abs,axiom,
    ! [Y: real] :
      ( aa(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,minus_minus(real,one_one(real)),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% sin_arccos_abs
tff(fact_3615_real__less__rsqrt,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less(real,aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Y)
     => aa(real,$o,ord_less(real,X),aa(real,real,sqrt,Y)) ) ).

% real_less_rsqrt
tff(fact_3616_sqrt__le__D,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,sqrt,X)),Y)
     => aa(real,$o,ord_less_eq(real,X),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% sqrt_le_D
tff(fact_3617_real__le__rsqrt,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Y)
     => aa(real,$o,ord_less_eq(real,X),aa(real,real,sqrt,Y)) ) ).

% real_le_rsqrt
tff(fact_3618_arccos__lbound,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
       => aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,arccos,Y)) ) ) ).

% arccos_lbound
tff(fact_3619_arccos__less__arccos,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,ord_less(real,X),Y)
       => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
         => aa(real,$o,ord_less(real,aa(real,real,arccos,Y)),aa(real,real,arccos,X)) ) ) ) ).

% arccos_less_arccos
tff(fact_3620_arccos__less__mono,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),X)),one_one(real))
     => ( aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( aa(real,$o,ord_less(real,aa(real,real,arccos,X)),aa(real,real,arccos,Y))
        <=> aa(real,$o,ord_less(real,Y),X) ) ) ) ).

% arccos_less_mono
tff(fact_3621_arccos__ubound,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
       => aa(real,$o,ord_less_eq(real,aa(real,real,arccos,Y)),pi) ) ) ).

% arccos_ubound
tff(fact_3622_tan__60,axiom,
    aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit1(one2)))) = aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit1(one2))) ).

% tan_60
tff(fact_3623_arccos__cos,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less_eq(real,X),pi)
       => ( aa(real,real,arccos,cos(real,X)) = X ) ) ) ).

% arccos_cos
tff(fact_3624_cos__arccos__abs,axiom,
    ! [Y: real] :
      ( aa(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_3625_real__le__lsqrt,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
       => ( aa(real,$o,ord_less_eq(real,X),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
         => aa(real,$o,ord_less_eq(real,aa(real,real,sqrt,X)),Y) ) ) ) ).

% real_le_lsqrt
tff(fact_3626_real__sqrt__unique,axiom,
    ! [Y: real,X: real] :
      ( ( aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))) = X )
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
       => ( aa(real,real,sqrt,X) = Y ) ) ) ).

% real_sqrt_unique
tff(fact_3627_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),U)
     => aa(real,$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_3628_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [X: real,Y: real] :
      ( ( aa(real,real,sqrt,aa(real,real,plus_plus(real,aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))) = X )
     => ( Y = zero_zero(real) ) ) ).

% real_sqrt_sum_squares_eq_cancel
tff(fact_3629_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [X: real,Y: real] :
      ( ( aa(real,real,sqrt,aa(real,real,plus_plus(real,aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))) = Y )
     => ( X = zero_zero(real) ) ) ).

% real_sqrt_sum_squares_eq_cancel2
tff(fact_3630_real__sqrt__sum__squares__ge1,axiom,
    ! [X: real,Y: real] : aa(real,$o,ord_less_eq(real,X),aa(real,real,sqrt,aa(real,real,plus_plus(real,aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% real_sqrt_sum_squares_ge1
tff(fact_3631_real__sqrt__sum__squares__ge2,axiom,
    ! [Y: real,X: real] : aa(real,$o,ord_less_eq(real,Y),aa(real,real,sqrt,aa(real,real,plus_plus(real,aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% real_sqrt_sum_squares_ge2
tff(fact_3632_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A3: real,C2: real,B3: real,D2: real] : aa(real,$o,ord_less_eq(real,aa(real,real,sqrt,aa(real,real,plus_plus(real,aa(nat,real,power_power(real,aa(real,real,plus_plus(real,A3),C2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,aa(real,real,plus_plus(real,B3),D2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(real,real,plus_plus(real,aa(real,real,sqrt,aa(real,real,plus_plus(real,aa(nat,real,power_power(real,A3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,B3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(real,real,sqrt,aa(real,real,plus_plus(real,aa(nat,real,power_power(real,C2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,D2),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ).

% real_sqrt_sum_squares_triangle_ineq
tff(fact_3633_sqrt__ge__absD,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),X)),aa(real,real,sqrt,Y))
     => aa(real,$o,ord_less_eq(real,aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Y) ) ).

% sqrt_ge_absD
tff(fact_3634_cos__45,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2))))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2)))),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% cos_45
tff(fact_3635_sin__45,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2))))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2)))),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% sin_45
tff(fact_3636_arccos__lt__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,ord_less(real,Y),one_one(real))
       => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,arccos,Y))
          & aa(real,$o,ord_less(real,aa(real,real,arccos,Y)),pi) ) ) ) ).

% arccos_lt_bounded
tff(fact_3637_arccos__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
       => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,arccos,Y))
          & aa(real,$o,ord_less_eq(real,aa(real,real,arccos,Y)),pi) ) ) ) ).

% arccos_bounded
tff(fact_3638_sin__arccos__nonzero,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,ord_less(real,X),one_one(real))
       => ( sin(real,aa(real,real,arccos,X)) != zero_zero(real) ) ) ) ).

% sin_arccos_nonzero
tff(fact_3639_arccos__cos2,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,X),zero_zero(real))
     => ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),pi)),X)
       => ( aa(real,real,arccos,cos(real,X)) = aa(real,real,uminus_uminus(real),X) ) ) ) ).

% arccos_cos2
tff(fact_3640_arccos__minus,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,ord_less_eq(real,X),one_one(real))
       => ( aa(real,real,arccos,aa(real,real,uminus_uminus(real),X)) = aa(real,real,minus_minus(real,pi),aa(real,real,arccos,X)) ) ) ) ).

% arccos_minus
tff(fact_3641_real__less__lsqrt,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
       => ( aa(real,$o,ord_less(real,X),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
         => aa(real,$o,ord_less(real,aa(real,real,sqrt,X)),Y) ) ) ) ).

% real_less_lsqrt
tff(fact_3642_sqrt__sum__squares__le__sum,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
       => aa(real,$o,ord_less_eq(real,aa(real,real,sqrt,aa(real,real,plus_plus(real,aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(real,real,plus_plus(real,X),Y)) ) ) ).

% sqrt_sum_squares_le_sum
tff(fact_3643_sqrt__even__pow2,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => ( aa(real,real,sqrt,aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),bit0(one2))),Na)) = aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% sqrt_even_pow2
tff(fact_3644_tan__30,axiom,
    aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit1(one2))))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit1(one2)))) ).

% tan_30
tff(fact_3645_real__sqrt__ge__abs1,axiom,
    ! [X: real,Y: real] : aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),X)),aa(real,real,sqrt,aa(real,real,plus_plus(real,aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% real_sqrt_ge_abs1
tff(fact_3646_real__sqrt__ge__abs2,axiom,
    ! [Y: real,X: real] : aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),Y)),aa(real,real,sqrt,aa(real,real,plus_plus(real,aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% real_sqrt_ge_abs2
tff(fact_3647_sqrt__sum__squares__le__sum__abs,axiom,
    ! [X: real,Y: real] : aa(real,$o,ord_less_eq(real,aa(real,real,sqrt,aa(real,real,plus_plus(real,aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(real,real,plus_plus(real,aa(real,real,abs_abs(real),X)),aa(real,real,abs_abs(real),Y))) ).

% sqrt_sum_squares_le_sum_abs
tff(fact_3648_ln__sqrt,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ( aa(real,real,ln_ln(real),aa(real,real,sqrt,X)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),X)),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% ln_sqrt
tff(fact_3649_cos__30,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit1(one2))))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit1(one2)))),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% cos_30
tff(fact_3650_sin__60,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit1(one2)))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit1(one2)))),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% sin_60
tff(fact_3651_arsinh__real__def,axiom,
    ! [X: real] : aa(real,real,arsinh(real),X) = aa(real,real,ln_ln(real),aa(real,real,plus_plus(real,X),aa(real,real,sqrt,aa(real,real,plus_plus(real,aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))))) ).

% arsinh_real_def
tff(fact_3652_complex__norm,axiom,
    ! [X: real,Y: real] : real_V7770717601297561774m_norm(complex,complex2(X,Y)) = aa(real,real,sqrt,aa(real,real,plus_plus(real,aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% complex_norm
tff(fact_3653_arccos,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
       => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,arccos,Y))
          & aa(real,$o,ord_less_eq(real,aa(real,real,arccos,Y)),pi)
          & ( cos(real,aa(real,real,arccos,Y)) = Y ) ) ) ) ).

% arccos
tff(fact_3654_arccos__minus__abs,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),X)),one_one(real))
     => ( aa(real,real,arccos,aa(real,real,uminus_uminus(real),X)) = aa(real,real,minus_minus(real,pi),aa(real,real,arccos,X)) ) ) ).

% arccos_minus_abs
tff(fact_3655_real__sqrt__power__even,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
       => ( aa(nat,real,power_power(real,aa(real,real,sqrt,X)),Na) = aa(nat,real,power_power(real,X),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_3656_arsinh__real__aux,axiom,
    ! [X: real] : aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,plus_plus(real,X),aa(real,real,sqrt,aa(real,real,plus_plus(real,aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))))) ).

% arsinh_real_aux
tff(fact_3657_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X: real,Y: real,Xa2: real,Ya: real] : aa(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,plus_plus(real,aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(real,real,plus_plus(real,aa(nat,real,power_power(real,Xa2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Ya),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ).

% real_sqrt_sum_squares_mult_ge_zero
tff(fact_3658_arith__geo__mean__sqrt,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
       => aa(real,$o,ord_less_eq(real,aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),X),Y))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,plus_plus(real,X),Y)),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_3659_powr__half__sqrt,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => ( powr(real,X,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,X) ) ) ).

% powr_half_sqrt
tff(fact_3660_tan__cot_H,axiom,
    ! [X: real] : aa(real,real,tan(real),aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),X)) = aa(real,real,cot(real),X) ).

% tan_cot'
tff(fact_3661_cos__x__y__le__one,axiom,
    ! [X: real,Y: real] : aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),X),aa(real,real,sqrt,aa(real,real,plus_plus(real,aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),one_one(real)) ).

% cos_x_y_le_one
tff(fact_3662_real__sqrt__sum__squares__less,axiom,
    ! [X: real,U: real,Y: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,abs_abs(real),X)),aa(real,real,aa(real,fun(real,real),divide_divide(real),U),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2)))))
     => ( aa(real,$o,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,ord_less(real,aa(real,real,sqrt,aa(real,real,plus_plus(real,aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),U) ) ) ).

% real_sqrt_sum_squares_less
tff(fact_3663_arcosh__real__def,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,one_one(real)),X)
     => ( aa(real,real,arcosh(real),X) = aa(real,real,ln_ln(real),aa(real,real,plus_plus(real,X),aa(real,real,sqrt,aa(real,real,minus_minus(real,aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))))) ) ) ).

% arcosh_real_def
tff(fact_3664_cos__arctan,axiom,
    ! [X: real] : cos(real,aa(real,real,arctan,X)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,sqrt,aa(real,real,plus_plus(real,one_one(real)),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% cos_arctan
tff(fact_3665_sin__arctan,axiom,
    ! [X: real] : sin(real,aa(real,real,arctan,X)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),X),aa(real,real,sqrt,aa(real,real,plus_plus(real,one_one(real)),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% sin_arctan
tff(fact_3666_sint__above__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),X: int] :
          ( aa(int,$o,ord_less_eq(int,aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,size_size(word(A),W)),one_one(nat)))),X)
         => aa(int,$o,ord_less(int,aa(word(A),int,ring_1_signed(A,int),W)),X) ) ) ).

% sint_above_size
tff(fact_3667_sint__below__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: int,W: word(A)] :
          ( aa(int,$o,ord_less_eq(int,X),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,size_size(word(A),W)),one_one(nat)))))
         => aa(int,$o,ord_less_eq(int,X),aa(word(A),int,ring_1_signed(A,int),W)) ) ) ).

% sint_below_size
tff(fact_3668_arccos__le__pi2,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
       => aa(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_3669_cot__gt__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ( aa(real,$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,ord_less(real,zero_zero(real)),aa(real,real,cot(real),X)) ) ) ).

% cot_gt_zero
tff(fact_3670_sqrt__sum__squares__half__less,axiom,
    ! [X: real,U: real,Y: real] :
      ( aa(real,$o,ord_less(real,X),aa(real,real,aa(real,fun(real,real),divide_divide(real),U),aa(num,real,numeral_numeral(real),bit0(one2))))
     => ( aa(real,$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,$o,ord_less_eq(real,zero_zero(real)),X)
         => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
           => aa(real,$o,ord_less(real,aa(real,real,sqrt,aa(real,real,plus_plus(real,aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),U) ) ) ) ) ).

% sqrt_sum_squares_half_less
tff(fact_3671_sin__cos__sqrt,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),sin(real,X))
     => ( sin(real,X) = aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,power_power(real,cos(real,X)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% sin_cos_sqrt
tff(fact_3672_arctan__half,axiom,
    ! [X: real] : aa(real,real,arctan,X) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),X),aa(real,real,plus_plus(real,one_one(real)),aa(real,real,sqrt,aa(real,real,plus_plus(real,one_one(real)),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) ).

% arctan_half
tff(fact_3673_cos__arcsin,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,ord_less_eq(real,X),one_one(real))
       => ( cos(real,aa(real,real,arcsin,X)) = aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ).

% cos_arcsin
tff(fact_3674_arcsin,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
       => ( aa(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,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_3675_arcsin__pi,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
       => ( aa(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,ord_less_eq(real,aa(real,real,arcsin,Y)),pi)
          & ( sin(real,aa(real,real,arcsin,Y)) = Y ) ) ) ) ).

% arcsin_pi
tff(fact_3676_arcsin__le__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,ord_less_eq(real,X),one_one(real))
       => ( aa(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,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,$o,ord_less_eq(real,aa(real,real,arcsin,X)),Y)
            <=> aa(real,$o,ord_less_eq(real,X),sin(real,Y)) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_3677_le__arcsin__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,ord_less_eq(real,X),one_one(real))
       => ( aa(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,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,$o,ord_less_eq(real,Y),aa(real,real,arcsin,X))
            <=> aa(real,$o,ord_less_eq(real,sin(real,Y)),X) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_3678_arcsin__0,axiom,
    aa(real,real,arcsin,zero_zero(real)) = zero_zero(real) ).

% arcsin_0
tff(fact_3679_sin__arcsin,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
       => ( sin(real,aa(real,real,arcsin,Y)) = Y ) ) ) ).

% sin_arcsin
tff(fact_3680_arcsin__1,axiom,
    aa(real,real,arcsin,one_one(real)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% arcsin_1
tff(fact_3681_arcsin__minus__1,axiom,
    aa(real,real,arcsin,aa(real,real,uminus_uminus(real),one_one(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)))) ).

% arcsin_minus_1
tff(fact_3682_scast__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ( aa(word(B),word(A),ring_1_signed(B,word(A)),zero_zero(word(B))) = zero_zero(word(A)) ) ) ).

% scast_0
tff(fact_3683_scast__n1,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ( aa(word(B),word(A),ring_1_signed(B,word(A)),aa(word(B),word(B),uminus_uminus(word(B)),one_one(word(B)))) = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ).

% scast_n1
tff(fact_3684_scast__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [W: word(B)] : aa(word(B),word(A),ring_1_signed(B,word(A)),W) = aa(int,word(A),ring_1_of_int(word(A)),aa(word(B),int,ring_1_signed(B,int),W)) ) ).

% scast_eq
tff(fact_3685_arcsin__minus,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,ord_less_eq(real,X),one_one(real))
       => ( aa(real,real,arcsin,aa(real,real,uminus_uminus(real),X)) = aa(real,real,uminus_uminus(real),aa(real,real,arcsin,X)) ) ) ) ).

% arcsin_minus
tff(fact_3686_arcsin__le__arcsin,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,ord_less_eq(real,X),Y)
       => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
         => aa(real,$o,ord_less_eq(real,aa(real,real,arcsin,X)),aa(real,real,arcsin,Y)) ) ) ) ).

% arcsin_le_arcsin
tff(fact_3687_arcsin__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),X)),one_one(real))
     => ( aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( ( aa(real,real,arcsin,X) = aa(real,real,arcsin,Y) )
        <=> ( X = Y ) ) ) ) ).

% arcsin_eq_iff
tff(fact_3688_arcsin__le__mono,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),X)),one_one(real))
     => ( aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( aa(real,$o,ord_less_eq(real,aa(real,real,arcsin,X)),aa(real,real,arcsin,Y))
        <=> aa(real,$o,ord_less_eq(real,X),Y) ) ) ) ).

% arcsin_le_mono
tff(fact_3689_arcsin__less__arcsin,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,ord_less(real,X),Y)
       => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
         => aa(real,$o,ord_less(real,aa(real,real,arcsin,X)),aa(real,real,arcsin,Y)) ) ) ) ).

% arcsin_less_arcsin
tff(fact_3690_arcsin__less__mono,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),X)),one_one(real))
     => ( aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( aa(real,$o,ord_less(real,aa(real,real,arcsin,X)),aa(real,real,arcsin,Y))
        <=> aa(real,$o,ord_less(real,X),Y) ) ) ) ).

% arcsin_less_mono
tff(fact_3691_cos__arcsin__nonzero,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,ord_less(real,X),one_one(real))
       => ( cos(real,aa(real,real,arcsin,X)) != zero_zero(real) ) ) ) ).

% cos_arcsin_nonzero
tff(fact_3692_arcsin__lt__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,ord_less(real,Y),one_one(real))
       => ( aa(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,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_3693_arcsin__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
       => ( aa(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,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_3694_arcsin__ubound,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
       => aa(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_3695_arcsin__lbound,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
       => aa(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_3696_arcsin__sin,axiom,
    ! [X: real] :
      ( aa(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,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))))
       => ( aa(real,real,arcsin,sin(real,X)) = X ) ) ) ).

% arcsin_sin
tff(fact_3697_gbinomial__code,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] :
          aa(nat,A,gbinomial(A,A3),K) = $ite(K = zero_zero(nat),one_one(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),set_fo6178422350223883121st_nat(A,aTP_Lamp_cy(A,fun(nat,fun(A,A)),A3),zero_zero(nat),aa(nat,nat,minus_minus(nat,K),one_one(nat)),one_one(A))),semiring_char_0_fact(A,K))) ) ).

% gbinomial_code
tff(fact_3698_log__base__10__eq1,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ( aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(bit1(bit0(one2))))),X) = 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),aa(real,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),X)) ) ) ).

% log_base_10_eq1
tff(fact_3699_flip__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : bit_se8732182000553998342ip_bit(A,zero_zero(nat),A3) = aa(A,A,plus_plus(A,aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% flip_bit_0
tff(fact_3700_take__bit__word__Bit1__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num,M: num] : bit_se2584673776208193580ke_bit(word(A),aa(num,nat,numeral_numeral(nat),Na),aa(num,word(A),numeral_numeral(word(A)),bit1(M))) = aa(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))),bit_se2584673776208193580ke_bit(word(A),pred_numeral(Na),aa(num,word(A),numeral_numeral(word(A)),M)))) ) ).

% take_bit_word_Bit1_eq
tff(fact_3701_nat__of__bool,axiom,
    ! [P: $o] : nat2(aa($o,int,zero_neq_one_of_bool(int),(P))) = aa($o,nat,zero_neq_one_of_bool(nat),(P)) ).

% nat_of_bool
tff(fact_3702_scast__id,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(word(A),word(A),ring_1_signed(A,word(A)),W) = W ) ).

% scast_id
tff(fact_3703_take__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : bit_se2584673776208193580ke_bit(A,Na,zero_zero(A)) = zero_zero(A) ) ).

% take_bit_of_0
tff(fact_3704_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_3705_of__bool__eq__0__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: $o] :
          ( ( aa($o,A,zero_neq_one_of_bool(A),(P)) = zero_zero(A) )
        <=> ~ (P) ) ) ).

% of_bool_eq_0_iff
tff(fact_3706_of__bool__less__eq__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [P: $o,Q: $o] :
          ( aa(A,$o,ord_less_eq(A,aa($o,A,zero_neq_one_of_bool(A),(P))),aa($o,A,zero_neq_one_of_bool(A),(Q)))
        <=> ( (P)
           => (Q) ) ) ) ).

% of_bool_less_eq_iff
tff(fact_3707_of__bool__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [P: $o,Q: $o] :
          ( aa(A,$o,ord_less(A,aa($o,A,zero_neq_one_of_bool(A),(P))),aa($o,A,zero_neq_one_of_bool(A),(Q)))
        <=> ( ~ (P)
            & (Q) ) ) ) ).

% of_bool_less_iff
tff(fact_3708_of__bool__eq_I2_J,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( aa($o,A,zero_neq_one_of_bool(A),$true) = one_one(A) ) ) ).

% of_bool_eq(2)
tff(fact_3709_of__bool__eq__1__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: $o] :
          ( ( aa($o,A,zero_neq_one_of_bool(A),(P)) = one_one(A) )
        <=> (P) ) ) ).

% of_bool_eq_1_iff
tff(fact_3710_of__nat__of__bool,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [P: $o] : aa(nat,A,semiring_1_of_nat(A),aa($o,nat,zero_neq_one_of_bool(nat),(P))) = aa($o,A,zero_neq_one_of_bool(A),(P)) ) ).

% of_nat_of_bool
tff(fact_3711_of__int__of__bool,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [P: $o] : aa(int,A,ring_1_of_int(A),aa($o,int,zero_neq_one_of_bool(int),(P))) = aa($o,A,zero_neq_one_of_bool(A),(P)) ) ).

% of_int_of_bool
tff(fact_3712_abs__bool__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P: $o] : aa(A,A,abs_abs(A),aa($o,A,zero_neq_one_of_bool(A),(P))) = aa($o,A,zero_neq_one_of_bool(A),(P)) ) ).

% abs_bool_eq
tff(fact_3713_gbinomial__1,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A3: A] : aa(nat,A,gbinomial(A,A3),one_one(nat)) = A3 ) ).

% gbinomial_1
tff(fact_3714_exp__less__mono,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less(real,X),Y)
     => aa(real,$o,ord_less(real,aa(real,real,exp(real),X)),aa(real,real,exp(real),Y)) ) ).

% exp_less_mono
tff(fact_3715_exp__less__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,exp(real),X)),aa(real,real,exp(real),Y))
    <=> aa(real,$o,ord_less(real,X),Y) ) ).

% exp_less_cancel_iff
tff(fact_3716_div__word__one,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),one_one(word(A))),W) = aa($o,word(A),zero_neq_one_of_bool(word(A)),W = one_one(word(A))) ) ).

% div_word_one
tff(fact_3717_take__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : bit_se2584673776208193580ke_bit(A,zero_zero(nat),A3) = zero_zero(A) ) ).

% take_bit_0
tff(fact_3718_take__bit__Suc__1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat] : bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Na),one_one(A)) = one_one(A) ) ).

% take_bit_Suc_1
tff(fact_3719_take__bit__numeral__1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num] : bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),L),one_one(A)) = one_one(A) ) ).

% take_bit_numeral_1
tff(fact_3720_zero__less__of__bool__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: $o] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa($o,A,zero_neq_one_of_bool(A),(P)))
        <=> (P) ) ) ).

% zero_less_of_bool_iff
tff(fact_3721_exp__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( aa(A,A,exp(A),zero_zero(A)) = one_one(A) ) ) ).

% exp_zero
tff(fact_3722_of__bool__less__one__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: $o] :
          ( aa(A,$o,ord_less(A,aa($o,A,zero_neq_one_of_bool(A),(P))),one_one(A))
        <=> ~ (P) ) ) ).

% of_bool_less_one_iff
tff(fact_3723_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_3724_of__bool__not__iff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [P: $o] : aa($o,A,zero_neq_one_of_bool(A),~ (P)) = aa(A,A,minus_minus(A,one_one(A)),aa($o,A,zero_neq_one_of_bool(A),(P))) ) ).

% of_bool_not_iff
tff(fact_3725_gbinomial__0_I2_J,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [K: nat] : aa(nat,A,gbinomial(A,zero_zero(A)),aa(nat,nat,suc,K)) = zero_zero(A) ) ).

% gbinomial_0(2)
tff(fact_3726_gbinomial__0_I1_J,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A3: A] : aa(nat,A,gbinomial(A,A3),zero_zero(nat)) = one_one(A) ) ).

% gbinomial_0(1)
tff(fact_3727_gbinomial__Suc0,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A3: A] : aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,zero_zero(nat))) = A3 ) ).

% gbinomial_Suc0
tff(fact_3728_exp__eq__one__iff,axiom,
    ! [X: real] :
      ( ( aa(real,real,exp(real),X) = one_one(real) )
    <=> ( X = zero_zero(real) ) ) ).

% exp_eq_one_iff
tff(fact_3729_div__word__by__minus__1__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),W),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) = aa($o,word(A),zero_neq_one_of_bool(word(A)),W = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) ) ).

% div_word_by_minus_1_eq
tff(fact_3730_take__bit__of__1__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat] :
          ( ( bit_se2584673776208193580ke_bit(A,Na,one_one(A)) = zero_zero(A) )
        <=> ( Na = zero_zero(nat) ) ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_3731_one__less__exp__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),aa(real,real,exp(real),X))
    <=> aa(real,$o,ord_less(real,zero_zero(real)),X) ) ).

% one_less_exp_iff
tff(fact_3732_exp__less__one__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,exp(real),X)),one_one(real))
    <=> aa(real,$o,ord_less(real,X),zero_zero(real)) ) ).

% exp_less_one_iff
tff(fact_3733_one__le__exp__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,one_one(real)),aa(real,real,exp(real),X))
    <=> aa(real,$o,ord_less_eq(real,zero_zero(real)),X) ) ).

% one_le_exp_iff
tff(fact_3734_exp__le__one__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,exp(real),X)),one_one(real))
    <=> aa(real,$o,ord_less_eq(real,X),zero_zero(real)) ) ).

% exp_le_one_iff
tff(fact_3735_exp__ln,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ( aa(real,real,exp(real),aa(real,real,ln_ln(real),X)) = X ) ) ).

% exp_ln
tff(fact_3736_exp__ln__iff,axiom,
    ! [X: real] :
      ( ( aa(real,real,exp(real),aa(real,real,ln_ln(real),X)) = X )
    <=> aa(real,$o,ord_less(real,zero_zero(real)),X) ) ).

% exp_ln_iff
tff(fact_3737_mod__word__one,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),one_one(word(A))),W) = aa(word(A),word(A),minus_minus(word(A),one_one(word(A))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),W),aa($o,word(A),zero_neq_one_of_bool(word(A)),W = one_one(word(A))))) ) ).

% mod_word_one
tff(fact_3738_odd__of__bool__self,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [P2: $o] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa($o,A,zero_neq_one_of_bool(A),(P2)))
        <=> (P2) ) ) ).

% odd_of_bool_self
tff(fact_3739_take__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : bit_se2584673776208193580ke_bit(A,Na,one_one(A)) = aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)) ) ).

% take_bit_of_1
tff(fact_3740_mod__word__by__minus__1__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),W),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),W),aa($o,word(A),zero_neq_one_of_bool(word(A)),aa(word(A),$o,ord_less(word(A),W),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))))) ) ).

% mod_word_by_minus_1_eq
tff(fact_3741_even__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2584673776208193580ke_bit(A,Na,A3))
        <=> ( ( Na = zero_zero(nat) )
            | aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3) ) ) ) ).

% even_take_bit_eq
tff(fact_3742_of__bool__half__eq__0,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [B3: $o] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa($o,A,zero_neq_one_of_bool(A),(B3))),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ).

% of_bool_half_eq_0
tff(fact_3743_take__bit__Suc__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,zero_zero(nat)),A3) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% take_bit_Suc_0
tff(fact_3744_take__bit__word__Bit0__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num,M: num] : bit_se2584673776208193580ke_bit(word(A),aa(num,nat,numeral_numeral(nat),Na),aa(num,word(A),numeral_numeral(word(A)),bit0(M))) = 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))),bit_se2584673776208193580ke_bit(word(A),pred_numeral(Na),aa(num,word(A),numeral_numeral(word(A)),M))) ) ).

% take_bit_word_Bit0_eq
tff(fact_3745_one__div__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Na: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa($o,A,zero_neq_one_of_bool(A),Na = zero_zero(nat)) ) ).

% one_div_2_pow_eq
tff(fact_3746_bits__1__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Na: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa($o,A,zero_neq_one_of_bool(A),Na = zero_zero(nat)) ) ).

% bits_1_div_exp
tff(fact_3747_take__bit__of__exp,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: nat,Na: nat] : bit_se2584673776208193580ke_bit(A,M,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,ord_less(nat,Na),M))),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) ) ).

% take_bit_of_exp
tff(fact_3748_take__bit__of__2,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat] : bit_se2584673776208193580ke_bit(A,Na,aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,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_3749_take__bit__word__minus__Bit0__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num,M: num] : bit_se2584673776208193580ke_bit(word(A),aa(num,nat,numeral_numeral(nat),Na),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(M)))) = 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))),bit_se2584673776208193580ke_bit(word(A),pred_numeral(Na),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),M)))) ) ).

% take_bit_word_minus_Bit0_eq
tff(fact_3750_one__mod__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Na: nat] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),one_one(A)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)) ) ).

% one_mod_2_pow_eq
tff(fact_3751_take__bit__of__nat,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,M: nat] : bit_se2584673776208193580ke_bit(A,Na,aa(nat,A,semiring_1_of_nat(A),M)) = aa(nat,A,semiring_1_of_nat(A),bit_se2584673776208193580ke_bit(nat,Na,M)) ) ).

% take_bit_of_nat
tff(fact_3752_of__bool__eq__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P2: $o,Q3: $o] :
          ( ( aa($o,A,zero_neq_one_of_bool(A),(P2)) = aa($o,A,zero_neq_one_of_bool(A),(Q3)) )
        <=> ( (P2)
          <=> (Q3) ) ) ) ).

% of_bool_eq_iff
tff(fact_3753_unsigned__take__bit__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_se359711467146920520ations(A) )
     => ! [Na: nat,W: word(B)] : aa(word(B),A,semiring_1_unsigned(B,A),bit_se2584673776208193580ke_bit(word(B),Na,W)) = bit_se2584673776208193580ke_bit(A,Na,aa(word(B),A,semiring_1_unsigned(B,A),W)) ) ).

% unsigned_take_bit_eq
tff(fact_3754_of__nat__gbinomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Na: nat,K: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,gbinomial(nat,Na),K)) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Na)),K) ) ).

% of_nat_gbinomial
tff(fact_3755_of__bool__conj,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [P: $o,Q: $o] :
          aa($o,A,zero_neq_one_of_bool(A),
            ( (P)
            & (Q) )) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),(P))),aa($o,A,zero_neq_one_of_bool(A),(Q))) ) ).

% of_bool_conj
tff(fact_3756_exp__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,exp(real),X)),aa(real,real,exp(real),Y))
     => aa(real,$o,ord_less(real,X),Y) ) ).

% exp_less_cancel
tff(fact_3757_exp__not__eq__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,A,exp(A),X) != zero_zero(A) ) ).

% exp_not_eq_zero
tff(fact_3758_zero__less__eq__of__bool,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: $o] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa($o,A,zero_neq_one_of_bool(A),(P))) ) ).

% zero_less_eq_of_bool
tff(fact_3759_not__exp__less__zero,axiom,
    ! [X: real] : ~ aa(real,$o,ord_less(real,aa(real,real,exp(real),X)),zero_zero(real)) ).

% not_exp_less_zero
tff(fact_3760_exp__gt__zero,axiom,
    ! [X: real] : aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,exp(real),X)) ).

% exp_gt_zero
tff(fact_3761_exp__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
     => ? [X4: real] : aa(real,real,exp(real),X4) = Y ) ).

% exp_total
tff(fact_3762_not__exp__le__zero,axiom,
    ! [X: real] : ~ aa(real,$o,ord_less_eq(real,aa(real,real,exp(real),X)),zero_zero(real)) ).

% not_exp_le_zero
tff(fact_3763_exp__ge__zero,axiom,
    ! [X: real] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,exp(real),X)) ).

% exp_ge_zero
tff(fact_3764_split__of__bool__asm,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: fun(A,$o),P2: $o] :
          ( aa(A,$o,P,aa($o,A,zero_neq_one_of_bool(A),(P2)))
        <=> ~ ( ( (P2)
                & ~ aa(A,$o,P,one_one(A)) )
              | ( ~ (P2)
                & ~ aa(A,$o,P,zero_zero(A)) ) ) ) ) ).

% split_of_bool_asm
tff(fact_3765_split__of__bool,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: fun(A,$o),P2: $o] :
          ( aa(A,$o,P,aa($o,A,zero_neq_one_of_bool(A),(P2)))
        <=> ( ( (P2)
             => aa(A,$o,P,one_one(A)) )
            & ( ~ (P2)
             => aa(A,$o,P,zero_zero(A)) ) ) ) ) ).

% split_of_bool
tff(fact_3766_of__bool__def,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P2: $o] :
          aa($o,A,zero_neq_one_of_bool(A),(P2)) = $ite((P2),one_one(A),zero_zero(A)) ) ).

% of_bool_def
tff(fact_3767_of__bool__less__eq__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: $o] : aa(A,$o,ord_less_eq(A,aa($o,A,zero_neq_one_of_bool(A),(P))),one_one(A)) ) ).

% of_bool_less_eq_one
tff(fact_3768_binomial__gbinomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Na: nat,K: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Na),K)) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Na)),K) ) ).

% binomial_gbinomial
tff(fact_3769_exp__gt__one,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => aa(real,$o,ord_less(real,one_one(real)),aa(real,real,exp(real),X)) ) ).

% exp_gt_one
tff(fact_3770_exp__ge__add__one__self,axiom,
    ! [X: real] : aa(real,$o,ord_less_eq(real,aa(real,real,plus_plus(real,one_one(real)),X)),aa(real,real,exp(real),X)) ).

% exp_ge_add_one_self
tff(fact_3771_gbinomial__Suc__Suc,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,plus_plus(A,A3),one_one(A))),aa(nat,nat,suc,K)) = aa(A,A,plus_plus(A,aa(nat,A,gbinomial(A,A3),K)),aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K))) ) ).

% gbinomial_Suc_Suc
tff(fact_3772_exp__minus__inverse,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,exp(A),X)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),X))) = one_one(A) ) ).

% exp_minus_inverse
tff(fact_3773_exp__of__nat2__mult,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Na: nat] : aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(nat,A,semiring_1_of_nat(A),Na))) = aa(nat,A,power_power(A,aa(A,A,exp(A),X)),Na) ) ).

% exp_of_nat2_mult
tff(fact_3774_exp__of__nat__mult,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Na: nat,X: A] : aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),X)) = aa(nat,A,power_power(A,aa(A,A,exp(A),X)),Na) ) ).

% exp_of_nat_mult
tff(fact_3775_gbinomial__of__nat__symmetric,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,ord_less_eq(nat,K),Na)
         => ( aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Na)),K) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Na)),aa(nat,nat,minus_minus(nat,Na),K)) ) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_3776_log__ln,axiom,
    ! [X: real] : aa(real,real,ln_ln(real),X) = aa(real,real,log(aa(real,real,exp(real),one_one(real))),X) ).

% log_ln
tff(fact_3777_exp__ge__add__one__self__aux,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => aa(real,$o,ord_less_eq(real,aa(real,real,plus_plus(real,one_one(real)),X)),aa(real,real,exp(real),X)) ) ).

% exp_ge_add_one_self_aux
tff(fact_3778_lemma__exp__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,one_one(real)),Y)
     => ? [X4: real] :
          ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X4)
          & aa(real,$o,ord_less_eq(real,X4),aa(real,real,minus_minus(real,Y),one_one(real)))
          & ( aa(real,real,exp(real),X4) = Y ) ) ) ).

% lemma_exp_total
tff(fact_3779_ln__ge__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less_eq(real,Y),aa(real,real,ln_ln(real),X))
      <=> aa(real,$o,ord_less_eq(real,aa(real,real,exp(real),Y)),X) ) ) ).

% ln_ge_iff
tff(fact_3780_gbinomial__addition__formula,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K)) = aa(A,A,plus_plus(A,aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A3),one_one(A))),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A3),one_one(A))),K)) ) ).

% gbinomial_addition_formula
tff(fact_3781_gbinomial__absorb__comp,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,A3),aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,A,gbinomial(A,A3),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A3),one_one(A))),K)) ) ).

% gbinomial_absorb_comp
tff(fact_3782_gbinomial__mult__1,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,gbinomial(A,A3),K)) = aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),K)),aa(nat,A,gbinomial(A,A3),K))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K)))) ) ).

% gbinomial_mult_1
tff(fact_3783_gbinomial__mult__1_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),K)),A3) = aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),K)),aa(nat,A,gbinomial(A,A3),K))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K)))) ) ).

% gbinomial_mult_1'
tff(fact_3784_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K: nat,A3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,semiring_1_of_nat(A),K)),A3)
         => aa(A,$o,ord_less_eq(A,aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,semiring_1_of_nat(A),K))),K)),aa(nat,A,gbinomial(A,A3),K)) ) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_3785_ln__x__over__x__mono,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,exp(real),one_one(real))),X)
     => ( aa(real,$o,ord_less_eq(real,X),Y)
       => aa(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),X)),X)) ) ) ).

% ln_x_over_x_mono
tff(fact_3786_powr__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [X: A,A3: A] :
          powr(A,X,A3) = $ite(X = zero_zero(A),zero_zero(A),aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,ln_ln(A),X)))) ) ).

% powr_def
tff(fact_3787_take__bit__Suc__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat,K: num] : 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),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_3788_take__bit__eq__mod,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A3: A] : bit_se2584673776208193580ke_bit(A,Na,A3) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) ) ).

% take_bit_eq_mod
tff(fact_3789_exp__le,axiom,
    aa(real,$o,ord_less_eq(real,aa(real,real,exp(real),one_one(real))),aa(num,real,numeral_numeral(real),bit1(one2))) ).

% exp_le
tff(fact_3790_of__bool__odd__eq__mod__2,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] : aa($o,A,zero_neq_one_of_bool(A),~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% of_bool_odd_eq_mod_2
tff(fact_3791_Suc__times__gbinomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,aa(A,A,plus_plus(A,A3),one_one(A))),aa(nat,nat,suc,K))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,plus_plus(A,A3),one_one(A))),aa(nat,A,gbinomial(A,A3),K)) ) ).

% Suc_times_gbinomial
tff(fact_3792_gbinomial__absorption,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K))) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A3),one_one(A))),K)) ) ).

% gbinomial_absorption
tff(fact_3793_gbinomial__trinomial__revision,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,M: nat,A3: A] :
          ( aa(nat,$o,ord_less_eq(nat,K),M)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),M)),aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),M)),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),K)),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A3),aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,nat,minus_minus(nat,M),K))) ) ) ) ).

% gbinomial_trinomial_revision
tff(fact_3794_exp__divide__power__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Na: nat,X: A] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
         => ( aa(nat,A,power_power(A,aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),aa(nat,A,semiring_1_of_nat(A),Na)))),Na) = aa(A,A,exp(A),X) ) ) ) ).

% exp_divide_power_eq
tff(fact_3795_take__bit__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A3: A] :
          ( ( bit_se2584673776208193580ke_bit(A,Na,A3) = zero_zero(A) )
        <=> aa(A,$o,dvd_dvd(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)),A3) ) ) ).

% take_bit_eq_0_iff
tff(fact_3796_bin__last__bintrunc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [L: nat,Na: A] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2584673776208193580ke_bit(A,L,Na))
        <=> ( aa(nat,$o,ord_less(nat,zero_zero(nat)),L)
            & ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na) ) ) ) ).

% bin_last_bintrunc
tff(fact_3797_take__bit__numeral__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: num] : bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),L),aa(num,A,numeral_numeral(A),bit0(K))) = aa(A,A,aa(A,fun(A,A),times_times(A),bit_se2584673776208193580ke_bit(A,pred_numeral(L),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% take_bit_numeral_bit0
tff(fact_3798_exp__half__le2,axiom,
    aa(real,$o,ord_less_eq(real,aa(real,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_3799_bits__induct,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [P: fun(A,$o),A3: A] :
          ( ! [A4: A] :
              ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A4),aa(num,A,numeral_numeral(A),bit0(one2))) = A4 )
             => aa(A,$o,P,A4) )
         => ( ! [A4: A,B5: $o] :
                ( aa(A,$o,P,A4)
               => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,plus_plus(A,aa($o,A,zero_neq_one_of_bool(A),(B5))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A4))),aa(num,A,numeral_numeral(A),bit0(one2))) = A4 )
                 => aa(A,$o,P,aa(A,A,plus_plus(A,aa($o,A,zero_neq_one_of_bool(A),(B5))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A4))) ) )
           => aa(A,$o,P,A3) ) ) ) ).

% bits_induct
tff(fact_3800_gbinomial__rec,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,plus_plus(A,A3),one_one(A))),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),K)),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,plus_plus(A,A3),one_one(A))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K)))) ) ).

% gbinomial_rec
tff(fact_3801_gbinomial__factors,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,plus_plus(A,A3),one_one(A))),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,plus_plus(A,A3),one_one(A))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K)))),aa(nat,A,gbinomial(A,A3),K)) ) ).

% gbinomial_factors
tff(fact_3802_gbinomial__negated__upper,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),K)),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,aa(A,A,minus_minus(A,aa(nat,A,semiring_1_of_nat(A),K)),A3)),one_one(A))),K)) ) ).

% gbinomial_negated_upper
tff(fact_3803_gbinomial__index__swap,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Na: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),K)),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Na))),one_one(A))),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),Na)),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A))),Na)) ) ).

% gbinomial_index_swap
tff(fact_3804_exp__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A] : aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Z)) = aa(nat,A,power_power(A,aa(A,A,exp(A),Z)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% exp_double
tff(fact_3805_exp__mod__exp,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,Na: nat] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,ord_less(nat,M),Na))),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M)) ) ).

% exp_mod_exp
tff(fact_3806_gbinomial__minus,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,uminus_uminus(A),A3)),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),K)),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,aa(A,A,plus_plus(A,A3),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A))),K)) ) ).

% gbinomial_minus
tff(fact_3807_gbinomial__reduce__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A3: A] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
         => ( aa(nat,A,gbinomial(A,A3),K) = aa(A,A,plus_plus(A,aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A3),one_one(A))),aa(nat,nat,minus_minus(nat,K),one_one(nat)))),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A3),one_one(A))),K)) ) ) ) ).

% gbinomial_reduce_nat
tff(fact_3808_gbinomial__pochhammer,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),K) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),K)),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),A3),K))),semiring_char_0_fact(A,K)) ) ).

% gbinomial_pochhammer
tff(fact_3809_gbinomial__pochhammer_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),K) = aa(A,A,aa(A,fun(A,A),divide_divide(A),comm_s3205402744901411588hammer(A,aa(A,A,plus_plus(A,aa(A,A,minus_minus(A,A3),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K)),semiring_char_0_fact(A,K)) ) ).

% gbinomial_pochhammer'
tff(fact_3810_exp__bound__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A] :
          ( aa(real,$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,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,exp(A),Z))),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% exp_bound_half
tff(fact_3811_take__bit__Suc__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat,K: num] : bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Na),aa(num,A,numeral_numeral(A),bit1(K))) = aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(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_3812_take__bit__Suc__minus__1__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat] : bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Na),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,minus_minus(A,aa(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_3813_take__bit__numeral__minus__1__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: num] : bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),K),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(num,nat,numeral_numeral(nat),K))),one_one(A)) ) ).

% take_bit_numeral_minus_1_eq
tff(fact_3814_take__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A3: A] : bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Na),A3) = aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),bit_se2584673776208193580ke_bit(A,Na,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% take_bit_Suc
tff(fact_3815_exp__bound,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less_eq(real,X),one_one(real))
       => aa(real,$o,ord_less_eq(real,aa(real,real,exp(real),X)),aa(real,real,plus_plus(real,aa(real,real,plus_plus(real,one_one(real)),X)),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% exp_bound
tff(fact_3816_stable__imp__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,Na: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) = A3 )
         => ( bit_se2584673776208193580ke_bit(A,Na,A3) = $ite(aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3),zero_zero(A),aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)),one_one(A))) ) ) ) ).

% stable_imp_take_bit_eq
tff(fact_3817_take__bit__numeral__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: num] : bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),L),aa(num,A,numeral_numeral(A),bit1(K))) = aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),bit_se2584673776208193580ke_bit(A,pred_numeral(L),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),bit0(one2)))),one_one(A)) ) ).

% take_bit_numeral_bit1
tff(fact_3818_real__exp__bound__lemma,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less_eq(real,X),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,ord_less_eq(real,aa(real,real,exp(real),X)),aa(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))),X))) ) ) ).

% real_exp_bound_lemma
tff(fact_3819_exp__div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Na: nat] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              aa($o,A,zero_neq_one_of_bool(A),
                ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M) != zero_zero(A) )
                & aa(nat,$o,ord_less_eq(nat,Na),M) ))),
            aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,minus_minus(nat,M),Na))) ) ).

% exp_div_exp_eq
tff(fact_3820_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [Na: nat,X: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(nat,real,semiring_1_of_nat(real),Na))),X)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
       => aa(real,$o,ord_less_eq(real,aa(nat,real,power_power(real,aa(real,real,plus_plus(real,one_one(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),X),aa(nat,real,semiring_1_of_nat(real),Na)))),Na)),aa(real,real,exp(real),X)) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_3821_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X: real,Na: nat] :
      ( aa(real,$o,ord_less_eq(real,X),aa(nat,real,semiring_1_of_nat(real),Na))
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
       => aa(real,$o,ord_less_eq(real,aa(nat,real,power_power(real,aa(real,real,minus_minus(real,one_one(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),X),aa(nat,real,semiring_1_of_nat(real),Na)))),Na)),aa(real,real,exp(real),aa(real,real,uminus_uminus(real),X))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_3822_gbinomial__absorption_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A3: A] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
         => ( aa(nat,A,gbinomial(A,A3),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A3),one_one(A))),aa(nat,nat,minus_minus(nat,K),one_one(nat)))) ) ) ) ).

% gbinomial_absorption'
tff(fact_3823_exp__bound__lemma,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A] :
          ( aa(real,$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,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,exp(A),Z))),aa(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_3824_exp__lower__Taylor__quadratic,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => aa(real,$o,ord_less_eq(real,aa(real,real,plus_plus(real,aa(real,real,plus_plus(real,one_one(real)),X)),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,exp(real),X)) ) ).

% exp_lower_Taylor_quadratic
tff(fact_3825_log__base__10__eq2,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ( aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(bit1(bit0(one2))))),X) = 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))))),aa(real,real,exp(real),one_one(real)))),aa(real,real,ln_ln(real),X)) ) ) ).

% log_base_10_eq2
tff(fact_3826_tanh__real__altdef,axiom,
    ! [X: real] : aa(real,real,tanh(real),X) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,one_one(real)),aa(real,real,exp(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,uminus_uminus(real),aa(num,real,numeral_numeral(real),bit0(one2)))),X)))),aa(real,real,plus_plus(real,one_one(real)),aa(real,real,exp(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,uminus_uminus(real),aa(num,real,numeral_numeral(real),bit0(one2)))),X)))) ).

% tanh_real_altdef
tff(fact_3827_take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A3: A] :
          bit_se2584673776208193580ke_bit(A,Na,A3) = $ite(Na = zero_zero(nat),zero_zero(A),aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),bit_se2584673776208193580ke_bit(A,aa(nat,nat,minus_minus(nat,Na),one_one(nat)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% take_bit_rec
tff(fact_3828_take__bit__word__minus__Bit1__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num,M: num] : bit_se2584673776208193580ke_bit(word(A),aa(num,nat,numeral_numeral(nat),Na),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit1(M)))) = aa(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))),bit_se2584673776208193580ke_bit(word(A),pred_numeral(Na),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),inc(M)))))) ) ).

% take_bit_word_minus_Bit1_eq
tff(fact_3829_and__int_Oelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Xa2) = Y )
     => ( Y = $ite(
            ( member(int,X,aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
            & member(int,Xa2,aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
            aa(int,int,uminus_uminus(int),
              aa($o,int,zero_neq_one_of_bool(int),
                ( ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),X)
                & ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Xa2) ))),
            aa(int,int,
              plus_plus(int,
                aa($o,int,zero_neq_one_of_bool(int),
                  ( ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),X)
                  & ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Xa2) ))),
              aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),X),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa2),aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ).

% and_int.elims
tff(fact_3830_and__int_Osimps,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = $ite(
        ( member(int,K,aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
        & member(int,L,aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
        aa(int,int,uminus_uminus(int),
          aa($o,int,zero_neq_one_of_bool(int),
            ( ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K)
            & ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),L) ))),
        aa(int,int,
          plus_plus(int,
            aa($o,int,zero_neq_one_of_bool(int),
              ( ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K)
              & ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),L) ))),
          aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ).

% and_int.simps
tff(fact_3831_arctan__def,axiom,
    ! [Y: real] : aa(real,real,arctan,Y) = the(real,aTP_Lamp_cz(real,fun(real,$o),Y)) ).

% arctan_def
tff(fact_3832_AND__twice,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [W: A,M: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),W),M)),M) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),W),M) ) ).

% AND_twice
tff(fact_3833_and__zero__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),zero_zero(A)) = zero_zero(A) ) ).

% and_zero_eq
tff(fact_3834_zero__and__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),zero_zero(A)),A3) = zero_zero(A) ) ).

% zero_and_eq
tff(fact_3835_bit_Oconj__zero__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),zero_zero(A)),X) = zero_zero(A) ) ).

% bit.conj_zero_left
tff(fact_3836_bit_Oconj__zero__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),zero_zero(A)) = zero_zero(A) ) ).

% bit.conj_zero_right
tff(fact_3837_and_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,uminus_uminus(A),one_one(A))),A3) = A3 ) ).

% and.left_neutral
tff(fact_3838_and_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),aa(A,A,uminus_uminus(A),one_one(A))) = A3 ) ).

% and.right_neutral
tff(fact_3839_bit_Oconj__one__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),aa(A,A,uminus_uminus(A),one_one(A))) = X ) ).

% bit.conj_one_right
tff(fact_3840_of__nat__nat__take__bit__eq,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Na: nat,K: int] : aa(nat,A,semiring_1_of_nat(A),nat2(bit_se2584673776208193580ke_bit(int,Na,K))) = aa(int,A,ring_1_of_int(A),bit_se2584673776208193580ke_bit(int,Na,K)) ) ).

% of_nat_nat_take_bit_eq
tff(fact_3841_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(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,ord_less_eq(int,zero_zero(int)),K)
        | aa(int,$o,ord_less_eq(int,zero_zero(int)),L) ) ) ).

% and_nonnegative_int_iff
tff(fact_3842_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(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,ord_less(int,K),zero_zero(int))
        & aa(int,$o,ord_less(int,L),zero_zero(int)) ) ) ).

% and_negative_int_iff
tff(fact_3843_pred__numeral__inc,axiom,
    ! [K: num] : pred_numeral(inc(K)) = aa(num,nat,numeral_numeral(nat),K) ).

% pred_numeral_inc
tff(fact_3844_and__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),bit1(X))),one_one(A)) = one_one(A) ) ).

% and_numerals(8)
tff(fact_3845_and__numerals_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),one_one(A)),aa(num,A,numeral_numeral(A),bit1(Y))) = one_one(A) ) ).

% and_numerals(2)
tff(fact_3846_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_3847_and__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),bit0(X))),one_one(A)) = zero_zero(A) ) ).

% and_numerals(5)
tff(fact_3848_and__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),bit0(X))),aa(num,A,numeral_numeral(A),bit0(Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y))) ) ).

% and_numerals(3)
tff(fact_3849_add__neg__numeral__special_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Na: num] : aa(A,A,plus_plus(A,aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(Na))) ) ).

% add_neg_numeral_special(5)
tff(fact_3850_add__neg__numeral__special_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,plus_plus(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(M))) ) ).

% add_neg_numeral_special(6)
tff(fact_3851_diff__numeral__special_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Na: num] : aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),Na)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(Na))) ) ).

% diff_numeral_special(5)
tff(fact_3852_diff__numeral__special_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,minus_minus(A,aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(num,A,numeral_numeral(A),inc(M)) ) ).

% diff_numeral_special(6)
tff(fact_3853_and__minus__numerals_I2_J,axiom,
    ! [Na: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit1(Na)))) = one_one(int) ).

% and_minus_numerals(2)
tff(fact_3854_and__minus__numerals_I6_J,axiom,
    ! [Na: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit1(Na)))),one_one(int)) = one_one(int) ).

% and_minus_numerals(6)
tff(fact_3855_take__bit__of__Suc__0,axiom,
    ! [Na: 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,ord_less(nat,zero_zero(nat)),Na)) ).

% take_bit_of_Suc_0
tff(fact_3856_and__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),bit0(X))),aa(num,A,numeral_numeral(A),bit1(Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y))) ) ).

% and_numerals(4)
tff(fact_3857_and__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),bit1(X))),aa(num,A,numeral_numeral(A),bit0(Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y))) ) ).

% and_numerals(6)
tff(fact_3858_and__minus__numerals_I5_J,axiom,
    ! [Na: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(Na)))),one_one(int)) = zero_zero(int) ).

% and_minus_numerals(5)
tff(fact_3859_and__minus__numerals_I1_J,axiom,
    ! [Na: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(Na)))) = zero_zero(int) ).

% and_minus_numerals(1)
tff(fact_3860_and__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),bit1(X))),aa(num,A,numeral_numeral(A),bit1(Y))) = aa(A,A,plus_plus(A,one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% and_numerals(7)
tff(fact_3861_signed__and__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_ri3973907225187159222ations(A) )
     => ! [V2: word(B),W: word(B)] : aa(word(B),A,ring_1_signed(B,A),aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),bit_se5824344872417868541ns_and(word(B)),V2),W)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(word(B),A,ring_1_signed(B,A),V2)),aa(word(B),A,ring_1_signed(B,A),W)) ) ).

% signed_and_eq
tff(fact_3862_num__induct,axiom,
    ! [P: fun(num,$o),X: num] :
      ( aa(num,$o,P,one2)
     => ( ! [X4: num] :
            ( aa(num,$o,P,X4)
           => aa(num,$o,P,inc(X4)) )
       => aa(num,$o,P,X) ) ) ).

% num_induct
tff(fact_3863_unsigned__and__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_se359711467146920520ations(A) )
     => ! [V2: word(B),W: word(B)] : aa(word(B),A,semiring_1_unsigned(B,A),aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),bit_se5824344872417868541ns_and(word(B)),V2),W)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(word(B),A,semiring_1_unsigned(B,A),V2)),aa(word(B),A,semiring_1_unsigned(B,A),W)) ) ).

% unsigned_and_eq
tff(fact_3864_of__nat__and__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),Na)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Na)) ) ).

% of_nat_and_eq
tff(fact_3865_nat__take__bit__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
     => ( nat2(bit_se2584673776208193580ke_bit(int,Na,K)) = bit_se2584673776208193580ke_bit(nat,Na,nat2(K)) ) ) ).

% nat_take_bit_eq
tff(fact_3866_take__bit__nat__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
     => ( bit_se2584673776208193580ke_bit(nat,Na,nat2(K)) = nat2(bit_se2584673776208193580ke_bit(int,Na,K)) ) ) ).

% take_bit_nat_eq
tff(fact_3867_add__inc,axiom,
    ! [X: num,Y: num] : aa(num,num,plus_plus(num,X),inc(Y)) = inc(aa(num,num,plus_plus(num,X),Y)) ).

% add_inc
tff(fact_3868_and__eq__minus__1__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),B3) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> ( ( A3 = aa(A,A,uminus_uminus(A),one_one(A)) )
            & ( B3 = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).

% and_eq_minus_1_iff
tff(fact_3869_AND__lower,axiom,
    ! [X: int,Y: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),X)
     => aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y)) ) ).

% AND_lower
tff(fact_3870_AND__upper1,axiom,
    ! [X: int,Y: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),X)
     => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y)),X) ) ).

% AND_upper1
tff(fact_3871_AND__upper2,axiom,
    ! [Y: int,X: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Y)
     => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y)),Y) ) ).

% AND_upper2
tff(fact_3872_AND__upper1_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Y)
     => ( aa(int,$o,ord_less_eq(int,Y),Z)
       => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Y),Ya)),Z) ) ) ).

% AND_upper1'
tff(fact_3873_AND__upper2_H,axiom,
    ! [Y: int,Z: int,X: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Y)
     => ( aa(int,$o,ord_less_eq(int,Y),Z)
       => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y)),Z) ) ) ).

% AND_upper2'
tff(fact_3874_take__bit__nonnegative,axiom,
    ! [Na: nat,K: int] : aa(int,$o,ord_less_eq(int,zero_zero(int)),bit_se2584673776208193580ke_bit(int,Na,K)) ).

% take_bit_nonnegative
tff(fact_3875_take__bit__int__less__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,ord_less_eq(int,bit_se2584673776208193580ke_bit(int,Na,K)),K)
    <=> aa(int,$o,ord_less_eq(int,zero_zero(int)),K) ) ).

% take_bit_int_less_eq_self_iff
tff(fact_3876_not__take__bit__negative,axiom,
    ! [Na: nat,K: int] : ~ aa(int,$o,ord_less(int,bit_se2584673776208193580ke_bit(int,Na,K)),zero_zero(int)) ).

% not_take_bit_negative
tff(fact_3877_take__bit__int__greater__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,ord_less(int,K),bit_se2584673776208193580ke_bit(int,Na,K))
    <=> aa(int,$o,ord_less(int,K),zero_zero(int)) ) ).

% take_bit_int_greater_self_iff
tff(fact_3878_uint__take__bit__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),bit_se2584673776208193580ke_bit(word(A),Na,W)) = bit_se2584673776208193580ke_bit(int,Na,aa(word(A),int,semiring_1_unsigned(A,int),W)) ) ).

% uint_take_bit_eq
tff(fact_3879_inc_Osimps_I1_J,axiom,
    inc(one2) = bit0(one2) ).

% inc.simps(1)
tff(fact_3880_inc_Osimps_I2_J,axiom,
    ! [X: num] : inc(bit0(X)) = bit1(X) ).

% inc.simps(2)
tff(fact_3881_inc_Osimps_I3_J,axiom,
    ! [X: num] : inc(bit1(X)) = bit0(inc(X)) ).

% inc.simps(3)
tff(fact_3882_add__One,axiom,
    ! [X: num] : aa(num,num,plus_plus(num,X),one2) = inc(X) ).

% add_One
tff(fact_3883_ln__real__def,axiom,
    ! [X: real] : aa(real,real,ln_ln(real),X) = the(real,aTP_Lamp_da(real,fun(real,$o),X)) ).

% ln_real_def
tff(fact_3884_suminf__def,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A)] : suminf(A,F2) = the(A,sums(A,F2)) ) ).

% suminf_def
tff(fact_3885_mult__inc,axiom,
    ! [X: num,Y: num] : aa(num,num,aa(num,fun(num,num),times_times(num),X),inc(Y)) = aa(num,num,plus_plus(num,aa(num,num,aa(num,fun(num,num),times_times(num),X),Y)),X) ).

% mult_inc
tff(fact_3886_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,X: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Y)
     => ( aa(int,$o,ord_less(int,Y),Z)
       => aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y)),Z) ) ) ).

% AND_upper2''
tff(fact_3887_AND__upper1_H_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Y)
     => ( aa(int,$o,ord_less(int,Y),Z)
       => aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Y),Ya)),Z) ) ) ).

% AND_upper1''
tff(fact_3888_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,ord_less(int,L),zero_zero(int))
     => aa(int,$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_3889_take__bit__decr__eq,axiom,
    ! [Na: nat,K: int] :
      ( ( bit_se2584673776208193580ke_bit(int,Na,K) != zero_zero(int) )
     => ( bit_se2584673776208193580ke_bit(int,Na,aa(int,int,minus_minus(int,K),one_one(int))) = aa(int,int,minus_minus(int,bit_se2584673776208193580ke_bit(int,Na,K)),one_one(int)) ) ) ).

% take_bit_decr_eq
tff(fact_3890_numeral__inc,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [X: num] : aa(num,A,numeral_numeral(A),inc(X)) = aa(A,A,plus_plus(A,aa(num,A,numeral_numeral(A),X)),one_one(A)) ) ).

% numeral_inc
tff(fact_3891_even__and__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),B3))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
            | aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),B3) ) ) ) ).

% even_and_iff
tff(fact_3892_ln__neg__is__const,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,X),zero_zero(real))
     => ( aa(real,real,ln_ln(real),X) = the(real,aTP_Lamp_db(real,$o)) ) ) ).

% ln_neg_is_const
tff(fact_3893_even__and__iff__int,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L))
    <=> ( aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K)
        | aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),L) ) ) ).

% even_and_iff_int
tff(fact_3894_the__elem__def,axiom,
    ! [A: $tType,X5: set(A)] : the_elem(A,X5) = the(A,aTP_Lamp_dc(set(A),fun(A,$o),X5)) ).

% the_elem_def
tff(fact_3895_one__and__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),one_one(A)),A3) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% one_and_eq
tff(fact_3896_and__one__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),one_one(A)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% and_one_eq
tff(fact_3897_take__bit__nat__eq__self__iff,axiom,
    ! [Na: nat,M: nat] :
      ( ( bit_se2584673776208193580ke_bit(nat,Na,M) = M )
    <=> aa(nat,$o,ord_less(nat,M),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ) ).

% take_bit_nat_eq_self_iff
tff(fact_3898_take__bit__nat__less__exp,axiom,
    ! [Na: nat,M: nat] : aa(nat,$o,ord_less(nat,bit_se2584673776208193580ke_bit(nat,Na,M)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ).

% take_bit_nat_less_exp
tff(fact_3899_take__bit__nat__eq__self,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less(nat,M),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
     => ( bit_se2584673776208193580ke_bit(nat,Na,M) = M ) ) ).

% take_bit_nat_eq_self
tff(fact_3900_take__bit__nat__def,axiom,
    ! [Na: nat,M: nat] : bit_se2584673776208193580ke_bit(nat,Na,M) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ).

% take_bit_nat_def
tff(fact_3901_take__bit__Suc__minus__bit1,axiom,
    ! [Na: nat,K: num] : 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,plus_plus(int,aa(int,int,aa(int,fun(int,int),times_times(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_3902_take__bit__int__less__exp,axiom,
    ! [Na: nat,K: int] : aa(int,$o,ord_less(int,bit_se2584673776208193580ke_bit(int,Na,K)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ).

% take_bit_int_less_exp
tff(fact_3903_take__bit__int__def,axiom,
    ! [Na: nat,K: int] : bit_se2584673776208193580ke_bit(int,Na,K) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ).

% take_bit_int_def
tff(fact_3904_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),L),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit1(K)))) = aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),times_times(int),bit_se2584673776208193580ke_bit(int,pred_numeral(L),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K))))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).

% take_bit_numeral_minus_bit1
tff(fact_3905_take__bit__nat__less__self__iff,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,bit_se2584673776208193580ke_bit(nat,Na,M)),M)
    <=> aa(nat,$o,ord_less_eq(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),M) ) ).

% take_bit_nat_less_self_iff
tff(fact_3906_arccos__def,axiom,
    ! [Y: real] : aa(real,real,arccos,Y) = the(real,aTP_Lamp_dd(real,fun(real,$o),Y)) ).

% arccos_def
tff(fact_3907_take__bit__Suc__minus__bit0,axiom,
    ! [Na: nat,K: num] : 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),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_3908_take__bit__int__less__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,ord_less(int,bit_se2584673776208193580ke_bit(int,Na,K)),K)
    <=> aa(int,$o,ord_less_eq(int,aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)),K) ) ).

% take_bit_int_less_self_iff
tff(fact_3909_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,ord_less_eq(int,K),bit_se2584673776208193580ke_bit(int,Na,K))
    <=> aa(int,$o,ord_less(int,K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_3910_and__int__rec,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = aa(int,int,
        plus_plus(int,
          aa($o,int,zero_neq_one_of_bool(int),
            ( ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K)
            & ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),L) ))),
        aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2)))))) ).

% and_int_rec
tff(fact_3911_take__bit__int__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( ( bit_se2584673776208193580ke_bit(int,Na,K) = K )
    <=> ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
        & aa(int,$o,ord_less(int,K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_3912_take__bit__int__eq__self,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
     => ( aa(int,$o,ord_less(int,K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))
       => ( bit_se2584673776208193580ke_bit(int,Na,K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_3913_take__bit__incr__eq,axiom,
    ! [Na: nat,K: int] :
      ( ( bit_se2584673776208193580ke_bit(int,Na,K) != aa(int,int,minus_minus(int,aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)),one_one(int)) )
     => ( bit_se2584673776208193580ke_bit(int,Na,aa(int,int,plus_plus(int,K),one_one(int))) = aa(int,int,plus_plus(int,one_one(int)),bit_se2584673776208193580ke_bit(int,Na,K)) ) ) ).

% take_bit_incr_eq
tff(fact_3914_take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),L),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),bit_se2584673776208193580ke_bit(int,pred_numeral(L),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% take_bit_numeral_minus_bit0
tff(fact_3915_take__bit__int__less__eq,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,ord_less_eq(int,aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)),K)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
       => aa(int,$o,ord_less_eq(int,bit_se2584673776208193580ke_bit(int,Na,K)),aa(int,int,minus_minus(int,K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))) ) ) ).

% take_bit_int_less_eq
tff(fact_3916_take__bit__int__greater__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,ord_less(int,K),zero_zero(int))
     => aa(int,$o,ord_less_eq(int,aa(int,int,plus_plus(int,K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))),bit_se2584673776208193580ke_bit(int,Na,K)) ) ).

% take_bit_int_greater_eq
tff(fact_3917_signed__take__bit__eq__take__bit__shift,axiom,
    ! [Na: nat,K: int] : bit_ri4674362597316999326ke_bit(int,Na,K) = aa(int,int,minus_minus(int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,Na),aa(int,int,plus_plus(int,K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_3918_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_de(real,$o)) ).

% pi_half
tff(fact_3919_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_de(real,$o))) ).

% pi_def
tff(fact_3920_take__bit__minus__small__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),K)
     => ( aa(int,$o,ord_less_eq(int,K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))
       => ( bit_se2584673776208193580ke_bit(int,Na,aa(int,int,uminus_uminus(int),K)) = aa(int,int,minus_minus(int,aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_3921_and__int__unfold,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = $ite(
        ( ( K = zero_zero(int) )
        | ( L = zero_zero(int) ) ),
        zero_zero(int),
        $ite(
          K = aa(int,int,uminus_uminus(int),one_one(int)),
          L,
          $ite(L = aa(int,int,uminus_uminus(int),one_one(int)),K,aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),times_times(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),modulo_modulo(int),L),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))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2))))))) ) ) ).

% and_int_unfold
tff(fact_3922_arcsin__def,axiom,
    ! [Y: real] : aa(real,real,arcsin,Y) = the(real,aTP_Lamp_df(real,fun(real,$o),Y)) ).

% arcsin_def
tff(fact_3923_the__equality,axiom,
    ! [A: $tType,P: fun(A,$o),A3: A] :
      ( aa(A,$o,P,A3)
     => ( ! [X4: A] :
            ( aa(A,$o,P,X4)
           => ( X4 = A3 ) )
       => ( the(A,P) = A3 ) ) ) ).

% the_equality
tff(fact_3924_the__eq__trivial,axiom,
    ! [A: $tType,A3: A] : the(A,aTP_Lamp_ad(A,fun(A,$o),A3)) = A3 ).

% the_eq_trivial
tff(fact_3925_the__sym__eq__trivial,axiom,
    ! [A: $tType,X: A] : the(A,aa(A,fun(A,$o),fequal(A),X)) = X ).

% the_sym_eq_trivial
tff(fact_3926_modulo__int__def,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L) = $ite(
        L = zero_zero(int),
        K,
        $ite(aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),nat2(aa(int,int,abs_abs(int),K))),nat2(aa(int,int,abs_abs(int),L))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),L)),aa($o,int,zero_neq_one_of_bool(int),~ aa(int,$o,dvd_dvd(int,L),K)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),nat2(aa(int,int,abs_abs(int),K))),nat2(aa(int,int,abs_abs(int),L))))))) ) ).

% modulo_int_def
tff(fact_3927_sgn__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : aa(A,A,sgn_sgn(A),aa(A,A,sgn_sgn(A),A3)) = aa(A,A,sgn_sgn(A),A3) ) ).

% sgn_sgn
tff(fact_3928_sgn__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ( aa(A,A,sgn_sgn(A),zero_zero(A)) = zero_zero(A) ) ) ).

% sgn_zero
tff(fact_3929_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_3930_sgn__one,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ( aa(A,A,sgn_sgn(A),one_one(A)) = one_one(A) ) ) ).

% sgn_one
tff(fact_3931_sgn__1,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( aa(A,A,sgn_sgn(A),one_one(A)) = one_one(A) ) ) ).

% sgn_1
tff(fact_3932_power__sgn,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,Na: nat] : aa(A,A,sgn_sgn(A),aa(nat,A,power_power(A,A3),Na)) = aa(nat,A,power_power(A,aa(A,A,sgn_sgn(A),A3)),Na) ) ).

% power_sgn
tff(fact_3933_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : aa(A,A,sgn_sgn(A),aa(A,A,uminus_uminus(A),A3)) = aa(A,A,uminus_uminus(A),aa(A,A,sgn_sgn(A),A3)) ) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_3934_sgn__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,sgn_sgn(A),A3)),zero_zero(A))
        <=> aa(A,$o,ord_less(A,A3),zero_zero(A)) ) ) ).

% sgn_less
tff(fact_3935_sgn__greater,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,sgn_sgn(A),A3))
        <=> aa(A,$o,ord_less(A,zero_zero(A)),A3) ) ) ).

% sgn_greater
tff(fact_3936_word__and__max,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) = X ) ).

% word_and_max
tff(fact_3937_word__bitwise__m1__simps_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),X) = X ) ).

% word_bitwise_m1_simps(2)
tff(fact_3938_sgn__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => ( aa(A,A,sgn_sgn(A),A3) = one_one(A) ) ) ) ).

% sgn_pos
tff(fact_3939_abs__sgn__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A3)) = one_one(A) ) ) ) ).

% abs_sgn_eq_1
tff(fact_3940_sgn__mult__self__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),A3)),aa(A,A,sgn_sgn(A),A3)) = aa($o,A,zero_neq_one_of_bool(A),A3 != zero_zero(A)) ) ).

% sgn_mult_self_eq
tff(fact_3941_sgn__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A3)) = aa($o,A,zero_neq_one_of_bool(A),A3 != zero_zero(A)) ) ).

% sgn_abs
tff(fact_3942_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : aa(A,A,sgn_sgn(A),aa(A,A,abs_abs(A),A3)) = aa($o,A,zero_neq_one_of_bool(A),A3 != zero_zero(A)) ) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_3943_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R3: int] :
      ( aa(int,$o,dvd_dvd(int,L),aa(int,int,aa(int,fun(int,int),times_times(int),K),aa(int,int,sgn_sgn(int),R3)))
    <=> ( aa(int,$o,dvd_dvd(int,L),K)
        | ( R3 = zero_zero(int) ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_3944_dvd__sgn__mult__iff,axiom,
    ! [L: int,R3: int,K: int] :
      ( aa(int,$o,dvd_dvd(int,L),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),R3)),K))
    <=> ( aa(int,$o,dvd_dvd(int,L),K)
        | ( R3 = zero_zero(int) ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_3945_mult__sgn__dvd__iff,axiom,
    ! [L: int,R3: int,K: int] :
      ( aa(int,$o,dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),L),aa(int,int,sgn_sgn(int),R3))),K)
    <=> ( aa(int,$o,dvd_dvd(int,L),K)
        & ( ( R3 = zero_zero(int) )
         => ( K = zero_zero(int) ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_3946_sgn__mult__dvd__iff,axiom,
    ! [R3: int,L: int,K: int] :
      ( aa(int,$o,dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),R3)),L)),K)
    <=> ( aa(int,$o,dvd_dvd(int,L),K)
        & ( ( R3 = zero_zero(int) )
         => ( K = zero_zero(int) ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_3947_sgn__uint__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(int,int,sgn_sgn(int),aa(word(A),int,semiring_1_unsigned(A,int),W)) = aa($o,int,zero_neq_one_of_bool(int),W != zero_zero(word(A))) ) ).

% sgn_uint_eq
tff(fact_3948_sgn__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,A3),zero_zero(A))
         => ( aa(A,A,sgn_sgn(A),A3) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% sgn_neg
tff(fact_3949_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,ord_less(nat,zero_zero(nat)),Na)) ) ).

% sgn_of_nat
tff(fact_3950_and__nat__numerals_I3_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(num,nat,numeral_numeral(nat),bit0(X))),aa(nat,nat,suc,zero_zero(nat))) = zero_zero(nat) ).

% and_nat_numerals(3)
tff(fact_3951_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_3952_word__no__log__defs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),A3)),aa(num,int,numeral_numeral(int),B3))) ) ).

% word_no_log_defs(3)
tff(fact_3953_and__nat__numerals_I4_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(num,nat,numeral_numeral(nat),bit1(X))),aa(nat,nat,suc,zero_zero(nat))) = one_one(nat) ).

% and_nat_numerals(4)
tff(fact_3954_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_3955_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_3956_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_3957_word__bitwise__1__simps_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),one_one(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),A3)),one_one(int))) ) ).

% word_bitwise_1_simps(4)
tff(fact_3958_word__bitwise__1__simps_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),one_one(word(A))),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(num,int,numeral_numeral(int),B3))) ) ).

% word_bitwise_1_simps(2)
tff(fact_3959_word__no__log__defs_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),A3)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3)))) ) ).

% word_no_log_defs(4)
tff(fact_3960_word__no__log__defs_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3))),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3))),aa(num,int,numeral_numeral(int),B3))) ) ).

% word_no_log_defs(5)
tff(fact_3961_word__no__log__defs_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3)))) ) ).

% word_no_log_defs(6)
tff(fact_3962_word__bitwise__1__simps_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3))),one_one(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3))),one_one(int))) ) ).

% word_bitwise_1_simps(5)
tff(fact_3963_word__bitwise__1__simps_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),one_one(word(A))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3)))) ) ).

% word_bitwise_1_simps(3)
tff(fact_3964_word__log__esimps_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),zero_zero(word(A))) = zero_zero(word(A)) ) ).

% word_log_esimps(1)
tff(fact_3965_word__log__esimps_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),zero_zero(word(A))),X) = zero_zero(word(A)) ) ).

% word_log_esimps(7)
tff(fact_3966_word__bw__assocs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A),Z: word(A)] : 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)),X),Y)),Z) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Y),Z)) ) ).

% word_bw_assocs(1)
tff(fact_3967_word__bw__comms_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),Y) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Y),X) ) ).

% word_bw_comms(1)
tff(fact_3968_word__bw__same_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),X) = X ) ).

% word_bw_same(1)
tff(fact_3969_word__bw__lcs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A),Z: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),Z)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Y),Z)) ) ).

% word_bw_lcs(1)
tff(fact_3970_word__and__le2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),Y: word(A)] : aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),A3),Y)),A3) ) ).

% word_and_le2
tff(fact_3971_word__and__le1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),A3: word(A)] : aa(word(A),$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),A3)),A3) ) ).

% word_and_le1
tff(fact_3972_sgn__mult,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A,B3: A] : aa(A,A,sgn_sgn(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),A3)),aa(A,A,sgn_sgn(A),B3)) ) ).

% sgn_mult
tff(fact_3973_same__sgn__sgn__add,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B3: A,A3: A] :
          ( ( aa(A,A,sgn_sgn(A),B3) = aa(A,A,sgn_sgn(A),A3) )
         => ( aa(A,A,sgn_sgn(A),aa(A,A,plus_plus(A,A3),B3)) = aa(A,A,sgn_sgn(A),A3) ) ) ) ).

% same_sgn_sgn_add
tff(fact_3974_sgn__eq__0__iff,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] :
          ( ( aa(A,A,sgn_sgn(A),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% sgn_eq_0_iff
tff(fact_3975_sgn__0__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( ( aa(A,A,sgn_sgn(A),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% sgn_0_0
tff(fact_3976_sgn__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] :
          ( ( aa(A,A,sgn_sgn(A),X) = zero_zero(A) )
        <=> ( X = zero_zero(A) ) ) ) ).

% sgn_zero_iff
tff(fact_3977_sgn__not__eq__imp,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B3: A,A3: A] :
          ( ( aa(A,A,sgn_sgn(A),B3) != aa(A,A,sgn_sgn(A),A3) )
         => ( ( aa(A,A,sgn_sgn(A),A3) != zero_zero(A) )
           => ( ( aa(A,A,sgn_sgn(A),B3) != zero_zero(A) )
             => ( aa(A,A,sgn_sgn(A),A3) = aa(A,A,uminus_uminus(A),aa(A,A,sgn_sgn(A),B3)) ) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_3978_sgn__minus__1,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( aa(A,A,sgn_sgn(A),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% sgn_minus_1
tff(fact_3979_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N: nat,L2: int] : K != aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L2)),aa(nat,int,semiring_1_of_nat(int),N)) ).

% int_sgnE
tff(fact_3980_same__sgn__abs__add,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B3: A,A3: A] :
          ( ( aa(A,A,sgn_sgn(A),B3) = aa(A,A,sgn_sgn(A),A3) )
         => ( aa(A,A,abs_abs(A),aa(A,A,plus_plus(A,A3),B3)) = aa(A,A,plus_plus(A,aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B3)) ) ) ) ).

% same_sgn_abs_add
tff(fact_3981_linordered__idom__class_Oabs__sgn,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [K: A] : aa(A,A,abs_abs(A),K) = aa(A,A,aa(A,fun(A,A),times_times(A),K),aa(A,A,sgn_sgn(A),K)) ) ).

% linordered_idom_class.abs_sgn
tff(fact_3982_abs__mult__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A3)),aa(A,A,sgn_sgn(A),A3)) = A3 ) ).

% abs_mult_sgn
tff(fact_3983_sgn__mult__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),A3)),aa(A,A,abs_abs(A),A3)) = A3 ) ).

% sgn_mult_abs
tff(fact_3984_mult__sgn__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),X)),aa(A,A,abs_abs(A),X)) = X ) ).

% mult_sgn_abs
tff(fact_3985_word__unat__and__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Na: nat,Y: word(A)] :
          ( ( aa(nat,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),X)),Na)
            | aa(nat,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),Na) )
         => aa(nat,$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)),X),Y))),Na) ) ) ).

% word_unat_and_lt
tff(fact_3986_word__and__max__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),A3: word(A)] :
          ( ( X = 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)),bit_se5824344872417868541ns_and(word(A)),A3),X) = A3 ) ) ) ).

% word_and_max_word
tff(fact_3987_uint__and,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: 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)),bit_se5824344872417868541ns_and(word(A)),X),Y)) = aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ).

% uint_and
tff(fact_3988_word__wi__log__defs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: int,B3: int] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(int,word(A),ring_1_of_int(word(A)),A3)),aa(int,word(A),ring_1_of_int(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),A3),B3)) ) ).

% word_wi_log_defs(2)
tff(fact_3989_sgn__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( ( aa(A,A,sgn_sgn(A),A3) = one_one(A) )
        <=> aa(A,$o,ord_less(A,zero_zero(A)),A3) ) ) ).

% sgn_1_pos
tff(fact_3990_abs__sgn__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A3)) = $ite(A3 = zero_zero(A),zero_zero(A),one_one(A)) ) ).

% abs_sgn_eq
tff(fact_3991_sgn__mod,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => ( ~ aa(int,$o,dvd_dvd(int,L),K)
       => ( aa(int,int,sgn_sgn(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L)) = aa(int,int,sgn_sgn(int),L) ) ) ) ).

% sgn_mod
tff(fact_3992_and__nat__def,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),Na) = nat2(aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),Na))) ).

% and_nat_def
tff(fact_3993_word__and__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),A3),B3) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(word(A),int,semiring_1_unsigned(A,int),A3)),aa(word(A),int,semiring_1_unsigned(A,int),B3))) ) ).

% word_and_def
tff(fact_3994_sgn__if,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          aa(A,A,sgn_sgn(A),X) = $ite(
            X = zero_zero(A),
            zero_zero(A),
            $ite(aa(A,$o,ord_less(A,zero_zero(A)),X),one_one(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ) ).

% sgn_if
tff(fact_3995_sgn__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( ( aa(A,A,sgn_sgn(A),A3) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> aa(A,$o,ord_less(A,A3),zero_zero(A)) ) ) ).

% sgn_1_neg
tff(fact_3996_zsgn__def,axiom,
    ! [I: int] :
      aa(int,int,sgn_sgn(int),I) = $ite(
        I = zero_zero(int),
        zero_zero(int),
        $ite(aa(int,$o,ord_less(int,zero_zero(int)),I),one_one(int),aa(int,int,uminus_uminus(int),one_one(int))) ) ).

% zsgn_def
tff(fact_3997_norm__sgn,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] :
          real_V7770717601297561774m_norm(A,aa(A,A,sgn_sgn(A),X)) = $ite(X = zero_zero(A),zero_zero(real),one_one(real)) ) ).

% norm_sgn
tff(fact_3998_div__sgn__abs__cancel,axiom,
    ! [V2: int,K: int,L: int] :
      ( ( V2 != 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),V2)),aa(int,int,abs_abs(int),K))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),V2)),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_3999_even__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] :
          ( aa(word(A),$o,even_word(A),A3)
        <=> ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),A3),one_one(word(A))) = zero_zero(word(A)) ) ) ) ).

% even_word_iff
tff(fact_4000_the1__equality,axiom,
    ! [A: $tType,P: fun(A,$o),A3: A] :
      ( ? [X2: A] :
          ( aa(A,$o,P,X2)
          & ! [Y3: A] :
              ( aa(A,$o,P,Y3)
             => ( Y3 = X2 ) ) )
     => ( aa(A,$o,P,A3)
       => ( the(A,P) = A3 ) ) ) ).

% the1_equality
tff(fact_4001_the1I2,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ? [X2: A] :
          ( aa(A,$o,P,X2)
          & ! [Y3: A] :
              ( aa(A,$o,P,Y3)
             => ( Y3 = X2 ) ) )
     => ( ! [X4: A] :
            ( aa(A,$o,P,X4)
           => aa(A,$o,Q,X4) )
       => aa(A,$o,Q,the(A,P)) ) ) ).

% the1I2
tff(fact_4002_If__def,axiom,
    ! [A: $tType,P: $o,X: A,Y: A] :
      $ite((P),X,Y) = the(A,aa(A,fun(A,$o),aa(A,fun(A,fun(A,$o)),aTP_Lamp_dg($o,fun(A,fun(A,fun(A,$o))),(P)),X),Y)) ).

% If_def
tff(fact_4003_theI2,axiom,
    ! [A: $tType,P: fun(A,$o),A3: A,Q: fun(A,$o)] :
      ( aa(A,$o,P,A3)
     => ( ! [X4: A] :
            ( aa(A,$o,P,X4)
           => ( X4 = A3 ) )
       => ( ! [X4: A] :
              ( aa(A,$o,P,X4)
             => aa(A,$o,Q,X4) )
         => aa(A,$o,Q,the(A,P)) ) ) ) ).

% theI2
tff(fact_4004_theI_H,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ? [X2: A] :
          ( aa(A,$o,P,X2)
          & ! [Y3: A] :
              ( aa(A,$o,P,Y3)
             => ( Y3 = X2 ) ) )
     => aa(A,$o,P,the(A,P)) ) ).

% theI'
tff(fact_4005_theI,axiom,
    ! [A: $tType,P: fun(A,$o),A3: A] :
      ( aa(A,$o,P,A3)
     => ( ! [X4: A] :
            ( aa(A,$o,P,X4)
           => ( X4 = A3 ) )
       => aa(A,$o,P,the(A,P)) ) ) ).

% theI
tff(fact_4006_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,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,dvd_dvd(int,L),K))) ) ) ) ).

% div_noneq_sgn_abs
tff(fact_4007_floor__real__def,axiom,
    ! [X: real] : archim6421214686448440834_floor(real,X) = the(int,aTP_Lamp_dh(real,fun(int,$o),X)) ).

% floor_real_def
tff(fact_4008_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,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_4009_and__nat__rec,axiom,
    ! [M: nat,Na: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),Na) = aa(nat,nat,
        plus_plus(nat,
          aa($o,nat,zero_neq_one_of_bool(nat),
            ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M)
            & ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) ))),
        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_rec
tff(fact_4010_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),
        $ite(aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(nat,$o,dvd_dvd(nat,Na),M)))))) ) ).

% divide_int_unfold
tff(fact_4011_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)),
        $ite(aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(nat,$o,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_4012_divide__int__def,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = $ite(
        L = zero_zero(int),
        zero_zero(int),
        $ite(aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),nat2(aa(int,int,abs_abs(int),K))),nat2(aa(int,int,abs_abs(int),L)))),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),nat2(aa(int,int,abs_abs(int),K))),nat2(aa(int,int,abs_abs(int),L)))),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(int,$o,dvd_dvd(int,L),K)))))) ) ).

% divide_int_def
tff(fact_4013_sgn__div__eq__sgn__mult,axiom,
    ! [A3: int,B3: int] :
      ( ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3) != zero_zero(int) )
     => ( aa(int,int,sgn_sgn(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3)) = aa(int,int,sgn_sgn(int),aa(int,int,aa(int,fun(int,int),times_times(int),A3),B3)) ) ) ).

% sgn_div_eq_sgn_mult
tff(fact_4014_and__mask__arith_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] :
          ( aa(nat,$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)),W),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)),W),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,size_size(word(A),W)),Na)))),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,size_size(word(A),W)),Na))) ) ) ) ).

% and_mask_arith'
tff(fact_4015_signed__take__bit__eq__take__bit__minus,axiom,
    ! [Na: nat,K: int] : bit_ri4674362597316999326ke_bit(int,Na,K) = aa(int,int,minus_minus(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,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_4016_num_Osize__gen_I3_J,axiom,
    ! [X32: num] : size_num(bit1(X32)) = aa(nat,nat,plus_plus(nat,size_num(X32)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size_gen(3)
tff(fact_4017_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_4018_sgn__le__0__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,sgn_sgn(real),X)),zero_zero(real))
    <=> aa(real,$o,ord_less_eq(real,X),zero_zero(real)) ) ).

% sgn_le_0_iff
tff(fact_4019_zero__le__sgn__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,sgn_sgn(real),X))
    <=> aa(real,$o,ord_less_eq(real,zero_zero(real)),X) ) ).

% zero_le_sgn_iff
tff(fact_4020_mask__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se2239418461657761734s_mask(A,zero_zero(nat)) = zero_zero(A) ) ) ).

% mask_0
tff(fact_4021_mask__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] :
          ( ( bit_se2239418461657761734s_mask(A,Na) = zero_zero(A) )
        <=> ( Na = zero_zero(nat) ) ) ) ).

% mask_eq_0_iff
tff(fact_4022_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_4023_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_4024_Word_Omask__Suc__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se2239418461657761734s_mask(A,aa(nat,nat,suc,zero_zero(nat))) = one_one(A) ) ) ).

% Word.mask_Suc_0
tff(fact_4025_take__bit__minus__one__eq__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat] : bit_se2584673776208193580ke_bit(A,Na,aa(A,A,uminus_uminus(A),one_one(A))) = bit_se2239418461657761734s_mask(A,Na) ) ).

% take_bit_minus_one_eq_mask
tff(fact_4026_signed__take__bit__nonnegative__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),bit_ri4674362597316999326ke_bit(int,Na,K))
    <=> ~ aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),Na) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_4027_signed__take__bit__negative__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,ord_less(int,bit_ri4674362597316999326ke_bit(int,Na,K)),zero_zero(int))
    <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),Na) ) ).

% signed_take_bit_negative_iff
tff(fact_4028_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W: 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(W)))),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),W))),Na) ) ).

% bit_minus_numeral_Bit0_Suc_iff
tff(fact_4029_bit__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W: num,Na: num] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),bit0(W))),aa(num,nat,numeral_numeral(nat),Na))
        <=> aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),W)),pred_numeral(Na)) ) ) ).

% bit_numeral_simps(2)
tff(fact_4030_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W: 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(W)))),aa(nat,nat,suc,Na))
    <=> ~ aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(num,int,numeral_numeral(int),W)),Na) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_4031_bit__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W: num,Na: num] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),bit1(W))),aa(num,nat,numeral_numeral(nat),Na))
        <=> aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),W)),pred_numeral(Na)) ) ) ).

% bit_numeral_simps(3)
tff(fact_4032_bit__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),zero_zero(nat))
        <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3) ) ) ).

% bit_0
tff(fact_4033_bit__minus__numeral__int_I1_J,axiom,
    ! [W: num,Na: num] :
      ( 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(W)))),aa(num,nat,numeral_numeral(nat),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),W))),pred_numeral(Na)) ) ).

% bit_minus_numeral_int(1)
tff(fact_4034_bit__minus__numeral__int_I2_J,axiom,
    ! [W: num,Na: num] :
      ( 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(W)))),aa(num,nat,numeral_numeral(nat),Na))
    <=> ~ aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(num,int,numeral_numeral(int),W)),pred_numeral(Na)) ) ).

% bit_minus_numeral_int(2)
tff(fact_4035_bit__mod__2__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))),Na)
        <=> ( ( Na = zero_zero(nat) )
            & ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3) ) ) ) ).

% bit_mod_2_iff
tff(fact_4036_bin__nth__minus__Bit0,axiom,
    ! [Na: nat,W: num] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(num,int,numeral_numeral(int),bit0(W))),Na)
      <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(num,int,numeral_numeral(int),W)),aa(nat,nat,minus_minus(nat,Na),one_one(nat))) ) ) ).

% bin_nth_minus_Bit0
tff(fact_4037_bin__nth__minus__Bit1,axiom,
    ! [Na: nat,W: num] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(num,int,numeral_numeral(int),bit1(W))),Na)
      <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(num,int,numeral_numeral(int),W)),aa(nat,nat,minus_minus(nat,Na),one_one(nat))) ) ) ).

% bin_nth_minus_Bit1
tff(fact_4038_bit__numeral__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),M)),Na)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),bit_se5641148757651400278ts_bit(nat),aa(num,nat,numeral_numeral(nat),M)),Na) ) ) ).

% bit_numeral_iff
tff(fact_4039_ex__mask__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ? [X4: nat] : bit_se2239418461657761734s_mask(word(A),X4) = one_one(word(A)) ) ).

% ex_mask_1
tff(fact_4040_test__bit__cong,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Xa2: A,Y: A,X: nat] :
          ( ( Xa2 = Y )
         => ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Xa2),X)
          <=> aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Y),X) ) ) ) ).

% test_bit_cong
tff(fact_4041_of__nat__mask__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : aa(nat,A,semiring_1_of_nat(A),bit_se2239418461657761734s_mask(nat,Na)) = bit_se2239418461657761734s_mask(A,Na) ) ).

% of_nat_mask_eq
tff(fact_4042_bit__of__nat__iff__bit,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(nat,A,semiring_1_of_nat(A),M)),Na)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),bit_se5641148757651400278ts_bit(nat),M),Na) ) ) ).

% bit_of_nat_iff_bit
tff(fact_4043_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_4044_bit__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: num] : ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),one_one(A)),aa(num,nat,numeral_numeral(nat),Na)) ) ).

% bit_numeral_simps(1)
tff(fact_4045_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_4046_bit__take__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A3: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),bit_se2584673776208193580ke_bit(A,M,A3)),Na)
        <=> ( aa(nat,$o,ord_less(nat,Na),M)
            & aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),Na) ) ) ) ).

% bit_take_bit_iff
tff(fact_4047_bit__of__bool__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [B3: $o,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa($o,A,zero_neq_one_of_bool(A),(B3))),Na)
        <=> ( (B3)
            & ( Na = zero_zero(nat) ) ) ) ) ).

% bit_of_bool_iff
tff(fact_4048_mask__twice2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat,X: word(A)] :
          ( aa(nat,$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)),X),bit_se2239418461657761734s_mask(word(A),M))),bit_se2239418461657761734s_mask(word(A),Na)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),bit_se2239418461657761734s_mask(word(A),Na)) ) ) ) ).

% mask_twice2
tff(fact_4049_and__mask__eq__iff__le__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Na: nat] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),Na)) = W )
        <=> aa(word(A),$o,ord_less_eq(word(A),W),bit_se2239418461657761734s_mask(word(A),Na)) ) ) ).

% and_mask_eq_iff_le_mask
tff(fact_4050_le__mask__imp__and__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Na: nat] :
          ( aa(word(A),$o,ord_less_eq(word(A),X),bit_se2239418461657761734s_mask(word(A),Na))
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),bit_se2239418461657761734s_mask(word(A),Na)) = X ) ) ) ).

% le_mask_imp_and_mask
tff(fact_4051_mask__eqs_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),Na: nat,B3: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(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)),A3),bit_se2239418461657761734s_mask(word(A),Na))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),B3),bit_se2239418461657761734s_mask(word(A),Na)))),bit_se2239418461657761734s_mask(word(A),Na)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),plus_plus(word(A),A3),B3)),bit_se2239418461657761734s_mask(word(A),Na)) ) ).

% mask_eqs(7)
tff(fact_4052_mask__eqs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A),Na: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),plus_plus(word(A),A3),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),B3),bit_se2239418461657761734s_mask(word(A),Na)))),bit_se2239418461657761734s_mask(word(A),Na)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),plus_plus(word(A),A3),B3)),bit_se2239418461657761734s_mask(word(A),Na)) ) ).

% mask_eqs(2)
tff(fact_4053_mask__eqs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),Na: nat,B3: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(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)),A3),bit_se2239418461657761734s_mask(word(A),Na))),B3)),bit_se2239418461657761734s_mask(word(A),Na)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),plus_plus(word(A),A3),B3)),bit_se2239418461657761734s_mask(word(A),Na)) ) ).

% mask_eqs(1)
tff(fact_4054_mask__eqs_I10_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),Na: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),A3),bit_se2239418461657761734s_mask(word(A),Na)))),bit_se2239418461657761734s_mask(word(A),Na)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),A3)),bit_se2239418461657761734s_mask(word(A),Na)) ) ).

% mask_eqs(10)
tff(fact_4055_mask__eqs_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),Na: nat,B3: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),A3),bit_se2239418461657761734s_mask(word(A),Na))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),B3),bit_se2239418461657761734s_mask(word(A),Na)))),bit_se2239418461657761734s_mask(word(A),Na)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),minus_minus(word(A),A3),B3)),bit_se2239418461657761734s_mask(word(A),Na)) ) ).

% mask_eqs(8)
tff(fact_4056_mask__eqs_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A),Na: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),minus_minus(word(A),A3),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),B3),bit_se2239418461657761734s_mask(word(A),Na)))),bit_se2239418461657761734s_mask(word(A),Na)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),minus_minus(word(A),A3),B3)),bit_se2239418461657761734s_mask(word(A),Na)) ) ).

% mask_eqs(4)
tff(fact_4057_mask__eqs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),Na: nat,B3: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),A3),bit_se2239418461657761734s_mask(word(A),Na))),B3)),bit_se2239418461657761734s_mask(word(A),Na)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),minus_minus(word(A),A3),B3)),bit_se2239418461657761734s_mask(word(A),Na)) ) ).

% mask_eqs(3)
tff(fact_4058_mask__power__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Na: nat,K: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(nat,word(A),power_power(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),bit_se2239418461657761734s_mask(word(A),Na))),K)),bit_se2239418461657761734s_mask(word(A),Na)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(nat,word(A),power_power(word(A),X),K)),bit_se2239418461657761734s_mask(word(A),Na)) ) ).

% mask_power_eq
tff(fact_4059_ucast__and__mask,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(B),Na: nat] : 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)),X),bit_se2239418461657761734s_mask(word(B),Na))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),X)),bit_se2239418461657761734s_mask(word(A),Na)) ) ).

% ucast_and_mask
tff(fact_4060_mask__eqs_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A),Na: nat] : 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)),times_times(word(A)),A3),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),B3),bit_se2239418461657761734s_mask(word(A),Na)))),bit_se2239418461657761734s_mask(word(A),Na)) = 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)),times_times(word(A)),A3),B3)),bit_se2239418461657761734s_mask(word(A),Na)) ) ).

% mask_eqs(5)
tff(fact_4061_mask__eqs_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: word(A),Na: nat,A3: word(A)] : 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)),times_times(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),B3),bit_se2239418461657761734s_mask(word(A),Na))),A3)),bit_se2239418461657761734s_mask(word(A),Na)) = 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)),times_times(word(A)),B3),A3)),bit_se2239418461657761734s_mask(word(A),Na)) ) ).

% mask_eqs(6)
tff(fact_4062_mask__eqs_I9_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),Na: nat,B3: word(A)] : 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)),times_times(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),A3),bit_se2239418461657761734s_mask(word(A),Na))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),B3),bit_se2239418461657761734s_mask(word(A),Na)))),bit_se2239418461657761734s_mask(word(A),Na)) = 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)),times_times(word(A)),A3),B3)),bit_se2239418461657761734s_mask(word(A),Na)) ) ).

% mask_eqs(9)
tff(fact_4063_less__eq__mask__iff__take__bit__eq__self,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Na: nat] :
          ( aa(word(A),$o,ord_less_eq(word(A),W),bit_se2239418461657761734s_mask(word(A),Na))
        <=> ( bit_se2584673776208193580ke_bit(word(A),Na,W) = W ) ) ) ).

% less_eq_mask_iff_take_bit_eq_self
tff(fact_4064_mask__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se2239418461657761734s_mask(A,one_one(nat)) = one_one(A) ) ) ).

% mask_1
tff(fact_4065_More__Word_Omask__Suc__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( bit_se2239418461657761734s_mask(word(A),aa(nat,nat,suc,zero_zero(nat))) = one_one(word(A)) ) ) ).

% More_Word.mask_Suc_0
tff(fact_4066_bit__not__int__iff_H,axiom,
    ! [K: int,Na: nat] :
      ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),K)),one_one(int))),Na)
    <=> ~ aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),Na) ) ).

% bit_not_int_iff'
tff(fact_4067_and__mask__wi,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: int,Na: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(int,word(A),ring_1_of_int(word(A)),I)),bit_se2239418461657761734s_mask(word(A),Na)) = aa(int,word(A),ring_1_of_int(word(A)),bit_se2584673776208193580ke_bit(int,Na,I)) ) ).

% and_mask_wi
tff(fact_4068_sgn__real__def,axiom,
    ! [A3: real] :
      aa(real,real,sgn_sgn(real),A3) = $ite(
        A3 = zero_zero(real),
        zero_zero(real),
        $ite(aa(real,$o,ord_less(real,zero_zero(real)),A3),one_one(real),aa(real,real,uminus_uminus(real),one_one(real))) ) ).

% sgn_real_def
tff(fact_4069_bit__imp__take__bit__positive,axiom,
    ! [Na: nat,M: nat,K: int] :
      ( aa(nat,$o,ord_less(nat,Na),M)
     => ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),Na)
       => aa(int,$o,ord_less(int,zero_zero(int)),bit_se2584673776208193580ke_bit(int,M,K)) ) ) ).

% bit_imp_take_bit_positive
tff(fact_4070_mask__bin,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] : bit_se2239418461657761734s_mask(word(A),Na) = aa(int,word(A),ring_1_of_int(word(A)),bit_se2584673776208193580ke_bit(int,Na,aa(int,int,uminus_uminus(int),one_one(int)))) ) ).

% mask_bin
tff(fact_4071_and__mask__bintr,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Na: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),Na)) = aa(int,word(A),ring_1_of_int(word(A)),bit_se2584673776208193580ke_bit(int,Na,aa(word(A),int,semiring_1_unsigned(A,int),W))) ) ).

% and_mask_bintr
tff(fact_4072_exp__eq__0__imp__not__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Na: nat,A3: A] :
          ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na) = zero_zero(A) )
         => ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),Na) ) ) ).

% exp_eq_0_imp_not_bit
tff(fact_4073_bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),aa(nat,nat,suc,Na))
        <=> aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))),Na) ) ) ).

% bit_Suc
tff(fact_4074_stable__imp__bit__iff__odd,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,Na: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) = A3 )
         => ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),Na)
          <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3) ) ) ) ).

% stable_imp_bit_iff_odd
tff(fact_4075_bit__iff__idd__imp__stable,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] :
          ( ! [N: nat] :
              ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),N)
            <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) = A3 ) ) ) ).

% bit_iff_idd_imp_stable
tff(fact_4076_word__1FF__is__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(num,word(A),numeral_numeral(word(A)),bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(one2))))))))) = bit_se2239418461657761734s_mask(word(A),aa(num,nat,numeral_numeral(nat),bit1(bit0(bit0(one2))))) ) ) ).

% word_1FF_is_mask
tff(fact_4077_word__FF__is__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(num,word(A),numeral_numeral(word(A)),bit1(bit1(bit1(bit1(bit1(bit1(bit1(one2)))))))) = bit_se2239418461657761734s_mask(word(A),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))) ) ) ).

% word_FF_is_mask
tff(fact_4078_sgn__power__injE,axiom,
    ! [A3: real,Na: nat,X: real,B3: real] :
      ( ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),A3)),aa(nat,real,power_power(real,aa(real,real,abs_abs(real),A3)),Na)) = X )
     => ( ( X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),B3)),aa(nat,real,power_power(real,aa(real,real,abs_abs(real),B3)),Na)) )
       => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
         => ( A3 = B3 ) ) ) ) ).

% sgn_power_injE
tff(fact_4079_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N: nat] :
          ( ! [M2: nat] :
              ( aa(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,ord_less(nat,zero_zero(nat)),N)
             => ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),aa(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_4080_and__mask__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: num,Na: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(num,word(A),numeral_numeral(word(A)),I)),bit_se2239418461657761734s_mask(word(A),Na)) = aa(int,word(A),ring_1_of_int(word(A)),bit_se2584673776208193580ke_bit(int,Na,aa(num,int,numeral_numeral(int),I))) ) ).

% and_mask_no
tff(fact_4081_num_Osize__gen_I1_J,axiom,
    size_num(one2) = zero_zero(nat) ).

% num.size_gen(1)
tff(fact_4082_bit__iff__odd,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),Na)
        <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na))) ) ) ).

% bit_iff_odd
tff(fact_4083_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,Na: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = zero_zero(A) )
        <=> ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),Na) ) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_4084_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2239418461657761734s_mask(A,Na))
        <=> ( Na = zero_zero(nat) ) ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_4085_mask__plus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] : aa(word(A),word(A),plus_plus(word(A),bit_se2239418461657761734s_mask(word(A),Na)),one_one(word(A))) = aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na) ) ).

% mask_plus_1
tff(fact_4086_less__mask__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Na: nat] :
          ( aa(word(A),$o,ord_less(word(A),X),aa(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)),X),bit_se2239418461657761734s_mask(word(A),Na)) = X ) ) ) ).

% less_mask_eq
tff(fact_4087_mask__eq__decr__exp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] : bit_se2239418461657761734s_mask(word(A),Na) = aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),one_one(word(A))) ) ).

% mask_eq_decr_exp
tff(fact_4088_bit__int__def,axiom,
    ! [K: int,Na: nat] :
      ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),Na)
    <=> ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))) ) ).

% bit_int_def
tff(fact_4089_even__bit__succ__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
         => ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,plus_plus(A,one_one(A)),A3)),Na)
          <=> ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),Na)
              | ( Na = zero_zero(nat) ) ) ) ) ) ).

% even_bit_succ_iff
tff(fact_4090_odd__bit__iff__bit__pred,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,Na: nat] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
         => ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),Na)
          <=> ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,minus_minus(A,A3),one_one(A))),Na)
              | ( Na = zero_zero(nat) ) ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_4091_mask__eq__exp__minus__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : bit_se2239418461657761734s_mask(A,Na) = aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)),one_one(A)) ) ).

% mask_eq_exp_minus_1
tff(fact_4092_mask__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Na: nat] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),Na)) = W )
        <=> aa(int,$o,ord_less(int,aa(word(A),int,semiring_1_unsigned(A,int),W)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ) ).

% mask_eq_iff
tff(fact_4093_and__mask__lt__2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Na: nat] : aa(int,$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)),W),bit_se2239418461657761734s_mask(word(A),Na)))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ).

% and_mask_lt_2p
tff(fact_4094_mask__Suc__rec,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] : bit_se2239418461657761734s_mask(word(A),aa(nat,nat,suc,Na)) = aa(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))),bit_se2239418461657761734s_mask(word(A),Na))),one_one(word(A))) ) ).

% mask_Suc_rec
tff(fact_4095_is__aligned__AND__less__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [U: word(A),Na: nat,V2: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),U),bit_se2239418461657761734s_mask(word(A),Na)) = zero_zero(word(A)) )
         => ( aa(word(A),$o,ord_less(word(A),V2),aa(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),V2) = zero_zero(word(A)) ) ) ) ) ).

% is_aligned_AND_less_0
tff(fact_4096_add__mask__fold,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Na: nat] : aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),plus_plus(word(A),X),aa(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),word(A),plus_plus(word(A),X),bit_se2239418461657761734s_mask(word(A),Na)) ) ).

% add_mask_fold
tff(fact_4097_bit__sum__mult__2__cases,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B3: A,Na: nat] :
          ( ! [J2: nat] : ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),aa(nat,nat,suc,J2))
         => ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,plus_plus(A,A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3))),Na)
          <=> $ite(Na = zero_zero(nat),~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3),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))),B3)),Na)) ) ) ) ).

% bit_sum_mult_2_cases
tff(fact_4098_bit__rec,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),Na)
        <=> $ite(Na = zero_zero(nat),~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,nat,minus_minus(nat,Na),one_one(nat)))) ) ) ).

% bit_rec
tff(fact_4099_and__mask__dvd__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] :
          ( aa(nat,$o,dvd_dvd(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),aa(word(A),nat,semiring_1_unsigned(A,nat),W))
        <=> ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),Na)) = zero_zero(word(A)) ) ) ) ).

% and_mask_dvd_nat
tff(fact_4100_and__mask__mod__2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Na: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),Na)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(word(A),int,semiring_1_unsigned(A,int),W)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))) ) ).

% and_mask_mod_2p
tff(fact_4101_and__mask__dvd,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] :
          ( aa(int,$o,dvd_dvd(int,aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)),aa(word(A),int,semiring_1_unsigned(A,int),W))
        <=> ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),Na)) = zero_zero(word(A)) ) ) ) ).

% and_mask_dvd
tff(fact_4102_mask__eq__iff__w2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] :
          ( aa(nat,$o,ord_less(nat,Na),size_size(word(A),W))
         => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),Na)) = W )
          <=> aa(word(A),$o,ord_less(word(A),W),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) ) ) ) ).

% mask_eq_iff_w2p
tff(fact_4103_and__mask__less__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,X: word(A)] :
          ( aa(nat,$o,ord_less(nat,Na),size_size(word(A),X))
         => aa(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)),X),bit_se2239418461657761734s_mask(word(A),Na))),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) ) ) ).

% and_mask_less_size
tff(fact_4104_word__and__mask__le__2pm1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Na: nat] : aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),Na))),aa(word(A),word(A),minus_minus(word(A),aa(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_4105_word__mod__2p__is__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,X: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),aa(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)),X),aa(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)),X),bit_se2239418461657761734s_mask(word(A),Na)) ) ) ) ).

% word_mod_2p_is_mask
tff(fact_4106_Bit__Operations_Oset__bit__eq,axiom,
    ! [Na: nat,K: int] : bit_se5668285175392031749et_bit(int,Na,K) = aa(int,int,plus_plus(int,K),aa(int,int,aa(int,fun(int,int),times_times(int),aa($o,int,zero_neq_one_of_bool(int),~ aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),Na))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))) ).

% Bit_Operations.set_bit_eq
tff(fact_4107_unset__bit__eq,axiom,
    ! [Na: nat,K: int] : bit_se2638667681897837118et_bit(int,Na,K) = aa(int,int,minus_minus(int,K),aa(int,int,aa(int,fun(int,int),times_times(int),aa($o,int,zero_neq_one_of_bool(int),aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),Na))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))) ).

% unset_bit_eq
tff(fact_4108_word__unat__mask__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,W: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,M),size_size(word(A),W))
         => aa(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)),W),bit_se2239418461657761734s_mask(word(A),M)))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) ) ) ).

% word_unat_mask_lt
tff(fact_4109_take__bit__Suc__from__most,axiom,
    ! [Na: nat,K: int] : bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,Na),K) = aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(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)))),bit_se2584673776208193580ke_bit(int,Na,K)) ).

% take_bit_Suc_from_most
tff(fact_4110_and__mask__arith,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Na: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),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)),W),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,size_size(word(A),W)),Na)))),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,size_size(word(A),W)),Na))) ) ).

% and_mask_arith
tff(fact_4111_mask__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: num] : bit_se2239418461657761734s_mask(A,aa(num,nat,numeral_numeral(nat),Na)) = aa(A,A,plus_plus(A,one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2239418461657761734s_mask(A,pred_numeral(Na)))) ) ).

% mask_numeral
tff(fact_4112_arctan__inverse,axiom,
    ! [X: real] :
      ( ( X != zero_zero(real) )
     => ( aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),X)) = aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),X)),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(real,real,arctan,X)) ) ) ).

% arctan_inverse
tff(fact_4113_num_Osize__gen_I2_J,axiom,
    ! [X22: num] : size_num(bit0(X22)) = aa(nat,nat,plus_plus(nat,size_num(X22)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size_gen(2)
tff(fact_4114_floor__rat__def,axiom,
    ! [X: rat] : archim6421214686448440834_floor(rat,X) = the(int,aTP_Lamp_di(rat,fun(int,$o),X)) ).

% floor_rat_def
tff(fact_4115_neg__mask__is__div_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] :
          ( aa(nat,$o,ord_less(nat,Na),size_size(word(A),W))
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),aa(word(A),word(A),bit_ri4277139882892585799ns_not(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)),W),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) ) ) ) ).

% neg_mask_is_div'
tff(fact_4116_VEBT__internal_OT__vebt__buildupi_H_Opelims,axiom,
    ! [X: nat,Y: int] :
      ( ( vEBT_V9176841429113362141ildupi(X) = Y )
     => ( accp(nat,vEBT_V3352910403632780892pi_rel,X)
       => ( ( ( X = zero_zero(nat) )
           => ( ( Y = one_one(int) )
             => ~ accp(nat,vEBT_V3352910403632780892pi_rel,zero_zero(nat)) ) )
         => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = one_one(int) )
               => ~ accp(nat,vEBT_V3352910403632780892pi_rel,aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [N: nat] :
                  ( ( X = aa(nat,nat,suc,aa(nat,nat,suc,N)) )
                 => ( ( Y = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N),aa(int,int,plus_plus(int,aa(num,int,numeral_numeral(int),bit1(one2))),aa(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,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,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,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,plus_plus(int,aa(num,int,numeral_numeral(int),bit1(one2))),aa(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,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,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,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)))))))))) )
                   => ~ accp(nat,vEBT_V3352910403632780892pi_rel,aa(nat,nat,suc,aa(nat,nat,suc,N))) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.pelims
tff(fact_4117_VEBT__internal_OTBOUND__buildupi,axiom,
    ! [Na: nat] :
      ( aa(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,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ) ).

% VEBT_internal.TBOUND_buildupi
tff(fact_4118_word__not__not,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),X)) = X ) ).

% word_not_not
tff(fact_4119_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_4120_bit_Oconj__cancel__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),aa(A,A,bit_ri4277139882892585799ns_not(A),X)) = zero_zero(A) ) ).

% bit.conj_cancel_right
tff(fact_4121_bit_Oconj__cancel__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),X) = zero_zero(A) ) ).

% bit.conj_cancel_left
tff(fact_4122_and__and__not,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : 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)),A3),B3)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),B3)) = zero_zero(word(A)) ) ).

% and_and_not
tff(fact_4123_word__and__not,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),X)) = zero_zero(word(A)) ) ).

% word_and_not
tff(fact_4124_mask__nat__positive__iff,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),bit_se2239418461657761734s_mask(nat,Na))
    <=> aa(nat,$o,ord_less(nat,zero_zero(nat)),Na) ) ).

% mask_nat_positive_iff
tff(fact_4125_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_4126_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_4127_NOT__mask__AND__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [W: 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),W),bit_se2239418461657761734s_mask(A,Na))),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,Na))) = zero_zero(A) ) ).

% NOT_mask_AND_mask
tff(fact_4128_minus__not__numeral__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: num] : aa(A,A,uminus_uminus(A),aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),Na))) = aa(num,A,numeral_numeral(A),inc(Na)) ) ).

% minus_not_numeral_eq
tff(fact_4129_word__bitwise__m1__simps_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) = zero_zero(word(A)) ) ) ).

% word_bitwise_m1_simps(1)
tff(fact_4130_word__add__not,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),word(A),plus_plus(word(A),X),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),X)) = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ).

% word_add_not
tff(fact_4131_even__not__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,bit_ri4277139882892585799ns_not(A),A3))
        <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3) ) ) ).

% even_not_iff
tff(fact_4132_not__one__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ( aa(A,A,bit_ri4277139882892585799ns_not(A),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% not_one_eq
tff(fact_4133_compl__of__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),one_one(word(A))) = aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))) ) ) ).

% compl_of_1
tff(fact_4134_shiftl__minus__1__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: num] : bit_Sh4282982442137083160shiftl(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(num,nat,numeral_numeral(nat),Na)) = aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,aa(num,nat,numeral_numeral(nat),Na))) ) ).

% shiftl_minus_1_numeral
tff(fact_4135_word__and__nth,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),Y)),Na)
        <=> ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),X),Na)
            & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),Na) ) ) ) ).

% word_and_nth
tff(fact_4136_sgn__rat__def,axiom,
    ! [A3: rat] :
      aa(rat,rat,sgn_sgn(rat),A3) = $ite(
        A3 = zero_zero(rat),
        zero_zero(rat),
        $ite(aa(rat,$o,ord_less(rat,zero_zero(rat)),A3),one_one(rat),aa(rat,rat,uminus_uminus(rat),one_one(rat))) ) ).

% sgn_rat_def
tff(fact_4137_word__exists__nth,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( ( W != zero_zero(word(A)) )
         => ? [X_12: nat] : aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),X_12) ) ) ).

% word_exists_nth
tff(fact_4138_nth__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] : ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),zero_zero(word(A))),Na) ) ).

% nth_0
tff(fact_4139_of__int__not__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: num] : aa(int,A,ring_1_of_int(A),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),K))) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),K)) ) ).

% of_int_not_numeral
tff(fact_4140_not__switch,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,X: A] :
          ( ( aa(A,A,bit_ri4277139882892585799ns_not(A),A3) = X )
         => ( A3 = aa(A,A,bit_ri4277139882892585799ns_not(A),X) ) ) ) ).

% not_switch
tff(fact_4141_less__eq__rat__def,axiom,
    ! [X: rat,Y: rat] :
      ( aa(rat,$o,ord_less_eq(rat,X),Y)
    <=> ( aa(rat,$o,ord_less(rat,X),Y)
        | ( X = Y ) ) ) ).

% less_eq_rat_def
tff(fact_4142_obtain__pos__sum,axiom,
    ! [R3: rat] :
      ( aa(rat,$o,ord_less(rat,zero_zero(rat)),R3)
     => ~ ! [S2: rat] :
            ( aa(rat,$o,ord_less(rat,zero_zero(rat)),S2)
           => ! [T2: rat] :
                ( aa(rat,$o,ord_less(rat,zero_zero(rat)),T2)
               => ( R3 != aa(rat,rat,plus_plus(rat,S2),T2) ) ) ) ) ).

% obtain_pos_sum
tff(fact_4143_abs__rat__def,axiom,
    ! [A3: rat] :
      aa(rat,rat,abs_abs(rat),A3) = $ite(aa(rat,$o,ord_less(rat,A3),zero_zero(rat)),aa(rat,rat,uminus_uminus(rat),A3),A3) ).

% abs_rat_def
tff(fact_4144_split__word__eq__on__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A),M: word(A)] :
          ( ( X = Y )
        <=> ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),M) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Y),M) )
            & ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),aa(word(A),word(A),bit_ri4277139882892585799ns_not(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)),M)) ) ) ) ) ).

% split_word_eq_on_mask
tff(fact_4145_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_4146_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_4147_mask__eq__x__eq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),W: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),W) = X )
        <=> ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),W)) = zero_zero(word(A)) ) ) ) ).

% mask_eq_x_eq_0
tff(fact_4148_mask__eq__0__eq__x,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),W: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),W) = zero_zero(word(A)) )
        <=> ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),W)) = X ) ) ) ).

% mask_eq_0_eq_x
tff(fact_4149_word__plus__and__or__coroll2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),W: word(A)] : aa(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)),X),W)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),W))) = X ) ).

% word_plus_and_or_coroll2
tff(fact_4150_mask__eqI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Na: nat,Y: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),bit_se2239418461657761734s_mask(word(A),Na)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Y),bit_se2239418461657761734s_mask(word(A),Na)) )
         => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),aa(word(A),word(A),bit_ri4277139882892585799ns_not(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)),Y),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),Na))) )
           => ( X = Y ) ) ) ) ).

% mask_eqI
tff(fact_4151_test__bit__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),Na)
         => aa(nat,$o,ord_less(nat,Na),size_size(word(A),W)) ) ) ).

% test_bit_size
tff(fact_4152_word__eqI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [U: word(A),V2: word(A)] :
          ( ! [N: nat] :
              ( aa(nat,$o,ord_less(nat,N),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)),V2),N) ) )
         => ( U = V2 ) ) ) ).

% word_eqI
tff(fact_4153_test__bit__over,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Na: nat] :
          ( aa(nat,$o,ord_less_eq(nat,size_size(word(A),X)),Na)
         => ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),X),Na) ) ) ).

% test_bit_over
tff(fact_4154_mask__nonnegative__int,axiom,
    ! [Na: nat] : aa(int,$o,ord_less_eq(int,zero_zero(int)),bit_se2239418461657761734s_mask(int,Na)) ).

% mask_nonnegative_int
tff(fact_4155_not__mask__negative__int,axiom,
    ! [Na: nat] : ~ aa(int,$o,ord_less(int,bit_se2239418461657761734s_mask(int,Na)),zero_zero(int)) ).

% not_mask_negative_int
tff(fact_4156_test__bit__def_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),X: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),A3),X)
        <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(word(A),int,semiring_1_unsigned(A,int),A3)),X) ) ) ).

% test_bit_def'
tff(fact_4157_word__test__bit__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),A3) = aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(word(A),int,semiring_1_unsigned(A,int),A3)) ) ).

% word_test_bit_def
tff(fact_4158_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_4159_minus__eq__not__plus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A] : aa(A,A,uminus_uminus(A),A3) = aa(A,A,plus_plus(A,aa(A,A,bit_ri4277139882892585799ns_not(A),A3)),one_one(A)) ) ).

% minus_eq_not_plus_1
tff(fact_4160_minus__eq__not__minus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A] : aa(A,A,uminus_uminus(A),A3) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,minus_minus(A,A3),one_one(A))) ) ).

% minus_eq_not_minus_1
tff(fact_4161_not__eq__complement,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),A3) = aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),A3)),one_one(A)) ) ).

% not_eq_complement
tff(fact_4162_mask__out__first__mask__some,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Na: nat,Y: word(A),M: nat] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),Na))) = Y )
         => ( aa(nat,$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)),X),aa(word(A),word(A),bit_ri4277139882892585799ns_not(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)),bit_se2239418461657761734s_mask(word(A),M))) ) ) ) ) ).

% mask_out_first_mask_some
tff(fact_4163_mask__lower__twice,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat,X: word(A)] :
          ( aa(nat,$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)),X),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),Na)))),aa(word(A),word(A),bit_ri4277139882892585799ns_not(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)),X),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),M))) ) ) ) ).

% mask_lower_twice
tff(fact_4164_NOT__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),X) = aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),uminus_uminus(word(A)),X)),one_one(word(A))) ) ).

% NOT_eq
tff(fact_4165_mask__AND__NOT__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Na: nat] : 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)),W),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),Na)))),bit_se2239418461657761734s_mask(word(A),Na)) = zero_zero(word(A)) ) ).

% mask_AND_NOT_mask
tff(fact_4166_AND__NOT__mask__plus__AND__mask__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Na: nat] : aa(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)),W),aa(word(A),word(A),bit_ri4277139882892585799ns_not(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)),W),bit_se2239418461657761734s_mask(word(A),Na))) = W ) ).

% AND_NOT_mask_plus_AND_mask_eq
tff(fact_4167_lsb__this__or__next,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),plus_plus(word(A),X),one_one(word(A)))),zero_zero(nat))
         => aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),X),zero_zero(nat)) ) ) ).

% lsb_this_or_next
tff(fact_4168_minus__numeral__inc__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: num] : aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(Na))) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),Na)) ) ).

% minus_numeral_inc_eq
tff(fact_4169_subtract__mask_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),Na: nat] : aa(word(A),word(A),minus_minus(word(A),P2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),P2),aa(word(A),word(A),bit_ri4277139882892585799ns_not(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)),P2),bit_se2239418461657761734s_mask(word(A),Na)) ) ).

% subtract_mask(2)
tff(fact_4170_subtract__mask_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),Na: nat] : aa(word(A),word(A),minus_minus(word(A),P2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),P2),bit_se2239418461657761734s_mask(word(A),Na))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),P2),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),Na))) ) ).

% subtract_mask(1)
tff(fact_4171_mask__out__sub__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Na: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),Na))) = aa(word(A),word(A),minus_minus(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),bit_se2239418461657761734s_mask(word(A),Na))) ) ).

% mask_out_sub_mask
tff(fact_4172_ucast__and__neg__mask,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(B),Na: nat] : 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)),X),aa(word(B),word(B),bit_ri4277139882892585799ns_not(word(B)),bit_se2239418461657761734s_mask(word(B),Na)))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),X)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),Na))) ) ).

% ucast_and_neg_mask
tff(fact_4173_word__leI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [U: word(A),V2: word(A)] :
          ( ! [N: nat] :
              ( aa(nat,$o,ord_less(nat,N),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)),V2),N) ) )
         => aa(word(A),$o,ord_less_eq(word(A),U),V2) ) ) ).

% word_leI
tff(fact_4174_nth__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,I: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),bit_se2239418461657761734s_mask(word(A),Na)),I)
        <=> ( aa(nat,$o,ord_less(nat,I),Na)
            & aa(nat,$o,ord_less(nat,I),size_size(word(A),bit_se2239418461657761734s_mask(word(A),Na))) ) ) ) ).

% nth_mask
tff(fact_4175_less__mask,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,zero_zero(nat))),Na)
     => aa(nat,$o,ord_less(nat,Na),bit_se2239418461657761734s_mask(nat,Na)) ) ).

% less_mask
tff(fact_4176_not__numeral__Bit0__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: num] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),bit0(Na))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit1(Na))) ) ).

% not_numeral_Bit0_eq
tff(fact_4177_take__bit__not__mask__eq__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,ord_less_eq(nat,M),Na)
         => ( bit_se2584673776208193580ke_bit(A,M,aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,Na))) = zero_zero(A) ) ) ) ).

% take_bit_not_mask_eq_0
tff(fact_4178_bit__nat__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),bit_se5641148757651400278ts_bit(nat),nat2(K)),Na)
    <=> ( aa(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_4179_multiple__mask__trivia,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,Na: nat,X: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,M),Na)
         => ( aa(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)),X),aa(word(A),word(A),bit_ri4277139882892585799ns_not(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)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),bit_se2239418461657761734s_mask(word(A),Na)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(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)),X),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),M))) ) ) ) ).

% multiple_mask_trivia
tff(fact_4180_and__mask__0__iff__le__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Na: nat] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),Na))) = zero_zero(word(A)) )
        <=> aa(word(A),$o,ord_less_eq(word(A),W),bit_se2239418461657761734s_mask(word(A),Na)) ) ) ).

% and_mask_0_iff_le_mask
tff(fact_4181_overflow__imp__lsb,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( ( aa(word(A),word(A),plus_plus(word(A),X),one_one(word(A))) = zero_zero(word(A)) )
         => aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),X),zero_zero(nat)) ) ) ).

% overflow_imp_lsb
tff(fact_4182_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_4183_test__bit__bin_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),Na)
        <=> ( aa(nat,$o,ord_less(nat,Na),size_size(word(A),W))
            & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(word(A),int,semiring_1_unsigned(A,int),W)),Na) ) ) ) ).

% test_bit_bin'
tff(fact_4184_take__bit__eq__mask__iff,axiom,
    ! [Na: nat,K: int] :
      ( ( bit_se2584673776208193580ke_bit(int,Na,K) = bit_se2239418461657761734s_mask(int,Na) )
    <=> ( bit_se2584673776208193580ke_bit(int,Na,aa(int,int,plus_plus(int,K),one_one(int))) = zero_zero(int) ) ) ).

% take_bit_eq_mask_iff
tff(fact_4185_bang__is__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),M: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),X),M)
         => aa(word(A),$o,ord_less_eq(word(A),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)),X) ) ) ).

% bang_is_le
tff(fact_4186_Suc__mask__eq__exp,axiom,
    ! [Na: nat] : aa(nat,nat,suc,bit_se2239418461657761734s_mask(nat,Na)) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) ).

% Suc_mask_eq_exp
tff(fact_4187_mask__nat__less__exp,axiom,
    ! [Na: nat] : aa(nat,$o,ord_less(nat,bit_se2239418461657761734s_mask(nat,Na)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ).

% mask_nat_less_exp
tff(fact_4188_bit__nat__def,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),bit_se5641148757651400278ts_bit(nat),M),Na)
    <=> ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ) ).

% bit_nat_def
tff(fact_4189_bit__not__iff__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A3)),Na)
        <=> ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na) != zero_zero(A) )
            & ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),Na) ) ) ) ).

% bit_not_iff_eq
tff(fact_4190_minus__exp__eq__not__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat] : aa(A,A,uminus_uminus(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,Na)) ) ).

% minus_exp_eq_not_mask
tff(fact_4191_NOT__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] : aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),Na)) = aa(word(A),word(A),uminus_uminus(word(A)),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) ) ).

% NOT_mask
tff(fact_4192_odd__iff__lsb,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,semiring_1_unsigned(A,nat),X))
        <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),X),zero_zero(nat)) ) ) ).

% odd_iff_lsb
tff(fact_4193_and__neq__0__is__nth,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Na: nat,X: word(A)] :
          ( ( Y = aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na) )
         => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),Y) != zero_zero(word(A)) )
          <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),X),Na) ) ) ) ).

% and_neq_0_is_nth
tff(fact_4194_nth__is__and__neq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),X),Na)
        <=> ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) != zero_zero(word(A)) ) ) ) ).

% nth_is_and_neq_0
tff(fact_4195_mask__half__int,axiom,
    ! [Na: nat] : aa(int,int,aa(int,fun(int,int),divide_divide(int),bit_se2239418461657761734s_mask(int,Na)),aa(num,int,numeral_numeral(int),bit0(one2))) = bit_se2239418461657761734s_mask(int,aa(nat,nat,minus_minus(nat,Na),one_one(nat))) ).

% mask_half_int
tff(fact_4196_mask__int__def,axiom,
    ! [Na: nat] : bit_se2239418461657761734s_mask(int,Na) = aa(int,int,minus_minus(int,aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)),one_one(int)) ).

% mask_int_def
tff(fact_4197_mask__nat__def,axiom,
    ! [Na: nat] : bit_se2239418461657761734s_mask(nat,Na) = aa(nat,nat,minus_minus(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),one_one(nat)) ).

% mask_nat_def
tff(fact_4198_neg__mask__is__div,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Na: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),aa(word(A),word(A),bit_ri4277139882892585799ns_not(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)),W),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) ) ).

% neg_mask_is_div
tff(fact_4199_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [Na: nat,K: int] :
      ( ( bit_se2584673776208193580ke_bit(int,Na,K) = bit_se2239418461657761734s_mask(int,Na) )
    <=> aa(int,$o,dvd_dvd(int,aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)),aa(int,int,plus_plus(int,K),one_one(int))) ) ).

% take_bit_eq_mask_iff_exp_dvd
tff(fact_4200_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,
    ! [X: nat,Y: nat] :
      ( ( vEBT_V8646137997579335489_i_l_d(X) = Y )
     => ( accp(nat,vEBT_V5144397997797733112_d_rel,X)
       => ( ( ( X = zero_zero(nat) )
           => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) )
             => ~ accp(nat,vEBT_V5144397997797733112_d_rel,zero_zero(nat)) ) )
         => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) )
               => ~ accp(nat,vEBT_V5144397997797733112_d_rel,aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [Va2: nat] :
                  ( ( X = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
                 => ( ( Y = $ite(
                          aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),
                          aa(nat,nat,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,plus_plus(nat,aa(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,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,plus_plus(nat,aa(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,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),vEBT_V8646137997579335489_i_l_d(half))) ) ) )
                   => ~ accp(nat,vEBT_V5144397997797733112_d_rel,aa(nat,nat,suc,aa(nat,nat,suc,Va2))) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.pelims
tff(fact_4201_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,
    ! [X: nat,Y: nat] :
      ( ( vEBT_V8346862874174094_d_u_p(X) = Y )
     => ( accp(nat,vEBT_V1247956027447740395_p_rel,X)
       => ( ( ( X = zero_zero(nat) )
           => ( ( Y = aa(num,nat,numeral_numeral(nat),bit1(one2)) )
             => ~ accp(nat,vEBT_V1247956027447740395_p_rel,zero_zero(nat)) ) )
         => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = aa(num,nat,numeral_numeral(nat),bit1(one2)) )
               => ~ accp(nat,vEBT_V1247956027447740395_p_rel,aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [Va2: nat] :
                  ( ( X = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
                 => ( ( Y = $ite(
                          aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),
                          aa(nat,nat,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,plus_plus(nat,aa(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,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),aa(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,plus_plus(nat,aa(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,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),aa(nat,nat,plus_plus(nat,vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) ) ) )
                   => ~ accp(nat,vEBT_V1247956027447740395_p_rel,aa(nat,nat,suc,aa(nat,nat,suc,Va2))) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.pelims
tff(fact_4202_VEBT__internal_OTb_Opelims,axiom,
    ! [X: nat,Y: int] :
      ( ( vEBT_VEBT_Tb(X) = Y )
     => ( accp(nat,vEBT_VEBT_Tb_rel2,X)
       => ( ( ( X = zero_zero(nat) )
           => ( ( Y = aa(num,int,numeral_numeral(int),bit1(one2)) )
             => ~ accp(nat,vEBT_VEBT_Tb_rel2,zero_zero(nat)) ) )
         => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = aa(num,int,numeral_numeral(int),bit1(one2)) )
               => ~ accp(nat,vEBT_VEBT_Tb_rel2,aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [N: nat] :
                  ( ( X = aa(nat,nat,suc,aa(nat,nat,suc,N)) )
                 => ( ( Y = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N),aa(int,int,plus_plus(int,aa(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,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,plus_plus(int,aa(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,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))))))))) )
                   => ~ accp(nat,vEBT_VEBT_Tb_rel2,aa(nat,nat,suc,aa(nat,nat,suc,N))) ) ) ) ) ) ) ).

% VEBT_internal.Tb.pelims
tff(fact_4203_VEBT__internal_OTb_H_Opelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( vEBT_VEBT_Tb2(X) = Y )
     => ( accp(nat,vEBT_VEBT_Tb_rel,X)
       => ( ( ( X = zero_zero(nat) )
           => ( ( Y = aa(num,nat,numeral_numeral(nat),bit1(one2)) )
             => ~ accp(nat,vEBT_VEBT_Tb_rel,zero_zero(nat)) ) )
         => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = aa(num,nat,numeral_numeral(nat),bit1(one2)) )
               => ~ accp(nat,vEBT_VEBT_Tb_rel,aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [N: nat] :
                  ( ( X = aa(nat,nat,suc,aa(nat,nat,suc,N)) )
                 => ( ( Y = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N),aa(nat,nat,plus_plus(nat,aa(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,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,plus_plus(nat,aa(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,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))))))))) )
                   => ~ accp(nat,vEBT_VEBT_Tb_rel,aa(nat,nat,suc,aa(nat,nat,suc,N))) ) ) ) ) ) ) ).

% VEBT_internal.Tb'.pelims
tff(fact_4204_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),K))
    <=> aa(int,$o,ord_less(int,K),zero_zero(int)) ) ).

% not_nonnegative_int_iff
tff(fact_4205_not__negative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,ord_less(int,aa(int,int,bit_ri4277139882892585799ns_not(int),K)),zero_zero(int))
    <=> aa(int,$o,ord_less_eq(int,zero_zero(int)),K) ) ).

% not_negative_int_iff
tff(fact_4206_word__no__log__defs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num] : aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),A3))) ) ).

% word_no_log_defs(1)
tff(fact_4207_word__of__int__not__numeral__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: num] : aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Bin))) = aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Bin))),one_one(word(A))) ) ).

% word_of_int_not_numeral_eq
tff(fact_4208_word__no__log__defs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num] : aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3)))) ) ).

% word_no_log_defs(2)
tff(fact_4209_test__bit__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [U: word(A),V2: word(A)] :
          ( ( aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),U) = aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),V2) )
        <=> ( U = V2 ) ) ) ).

% test_bit_eq_iff
tff(fact_4210_word__eqD,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [U: word(A),V2: word(A),X: nat] :
          ( ( U = V2 )
         => ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),U),X)
          <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),V2),X) ) ) ) ).

% word_eqD
tff(fact_4211_bang__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ( X = Y )
        <=> ! [N3: nat] :
              ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),X),N3)
            <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),N3) ) ) ) ).

% bang_eq
tff(fact_4212_word__wi__log__defs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: int] : aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(int,word(A),ring_1_of_int(word(A)),A3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_ri4277139882892585799ns_not(int),A3)) ) ).

% word_wi_log_defs(1)
tff(fact_4213_not__int__def,axiom,
    ! [K: int] : aa(int,int,bit_ri4277139882892585799ns_not(int),K) = aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),K)),one_one(int)) ).

% not_int_def
tff(fact_4214_and__not__numerals_I1_J,axiom,
    aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = zero_zero(int) ).

% and_not_numerals(1)
tff(fact_4215_word__not__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),A3) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(word(A),int,semiring_1_unsigned(A,int),A3))) ) ).

% word_not_def
tff(fact_4216_not__int__div__2,axiom,
    ! [K: int] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,bit_ri4277139882892585799ns_not(int),K)),aa(num,int,numeral_numeral(int),bit0(one2))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))) ).

% not_int_div_2
tff(fact_4217_even__not__iff__int,axiom,
    ! [K: int] :
      ( aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,bit_ri4277139882892585799ns_not(int),K))
    <=> ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K) ) ).

% even_not_iff_int
tff(fact_4218_and__not__numerals_I4_J,axiom,
    ! [M: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),bit0(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(num,int,numeral_numeral(int),bit0(M)) ).

% and_not_numerals(4)
tff(fact_4219_and__not__numerals_I2_J,axiom,
    ! [Na: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(Na)))) = one_one(int) ).

% and_not_numerals(2)
tff(fact_4220_bit__minus__int__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(int,int,uminus_uminus(int),K)),Na)
    <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,minus_minus(int,K),one_one(int)))),Na) ) ).

% bit_minus_int_iff
tff(fact_4221_and__not__numerals_I5_J,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),bit0(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(Na)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Na)))) ).

% and_not_numerals(5)
tff(fact_4222_and__not__numerals_I7_J,axiom,
    ! [M: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),bit1(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(num,int,numeral_numeral(int),bit0(M)) ).

% and_not_numerals(7)
tff(fact_4223_and__not__numerals_I3_J,axiom,
    ! [Na: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit1(Na)))) = zero_zero(int) ).

% and_not_numerals(3)
tff(fact_4224_and__not__numerals_I9_J,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),bit1(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit1(Na)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Na)))) ).

% and_not_numerals(9)
tff(fact_4225_and__not__numerals_I6_J,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),bit0(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit1(Na)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Na)))) ).

% and_not_numerals(6)
tff(fact_4226_and__not__numerals_I8_J,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),bit1(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(Na)))) = aa(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),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Na))))) ).

% and_not_numerals(8)
tff(fact_4227_not__int__rec,axiom,
    ! [K: int] : aa(int,int,bit_ri4277139882892585799ns_not(int),K) = aa(int,int,plus_plus(int,aa($o,int,zero_neq_one_of_bool(int),aa(int,$o,dvd_dvd(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(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))))) ).

% not_int_rec
tff(fact_4228_VEBT__internal_OT__vebt__buildupi_Opelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( vEBT_V441764108873111860ildupi(X) = Y )
     => ( accp(nat,vEBT_V2957053500504383685pi_rel,X)
       => ( ( ( X = zero_zero(nat) )
           => ( ( Y = aa(nat,nat,suc,zero_zero(nat)) )
             => ~ accp(nat,vEBT_V2957053500504383685pi_rel,zero_zero(nat)) ) )
         => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = aa(nat,nat,suc,zero_zero(nat)) )
               => ~ accp(nat,vEBT_V2957053500504383685pi_rel,aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [N: nat] :
                  ( ( X = aa(nat,nat,suc,aa(nat,nat,suc,N)) )
                 => ( ( Y = $ite(aa(nat,$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,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,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,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,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,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,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,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,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)))))))))))) )
                   => ~ accp(nat,vEBT_V2957053500504383685pi_rel,aa(nat,nat,suc,aa(nat,nat,suc,N))) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.pelims
tff(fact_4229_int__not__code_I1_J,axiom,
    aa(int,int,bit_ri4277139882892585799ns_not(int),zero_zero(int)) = aa(int,int,uminus_uminus(int),one_one(int)) ).

% int_not_code(1)
tff(fact_4230_xor__int__unfold,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = $ite(
        K = aa(int,int,uminus_uminus(int),one_one(int)),
        aa(int,int,bit_ri4277139882892585799ns_not(int),L),
        $ite(
          L = aa(int,int,uminus_uminus(int),one_one(int)),
          aa(int,int,bit_ri4277139882892585799ns_not(int),K),
          $ite(
            K = zero_zero(int),
            L,
            $ite(L = zero_zero(int),K,aa(int,int,plus_plus(int,aa(int,int,abs_abs(int),aa(int,int,minus_minus(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),modulo_modulo(int),L),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))),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2))))))) ) ) ) ).

% xor_int_unfold
tff(fact_4231_httI__TBOUND,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),Ta: nat] :
      ( hoare_hoare_triple(A,P,C2,Q)
     => ( time_TBOUND(A,C2,Ta)
       => time_htt(A,P,C2,Q,Ta) ) ) ).

% httI_TBOUND
tff(fact_4232_xor_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),zero_zero(A)) = A3 ) ).

% xor.right_neutral
tff(fact_4233_xor_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),zero_zero(A)),A3) = A3 ) ).

% xor.left_neutral
tff(fact_4234_xor__self__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),A3) = zero_zero(A) ) ).

% xor_self_eq
tff(fact_4235_bit_Oxor__self,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),X) = zero_zero(A) ) ).

% bit.xor_self
tff(fact_4236_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(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,ord_less_eq(int,zero_zero(int)),K)
      <=> aa(int,$o,ord_less_eq(int,zero_zero(int)),L) ) ) ).

% xor_nonnegative_int_iff
tff(fact_4237_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(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,ord_less(int,K),zero_zero(int))
        <=> aa(int,$o,ord_less(int,L),zero_zero(int)) ) ) ).

% xor_negative_int_iff
tff(fact_4238_bit_Oxor__cancel__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),aa(A,A,bit_ri4277139882892585799ns_not(A),X)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.xor_cancel_right
tff(fact_4239_bit_Oxor__cancel__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),X) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.xor_cancel_left
tff(fact_4240_bit_Oxor__one__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,bit_ri4277139882892585799ns_not(A),X) ) ).

% bit.xor_one_right
tff(fact_4241_bit_Oxor__one__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,uminus_uminus(A),one_one(A))),X) = aa(A,A,bit_ri4277139882892585799ns_not(A),X) ) ).

% bit.xor_one_left
tff(fact_4242_xor__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),bit0(X))),aa(num,A,numeral_numeral(A),bit0(Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y))) ) ).

% xor_numerals(3)
tff(fact_4243_xor__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),bit1(X))),one_one(A)) = aa(num,A,numeral_numeral(A),bit0(X)) ) ).

% xor_numerals(8)
tff(fact_4244_xor__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),bit0(X))),one_one(A)) = aa(num,A,numeral_numeral(A),bit1(X)) ) ).

% xor_numerals(5)
tff(fact_4245_xor__numerals_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),one_one(A)),aa(num,A,numeral_numeral(A),bit1(Y))) = aa(num,A,numeral_numeral(A),bit0(Y)) ) ).

% xor_numerals(2)
tff(fact_4246_xor__numerals_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(Y))) = aa(num,A,numeral_numeral(A),bit1(Y)) ) ).

% xor_numerals(1)
tff(fact_4247_xor__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),bit1(X))),aa(num,A,numeral_numeral(A),bit1(Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y))) ) ).

% xor_numerals(7)
tff(fact_4248_xor__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),bit1(X))),aa(num,A,numeral_numeral(A),bit0(Y))) = aa(A,A,plus_plus(A,one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% xor_numerals(6)
tff(fact_4249_xor__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),bit0(X))),aa(num,A,numeral_numeral(A),bit1(Y))) = aa(A,A,plus_plus(A,one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% xor_numerals(4)
tff(fact_4250_int__xor__code_I2_J,axiom,
    ! [I: int] : aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),I),zero_zero(int)) = I ).

% int_xor_code(2)
tff(fact_4251_int__xor__code_I1_J,axiom,
    ! [J: int] : aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),zero_zero(int)),J) = J ).

% int_xor_code(1)
tff(fact_4252_of__nat__xor__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),Na)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Na)) ) ).

% of_nat_xor_eq
tff(fact_4253_unsigned__xor__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_se359711467146920520ations(A) )
     => ! [V2: word(B),W: word(B)] : aa(word(B),A,semiring_1_unsigned(B,A),aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),bit_se5824344971392196577ns_xor(word(B)),V2),W)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(word(B),A,semiring_1_unsigned(B,A),V2)),aa(word(B),A,semiring_1_unsigned(B,A),W)) ) ).

% unsigned_xor_eq
tff(fact_4254_signed__xor__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_ri3973907225187159222ations(A) )
     => ! [V2: word(B),W: word(B)] : aa(word(B),A,ring_1_signed(B,A),aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),bit_se5824344971392196577ns_xor(word(B)),V2),W)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(word(B),A,ring_1_signed(B,A),V2)),aa(word(B),A,ring_1_signed(B,A),W)) ) ).

% signed_xor_eq
tff(fact_4255_XOR__lower,axiom,
    ! [X: int,Y: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),X)
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Y)
       => aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),X),Y)) ) ) ).

% XOR_lower
tff(fact_4256_even__xor__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),B3))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
          <=> aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),B3) ) ) ) ).

% even_xor_iff
tff(fact_4257_XOR__upper,axiom,
    ! [X: int,Na: nat,Y: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),X)
     => ( aa(int,$o,ord_less(int,X),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))
       => ( aa(int,$o,ord_less(int,Y),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))
         => aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),X),Y)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ) ) ).

% XOR_upper
tff(fact_4258_test__bit__int__code_I1_J,axiom,
    ! [Na: nat] : ~ aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),zero_zero(int)),Na) ).

% test_bit_int_code(1)
tff(fact_4259_int__and__code_I1_J,axiom,
    ! [J: int] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),zero_zero(int)),J) = zero_zero(int) ).

% int_and_code(1)
tff(fact_4260_int__and__code_I2_J,axiom,
    ! [I: int] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),I),zero_zero(int)) = zero_zero(int) ).

% int_and_code(2)
tff(fact_4261_xor__int__rec,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = aa(int,int,
        plus_plus(int,
          aa($o,int,zero_neq_one_of_bool(int),
            ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K) != ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),L))),
        aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2)))))) ).

% xor_int_rec
tff(fact_4262_one__xor__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),one_one(A)),A3) = aa(A,A,minus_minus(A,aa(A,A,plus_plus(A,A3),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)))),aa($o,A,zero_neq_one_of_bool(A),~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3))) ) ).

% one_xor_eq
tff(fact_4263_xor__one__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),one_one(A)) = aa(A,A,minus_minus(A,aa(A,A,plus_plus(A,A3),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)))),aa($o,A,zero_neq_one_of_bool(A),~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3))) ) ).

% xor_one_eq
tff(fact_4264_htt__htD,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),Ta: nat] :
      ( time_htt(A,P,C2,Q,Ta)
     => hoare_hoare_triple(A,P,C2,Q) ) ).

% htt_htD
tff(fact_4265_neg__mask__add__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Na: nat] : aa(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)),X),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),Na)))),aa(word(A),word(A),minus_minus(word(A),aa(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),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),X),bit_se2239418461657761734s_mask(word(A),Na)) ) ).

% neg_mask_add_mask
tff(fact_4266_monoseq__minus,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: fun(nat,A)] :
          ( topological_monoseq(A,A3)
         => topological_monoseq(A,aTP_Lamp_dj(fun(nat,A),fun(nat,A),A3)) ) ) ).

% monoseq_minus
tff(fact_4267_word__bitwise__1__simps_I13_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3))),one_one(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3))),one_one(int))) ) ).

% word_bitwise_1_simps(13)
tff(fact_4268_word__bitwise__1__simps_I11_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),one_one(word(A))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3)))) ) ).

% word_bitwise_1_simps(11)
tff(fact_4269_or_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),zero_zero(A)) = A3 ) ).

% or.right_neutral
tff(fact_4270_or_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),zero_zero(A)),A3) = A3 ) ).

% or.left_neutral
tff(fact_4271_word__ao__absorbs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Y),X)) = X ) ).

% word_ao_absorbs(1)
tff(fact_4272_word__ao__absorbs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Y),X)) = X ) ).

% word_ao_absorbs(2)
tff(fact_4273_word__ao__absorbs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),X),Y)) = X ) ).

% word_ao_absorbs(3)
tff(fact_4274_word__ao__absorbs_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Y),X)),X) = X ) ).

% word_ao_absorbs(4)
tff(fact_4275_word__ao__absorbs_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A)] : 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)),Y),X)),X) = X ) ).

% word_ao_absorbs(5)
tff(fact_4276_word__ao__absorbs_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),Y)) = X ) ).

% word_ao_absorbs(6)
tff(fact_4277_word__ao__absorbs_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] : 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)),X),Y)),X) = X ) ).

% word_ao_absorbs(7)
tff(fact_4278_word__ao__absorbs_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),Y)),X) = X ) ).

% word_ao_absorbs(8)
tff(fact_4279_bit_Odisj__one__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,uminus_uminus(A),one_one(A))),X) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_one_left
tff(fact_4280_bit_Odisj__one__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_one_right
tff(fact_4281_word__plus__and__or,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] : aa(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)),X),Y)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),X),Y)) = aa(word(A),word(A),plus_plus(word(A),X),Y) ) ).

% word_plus_and_or
tff(fact_4282_word__bitwise__m1__simps_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),X) = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ).

% word_bitwise_m1_simps(4)
tff(fact_4283_word__or__max,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),X),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ).

% word_or_max
tff(fact_4284_or__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),bit1(X))),one_one(A)) = aa(num,A,numeral_numeral(A),bit1(X)) ) ).

% or_numerals(8)
tff(fact_4285_or__numerals_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),one_one(A)),aa(num,A,numeral_numeral(A),bit1(Y))) = aa(num,A,numeral_numeral(A),bit1(Y)) ) ).

% or_numerals(2)
tff(fact_4286_bit_Odisj__cancel__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),X) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_cancel_left
tff(fact_4287_bit_Odisj__cancel__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),aa(A,A,bit_ri4277139882892585799ns_not(A),X)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_cancel_right
tff(fact_4288_word__or__not,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),X),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),X)) = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ).

% word_or_not
tff(fact_4289_word__bitwise__m1__simps_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),X),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) = aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),X) ) ).

% word_bitwise_m1_simps(7)
tff(fact_4290_word__bitwise__m1__simps_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),X) = aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),X) ) ).

% word_bitwise_m1_simps(6)
tff(fact_4291_or__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),bit0(X))),aa(num,A,numeral_numeral(A),bit0(Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y))) ) ).

% or_numerals(3)
tff(fact_4292_or__numerals_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(Y))) = aa(num,A,numeral_numeral(A),bit1(Y)) ) ).

% or_numerals(1)
tff(fact_4293_or__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),bit0(X))),one_one(A)) = aa(num,A,numeral_numeral(A),bit1(X)) ) ).

% or_numerals(5)
tff(fact_4294_word__no__log__defs_I11_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(num,int,numeral_numeral(int),A3)),aa(num,int,numeral_numeral(int),B3))) ) ).

% word_no_log_defs(11)
tff(fact_4295_xor__nat__numerals_I4_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(num,nat,numeral_numeral(nat),bit1(X))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),bit0(X)) ).

% xor_nat_numerals(4)
tff(fact_4296_xor__nat__numerals_I3_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(num,nat,numeral_numeral(nat),bit0(X))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),bit1(X)) ).

% xor_nat_numerals(3)
tff(fact_4297_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_4298_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_4299_word__bitwise__1__simps_I10_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),one_one(word(A))),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),one_one(int)),aa(num,int,numeral_numeral(int),B3))) ) ).

% word_bitwise_1_simps(10)
tff(fact_4300_word__bitwise__1__simps_I12_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),one_one(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(num,int,numeral_numeral(int),A3)),one_one(int))) ) ).

% word_bitwise_1_simps(12)
tff(fact_4301_word__no__log__defs_I14_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3)))) ) ).

% word_no_log_defs(14)
tff(fact_4302_word__no__log__defs_I13_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3))),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3))),aa(num,int,numeral_numeral(int),B3))) ) ).

% word_no_log_defs(13)
tff(fact_4303_word__no__log__defs_I12_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(num,int,numeral_numeral(int),A3)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3)))) ) ).

% word_no_log_defs(12)
tff(fact_4304_or__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),bit0(X))),aa(num,A,numeral_numeral(A),bit1(Y))) = aa(A,A,plus_plus(A,one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% or_numerals(4)
tff(fact_4305_or__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),bit1(X))),aa(num,A,numeral_numeral(A),bit0(Y))) = aa(A,A,plus_plus(A,one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% or_numerals(6)
tff(fact_4306_or__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),bit1(X))),aa(num,A,numeral_numeral(A),bit1(Y))) = aa(A,A,plus_plus(A,one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% or_numerals(7)
tff(fact_4307_word__xor__nth,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),X),Y)),Na)
        <=> ~ ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),X),Na)
            <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),Na) ) ) ) ).

% word_xor_nth
tff(fact_4308_word__xor__and__or,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),X),Y) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),aa(word(A),word(A),bit_ri4277139882892585799ns_not(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),bit_ri4277139882892585799ns_not(word(A)),X)),Y)) ) ).

% word_xor_and_or
tff(fact_4309_word__log__esimps_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),X),zero_zero(word(A))) = X ) ).

% word_log_esimps(5)
tff(fact_4310_word__log__esimps_I11_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),zero_zero(word(A))),X) = X ) ).

% word_log_esimps(11)
tff(fact_4311_word__bw__same_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),X),X) = zero_zero(word(A)) ) ).

% word_bw_same(3)
tff(fact_4312_word__bw__lcs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A),Z: 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_se5824344971392196577ns_xor(word(A)),X),Z)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Y),Z)) ) ).

% word_bw_lcs(3)
tff(fact_4313_word__bw__comms_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),X),Y) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Y),X) ) ).

% word_bw_comms(3)
tff(fact_4314_word__bw__assocs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A),Z: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),X),Y)),Z) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Y),Z)) ) ).

% word_bw_assocs(3)
tff(fact_4315_swap__with__xor,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),A3: word(A),B3: word(A),Y: word(A),Z: word(A)] :
          ( ( X = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),A3),B3) )
         => ( ( Y = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),B3),X) )
           => ( ( Z = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),X),Y) )
             => ( ( Z = B3 )
                & ( Y = A3 ) ) ) ) ) ) ).

% swap_with_xor
tff(fact_4316_le__word__or1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [C2: word(A),Y: word(A)] : aa(word(A),$o,ord_less_eq(word(A),C2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Y),C2)) ) ).

% le_word_or1
tff(fact_4317_le__word__or2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] : aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),X),Y)) ) ).

% le_word_or2
tff(fact_4318_word__combine__masks,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),M: word(A),Z: word(A),M6: word(A),Z4: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),M) = Z )
         => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),M6) = Z4 )
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),M),M6)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Z),Z4) ) ) ) ) ).

% word_combine_masks
tff(fact_4319_word__oa__dist2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A),Z: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Y),Z)) = 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)),X),Y)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),X),Z)) ) ).

% word_oa_dist2
tff(fact_4320_word__ao__equiv,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),W3: word(A)] :
          ( ( W = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),W),W3) )
        <=> ( W3 = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),W3) ) ) ) ).

% word_ao_equiv
tff(fact_4321_word__ao__dist2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A),Z: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Y),Z)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),Y)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),Z)) ) ).

% word_ao_dist2
tff(fact_4322_word__oa__dist,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A),Z: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),Y)),Z) = 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)),X),Z)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Y),Z)) ) ).

% word_oa_dist
tff(fact_4323_word__ao__dist,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A),Z: word(A)] : 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)),X),Y)),Z) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),Z)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Y),Z)) ) ).

% word_ao_dist
tff(fact_4324_leoa,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),X: word(A),Y: word(A)] :
          ( ( W = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),X),Y) )
         => ( Y = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),Y) ) ) ) ).

% leoa
tff(fact_4325_leao,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W3: word(A),X6: word(A),Y5: word(A)] :
          ( ( W3 = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X6),Y5) )
         => ( X6 = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),X6),W3) ) ) ) ).

% leao
tff(fact_4326_signed__or__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_ri3973907225187159222ations(A) )
     => ! [V2: word(B),W: word(B)] : aa(word(B),A,ring_1_signed(B,A),aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),bit_se1065995026697491101ons_or(word(B)),V2),W)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(word(B),A,ring_1_signed(B,A),V2)),aa(word(B),A,ring_1_signed(B,A),W)) ) ).

% signed_or_eq
tff(fact_4327_unsigned__or__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_se359711467146920520ations(A) )
     => ! [V2: word(B),W: word(B)] : aa(word(B),A,semiring_1_unsigned(B,A),aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),bit_se1065995026697491101ons_or(word(B)),V2),W)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(word(B),A,semiring_1_unsigned(B,A),V2)),aa(word(B),A,semiring_1_unsigned(B,A),W)) ) ).

% unsigned_or_eq
tff(fact_4328_word__bw__assocs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A),Z: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),X),Y)),Z) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Y),Z)) ) ).

% word_bw_assocs(2)
tff(fact_4329_word__bw__comms_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),X),Y) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Y),X) ) ).

% word_bw_comms(2)
tff(fact_4330_word__bw__same_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),X),X) = X ) ).

% word_bw_same(2)
tff(fact_4331_word__bw__lcs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A),Z: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),X),Z)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Y),Z)) ) ).

% word_bw_lcs(2)
tff(fact_4332_of__nat__or__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),Na)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Na)) ) ).

% of_nat_or_eq
tff(fact_4333_ucast__or__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [X: word(B),Y: word(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)),bit_se1065995026697491101ons_or(word(B)),X),Y)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),X)),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Y)) ) ).

% ucast_or_distrib
tff(fact_4334_word__log__esimps_I9_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),zero_zero(word(A))),X) = X ) ).

% word_log_esimps(9)
tff(fact_4335_word__log__esimps_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),X),zero_zero(word(A))) = X ) ).

% word_log_esimps(3)
tff(fact_4336_word__or__zero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),A3),B3) = zero_zero(word(A)) )
        <=> ( ( A3 = zero_zero(word(A)) )
            & ( B3 = zero_zero(word(A)) ) ) ) ) ).

% word_or_zero
tff(fact_4337_bit_Odisj__zero__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),zero_zero(A)) = X ) ).

% bit.disj_zero_right
tff(fact_4338_or__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),B3) = zero_zero(A) )
        <=> ( ( A3 = zero_zero(A) )
            & ( B3 = zero_zero(A) ) ) ) ) ).

% or_eq_0_iff
tff(fact_4339_word__or__nth,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),X),Y)),Na)
        <=> ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),X),Na)
            | aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),Na) ) ) ) ).

% word_or_nth
tff(fact_4340_word__ao__nth,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A),Na: nat] :
          ( ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),X),Y)),Na)
          <=> ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),X),Na)
              | aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),Na) ) )
          & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),Y)),Na)
          <=> ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),X),Na)
              & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),Na) ) ) ) ) ).

% word_ao_nth
tff(fact_4341_uint__xor,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: 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)),bit_se5824344971392196577ns_xor(word(A)),X),Y)) = aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ).

% uint_xor
tff(fact_4342_word__not__dist_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] : aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),X),Y)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),X)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Y)) ) ).

% word_not_dist(1)
tff(fact_4343_word__not__dist_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] : aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),Y)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),X)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Y)) ) ).

% word_not_dist(2)
tff(fact_4344_word__wi__log__defs_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: int,B3: int] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),aa(int,word(A),ring_1_of_int(word(A)),A3)),aa(int,word(A),ring_1_of_int(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),A3),B3)) ) ).

% word_wi_log_defs(4)
tff(fact_4345_word__ops__nth__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,X: word(A),Y: word(A)] :
          ( aa(nat,$o,ord_less(nat,Na),size_size(word(A),X))
         => ( ( 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)),X),Y)),Na)
            <=> ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),X),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)),X),Y)),Na)
            <=> ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),X),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)),X),Y)),Na)
            <=> ~ ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),X),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)),X)),Na)
            <=> ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),X),Na) ) ) ) ) ).

% word_ops_nth_size
tff(fact_4346_word__plus__and__or__coroll,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),Y) = zero_zero(word(A)) )
         => ( aa(word(A),word(A),plus_plus(word(A),X),Y) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),X),Y) ) ) ) ).

% word_plus_and_or_coroll
tff(fact_4347_word__xor__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),A3),B3) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(word(A),int,semiring_1_unsigned(A,int),A3)),aa(word(A),int,semiring_1_unsigned(A,int),B3))) ) ).

% word_xor_def
tff(fact_4348_xor__nat__def,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),Na) = nat2(aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),Na))) ).

% xor_nat_def
tff(fact_4349_or__not__mask__nop,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Na: nat] : 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)),X),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),Na)))),bit_se2239418461657761734s_mask(word(A),Na)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),bit_se2239418461657761734s_mask(word(A),Na)) ) ).

% or_not_mask_nop
tff(fact_4350_mask__or__not__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Na: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),bit_se2239418461657761734s_mask(word(A),Na))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),Na)))) = X ) ).

% mask_or_not_mask
tff(fact_4351_even__or__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),B3))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
            & aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),B3) ) ) ) ).

% even_or_iff
tff(fact_4352_bit_Ocomplement__unique,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),X) = zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),X) = aa(A,A,uminus_uminus(A),one_one(A)) )
           => ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),Y) = zero_zero(A) )
             => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),Y) = aa(A,A,uminus_uminus(A),one_one(A)) )
               => ( X = Y ) ) ) ) ) ) ).

% bit.complement_unique
tff(fact_4353_mask__subsume,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat,X: word(A),Y: word(A)] :
          ( aa(nat,$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)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Y),bit_se2239418461657761734s_mask(word(A),Na)))),aa(word(A),word(A),bit_ri4277139882892585799ns_not(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)),X),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),M))) ) ) ) ).

% mask_subsume
tff(fact_4354_bit_Ocompl__unique,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),Y) = zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),Y) = aa(A,A,uminus_uminus(A),one_one(A)) )
           => ( aa(A,A,bit_ri4277139882892585799ns_not(A),X) = Y ) ) ) ) ).

% bit.compl_unique
tff(fact_4355_mask__Suc__exp,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : bit_se2239418461657761734s_mask(A,aa(nat,nat,suc,Na)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)),bit_se2239418461657761734s_mask(A,Na)) ) ).

% mask_Suc_exp
tff(fact_4356_mask__Suc__double,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : 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))),bit_se2239418461657761734s_mask(A,Na))) ) ).

% mask_Suc_double
tff(fact_4357_or__one__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),one_one(A)) = aa(A,A,plus_plus(A,A3),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3))) ) ).

% or_one_eq
tff(fact_4358_one__or__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),one_one(A)),A3) = aa(A,A,plus_plus(A,A3),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3))) ) ).

% one_or_eq
tff(fact_4359_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,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(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_4360_xor__nat__rec,axiom,
    ! [M: nat,Na: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),Na) = aa(nat,nat,
        plus_plus(nat,
          aa($o,nat,zero_neq_one_of_bool(nat),
            ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M) != ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))),
        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_rec
tff(fact_4361_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,minus_minus(nat,aa(nat,nat,plus_plus(nat,Na),aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$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,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ).

% xor_Suc_0_eq
tff(fact_4362_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,minus_minus(nat,aa(nat,nat,plus_plus(nat,Na),aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$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,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ).

% Suc_0_xor_eq
tff(fact_4363_word__ops__lsb,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),X),Y)),zero_zero(nat))
          <=> ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),X),zero_zero(nat))
              | aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),zero_zero(nat)) ) )
          & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),Y)),zero_zero(nat))
          <=> ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),X),zero_zero(nat))
              & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),zero_zero(nat)) ) )
          & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),X),Y)),zero_zero(nat))
          <=> ~ ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),X),zero_zero(nat))
              <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),zero_zero(nat)) ) )
          & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),X)),zero_zero(nat))
          <=> ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),X),zero_zero(nat)) ) ) ) ).

% word_ops_lsb
tff(fact_4364_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_4365_word__bitwise__1__simps_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),one_one(word(A))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3)))) ) ).

% word_bitwise_1_simps(7)
tff(fact_4366_word__bitwise__1__simps_I9_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3))),one_one(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3))),one_one(int))) ) ).

% word_bitwise_1_simps(9)
tff(fact_4367_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(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,ord_less_eq(int,zero_zero(int)),K)
        & aa(int,$o,ord_less_eq(int,zero_zero(int)),L) ) ) ).

% or_nonnegative_int_iff
tff(fact_4368_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(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,ord_less(int,K),zero_zero(int))
        | aa(int,$o,ord_less(int,L),zero_zero(int)) ) ) ).

% or_negative_int_iff
tff(fact_4369_or__minus__numerals_I6_J,axiom,
    ! [Na: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit1(Na)))),one_one(int)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit1(Na))) ).

% or_minus_numerals(6)
tff(fact_4370_or__minus__numerals_I2_J,axiom,
    ! [Na: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit1(Na)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit1(Na))) ).

% or_minus_numerals(2)
tff(fact_4371_or__nat__numerals_I4_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(num,nat,numeral_numeral(nat),bit1(X))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),bit1(X)) ).

% or_nat_numerals(4)
tff(fact_4372_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_4373_word__no__log__defs_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),A3)),aa(num,int,numeral_numeral(int),B3))) ) ).

% word_no_log_defs(7)
tff(fact_4374_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_4375_or__nat__numerals_I3_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(num,nat,numeral_numeral(nat),bit0(X))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),bit1(X)) ).

% or_nat_numerals(3)
tff(fact_4376_word__bitwise__1__simps_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),one_one(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),A3)),one_one(int))) ) ).

% word_bitwise_1_simps(8)
tff(fact_4377_word__bitwise__1__simps_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),one_one(word(A))),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(num,int,numeral_numeral(int),B3))) ) ).

% word_bitwise_1_simps(6)
tff(fact_4378_word__no__log__defs_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),A3)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3)))) ) ).

% word_no_log_defs(8)
tff(fact_4379_word__no__log__defs_I9_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3))),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3))),aa(num,int,numeral_numeral(int),B3))) ) ).

% word_no_log_defs(9)
tff(fact_4380_word__no__log__defs_I10_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3)))) ) ).

% word_no_log_defs(10)
tff(fact_4381_and__minus__minus__numerals,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,minus_minus(int,aa(num,int,numeral_numeral(int),M)),one_one(int))),aa(int,int,minus_minus(int,aa(num,int,numeral_numeral(int),Na)),one_one(int)))) ).

% and_minus_minus_numerals
tff(fact_4382_or__minus__minus__numerals,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,minus_minus(int,aa(num,int,numeral_numeral(int),M)),one_one(int))),aa(int,int,minus_minus(int,aa(num,int,numeral_numeral(int),Na)),one_one(int)))) ).

% or_minus_minus_numerals
tff(fact_4383_int__or__code_I2_J,axiom,
    ! [I: int] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),I),zero_zero(int)) = I ).

% int_or_code(2)
tff(fact_4384_int__or__code_I1_J,axiom,
    ! [J: int] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),zero_zero(int)),J) = J ).

% int_or_code(1)
tff(fact_4385_or__nat__def,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),Na) = nat2(aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),Na))) ).

% or_nat_def
tff(fact_4386_abstract__boolean__algebra__sym__diff_Oxor__def,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A,Y: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Xor,X),Y) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,X),aa(A,A,Compl,Y))),aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,Compl,X)),Y)) ) ) ).

% abstract_boolean_algebra_sym_diff.xor_def
tff(fact_4387_abstract__boolean__algebra__sym__diff_Oxor__def2,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A,Y: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Xor,X),Y) = aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,aa(A,fun(A,A),Disj,X),Y)),aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,Compl,X)),aa(A,A,Compl,Y))) ) ) ).

% abstract_boolean_algebra_sym_diff.xor_def2
tff(fact_4388_abstract__boolean__algebra__sym__diff_Oxor__self,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Xor,X),X) = Zero ) ) ).

% abstract_boolean_algebra_sym_diff.xor_self
tff(fact_4389_abstract__boolean__algebra__sym__diff_Oxor__one__left,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Xor,One),X) = aa(A,A,Compl,X) ) ) ).

% abstract_boolean_algebra_sym_diff.xor_one_left
tff(fact_4390_abstract__boolean__algebra__sym__diff_Oxor__left__self,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A,Y: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Xor,X),aa(A,A,aa(A,fun(A,A),Xor,X),Y)) = Y ) ) ).

% abstract_boolean_algebra_sym_diff.xor_left_self
tff(fact_4391_abstract__boolean__algebra__sym__diff_Oxor__one__right,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Xor,X),One) = aa(A,A,Compl,X) ) ) ).

% abstract_boolean_algebra_sym_diff.xor_one_right
tff(fact_4392_abstract__boolean__algebra__sym__diff_Oxor__compl__left,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A,Y: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Xor,aa(A,A,Compl,X)),Y) = aa(A,A,Compl,aa(A,A,aa(A,fun(A,A),Xor,X),Y)) ) ) ).

% abstract_boolean_algebra_sym_diff.xor_compl_left
tff(fact_4393_abstract__boolean__algebra__sym__diff_Oxor__cancel__left,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Xor,aa(A,A,Compl,X)),X) = One ) ) ).

% abstract_boolean_algebra_sym_diff.xor_cancel_left
tff(fact_4394_abstract__boolean__algebra__sym__diff_Oxor__compl__right,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A,Y: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Xor,X),aa(A,A,Compl,Y)) = aa(A,A,Compl,aa(A,A,aa(A,fun(A,A),Xor,X),Y)) ) ) ).

% abstract_boolean_algebra_sym_diff.xor_compl_right
tff(fact_4395_abstract__boolean__algebra__sym__diff_Oconj__xor__distrib,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A,Y: A,Z: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Conj,X),aa(A,A,aa(A,fun(A,A),Xor,Y),Z)) = aa(A,A,aa(A,fun(A,A),Xor,aa(A,A,aa(A,fun(A,A),Conj,X),Y)),aa(A,A,aa(A,fun(A,A),Conj,X),Z)) ) ) ).

% abstract_boolean_algebra_sym_diff.conj_xor_distrib
tff(fact_4396_abstract__boolean__algebra__sym__diff_Oxor__cancel__right,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Xor,X),aa(A,A,Compl,X)) = One ) ) ).

% abstract_boolean_algebra_sym_diff.xor_cancel_right
tff(fact_4397_abstract__boolean__algebra__sym__diff_Oconj__xor__distrib2,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),Y: A,Z: A,X: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,aa(A,fun(A,A),Xor,Y),Z)),X) = aa(A,A,aa(A,fun(A,A),Xor,aa(A,A,aa(A,fun(A,A),Conj,Y),X)),aa(A,A,aa(A,fun(A,A),Conj,Z),X)) ) ) ).

% abstract_boolean_algebra_sym_diff.conj_xor_distrib2
tff(fact_4398_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),L)
     => aa(int,$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_4399_OR__lower,axiom,
    ! [X: int,Y: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),X)
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Y)
       => aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),X),Y)) ) ) ).

% OR_lower
tff(fact_4400_uint__or,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: 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)),bit_se1065995026697491101ons_or(word(A)),X),Y)) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ).

% uint_or
tff(fact_4401_word__wi__log__defs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: int,B3: int] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(int,word(A),ring_1_of_int(word(A)),A3)),aa(int,word(A),ring_1_of_int(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),A3),B3)) ) ).

% word_wi_log_defs(3)
tff(fact_4402_or__not__numerals_I1_J,axiom,
    aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(int,int,bit_ri4277139882892585799ns_not(int),zero_zero(int)) ).

% or_not_numerals(1)
tff(fact_4403_word__or__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),A3),B3) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(word(A),int,semiring_1_unsigned(A,int),A3)),aa(word(A),int,semiring_1_unsigned(A,int),B3))) ) ).

% word_or_def
tff(fact_4404_or__not__numerals_I2_J,axiom,
    ! [Na: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(Na)))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(Na))) ).

% or_not_numerals(2)
tff(fact_4405_or__not__numerals_I4_J,axiom,
    ! [M: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),bit0(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int)) ).

% or_not_numerals(4)
tff(fact_4406_or__not__numerals_I3_J,axiom,
    ! [Na: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit1(Na)))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(Na))) ).

% or_not_numerals(3)
tff(fact_4407_or__not__numerals_I7_J,axiom,
    ! [M: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),bit1(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(int,int,bit_ri4277139882892585799ns_not(int),zero_zero(int)) ).

% or_not_numerals(7)
tff(fact_4408_or__not__numerals_I6_J,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),bit0(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit1(Na)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Na)))) ).

% or_not_numerals(6)
tff(fact_4409_OR__upper,axiom,
    ! [X: int,Na: nat,Y: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),X)
     => ( aa(int,$o,ord_less(int,X),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))
       => ( aa(int,$o,ord_less(int,Y),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))
         => aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),X),Y)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ) ) ).

% OR_upper
tff(fact_4410_or__not__numerals_I5_J,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),bit0(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(Na)))) = aa(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),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Na))))) ).

% or_not_numerals(5)
tff(fact_4411_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,plus_plus(nat,Na),aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ).

% or_Suc_0_eq
tff(fact_4412_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,plus_plus(nat,Na),aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ).

% Suc_0_or_eq
tff(fact_4413_or__nat__rec,axiom,
    ! [M: nat,Na: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),Na) = aa(nat,nat,
        plus_plus(nat,
          aa($o,nat,zero_neq_one_of_bool(nat),
            ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M)
            | ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) ))),
        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_rec
tff(fact_4414_or__not__numerals_I9_J,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),bit1(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit1(Na)))) = aa(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),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Na))))) ).

% or_not_numerals(9)
tff(fact_4415_or__not__numerals_I8_J,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),bit1(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(Na)))) = aa(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),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Na))))) ).

% or_not_numerals(8)
tff(fact_4416_or__int__rec,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = aa(int,int,
        plus_plus(int,
          aa($o,int,zero_neq_one_of_bool(int),
            ( ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K)
            | ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),L) ))),
        aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2)))))) ).

% or_int_rec
tff(fact_4417_word__lsb__neg__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: num] :
          ( aa(word(A),$o,least_8051144512741203767sb_lsb(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Bin)))
        <=> ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Bin))) ) ) ).

% word_lsb_neg_numeral
tff(fact_4418_lsb__odd,axiom,
    ! [A: $tType] :
      ( least_6119777620449941438nt_lsb(A)
     => ! [X2: A] :
          ( aa(A,$o,least_8051144512741203767sb_lsb(A),X2)
        <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),X2) ) ) ).

% lsb_odd
tff(fact_4419_or__int__unfold,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = $ite(
        ( ( K = aa(int,int,uminus_uminus(int),one_one(int)) )
        | ( L = aa(int,int,uminus_uminus(int),one_one(int)) ) ),
        aa(int,int,uminus_uminus(int),one_one(int)),
        $ite(
          K = zero_zero(int),
          L,
          $ite(L = zero_zero(int),K,aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),ord_max(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),modulo_modulo(int),L),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))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2))))))) ) ) ).

% or_int_unfold
tff(fact_4420_word__lsb__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: num] :
          ( aa(word(A),$o,least_8051144512741203767sb_lsb(word(A)),aa(num,word(A),numeral_numeral(word(A)),Bin))
        <=> ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(num,int,numeral_numeral(int),Bin)) ) ) ).

% word_lsb_numeral
tff(fact_4421_int__lsb__numeral_I1_J,axiom,
    ~ aa(int,$o,least_8051144512741203767sb_lsb(int),zero_zero(int)) ).

% int_lsb_numeral(1)
tff(fact_4422_int__lsb__numeral_I2_J,axiom,
    aa(int,$o,least_8051144512741203767sb_lsb(int),one_one(int)) ).

% int_lsb_numeral(2)
tff(fact_4423_max_Oright__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3)),B3) = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) ) ).

% max.right_idem
tff(fact_4424_max_Oleft__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),A3),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) ) ).

% max.left_idem
tff(fact_4425_max_Oidem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),A3),A3) = A3 ) ).

% max.idem
tff(fact_4426_int__lsb__numeral_I6_J,axiom,
    ! [W: num] : ~ aa(int,$o,least_8051144512741203767sb_lsb(int),aa(num,int,numeral_numeral(int),bit0(W))) ).

% int_lsb_numeral(6)
tff(fact_4427_int__lsb__numeral_I3_J,axiom,
    aa(int,$o,least_8051144512741203767sb_lsb(int),aa(num,int,numeral_numeral(int),one2)) ).

% int_lsb_numeral(3)
tff(fact_4428_int__lsb__numeral_I7_J,axiom,
    ! [W: num] : aa(int,$o,least_8051144512741203767sb_lsb(int),aa(num,int,numeral_numeral(int),bit1(W))) ).

% int_lsb_numeral(7)
tff(fact_4429_max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,C2: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),ord_max(A),B3),C2)),A3)
        <=> ( aa(A,$o,ord_less_eq(A,B3),A3)
            & aa(A,$o,ord_less_eq(A,C2),A3) ) ) ) ).

% max.bounded_iff
tff(fact_4430_max_Oabsorb2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) = B3 ) ) ) ).

% max.absorb2
tff(fact_4431_max_Oabsorb1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) = A3 ) ) ) ).

% max.absorb1
tff(fact_4432_max__less__iff__conj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Z: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),Z)
        <=> ( aa(A,$o,ord_less(A,X),Z)
            & aa(A,$o,ord_less(A,Y),Z) ) ) ) ).

% max_less_iff_conj
tff(fact_4433_max_Oabsorb4,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) = B3 ) ) ) ).

% max.absorb4
tff(fact_4434_max_Oabsorb3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,B3),A3)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) = A3 ) ) ) ).

% max.absorb3
tff(fact_4435_max__bot,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),bot_bot(A)),X) = X ) ).

% max_bot
tff(fact_4436_max__bot2,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),X),bot_bot(A)) = X ) ).

% max_bot2
tff(fact_4437_int__lsb__numeral_I4_J,axiom,
    aa(int,$o,least_8051144512741203767sb_lsb(int),aa(int,int,uminus_uminus(int),one_one(int))) ).

% int_lsb_numeral(4)
tff(fact_4438_of__bool__or__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: $o,Q: $o] :
          aa($o,A,zero_neq_one_of_bool(A),
            ( (P)
            | (Q) )) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa($o,A,zero_neq_one_of_bool(A),(P))),aa($o,A,zero_neq_one_of_bool(A),(Q))) ) ).

% of_bool_or_iff
tff(fact_4439_max__0__1_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),X)),zero_zero(A)) = aa(num,A,numeral_numeral(A),X) ) ).

% max_0_1(4)
tff(fact_4440_max__0__1_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),zero_zero(A)),aa(num,A,numeral_numeral(A),X)) = aa(num,A,numeral_numeral(A),X) ) ).

% max_0_1(3)
tff(fact_4441_max__number__of_I1_J,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ord(A) )
     => ! [U: num,V2: 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),V2)) = $ite(aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)),aa(num,A,numeral_numeral(A),V2),aa(num,A,numeral_numeral(A),U)) ) ).

% max_number_of(1)
tff(fact_4442_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_4443_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_4444_max__0__1_I6_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),X)),one_one(A)) = aa(num,A,numeral_numeral(A),X) ) ).

% max_0_1(6)
tff(fact_4445_max__0__1_I5_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),one_one(A)),aa(num,A,numeral_numeral(A),X)) = aa(num,A,numeral_numeral(A),X) ) ).

% max_0_1(5)
tff(fact_4446_int__lsb__numeral_I8_J,axiom,
    ! [W: num] : ~ aa(int,$o,least_8051144512741203767sb_lsb(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(W)))) ).

% int_lsb_numeral(8)
tff(fact_4447_int__lsb__numeral_I5_J,axiom,
    aa(int,$o,least_8051144512741203767sb_lsb(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),one2))) ).

% int_lsb_numeral(5)
tff(fact_4448_int__lsb__numeral_I9_J,axiom,
    ! [W: num] : aa(int,$o,least_8051144512741203767sb_lsb(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit1(W)))) ).

% int_lsb_numeral(9)
tff(fact_4449_max__number__of_I4_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: 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),V2))) = $ite(aa(A,$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),V2))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))) ) ).

% max_number_of(4)
tff(fact_4450_max__number__of_I3_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: 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),V2)) = $ite(aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V2)),aa(num,A,numeral_numeral(A),V2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))) ) ).

% max_number_of(3)
tff(fact_4451_max__number__of_I2_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: 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),V2))) = $ite(aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2)),aa(num,A,numeral_numeral(A),U)) ) ).

% max_number_of(2)
tff(fact_4452_max__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X2: A,Xa: A] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),X2),Xa) = $ite(aa(A,$o,ord_less_eq(A,X2),Xa),Xa,X2) ) ).

% max_def_raw
tff(fact_4453_of__int__max,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: int,Y: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),ord_max(int),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(int,A,ring_1_of_int(A),X)),aa(int,A,ring_1_of_int(A),Y)) ) ).

% of_int_max
tff(fact_4454_max_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,C2),B3)
         => aa(A,$o,ord_less(A,C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3)) ) ) ).

% max.strict_coboundedI2
tff(fact_4455_max_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,C2),A3)
         => aa(A,$o,ord_less(A,C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3)) ) ) ).

% max.strict_coboundedI1
tff(fact_4456_max_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,B3),A3)
        <=> ( ( A3 = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) )
            & ( A3 != B3 ) ) ) ) ).

% max.strict_order_iff
tff(fact_4457_max_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,C2: A,A3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),ord_max(A),B3),C2)),A3)
         => ~ ( aa(A,$o,ord_less(A,B3),A3)
             => ~ aa(A,$o,ord_less(A,C2),A3) ) ) ) ).

% max.strict_boundedE
tff(fact_4458_less__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,X: A,Y: A] :
          ( aa(A,$o,ord_less(A,Z),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y))
        <=> ( aa(A,$o,ord_less(A,Z),X)
            | aa(A,$o,ord_less(A,Z),Y) ) ) ) ).

% less_max_iff_disj
tff(fact_4459_max_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,C2),B3)
         => aa(A,$o,ord_less_eq(A,C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3)) ) ) ).

% max.coboundedI2
tff(fact_4460_max_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,C2),A3)
         => aa(A,$o,ord_less_eq(A,C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3)) ) ) ).

% max.coboundedI1
tff(fact_4461_max_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
        <=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) = B3 ) ) ) ).

% max.absorb_iff2
tff(fact_4462_max_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
        <=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) = A3 ) ) ) ).

% max.absorb_iff1
tff(fact_4463_le__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,X: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,Z),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y))
        <=> ( aa(A,$o,ord_less_eq(A,Z),X)
            | aa(A,$o,ord_less_eq(A,Z),Y) ) ) ) ).

% le_max_iff_disj
tff(fact_4464_max_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] : aa(A,$o,ord_less_eq(A,B3),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3)) ) ).

% max.cobounded2
tff(fact_4465_max_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] : aa(A,$o,ord_less_eq(A,A3),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3)) ) ).

% max.cobounded1
tff(fact_4466_max_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
        <=> ( A3 = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) ) ) ) ).

% max.order_iff
tff(fact_4467_max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
         => ( aa(A,$o,ord_less_eq(A,C2),A3)
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),ord_max(A),B3),C2)),A3) ) ) ) ).

% max.boundedI
tff(fact_4468_max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,C2: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),ord_max(A),B3),C2)),A3)
         => ~ ( aa(A,$o,ord_less_eq(A,B3),A3)
             => ~ aa(A,$o,ord_less_eq(A,C2),A3) ) ) ) ).

% max.boundedE
tff(fact_4469_max_OorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) )
         => aa(A,$o,ord_less_eq(A,B3),A3) ) ) ).

% max.orderI
tff(fact_4470_max_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
         => ( A3 = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) ) ) ) ).

% max.orderE
tff(fact_4471_max_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A3: A,D2: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,C2),A3)
         => ( aa(A,$o,ord_less_eq(A,D2),B3)
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),ord_max(A),C2),D2)),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3)) ) ) ) ).

% max.mono
tff(fact_4472_max__absorb2,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,X),Y)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y) = Y ) ) ) ).

% max_absorb2
tff(fact_4473_max__absorb1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,ord_less_eq(A,Y),X)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y) = X ) ) ) ).

% max_absorb1
tff(fact_4474_max__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B3: A] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) = $ite(aa(A,$o,ord_less_eq(A,A3),B3),B3,A3) ) ).

% max_def
tff(fact_4475_sup__max,axiom,
    ! [A: $tType] :
      ( ( semilattice_sup(A)
        & linorder(A) )
     => ( sup_sup(A) = ord_max(A) ) ) ).

% sup_max
tff(fact_4476_sup__int__def,axiom,
    sup_sup(int) = ord_max(int) ).

% sup_int_def
tff(fact_4477_max_Oleft__commute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),B3),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C2)) = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),aa(A,A,aa(A,fun(A,A),ord_max(A),B3),C2)) ) ).

% max.left_commute
tff(fact_4478_max_Ocommute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) = aa(A,A,aa(A,fun(A,A),ord_max(A),B3),A3) ) ).

% max.commute
tff(fact_4479_max_Oassoc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3)),C2) = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),aa(A,A,aa(A,fun(A,A),ord_max(A),B3),C2)) ) ).

% max.assoc
tff(fact_4480_of__nat__max,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: nat,Y: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,semiring_1_of_nat(A),Y)) ) ).

% of_nat_max
tff(fact_4481_max__diff__distrib__left,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A,Y: A,Z: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),Z) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,minus_minus(A,X),Z)),aa(A,A,minus_minus(A,Y),Z)) ) ).

% max_diff_distrib_left
tff(fact_4482_max__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [X: A,Y: A,Z: A] : aa(A,A,plus_plus(A,X),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,plus_plus(A,X),Y)),aa(A,A,plus_plus(A,X),Z)) ) ).

% max_add_distrib_right
tff(fact_4483_max__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [X: A,Y: A,Z: A] : aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),Z) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,plus_plus(A,X),Z)),aa(A,A,plus_plus(A,Y),Z)) ) ).

% max_add_distrib_left
tff(fact_4484_lsb__int__def,axiom,
    ! [I: int] :
      ( aa(int,$o,least_8051144512741203767sb_lsb(int),I)
    <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),I),zero_zero(nat)) ) ).

% lsb_int_def
tff(fact_4485_word__lsb__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( aa(word(A),$o,least_8051144512741203767sb_lsb(word(A)),W)
        <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),zero_zero(nat)) ) ) ).

% word_lsb_alt
tff(fact_4486_word__lsb__1__0,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ( aa(word(A),$o,least_8051144512741203767sb_lsb(word(A)),one_one(word(A)))
        & ~ aa(word(B),$o,least_8051144512741203767sb_lsb(word(B)),zero_zero(word(B))) ) ) ).

% word_lsb_1_0
tff(fact_4487_bin__last__conv__lsb,axiom,
    aTP_Lamp_dk(int,$o) = least_8051144512741203767sb_lsb(int) ).

% bin_last_conv_lsb
tff(fact_4488_lsb__word__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X2: word(A)] :
          ( aa(word(A),$o,least_8051144512741203767sb_lsb(word(A)),X2)
        <=> ~ aa(word(A),$o,dvd_dvd(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),X2) ) ) ).

% lsb_word_eq
tff(fact_4489_word__lsb__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] :
          ( aa(word(A),$o,least_8051144512741203767sb_lsb(word(A)),A3)
        <=> ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(word(A),int,semiring_1_unsigned(A,int),A3)) ) ) ).

% word_lsb_def
tff(fact_4490_word__lsb__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( aa(word(A),$o,least_8051144512741203767sb_lsb(word(A)),W)
        <=> ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),W)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(nat) ) ) ) ).

% word_lsb_nat
tff(fact_4491_word__lsb__int,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( aa(word(A),$o,least_8051144512741203767sb_lsb(word(A)),W)
        <=> ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(word(A),int,semiring_1_unsigned(A,int),W)),aa(num,int,numeral_numeral(int),bit0(one2))) = one_one(int) ) ) ) ).

% word_lsb_int
tff(fact_4492_exp__two__pi__i,axiom,
    aa(complex,complex,exp(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(num,complex,numeral_numeral(complex),bit0(one2))),aa(real,complex,real_Vector_of_real(complex),pi))),imaginary_unit)) = one_one(complex) ).

% exp_two_pi_i
tff(fact_4493_exp__two__pi__i_H,axiom,
    aa(complex,complex,exp(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(real,complex,real_Vector_of_real(complex),pi)),aa(num,complex,numeral_numeral(complex),bit0(one2))))) = one_one(complex) ).

% exp_two_pi_i'
tff(fact_4494_powr__real__of__int,axiom,
    ! [X: real,Na: int] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ( powr(real,X,aa(int,real,ring_1_of_int(real),Na)) = $ite(aa(int,$o,ord_less_eq(int,zero_zero(int)),Na),aa(nat,real,power_power(real,X),nat2(Na)),aa(real,real,inverse_inverse(real),aa(nat,real,power_power(real,X),nat2(aa(int,int,uminus_uminus(int),Na))))) ) ) ).

% powr_real_of_int
tff(fact_4495_sofl__test,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ( aa(int,int,plus_plus(int,aa(word(A),int,ring_1_signed(A,int),X)),aa(word(A),int,ring_1_signed(A,int),Y)) = aa(word(A),int,ring_1_signed(A,int),aa(word(A),word(A),plus_plus(word(A),X),Y)) )
        <=> ( aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),aa(nat,nat,minus_minus(nat,size_size(word(A),X)),one_one(nat))),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)),aa(word(A),word(A),plus_plus(word(A),X),Y)),X)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),aa(word(A),word(A),plus_plus(word(A),X),Y)),Y))) = zero_zero(word(A)) ) ) ) ).

% sofl_test
tff(fact_4496_max__enat__simps_I2_J,axiom,
    ! [Q3: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),ord_max(extended_enat),Q3),zero_zero(extended_enat)) = Q3 ).

% max_enat_simps(2)
tff(fact_4497_max__enat__simps_I3_J,axiom,
    ! [Q3: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),ord_max(extended_enat),zero_zero(extended_enat)),Q3) = Q3 ).

% max_enat_simps(3)
tff(fact_4498_inverse__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ( aa(A,A,inverse_inverse(A),zero_zero(A)) = zero_zero(A) ) ) ).

% inverse_zero
tff(fact_4499_inverse__nonzero__iff__nonzero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( aa(A,A,inverse_inverse(A),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% inverse_nonzero_iff_nonzero
tff(fact_4500_inverse__eq__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A] :
          ( ( aa(A,A,inverse_inverse(A),X) = one_one(A) )
        <=> ( X = one_one(A) ) ) ) ).

% inverse_eq_1_iff
tff(fact_4501_inverse__1,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ( aa(A,A,inverse_inverse(A),one_one(A)) = one_one(A) ) ) ).

% inverse_1
tff(fact_4502_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_4503_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_4504_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_4505_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_4506_max__nat_Oright__neutral,axiom,
    ! [A3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A3),zero_zero(nat)) = A3 ).

% max_nat.right_neutral
tff(fact_4507_max__nat_Oneutr__eq__iff,axiom,
    ! [A3: nat,B3: nat] :
      ( ( zero_zero(nat) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A3),B3) )
    <=> ( ( A3 = zero_zero(nat) )
        & ( B3 = zero_zero(nat) ) ) ) ).

% max_nat.neutr_eq_iff
tff(fact_4508_max__nat_Oleft__neutral,axiom,
    ! [A3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),zero_zero(nat)),A3) = A3 ).

% max_nat.left_neutral
tff(fact_4509_max__nat_Oeq__neutr__iff,axiom,
    ! [A3: nat,B3: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A3),B3) = zero_zero(nat) )
    <=> ( ( A3 = zero_zero(nat) )
        & ( B3 = zero_zero(nat) ) ) ) ).

% max_nat.eq_neutr_iff
tff(fact_4510_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,inverse_inverse(A),A3)),zero_zero(A))
        <=> aa(A,$o,ord_less_eq(A,A3),zero_zero(A)) ) ) ).

% inverse_nonpositive_iff_nonpositive
tff(fact_4511_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,inverse_inverse(A),A3))
        <=> aa(A,$o,ord_less_eq(A,zero_zero(A)),A3) ) ) ).

% inverse_nonnegative_iff_nonnegative
tff(fact_4512_inverse__less__iff__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),B3)
           => ( aa(A,$o,ord_less(A,aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3))
            <=> aa(A,$o,ord_less(A,B3),A3) ) ) ) ) ).

% inverse_less_iff_less
tff(fact_4513_inverse__less__iff__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),zero_zero(A))
         => ( aa(A,$o,ord_less(A,B3),zero_zero(A))
           => ( aa(A,$o,ord_less(A,aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3))
            <=> aa(A,$o,ord_less(A,B3),A3) ) ) ) ) ).

% inverse_less_iff_less_neg
tff(fact_4514_inverse__negative__iff__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,inverse_inverse(A),A3)),zero_zero(A))
        <=> aa(A,$o,ord_less(A,A3),zero_zero(A)) ) ) ).

% inverse_negative_iff_negative
tff(fact_4515_inverse__positive__iff__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,inverse_inverse(A),A3))
        <=> aa(A,$o,ord_less(A,zero_zero(A)),A3) ) ) ).

% inverse_positive_iff_positive
tff(fact_4516_drop__bit__of__bool,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,B3: $o] :
          aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),aa($o,A,zero_neq_one_of_bool(A),(B3))) = aa($o,A,zero_neq_one_of_bool(A),
            ( ( Na = zero_zero(nat) )
            & (B3) )) ) ).

% drop_bit_of_bool
tff(fact_4517_norm__ii,axiom,
    real_V7770717601297561774m_norm(complex,imaginary_unit) = one_one(real) ).

% norm_ii
tff(fact_4518_inverse__le__iff__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),B3)
           => ( aa(A,$o,ord_less_eq(A,aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3))
            <=> aa(A,$o,ord_less_eq(A,B3),A3) ) ) ) ) ).

% inverse_le_iff_le
tff(fact_4519_inverse__le__iff__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),zero_zero(A))
         => ( aa(A,$o,ord_less(A,B3),zero_zero(A))
           => ( aa(A,$o,ord_less_eq(A,aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3))
            <=> aa(A,$o,ord_less_eq(A,B3),A3) ) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_4520_right__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,inverse_inverse(A),A3)) = one_one(A) ) ) ) ).

% right_inverse
tff(fact_4521_left__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A3)),A3) = one_one(A) ) ) ) ).

% left_inverse
tff(fact_4522_inverse__eq__divide__numeral,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W: num] : aa(A,A,inverse_inverse(A),aa(num,A,numeral_numeral(A),W)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),W)) ) ).

% inverse_eq_divide_numeral
tff(fact_4523_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_4524_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_4525_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_4526_max__Suc__numeral,axiom,
    ! [Na: nat,K: num] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,Na)),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Na),pred_numeral(K))) ).

% max_Suc_numeral
tff(fact_4527_max__numeral__Suc,axiom,
    ! [K: num,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Na)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),pred_numeral(K)),Na)) ).

% max_numeral_Suc
tff(fact_4528_i__squared,axiom,
    aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),imaginary_unit) = aa(complex,complex,uminus_uminus(complex),one_one(complex)) ).

% i_squared
tff(fact_4529_inverse__eq__divide__neg__numeral,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W: num] : aa(A,A,inverse_inverse(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) ) ).

% inverse_eq_divide_neg_numeral
tff(fact_4530_drop__bit__numeral__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: num] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),aa(num,nat,numeral_numeral(nat),L)),aa(num,A,numeral_numeral(A),bit0(K))) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),pred_numeral(L)),aa(num,A,numeral_numeral(A),K)) ) ).

% drop_bit_numeral_bit0
tff(fact_4531_drop__bit__numeral__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: num] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),aa(num,nat,numeral_numeral(nat),L)),aa(num,A,numeral_numeral(A),bit1(K))) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),pred_numeral(L)),aa(num,A,numeral_numeral(A),K)) ) ).

% drop_bit_numeral_bit1
tff(fact_4532_power2__i,axiom,
    aa(nat,complex,power_power(complex,imaginary_unit),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(complex,complex,uminus_uminus(complex),one_one(complex)) ).

% power2_i
tff(fact_4533_i__even__power,axiom,
    ! [Na: nat] : aa(nat,complex,power_power(complex,imaginary_unit),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(nat,complex,power_power(complex,aa(complex,complex,uminus_uminus(complex),one_one(complex))),Na) ).

% i_even_power
tff(fact_4534_exp__pi__i,axiom,
    aa(complex,complex,exp(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(real,complex,real_Vector_of_real(complex),pi)),imaginary_unit)) = aa(complex,complex,uminus_uminus(complex),one_one(complex)) ).

% exp_pi_i
tff(fact_4535_exp__pi__i_H,axiom,
    aa(complex,complex,exp(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),aa(real,complex,real_Vector_of_real(complex),pi))) = aa(complex,complex,uminus_uminus(complex),one_one(complex)) ).

% exp_pi_i'
tff(fact_4536_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_4537_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_4538_sup__nat__def,axiom,
    sup_sup(nat) = ord_max(nat) ).

% sup_nat_def
tff(fact_4539_nat__add__max__left,axiom,
    ! [M: nat,Na: nat,Q3: nat] : aa(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,plus_plus(nat,M),Q3)),aa(nat,nat,plus_plus(nat,Na),Q3)) ).

% nat_add_max_left
tff(fact_4540_nat__add__max__right,axiom,
    ! [M: nat,Na: nat,Q3: nat] : aa(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,plus_plus(nat,M),Na)),aa(nat,nat,plus_plus(nat,M),Q3)) ).

% nat_add_max_right
tff(fact_4541_complex__i__not__one,axiom,
    imaginary_unit != one_one(complex) ).

% complex_i_not_one
tff(fact_4542_nonzero__norm__inverse,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,aa(A,A,inverse_inverse(A),A3)) = aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,A3)) ) ) ) ).

% nonzero_norm_inverse
tff(fact_4543_nonzero__of__real__inverse,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [X: real] :
          ( ( X != zero_zero(real) )
         => ( aa(real,A,real_Vector_of_real(A),aa(real,real,inverse_inverse(real),X)) = aa(A,A,inverse_inverse(A),aa(real,A,real_Vector_of_real(A),X)) ) ) ) ).

% nonzero_of_real_inverse
tff(fact_4544_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_4545_inverse__zero__imp__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( aa(A,A,inverse_inverse(A),A3) = zero_zero(A) )
         => ( A3 = zero_zero(A) ) ) ) ).

% inverse_zero_imp_zero
tff(fact_4546_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,inverse_inverse(A),A3) = aa(A,A,inverse_inverse(A),B3) )
         => ( ( A3 != zero_zero(A) )
           => ( ( B3 != zero_zero(A) )
             => ( A3 = B3 ) ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
tff(fact_4547_nonzero__inverse__inverse__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),aa(A,A,inverse_inverse(A),A3)) = A3 ) ) ) ).

% nonzero_inverse_inverse_eq
tff(fact_4548_nonzero__imp__inverse__nonzero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),A3) != zero_zero(A) ) ) ) ).

% nonzero_imp_inverse_nonzero
tff(fact_4549_of__nat__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: nat,Na: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se4197421643247451524op_bit(nat),M),Na)) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),M),aa(nat,A,semiring_1_of_nat(A),Na)) ) ).

% of_nat_drop_bit
tff(fact_4550_drop__bit__of__nat,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat,M: nat] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),aa(nat,A,semiring_1_of_nat(A),M)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se4197421643247451524op_bit(nat),Na),M)) ) ).

% drop_bit_of_nat
tff(fact_4551_power__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,Na: nat] : aa(nat,A,power_power(A,aa(A,A,inverse_inverse(A),A3)),Na) = aa(A,A,inverse_inverse(A),aa(nat,A,power_power(A,A3),Na)) ) ).

% power_inverse
tff(fact_4552_real__sqrt__inverse,axiom,
    ! [X: real] : aa(real,real,sqrt,aa(real,real,inverse_inverse(real),X)) = aa(real,real,inverse_inverse(real),aa(real,real,sqrt,X)) ).

% real_sqrt_inverse
tff(fact_4553_complex__i__not__zero,axiom,
    imaginary_unit != zero_zero(complex) ).

% complex_i_not_zero
tff(fact_4554_norm__inverse__le__norm,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [R3: real,X: A] :
          ( aa(real,$o,ord_less_eq(real,R3),real_V7770717601297561774m_norm(A,X))
         => ( aa(real,$o,ord_less(real,zero_zero(real)),R3)
           => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,inverse_inverse(A),X))),aa(real,real,inverse_inverse(real),R3)) ) ) ) ).

% norm_inverse_le_norm
tff(fact_4555_inverse__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3))
         => ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
           => aa(A,$o,ord_less(A,B3),A3) ) ) ) ).

% inverse_less_imp_less
tff(fact_4556_less__imp__inverse__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
           => aa(A,$o,ord_less(A,aa(A,A,inverse_inverse(A),B3)),aa(A,A,inverse_inverse(A),A3)) ) ) ) ).

% less_imp_inverse_less
tff(fact_4557_inverse__less__imp__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3))
         => ( aa(A,$o,ord_less(A,B3),zero_zero(A))
           => aa(A,$o,ord_less(A,B3),A3) ) ) ) ).

% inverse_less_imp_less_neg
tff(fact_4558_less__imp__inverse__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(A,$o,ord_less(A,B3),zero_zero(A))
           => aa(A,$o,ord_less(A,aa(A,A,inverse_inverse(A),B3)),aa(A,A,inverse_inverse(A),A3)) ) ) ) ).

% less_imp_inverse_less_neg
tff(fact_4559_inverse__negative__imp__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,inverse_inverse(A),A3)),zero_zero(A))
         => ( ( A3 != zero_zero(A) )
           => aa(A,$o,ord_less(A,A3),zero_zero(A)) ) ) ) ).

% inverse_negative_imp_negative
tff(fact_4560_inverse__positive__imp__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,inverse_inverse(A),A3))
         => ( ( A3 != zero_zero(A) )
           => aa(A,$o,ord_less(A,zero_zero(A)),A3) ) ) ) ).

% inverse_positive_imp_positive
tff(fact_4561_negative__imp__inverse__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,A3),zero_zero(A))
         => aa(A,$o,ord_less(A,aa(A,A,inverse_inverse(A),A3)),zero_zero(A)) ) ) ).

% negative_imp_inverse_negative
tff(fact_4562_positive__imp__inverse__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,inverse_inverse(A),A3)) ) ) ).

% positive_imp_inverse_positive
tff(fact_4563_nonzero__inverse__mult__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B3 != zero_zero(A) )
           => ( aa(A,A,inverse_inverse(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),B3)),aa(A,A,inverse_inverse(A),A3)) ) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_4564_inverse__numeral__1,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ( aa(A,A,inverse_inverse(A),aa(num,A,numeral_numeral(A),one2)) = aa(num,A,numeral_numeral(A),one2) ) ) ).

% inverse_numeral_1
tff(fact_4565_inverse__unique,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3) = one_one(A) )
         => ( aa(A,A,inverse_inverse(A),A3) = B3 ) ) ) ).

% inverse_unique
tff(fact_4566_nonzero__inverse__minus__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),aa(A,A,uminus_uminus(A),A3)) = aa(A,A,uminus_uminus(A),aa(A,A,inverse_inverse(A),A3)) ) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_4567_inverse__eq__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] : aa(A,A,inverse_inverse(A),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3) ) ).

% inverse_eq_divide
tff(fact_4568_power__mult__power__inverse__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: nat,Na: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,X),M)),aa(nat,A,power_power(A,aa(A,A,inverse_inverse(A),X)),Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,inverse_inverse(A),X)),Na)),aa(nat,A,power_power(A,X),M)) ) ).

% power_mult_power_inverse_commute
tff(fact_4569_power__mult__inverse__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,X),M)),aa(A,A,inverse_inverse(A),X)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),X)),aa(nat,A,power_power(A,X),M)) ) ).

% power_mult_inverse_distrib
tff(fact_4570_mult__inverse__of__nat__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa2: nat,X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Xa2))),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Xa2))) ) ).

% mult_inverse_of_nat_commute
tff(fact_4571_nonzero__abs__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),aa(A,A,inverse_inverse(A),A3)) = aa(A,A,inverse_inverse(A),aa(A,A,abs_abs(A),A3)) ) ) ) ).

% nonzero_abs_inverse
tff(fact_4572_mult__inverse__of__int__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa2: int,X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(int,A,ring_1_of_int(A),Xa2))),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(A,A,inverse_inverse(A),aa(int,A,ring_1_of_int(A),Xa2))) ) ).

% mult_inverse_of_int_commute
tff(fact_4573_nat__minus__add__max,axiom,
    ! [Na: nat,M: nat] : aa(nat,nat,plus_plus(nat,aa(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_4574_divide__real__def,axiom,
    ! [X: real,Y: real] : aa(real,real,aa(real,fun(real,real),divide_divide(real),X),Y) = aa(real,real,aa(real,fun(real,real),times_times(real),X),aa(real,real,inverse_inverse(real),Y)) ).

% divide_real_def
tff(fact_4575_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A3: A] :
          ( ( bit_se2584673776208193580ke_bit(A,Na,A3) = A3 )
        <=> ( aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),A3) = zero_zero(A) ) ) ) ).

% take_bit_eq_self_iff_drop_bit_eq_0
tff(fact_4576_drop__bit__mask__eq,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: nat,Na: nat] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),M),bit_se2239418461657761734s_mask(A,Na)) = bit_se2239418461657761734s_mask(A,aa(nat,nat,minus_minus(nat,Na),M)) ) ).

% drop_bit_mask_eq
tff(fact_4577_exp__fdiffs,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X2: nat] : aa(nat,A,diffs(A,aTP_Lamp_dl(nat,A)),X2) = aa(A,A,inverse_inverse(A),semiring_char_0_fact(A,X2)) ) ).

% exp_fdiffs
tff(fact_4578_inverse__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3))
         => ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
           => aa(A,$o,ord_less_eq(A,B3),A3) ) ) ) ).

% inverse_le_imp_le
tff(fact_4579_le__imp__inverse__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
           => aa(A,$o,ord_less_eq(A,aa(A,A,inverse_inverse(A),B3)),aa(A,A,inverse_inverse(A),A3)) ) ) ) ).

% le_imp_inverse_le
tff(fact_4580_inverse__le__imp__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3))
         => ( aa(A,$o,ord_less(A,B3),zero_zero(A))
           => aa(A,$o,ord_less_eq(A,B3),A3) ) ) ) ).

% inverse_le_imp_le_neg
tff(fact_4581_le__imp__inverse__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less(A,B3),zero_zero(A))
           => aa(A,$o,ord_less_eq(A,aa(A,A,inverse_inverse(A),B3)),aa(A,A,inverse_inverse(A),A3)) ) ) ) ).

% le_imp_inverse_le_neg
tff(fact_4582_inverse__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,inverse_inverse(A),X)),one_one(A))
        <=> ( aa(A,$o,ord_less_eq(A,X),zero_zero(A))
            | aa(A,$o,ord_less_eq(A,one_one(A)),X) ) ) ) ).

% inverse_le_1_iff
tff(fact_4583_one__less__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less(A,A3),one_one(A))
           => aa(A,$o,ord_less(A,one_one(A)),aa(A,A,inverse_inverse(A),A3)) ) ) ) ).

% one_less_inverse
tff(fact_4584_one__less__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( aa(A,$o,ord_less(A,one_one(A)),aa(A,A,inverse_inverse(A),X))
        <=> ( aa(A,$o,ord_less(A,zero_zero(A)),X)
            & aa(A,$o,ord_less(A,X),one_one(A)) ) ) ) ).

% one_less_inverse_iff
tff(fact_4585_division__ring__inverse__add,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B3 != zero_zero(A) )
           => ( aa(A,A,plus_plus(A,aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,plus_plus(A,A3),B3))),aa(A,A,inverse_inverse(A),B3)) ) ) ) ) ).

% division_ring_inverse_add
tff(fact_4586_inverse__add,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B3 != zero_zero(A) )
           => ( aa(A,A,plus_plus(A,aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,plus_plus(A,A3),B3)),aa(A,A,inverse_inverse(A),A3))),aa(A,A,inverse_inverse(A),B3)) ) ) ) ) ).

% inverse_add
tff(fact_4587_field__class_Ofield__inverse,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A3)),A3) = one_one(A) ) ) ) ).

% field_class.field_inverse
tff(fact_4588_division__ring__inverse__diff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B3 != zero_zero(A) )
           => ( aa(A,A,minus_minus(A,aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,minus_minus(A,B3),A3))),aa(A,A,inverse_inverse(A),B3)) ) ) ) ) ).

% division_ring_inverse_diff
tff(fact_4589_nonzero__inverse__eq__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3) ) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_4590_max__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A),C2: word(A)] : 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)),ord_max(word(A)),A3),B3)),C2)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),ord_max(word(A)),A3),B3))),aa(word(A),nat,semiring_1_unsigned(A,nat),C2)) ) ).

% max_lt
tff(fact_4591_bit__word__iff__drop__bit__and,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),A3),Na)
        <=> ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),Na),A3)),one_one(word(A))) = one_one(word(A)) ) ) ) ).

% bit_word_iff_drop_bit_and
tff(fact_4592_inverse__powr,axiom,
    ! [Y: real,A3: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
     => ( powr(real,aa(real,real,inverse_inverse(real),Y),A3) = aa(real,real,inverse_inverse(real),powr(real,Y,A3)) ) ) ).

% inverse_powr
tff(fact_4593_and__not__mask__twice,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Na: nat,M: nat] : 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)),W),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),Na)))),aa(word(A),word(A),bit_ri4277139882892585799ns_not(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)),W),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),Na)))) ) ).

% and_not_mask_twice
tff(fact_4594_mask__lower__twice2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),Na: nat,M: nat] : 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)),A3),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),Na)))),aa(word(A),word(A),bit_ri4277139882892585799ns_not(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)),A3),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Na),M)))) ) ).

% mask_lower_twice2
tff(fact_4595_neg__mask__combine,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: nat,B3: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),A3))),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),B3))) = aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A3),B3))) ) ).

% neg_mask_combine
tff(fact_4596_neg__mask__twice,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Na: nat,M: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),Na))),aa(word(A),word(A),bit_ri4277139882892585799ns_not(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)),X),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Na),M)))) ) ).

% neg_mask_twice
tff(fact_4597_bit__iff__and__drop__bit__eq__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),Na)
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),A3)),one_one(A)) = one_one(A) ) ) ) ).

% bit_iff_and_drop_bit_eq_1
tff(fact_4598_Complex__eq__i,axiom,
    ! [X: real,Y: real] :
      ( ( complex2(X,Y) = imaginary_unit )
    <=> ( ( X = zero_zero(real) )
        & ( Y = one_one(real) ) ) ) ).

% Complex_eq_i
tff(fact_4599_imaginary__unit_Ocode,axiom,
    imaginary_unit = complex2(zero_zero(real),one_one(real)) ).

% imaginary_unit.code
tff(fact_4600_one__le__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less_eq(A,A3),one_one(A))
           => aa(A,$o,ord_less_eq(A,one_one(A)),aa(A,A,inverse_inverse(A),A3)) ) ) ) ).

% one_le_inverse
tff(fact_4601_inverse__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,inverse_inverse(A),X)),one_one(A))
        <=> ( aa(A,$o,ord_less_eq(A,X),zero_zero(A))
            | aa(A,$o,ord_less(A,one_one(A)),X) ) ) ) ).

% inverse_less_1_iff
tff(fact_4602_one__le__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( aa(A,$o,ord_less_eq(A,one_one(A)),aa(A,A,inverse_inverse(A),X))
        <=> ( aa(A,$o,ord_less(A,zero_zero(A)),X)
            & aa(A,$o,ord_less_eq(A,X),one_one(A)) ) ) ) ).

% one_le_inverse_iff
tff(fact_4603_inverse__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))
             => aa(A,$o,ord_less_eq(A,B3),A3) )
            & ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),zero_zero(A))
             => aa(A,$o,ord_less_eq(A,A3),B3) ) ) ) ) ).

% inverse_le_iff
tff(fact_4604_inverse__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))
             => aa(A,$o,ord_less(A,B3),A3) )
            & ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),zero_zero(A))
             => aa(A,$o,ord_less(A,A3),B3) ) ) ) ) ).

% inverse_less_iff
tff(fact_4605_inverse__diff__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B3 != zero_zero(A) )
           => ( aa(A,A,minus_minus(A,aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3)) = 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),A3)),aa(A,A,minus_minus(A,A3),B3))),aa(A,A,inverse_inverse(A),B3))) ) ) ) ) ).

% inverse_diff_inverse
tff(fact_4606_reals__Archimedean,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),X)
         => ? [N: nat] : aa(A,$o,ord_less(A,aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N)))),X) ) ) ).

% reals_Archimedean
tff(fact_4607_complex__of__real__i,axiom,
    ! [R3: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(real,complex,real_Vector_of_real(complex),R3)),imaginary_unit) = complex2(zero_zero(real),R3) ).

% complex_of_real_i
tff(fact_4608_i__complex__of__real,axiom,
    ! [R3: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),aa(real,complex,real_Vector_of_real(complex),R3)) = complex2(zero_zero(real),R3) ).

% i_complex_of_real
tff(fact_4609_forall__pos__mono__1,axiom,
    ! [P: fun(real,$o),E2: real] :
      ( ! [D6: real,E: real] :
          ( aa(real,$o,ord_less(real,D6),E)
         => ( aa(real,$o,P,D6)
           => aa(real,$o,P,E) ) )
     => ( ! [N: nat] : aa(real,$o,P,aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N))))
       => ( aa(real,$o,ord_less(real,zero_zero(real)),E2)
         => aa(real,$o,P,E2) ) ) ) ).

% forall_pos_mono_1
tff(fact_4610_real__arch__inverse,axiom,
    ! [E2: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),E2)
    <=> ? [N3: nat] :
          ( ( N3 != zero_zero(nat) )
          & aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N3)))
          & aa(real,$o,ord_less(real,aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N3))),E2) ) ) ).

% real_arch_inverse
tff(fact_4611_forall__pos__mono,axiom,
    ! [P: fun(real,$o),E2: real] :
      ( ! [D6: real,E: real] :
          ( aa(real,$o,ord_less(real,D6),E)
         => ( aa(real,$o,P,D6)
           => aa(real,$o,P,E) ) )
     => ( ! [N: nat] :
            ( ( N != zero_zero(nat) )
           => aa(real,$o,P,aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N))) )
       => ( aa(real,$o,ord_less(real,zero_zero(real)),E2)
         => aa(real,$o,P,E2) ) ) ) ).

% forall_pos_mono
tff(fact_4612_sqrt__divide__self__eq,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => ( aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,X)),X) = aa(real,real,inverse_inverse(real),aa(real,real,sqrt,X)) ) ) ).

% sqrt_divide_self_eq
tff(fact_4613_drop__bit__half,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),A3)),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% drop_bit_half
tff(fact_4614_stable__imp__drop__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,Na: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) = A3 )
         => ( aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),A3) = A3 ) ) ) ).

% stable_imp_drop_bit_eq
tff(fact_4615_ln__inverse,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ( aa(real,real,ln_ln(real),aa(real,real,inverse_inverse(real),X)) = aa(real,real,uminus_uminus(real),aa(real,real,ln_ln(real),X)) ) ) ).

% ln_inverse
tff(fact_4616_summable__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : summable(A,aTP_Lamp_dm(A,fun(nat,A),X)) ) ).

% summable_exp
tff(fact_4617_ex__inverse__of__nat__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),X)
         => ? [N: nat] :
              ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
              & aa(A,$o,ord_less(A,aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),N))),X) ) ) ) ).

% ex_inverse_of_nat_less
tff(fact_4618_power__diff__conv__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: nat,Na: nat] :
          ( ( X != zero_zero(A) )
         => ( aa(nat,$o,ord_less_eq(nat,M),Na)
           => ( aa(nat,A,power_power(A,X),aa(nat,nat,minus_minus(nat,Na),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,X),Na)),aa(nat,A,power_power(A,aa(A,A,inverse_inverse(A),X)),M)) ) ) ) ) ).

% power_diff_conv_inverse
tff(fact_4619_log__inverse,axiom,
    ! [A3: real,X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
     => ( ( A3 != one_one(real) )
       => ( aa(real,$o,ord_less(real,zero_zero(real)),X)
         => ( aa(real,real,log(A3),aa(real,real,inverse_inverse(real),X)) = aa(real,real,uminus_uminus(real),aa(real,real,log(A3),X)) ) ) ) ) ).

% log_inverse
tff(fact_4620_drop__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),aa(nat,nat,suc,Na)),A3) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% drop_bit_Suc
tff(fact_4621_drop__bit__eq__div,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) ) ).

% drop_bit_eq_div
tff(fact_4622_bit__iff__odd__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),Na)
        <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),A3)) ) ) ).

% bit_iff_odd_drop_bit
tff(fact_4623_even__drop__bit__iff__not__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),A3))
        <=> ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),Na) ) ) ).

% even_drop_bit_iff_not_bit
tff(fact_4624_exp__plus__inverse__exp,axiom,
    ! [X: real] : aa(real,$o,ord_less_eq(real,aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,plus_plus(real,aa(real,real,exp(real),X)),aa(real,real,inverse_inverse(real),aa(real,real,exp(real),X)))) ).

% exp_plus_inverse_exp
tff(fact_4625_bit__twiddle__max,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),X),
            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)),X),Y)),
              $ite(aa(word(A),$o,ord_less(word(A),X),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)),X),Y) ) ).

% bit_twiddle_max
tff(fact_4626_cmod__unit__one,axiom,
    ! [A3: real] : real_V7770717601297561774m_norm(complex,aa(complex,complex,plus_plus(complex,aa(real,complex,real_Vector_of_real(complex),cos(real,A3))),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),aa(real,complex,real_Vector_of_real(complex),sin(real,A3))))) = one_one(real) ).

% cmod_unit_one
tff(fact_4627_plus__inverse__ge__2,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => aa(real,$o,ord_less_eq(real,aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,plus_plus(real,X),aa(real,real,inverse_inverse(real),X))) ) ).

% plus_inverse_ge_2
tff(fact_4628_real__inv__sqrt__pow2,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ( aa(nat,real,power_power(real,aa(real,real,inverse_inverse(real),aa(real,real,sqrt,X))),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(real,real,inverse_inverse(real),X) ) ) ).

% real_inv_sqrt_pow2
tff(fact_4629_tan__cot,axiom,
    ! [X: real] : aa(real,real,tan(real),aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),X)) = aa(real,real,inverse_inverse(real),aa(real,real,tan(real),X)) ).

% tan_cot
tff(fact_4630_drop__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A3: A] :
          aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),A3) = $ite(Na = zero_zero(nat),A3,aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),aa(nat,nat,minus_minus(nat,Na),one_one(nat))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% drop_bit_rec
tff(fact_4631_real__le__x__sinh,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => aa(real,$o,ord_less_eq(real,X),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,aa(real,real,exp(real),X)),aa(real,real,inverse_inverse(real),aa(real,real,exp(real),X)))),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ).

% real_le_x_sinh
tff(fact_4632_real__le__abs__sinh,axiom,
    ! [X: real] : aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),X)),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,aa(real,real,exp(real),X)),aa(real,real,inverse_inverse(real),aa(real,real,exp(real),X)))),aa(num,real,numeral_numeral(real),bit0(one2))))) ).

% real_le_abs_sinh
tff(fact_4633_tan__sec,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( aa(A,A,plus_plus(A,one_one(A)),aa(nat,A,power_power(A,aa(A,A,tan(A),X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(nat,A,power_power(A,aa(A,A,inverse_inverse(A),cos(A,X))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ) ) ).

% tan_sec
tff(fact_4634_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,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_4635_Arg__minus__ii,axiom,
    arg(aa(complex,complex,uminus_uminus(complex),imaginary_unit)) = 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))) ).

% Arg_minus_ii
tff(fact_4636_csqrt__ii,axiom,
    csqrt(imaginary_unit) = aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),aa(complex,complex,plus_plus(complex,one_one(complex)),imaginary_unit)),aa(real,complex,real_Vector_of_real(complex),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2))))) ).

% csqrt_ii
tff(fact_4637_Arg__ii,axiom,
    arg(imaginary_unit) = aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% Arg_ii
tff(fact_4638_cis__minus__pi__half,axiom,
    cis(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(complex,complex,uminus_uminus(complex),imaginary_unit) ).

% cis_minus_pi_half
tff(fact_4639_drop__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(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,ord_less_eq(int,zero_zero(int)),K) ) ).

% drop_bit_nonnegative_int_iff
tff(fact_4640_drop__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(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,ord_less(int,K),zero_zero(int)) ) ).

% drop_bit_negative_int_iff
tff(fact_4641_drop__bit__minus__one,axiom,
    ! [Na: nat] : aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),Na),aa(int,int,uminus_uminus(int),one_one(int))) = aa(int,int,uminus_uminus(int),one_one(int)) ).

% drop_bit_minus_one
tff(fact_4642_csqrt__eq__0,axiom,
    ! [Z: complex] :
      ( ( csqrt(Z) = zero_zero(complex) )
    <=> ( Z = zero_zero(complex) ) ) ).

% csqrt_eq_0
tff(fact_4643_csqrt__0,axiom,
    csqrt(zero_zero(complex)) = zero_zero(complex) ).

% csqrt_0
tff(fact_4644_csqrt__eq__1,axiom,
    ! [Z: complex] :
      ( ( csqrt(Z) = one_one(complex) )
    <=> ( Z = one_one(complex) ) ) ).

% csqrt_eq_1
tff(fact_4645_csqrt__1,axiom,
    csqrt(one_one(complex)) = one_one(complex) ).

% csqrt_1
tff(fact_4646_norm__cis,axiom,
    ! [A3: real] : real_V7770717601297561774m_norm(complex,cis(A3)) = one_one(real) ).

% norm_cis
tff(fact_4647_cis__zero,axiom,
    cis(zero_zero(real)) = one_one(complex) ).

% cis_zero
tff(fact_4648_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_4649_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_4650_cis__pi,axiom,
    cis(pi) = aa(complex,complex,uminus_uminus(complex),one_one(complex)) ).

% cis_pi
tff(fact_4651_drop__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(num,nat,numeral_numeral(nat),L)),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),pred_numeral(L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))) ).

% drop_bit_numeral_minus_bit0
tff(fact_4652_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_4653_power2__csqrt,axiom,
    ! [Z: complex] : aa(nat,complex,power_power(complex,csqrt(Z)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = Z ).

% power2_csqrt
tff(fact_4654_drop__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(num,nat,numeral_numeral(nat),L)),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),pred_numeral(L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K)))) ).

% drop_bit_numeral_minus_bit1
tff(fact_4655_cis__pi__half,axiom,
    cis(aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) = imaginary_unit ).

% cis_pi_half
tff(fact_4656_cis__2pi,axiom,
    cis(aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)) = one_one(complex) ).

% cis_2pi
tff(fact_4657_sup__enat__def,axiom,
    sup_sup(extended_enat) = ord_max(extended_enat) ).

% sup_enat_def
tff(fact_4658_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_4659_cis__Arg,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero(complex) )
     => ( cis(arg(Z)) = aa(complex,complex,sgn_sgn(complex),Z) ) ) ).

% cis_Arg
tff(fact_4660_cis__neq__zero,axiom,
    ! [A3: real] : cis(A3) != zero_zero(complex) ).

% cis_neq_zero
tff(fact_4661_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_4662_unat__drop__bit__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),Na),W)) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se4197421643247451524op_bit(nat),Na),aa(word(A),nat,semiring_1_unsigned(A,nat),W)) ) ).

% unat_drop_bit_eq
tff(fact_4663_DeMoivre,axiom,
    ! [A3: real,Na: nat] : aa(nat,complex,power_power(complex,cis(A3)),Na) = cis(aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),A3)) ).

% DeMoivre
tff(fact_4664_cis__Arg__unique,axiom,
    ! [Z: complex,X: real] :
      ( ( aa(complex,complex,sgn_sgn(complex),Z) = cis(X) )
     => ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),pi)),X)
       => ( aa(real,$o,ord_less_eq(real,X),pi)
         => ( arg(Z) = X ) ) ) ) ).

% cis_Arg_unique
tff(fact_4665_Arg__correct,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero(complex) )
     => ( ( aa(complex,complex,sgn_sgn(complex),Z) = cis(arg(Z)) )
        & aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),pi)),arg(Z))
        & aa(real,$o,ord_less_eq(real,arg(Z)),pi) ) ) ).

% Arg_correct
tff(fact_4666_Arg__zero,axiom,
    arg(zero_zero(complex)) = zero_zero(real) ).

% Arg_zero
tff(fact_4667_bin__rest__code,axiom,
    ! [I: int] : aa(int,int,aa(int,fun(int,int),divide_divide(int),I),aa(num,int,numeral_numeral(int),bit0(one2))) = aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),one_one(nat)),I) ).

% bin_rest_code
tff(fact_4668_drop__bit__int__def,axiom,
    ! [Na: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),Na),K) = aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ).

% drop_bit_int_def
tff(fact_4669_drop__bit__nat__def,axiom,
    ! [Na: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se4197421643247451524op_bit(nat),Na),M) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ).

% drop_bit_nat_def
tff(fact_4670_of__real__sqrt,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => ( aa(real,complex,real_Vector_of_real(complex),aa(real,real,sqrt,X)) = csqrt(aa(real,complex,real_Vector_of_real(complex),X)) ) ) ).

% of_real_sqrt
tff(fact_4671_Arg__bounded,axiom,
    ! [Z: complex] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),pi)),arg(Z))
      & aa(real,$o,ord_less_eq(real,arg(Z)),pi) ) ).

% Arg_bounded
tff(fact_4672_complex__inverse,axiom,
    ! [A3: real,B3: real] : aa(complex,complex,inverse_inverse(complex),complex2(A3,B3)) = complex2(aa(real,real,aa(real,fun(real,real),divide_divide(real),A3),aa(real,real,plus_plus(real,aa(nat,real,power_power(real,A3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,B3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),B3)),aa(real,real,plus_plus(real,aa(nat,real,power_power(real,A3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,B3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% complex_inverse
tff(fact_4673_exp__first__two__terms,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,A,exp(A),X) = aa(A,A,plus_plus(A,aa(A,A,plus_plus(A,one_one(A)),X)),suminf(A,aTP_Lamp_dn(A,fun(nat,A),X))) ) ).

% exp_first_two_terms
tff(fact_4674_shiftr__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: num,Na: num] : bit_Sh4282982442137083166shiftr(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)),aa(num,nat,numeral_numeral(nat),Na)) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),aa(num,nat,numeral_numeral(nat),Na)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) ) ).

% shiftr_minus_numeral_numeral
tff(fact_4675_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_4676_scaleR__zero__right,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: real] : aa(A,A,real_V8093663219630862766scaleR(A,A3),zero_zero(A)) = zero_zero(A) ) ).

% scaleR_zero_right
tff(fact_4677_scaleR__cancel__right,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: real,X: A,B3: real] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,A3),X) = aa(A,A,real_V8093663219630862766scaleR(A,B3),X) )
        <=> ( ( A3 = B3 )
            | ( X = zero_zero(A) ) ) ) ) ).

% scaleR_cancel_right
tff(fact_4678_scaleR__cancel__left,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: real,X: A,Y: A] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,A3),X) = aa(A,A,real_V8093663219630862766scaleR(A,A3),Y) )
        <=> ( ( X = Y )
            | ( A3 = zero_zero(real) ) ) ) ) ).

% scaleR_cancel_left
tff(fact_4679_scaleR__one,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A] : aa(A,A,real_V8093663219630862766scaleR(A,one_one(real)),X) = X ) ).

% scaleR_one
tff(fact_4680_shiftr__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : bit_Sh4282982442137083166shiftr(A,zero_zero(A),Na) = zero_zero(A) ) ).

% shiftr_0
tff(fact_4681_shiftr__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : bit_Sh4282982442137083166shiftr(A,A3,zero_zero(nat)) = A3 ) ).

% shiftr_of_0
tff(fact_4682_scaleR__zero__left,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A] : aa(A,A,real_V8093663219630862766scaleR(A,zero_zero(real)),X) = zero_zero(A) ) ).

% scaleR_zero_left
tff(fact_4683_scaleR__eq__0__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: real,X: A] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,A3),X) = zero_zero(A) )
        <=> ( ( A3 = zero_zero(real) )
            | ( X = zero_zero(A) ) ) ) ) ).

% scaleR_eq_0_iff
tff(fact_4684_scaleR__eq__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B3: A,U: real,A3: A] :
          ( ( aa(A,A,plus_plus(A,B3),aa(A,A,real_V8093663219630862766scaleR(A,U),A3)) = aa(A,A,plus_plus(A,A3),aa(A,A,real_V8093663219630862766scaleR(A,U),B3)) )
        <=> ( ( A3 = B3 )
            | ( U = one_one(real) ) ) ) ) ).

% scaleR_eq_iff
tff(fact_4685_scaleR__power,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [X: real,Y: A,Na: nat] : aa(nat,A,power_power(A,aa(A,A,real_V8093663219630862766scaleR(A,X),Y)),Na) = aa(A,A,real_V8093663219630862766scaleR(A,aa(nat,real,power_power(real,X),Na)),aa(nat,A,power_power(A,Y),Na)) ) ).

% scaleR_power
tff(fact_4686_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_4687_scaleR__minus1__left,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,uminus_uminus(real),one_one(real))),X) = aa(A,A,uminus_uminus(A),X) ) ).

% scaleR_minus1_left
tff(fact_4688_scaleR__collapse,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [U: real,A3: A] : aa(A,A,plus_plus(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,minus_minus(real,one_one(real)),U)),A3)),aa(A,A,real_V8093663219630862766scaleR(A,U),A3)) = A3 ) ).

% scaleR_collapse
tff(fact_4689_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_4690_shiftr__numeral__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: num,Na: num] : bit_Sh4282982442137083166shiftr(A,aa(num,A,numeral_numeral(A),M),aa(num,nat,numeral_numeral(nat),Na)) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),aa(num,nat,numeral_numeral(nat),Na)),aa(num,A,numeral_numeral(A),M)) ) ).

% shiftr_numeral_numeral
tff(fact_4691_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_4692_scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [U: num,W: num,A3: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(num,real,numeral_numeral(real),U)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),A3)) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),U)),aa(num,real,numeral_numeral(real),W))),A3) ) ).

% scaleR_times
tff(fact_4693_inverse__scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [V2: num,W: num,A3: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),V2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),A3)) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),W)),aa(num,real,numeral_numeral(real),V2))),A3) ) ).

% inverse_scaleR_times
tff(fact_4694_fraction__scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [U: num,V2: num,W: num,A3: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),U)),aa(num,real,numeral_numeral(real),V2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),A3)) = aa(A,A,real_V8093663219630862766scaleR(A,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),U)),aa(num,real,numeral_numeral(real),W))),aa(num,real,numeral_numeral(real),V2))),A3) ) ).

% fraction_scaleR_times
tff(fact_4695_scaleR__half__double,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(A,A,plus_plus(A,A3),A3)) = A3 ) ).

% scaleR_half_double
tff(fact_4696_summable__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),R3: real] :
          ( summable(A,X5)
         => summable(A,aa(real,fun(nat,A),aTP_Lamp_do(fun(nat,A),fun(real,fun(nat,A)),X5),R3)) ) ) ).

% summable_scaleR_right
tff(fact_4697_scaleR__left__imp__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: real,X: A,Y: A] :
          ( ( A3 != zero_zero(real) )
         => ( ( aa(A,A,real_V8093663219630862766scaleR(A,A3),X) = aa(A,A,real_V8093663219630862766scaleR(A,A3),Y) )
           => ( X = Y ) ) ) ) ).

% scaleR_left_imp_eq
tff(fact_4698_scaleR__right__imp__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,A3: real,B3: real] :
          ( ( X != zero_zero(A) )
         => ( ( aa(A,A,real_V8093663219630862766scaleR(A,A3),X) = aa(A,A,real_V8093663219630862766scaleR(A,B3),X) )
           => ( A3 = B3 ) ) ) ) ).

% scaleR_right_imp_eq
tff(fact_4699_sums__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),A3: A,R3: real] :
          ( aa(A,$o,sums(A,X5),A3)
         => aa(A,$o,sums(A,aa(real,fun(nat,A),aTP_Lamp_do(fun(nat,A),fun(real,fun(nat,A)),X5),R3)),aa(A,A,real_V8093663219630862766scaleR(A,R3),A3)) ) ) ).

% sums_scaleR_right
tff(fact_4700_of__real__def,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [R3: real] : aa(real,A,real_Vector_of_real(A),R3) = aa(A,A,real_V8093663219630862766scaleR(A,R3),one_one(A)) ) ).

% of_real_def
tff(fact_4701_suminf__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),R3: real] :
          ( summable(A,X5)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,R3),suminf(A,X5)) = suminf(A,aa(real,fun(nat,A),aTP_Lamp_do(fun(nat,A),fun(real,fun(nat,A)),X5),R3)) ) ) ) ).

% suminf_scaleR_right
tff(fact_4702_summable__scaleR__left,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,real),X: A] :
          ( summable(real,X5)
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_dp(fun(nat,real),fun(A,fun(nat,A)),X5),X)) ) ) ).

% summable_scaleR_left
tff(fact_4703_sums__scaleR__left,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,real),A3: real,X: A] :
          ( aa(real,$o,sums(real,X5),A3)
         => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_dp(fun(nat,real),fun(A,fun(nat,A)),X5),X)),aa(A,A,real_V8093663219630862766scaleR(A,A3),X)) ) ) ).

% sums_scaleR_left
tff(fact_4704_scaleR__right__mono__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [B3: real,A3: real,C2: A] :
          ( aa(real,$o,ord_less_eq(real,B3),A3)
         => ( aa(A,$o,ord_less_eq(A,C2),zero_zero(A))
           => aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,A3),C2)),aa(A,A,real_V8093663219630862766scaleR(A,B3),C2)) ) ) ) ).

% scaleR_right_mono_neg
tff(fact_4705_scaleR__right__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B3: real,X: A] :
          ( aa(real,$o,ord_less_eq(real,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),X)
           => aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,A3),X)),aa(A,A,real_V8093663219630862766scaleR(A,B3),X)) ) ) ) ).

% scaleR_right_mono
tff(fact_4706_scaleR__le__cancel__left,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B3))
        <=> ( ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
             => aa(A,$o,ord_less_eq(A,A3),B3) )
            & ( aa(real,$o,ord_less(real,C2),zero_zero(real))
             => aa(A,$o,ord_less_eq(A,B3),A3) ) ) ) ) ).

% scaleR_le_cancel_left
tff(fact_4707_scaleR__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A3: A,B3: A] :
          ( aa(real,$o,ord_less(real,C2),zero_zero(real))
         => ( aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B3))
          <=> aa(A,$o,ord_less_eq(A,B3),A3) ) ) ) ).

% scaleR_le_cancel_left_neg
tff(fact_4708_scaleR__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A3: A,B3: A] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
         => ( aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B3))
          <=> aa(A,$o,ord_less_eq(A,A3),B3) ) ) ) ).

% scaleR_le_cancel_left_pos
tff(fact_4709_scaleR__left__mono__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [B3: A,A3: A,C2: real] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
         => ( aa(real,$o,ord_less_eq(real,C2),zero_zero(real))
           => aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B3)) ) ) ) ).

% scaleR_left_mono_neg
tff(fact_4710_scaleR__left__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [X: A,Y: A,A3: real] :
          ( aa(A,$o,ord_less_eq(A,X),Y)
         => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),A3)
           => aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,A3),X)),aa(A,A,real_V8093663219630862766scaleR(A,A3),Y)) ) ) ) ).

% scaleR_left_mono
tff(fact_4711_vector__fraction__eq__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [U: real,V2: real,A3: A,X: A] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),U),V2)),A3) = X )
        <=> $ite(V2 = zero_zero(real),X = zero_zero(A),aa(A,A,real_V8093663219630862766scaleR(A,U),A3) = aa(A,A,real_V8093663219630862766scaleR(A,V2),X)) ) ) ).

% vector_fraction_eq_iff
tff(fact_4712_eq__vector__fraction__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,U: real,V2: real,A3: A] :
          ( ( X = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),U),V2)),A3) )
        <=> $ite(V2 = zero_zero(real),X = zero_zero(A),aa(A,A,real_V8093663219630862766scaleR(A,V2),X) = aa(A,A,real_V8093663219630862766scaleR(A,U),A3)) ) ) ).

% eq_vector_fraction_iff
tff(fact_4713_zero__le__scaleR__iff,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A3),B3))
        <=> ( ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
              & aa(A,$o,ord_less_eq(A,zero_zero(A)),B3) )
            | ( aa(real,$o,ord_less(real,A3),zero_zero(real))
              & aa(A,$o,ord_less_eq(A,B3),zero_zero(A)) )
            | ( A3 = zero_zero(real) ) ) ) ) ).

% zero_le_scaleR_iff
tff(fact_4714_scaleR__le__0__iff,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,A3),B3)),zero_zero(A))
        <=> ( ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
              & aa(A,$o,ord_less_eq(A,B3),zero_zero(A)) )
            | ( aa(real,$o,ord_less(real,A3),zero_zero(real))
              & aa(A,$o,ord_less_eq(A,zero_zero(A)),B3) )
            | ( A3 = zero_zero(real) ) ) ) ) ).

% scaleR_le_0_iff
tff(fact_4715_scaleR__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B3: A] :
          ( aa(real,$o,ord_less_eq(real,A3),zero_zero(real))
         => ( aa(A,$o,ord_less_eq(A,B3),zero_zero(A))
           => aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A3),B3)) ) ) ) ).

% scaleR_nonpos_nonpos
tff(fact_4716_scaleR__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,X: A] :
          ( aa(real,$o,ord_less_eq(real,A3),zero_zero(real))
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),X)
           => aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,A3),X)),zero_zero(A)) ) ) ) ).

% scaleR_nonpos_nonneg
tff(fact_4717_scaleR__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,X: A] :
          ( aa(real,$o,ord_less_eq(real,zero_zero(real)),A3)
         => ( aa(A,$o,ord_less_eq(A,X),zero_zero(A))
           => aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,A3),X)),zero_zero(A)) ) ) ) ).

% scaleR_nonneg_nonpos
tff(fact_4718_scaleR__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,X: A] :
          ( aa(real,$o,ord_less_eq(real,zero_zero(real)),A3)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),X)
           => aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A3),X)) ) ) ) ).

% scaleR_nonneg_nonneg
tff(fact_4719_split__scaleR__pos__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B3: A] :
          ( ( ( aa(real,$o,ord_less_eq(real,zero_zero(real)),A3)
              & aa(A,$o,ord_less_eq(A,zero_zero(A)),B3) )
            | ( aa(real,$o,ord_less_eq(real,A3),zero_zero(real))
              & aa(A,$o,ord_less_eq(A,B3),zero_zero(A)) ) )
         => aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A3),B3)) ) ) ).

% split_scaleR_pos_le
tff(fact_4720_split__scaleR__neg__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,X: A] :
          ( ( ( aa(real,$o,ord_less_eq(real,zero_zero(real)),A3)
              & aa(A,$o,ord_less_eq(A,X),zero_zero(A)) )
            | ( aa(real,$o,ord_less_eq(real,A3),zero_zero(real))
              & aa(A,$o,ord_less_eq(A,zero_zero(A)),X) ) )
         => aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,A3),X)),zero_zero(A)) ) ) ).

% split_scaleR_neg_le
tff(fact_4721_scaleR__mono_H,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B3: real,C2: A,D2: A] :
          ( aa(real,$o,ord_less_eq(real,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,C2),D2)
           => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),A3)
             => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
               => aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,A3),C2)),aa(A,A,real_V8093663219630862766scaleR(A,B3),D2)) ) ) ) ) ) ).

% scaleR_mono'
tff(fact_4722_scaleR__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B3: real,X: A,Y: A] :
          ( aa(real,$o,ord_less_eq(real,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,X),Y)
           => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),B3)
             => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),X)
               => aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,A3),X)),aa(A,A,real_V8093663219630862766scaleR(A,B3),Y)) ) ) ) ) ) ).

% scaleR_mono
tff(fact_4723_suminf__scaleR__left,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,real),X: A] :
          ( summable(real,X5)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,suminf(real,X5)),X) = suminf(A,aa(A,fun(nat,A),aTP_Lamp_dp(fun(nat,real),fun(A,fun(nat,A)),X5),X)) ) ) ) ).

% suminf_scaleR_left
tff(fact_4724_scaleR__2,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(num,real,numeral_numeral(real),bit0(one2))),X) = aa(A,A,plus_plus(A,X),X) ) ).

% scaleR_2
tff(fact_4725_scaleR__left__le__one__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [X: A,A3: real] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),X)
         => ( aa(real,$o,ord_less_eq(real,A3),one_one(real))
           => aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,A3),X)),X) ) ) ) ).

% scaleR_left_le_one_le
tff(fact_4726_real__vector__eq__affinity,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [M: real,Y: A,X: A,C2: A] :
          ( ( M != zero_zero(real) )
         => ( ( Y = aa(A,A,plus_plus(A,aa(A,A,real_V8093663219630862766scaleR(A,M),X)),C2) )
          <=> ( aa(A,A,minus_minus(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),M)),Y)),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),M)),C2)) = X ) ) ) ) ).

% real_vector_eq_affinity
tff(fact_4727_real__vector__affinity__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [M: real,X: A,C2: A,Y: A] :
          ( ( M != zero_zero(real) )
         => ( ( aa(A,A,plus_plus(A,aa(A,A,real_V8093663219630862766scaleR(A,M),X)),C2) = Y )
          <=> ( X = aa(A,A,minus_minus(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),M)),Y)),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),M)),C2)) ) ) ) ) ).

% real_vector_affinity_eq
tff(fact_4728_neg__le__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A3: A,B3: A] :
          ( aa(real,$o,ord_less(real,C2),zero_zero(real))
         => ( aa(A,$o,ord_less_eq(A,A3),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B3))
          <=> aa(A,$o,ord_less_eq(A,B3),aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)) ) ) ) ).

% neg_le_divideR_eq
tff(fact_4729_neg__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B3: A,A3: A] :
          ( aa(real,$o,ord_less(real,C2),zero_zero(real))
         => ( aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B3)),A3)
          <=> aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)),B3) ) ) ) ).

% neg_divideR_le_eq
tff(fact_4730_pos__le__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A3: A,B3: A] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
         => ( aa(A,$o,ord_less_eq(A,A3),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B3))
          <=> aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)),B3) ) ) ) ).

% pos_le_divideR_eq
tff(fact_4731_pos__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B3: A,A3: A] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
         => ( aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B3)),A3)
          <=> aa(A,$o,ord_less_eq(A,B3),aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)) ) ) ) ).

% pos_divideR_le_eq
tff(fact_4732_pos__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B3: A,A3: A] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
         => ( aa(A,$o,ord_less(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B3)),A3)
          <=> aa(A,$o,ord_less(A,B3),aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)) ) ) ) ).

% pos_divideR_less_eq
tff(fact_4733_pos__less__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A3: A,B3: A] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
         => ( aa(A,$o,ord_less(A,A3),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B3))
          <=> aa(A,$o,ord_less(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)),B3) ) ) ) ).

% pos_less_divideR_eq
tff(fact_4734_neg__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B3: A,A3: A] :
          ( aa(real,$o,ord_less(real,C2),zero_zero(real))
         => ( aa(A,$o,ord_less(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B3)),A3)
          <=> aa(A,$o,ord_less(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)),B3) ) ) ) ).

% neg_divideR_less_eq
tff(fact_4735_neg__less__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A3: A,B3: A] :
          ( aa(real,$o,ord_less(real,C2),zero_zero(real))
         => ( aa(A,$o,ord_less(A,A3),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B3))
          <=> aa(A,$o,ord_less(A,B3),aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)) ) ) ) ).

% neg_less_divideR_eq
tff(fact_4736_nonzero__inverse__scaleR__distrib,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [A3: real,X: A] :
          ( ( A3 != zero_zero(real) )
         => ( ( X != zero_zero(A) )
           => ( aa(A,A,inverse_inverse(A),aa(A,A,real_V8093663219630862766scaleR(A,A3),X)) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),A3)),aa(A,A,inverse_inverse(A),X)) ) ) ) ) ).

% nonzero_inverse_scaleR_distrib
tff(fact_4737_summable__exp__generic,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : summable(A,aTP_Lamp_dq(A,fun(nat,A),X)) ) ).

% summable_exp_generic
tff(fact_4738_sin__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,$o,sums(A,aTP_Lamp_dr(A,fun(nat,A),X)),sin(A,X)) ) ).

% sin_converges
tff(fact_4739_sin__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X2: A] : sin(A,X2) = suminf(A,aTP_Lamp_dr(A,fun(nat,A),X2)) ) ).

% sin_def
tff(fact_4740_cos__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,$o,sums(A,aTP_Lamp_ds(A,fun(nat,A),X)),cos(A,X)) ) ).

% cos_converges
tff(fact_4741_cos__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X2: A] : cos(A,X2) = suminf(A,aTP_Lamp_ds(A,fun(nat,A),X2)) ) ).

% cos_def
tff(fact_4742_summable__norm__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : summable(real,aTP_Lamp_dt(A,fun(nat,real),X)) ) ).

% summable_norm_sin
tff(fact_4743_summable__norm__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : summable(real,aTP_Lamp_du(A,fun(nat,real),X)) ) ).

% summable_norm_cos
tff(fact_4744_pos__le__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A3: A,B3: A] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
         => ( aa(A,$o,ord_less_eq(A,A3),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B3)))
          <=> aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)),aa(A,A,uminus_uminus(A),B3)) ) ) ) ).

% pos_le_minus_divideR_eq
tff(fact_4745_pos__minus__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B3: A,A3: A] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
         => ( aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B3))),A3)
          <=> aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),B3)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)) ) ) ) ).

% pos_minus_divideR_le_eq
tff(fact_4746_neg__le__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A3: A,B3: A] :
          ( aa(real,$o,ord_less(real,C2),zero_zero(real))
         => ( aa(A,$o,ord_less_eq(A,A3),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B3)))
          <=> aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),B3)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)) ) ) ) ).

% neg_le_minus_divideR_eq
tff(fact_4747_neg__minus__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B3: A,A3: A] :
          ( aa(real,$o,ord_less(real,C2),zero_zero(real))
         => ( aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B3))),A3)
          <=> aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)),aa(A,A,uminus_uminus(A),B3)) ) ) ) ).

% neg_minus_divideR_le_eq
tff(fact_4748_pos__less__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A3: A,B3: A] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
         => ( aa(A,$o,ord_less(A,A3),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B3)))
          <=> aa(A,$o,ord_less(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)),aa(A,A,uminus_uminus(A),B3)) ) ) ) ).

% pos_less_minus_divideR_eq
tff(fact_4749_pos__minus__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B3: A,A3: A] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
         => ( aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B3))),A3)
          <=> aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),B3)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)) ) ) ) ).

% pos_minus_divideR_less_eq
tff(fact_4750_neg__less__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A3: A,B3: A] :
          ( aa(real,$o,ord_less(real,C2),zero_zero(real))
         => ( aa(A,$o,ord_less(A,A3),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B3)))
          <=> aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),B3)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)) ) ) ) ).

% neg_less_minus_divideR_eq
tff(fact_4751_neg__minus__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B3: A,A3: A] :
          ( aa(real,$o,ord_less(real,C2),zero_zero(real))
         => ( aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B3))),A3)
          <=> aa(A,$o,ord_less(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)),aa(A,A,uminus_uminus(A),B3)) ) ) ) ).

% neg_minus_divideR_less_eq
tff(fact_4752_exp__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,$o,sums(A,aTP_Lamp_dq(A,fun(nat,A),X)),aa(A,A,exp(A),X)) ) ).

% exp_converges
tff(fact_4753_exp__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X2: A] : aa(A,A,exp(A),X2) = suminf(A,aTP_Lamp_dq(A,fun(nat,A),X2)) ) ).

% exp_def
tff(fact_4754_summable__norm__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : summable(real,aTP_Lamp_dv(A,fun(nat,real),X)) ) ).

% summable_norm_exp
tff(fact_4755_sin__minus__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,$o,sums(A,aTP_Lamp_dw(A,fun(nat,A),X)),sin(A,X)) ) ).

% sin_minus_converges
tff(fact_4756_cos__minus__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,$o,sums(A,aTP_Lamp_dx(A,fun(nat,A),X)),cos(A,X)) ) ).

% cos_minus_converges
tff(fact_4757_exp__first__term,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,A,exp(A),X) = aa(A,A,plus_plus(A,one_one(A)),suminf(A,aTP_Lamp_dy(A,fun(nat,A),X))) ) ).

% exp_first_term
tff(fact_4758_shiftr__eq__div,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [X: A,Na: nat] : bit_Sh4282982442137083166shiftr(A,X,Na) = aa(A,A,aa(A,fun(A,A),divide_divide(A),X),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) ) ).

% shiftr_eq_div
tff(fact_4759_sinh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,$o,sums(A,aTP_Lamp_dz(A,fun(nat,A),X)),sinh(A,X)) ) ).

% sinh_converges
tff(fact_4760_cosh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,$o,sums(A,aTP_Lamp_ea(A,fun(nat,A),X)),cosh(A,X)) ) ).

% cosh_converges
tff(fact_4761_sin__x__sin__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_ec(A,fun(A,fun(nat,A)),X),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,X)),sin(A,Y))) ) ).

% sin_x_sin_y
tff(fact_4762_sinh__real__zero__iff,axiom,
    ! [X: real] :
      ( ( sinh(real,X) = zero_zero(real) )
    <=> ( X = zero_zero(real) ) ) ).

% sinh_real_zero_iff
tff(fact_4763_sinh__real__less__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less(real,sinh(real,X)),sinh(real,Y))
    <=> aa(real,$o,ord_less(real,X),Y) ) ).

% sinh_real_less_iff
tff(fact_4764_sinh__real__neg__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,sinh(real,X)),zero_zero(real))
    <=> aa(real,$o,ord_less(real,X),zero_zero(real)) ) ).

% sinh_real_neg_iff
tff(fact_4765_sinh__real__pos__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),sinh(real,X))
    <=> aa(real,$o,ord_less(real,zero_zero(real)),X) ) ).

% sinh_real_pos_iff
tff(fact_4766_sinh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),sinh(real,X))
    <=> aa(real,$o,ord_less_eq(real,zero_zero(real)),X) ) ).

% sinh_real_nonneg_iff
tff(fact_4767_sinh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,sinh(real,X)),zero_zero(real))
    <=> aa(real,$o,ord_less_eq(real,X),zero_zero(real)) ) ).

% sinh_real_nonpos_iff
tff(fact_4768_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_4769_of__nat__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [F2: fun(B,nat),A2: set(B)] : aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,groups7311177749621191930dd_sum(B,nat,F2),A2)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_ed(fun(B,nat),fun(B,A),F2)),A2) ) ).

% of_nat_sum
tff(fact_4770_of__int__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(A)
     => ! [F2: fun(B,int),A2: set(B)] : aa(int,A,ring_1_of_int(A),aa(set(B),int,groups7311177749621191930dd_sum(B,int,F2),A2)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_ee(fun(B,int),fun(B,A),F2)),A2) ) ).

% of_int_sum
tff(fact_4771_of__real__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [F2: fun(B,real),Sa: set(B)] : aa(real,A,real_Vector_of_real(A),aa(set(B),real,groups7311177749621191930dd_sum(B,real,F2),Sa)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_ef(fun(B,real),fun(B,A),F2)),Sa) ) ).

% of_real_sum
tff(fact_4772_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_4773_atMost__0,axiom,
    set_ord_atMost(nat,zero_zero(nat)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),bot_bot(set(nat))) ).

% atMost_0
tff(fact_4774_scaleR__left_Osum,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [G: fun(B,real),A2: set(B),X: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(set(B),real,groups7311177749621191930dd_sum(B,real,G),A2)),X) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_eg(fun(B,real),fun(A,fun(B,A)),G),X)),A2) ) ).

% scaleR_left.sum
tff(fact_4775_scaleR__right_Osum,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: real,G: fun(B,A),A2: set(B)] : aa(A,A,real_V8093663219630862766scaleR(A,A3),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A2)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_eh(real,fun(fun(B,A),fun(B,A)),A3),G)),A2) ) ).

% scaleR_right.sum
tff(fact_4776_scaleR__sum__left,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [F2: fun(B,real),A2: set(B),X: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(set(B),real,groups7311177749621191930dd_sum(B,real,F2),A2)),X) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_eg(fun(B,real),fun(A,fun(B,A)),F2),X)),A2) ) ).

% scaleR_sum_left
tff(fact_4777_scaleR__sum__right,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: real,F2: fun(B,A),A2: set(B)] : aa(A,A,real_V8093663219630862766scaleR(A,A3),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A2)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_eh(real,fun(fun(B,A),fun(B,A)),A3),F2)),A2) ) ).

% scaleR_sum_right
tff(fact_4778_atMost__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : set_ord_atMost(A,U) = collect(A,aTP_Lamp_ei(A,fun(A,$o),U)) ) ).

% atMost_def
tff(fact_4779_bounded__imp__summable,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linord2810124833399127020strict(A)
        & topolo1944317154257567458pology(A) )
     => ! [A3: fun(nat,A),B2: A] :
          ( ! [N: nat] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,A3,N))
         => ( ! [N: nat] : aa(A,$o,ord_less_eq(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,A3),set_ord_atMost(nat,N))),B2)
           => summable(A,A3) ) ) ) ).

% bounded_imp_summable
tff(fact_4780_norm__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(B,A),A2: set(B)] : aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A2))),aa(set(B),real,groups7311177749621191930dd_sum(B,real,aTP_Lamp_ej(fun(B,A),fun(B,real),F2)),A2)) ) ).

% norm_sum
tff(fact_4781_sinh__less__cosh__real,axiom,
    ! [X: real] : aa(real,$o,ord_less(real,sinh(real,X)),cosh(real,X)) ).

% sinh_less_cosh_real
tff(fact_4782_not__empty__eq__Iic__eq__empty,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [H: A] : bot_bot(set(A)) != set_ord_atMost(A,H) ) ).

% not_empty_eq_Iic_eq_empty
tff(fact_4783_summable__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [I5: set(A),F2: fun(A,fun(nat,B))] :
          ( ! [I3: A] :
              ( member(A,I3,I5)
             => summable(B,aa(A,fun(nat,B),F2,I3)) )
         => summable(B,aa(fun(A,fun(nat,B)),fun(nat,B),aTP_Lamp_el(set(A),fun(fun(A,fun(nat,B)),fun(nat,B)),I5),F2)) ) ) ).

% summable_sum
tff(fact_4784_sums__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [I5: set(A),F2: fun(A,fun(nat,B)),X: fun(A,B)] :
          ( ! [I3: A] :
              ( member(A,I3,I5)
             => aa(B,$o,sums(B,aa(A,fun(nat,B),F2,I3)),aa(A,B,X,I3)) )
         => aa(B,$o,sums(B,aa(fun(A,fun(nat,B)),fun(nat,B),aTP_Lamp_el(set(A),fun(fun(A,fun(nat,B)),fun(nat,B)),I5),F2)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,X),I5)) ) ) ).

% sums_sum
tff(fact_4785_mod__sum__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [F2: fun(B,A),A3: A,A2: set(B)] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_em(fun(B,A),fun(A,fun(B,A)),F2),A3)),A2)),A3) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A2)),A3) ) ).

% mod_sum_eq
tff(fact_4786_sum__choose__upper,axiom,
    ! [M: nat,Na: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_en(nat,fun(nat,nat),M)),set_ord_atMost(nat,Na)) = aa(nat,nat,binomial(aa(nat,nat,suc,Na)),aa(nat,nat,suc,M)) ).

% sum_choose_upper
tff(fact_4787_sum__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(nat,A),I: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_eo(fun(nat,A),fun(nat,A),F2)),set_ord_atMost(nat,I)) = aa(A,A,minus_minus(A,aa(nat,A,F2,zero_zero(nat))),aa(nat,A,F2,aa(nat,nat,suc,I))) ) ).

% sum_telescope
tff(fact_4788_cosh__real__nonzero,axiom,
    ! [X: real] : cosh(real,X) != zero_zero(real) ).

% cosh_real_nonzero
tff(fact_4789_sum_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Na: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_ord_atMost(nat,aa(nat,nat,suc,Na))) = aa(A,A,plus_plus(A,aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ep(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,Na))) ) ).

% sum.atMost_Suc_shift
tff(fact_4790_choose__rising__sum_I2_J,axiom,
    ! [Na: nat,M: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_eq(nat,fun(nat,nat),Na)),set_ord_atMost(nat,M)) = aa(nat,nat,binomial(aa(nat,nat,plus_plus(nat,aa(nat,nat,plus_plus(nat,Na),M)),one_one(nat))),M) ).

% choose_rising_sum(2)
tff(fact_4791_choose__rising__sum_I1_J,axiom,
    ! [Na: nat,M: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_eq(nat,fun(nat,nat),Na)),set_ord_atMost(nat,M)) = aa(nat,nat,binomial(aa(nat,nat,plus_plus(nat,aa(nat,nat,plus_plus(nat,Na),M)),one_one(nat))),aa(nat,nat,plus_plus(nat,Na),one_one(nat))) ).

% choose_rising_sum(1)
tff(fact_4792_sum__choose__lower,axiom,
    ! [R3: nat,Na: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_er(nat,fun(nat,nat),R3)),set_ord_atMost(nat,Na)) = aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,plus_plus(nat,R3),Na))),Na) ).

% sum_choose_lower
tff(fact_4793_polyfun__eq__coeffs,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Na: nat,D2: fun(nat,A)] :
          ( ! [X3: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_es(fun(nat,A),fun(A,fun(nat,A)),C2),X3)),set_ord_atMost(nat,Na)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_es(fun(nat,A),fun(A,fun(nat,A)),D2),X3)),set_ord_atMost(nat,Na))
        <=> ! [I4: nat] :
              ( aa(nat,$o,ord_less_eq(nat,I4),Na)
             => ( aa(nat,A,C2,I4) = aa(nat,A,D2,I4) ) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_4794_polyfun__eq__0,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Na: nat] :
          ( ! [X3: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_es(fun(nat,A),fun(A,fun(nat,A)),C2),X3)),set_ord_atMost(nat,Na)) = zero_zero(A)
        <=> ! [I4: nat] :
              ( aa(nat,$o,ord_less_eq(nat,I4),Na)
             => ( aa(nat,A,C2,I4) = zero_zero(A) ) ) ) ) ).

% polyfun_eq_0
tff(fact_4795_zero__polynom__imp__zero__coeffs,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_mult(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [C2: fun(nat,A),Na: nat,K: nat] :
          ( ! [W2: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_et(fun(nat,A),fun(A,fun(nat,A)),C2),W2)),set_ord_atMost(nat,Na)) = zero_zero(A)
         => ( aa(nat,$o,ord_less_eq(nat,K),Na)
           => ( aa(nat,A,C2,K) = zero_zero(A) ) ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_4796_gbinomial__parallel__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,Na: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_eu(A,fun(nat,A),A3)),set_ord_atMost(nat,Na)) = aa(nat,A,gbinomial(A,aa(A,A,plus_plus(A,aa(A,A,plus_plus(A,A3),aa(nat,A,semiring_1_of_nat(A),Na))),one_one(A))),Na) ) ).

% gbinomial_parallel_sum
tff(fact_4797_sum__choose__diagonal,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),Na)
     => ( aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_ev(nat,fun(nat,fun(nat,nat)),M),Na)),set_ord_atMost(nat,M)) = aa(nat,nat,binomial(aa(nat,nat,suc,Na)),M) ) ) ).

% sum_choose_diagonal
tff(fact_4798_vandermonde,axiom,
    ! [M: nat,Na: nat,R3: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_ew(nat,fun(nat,fun(nat,fun(nat,nat))),M),Na),R3)),set_ord_atMost(nat,R3)) = aa(nat,nat,binomial(aa(nat,nat,plus_plus(nat,M),Na)),R3) ).

% vandermonde
tff(fact_4799_cosh__real__pos,axiom,
    ! [X: real] : aa(real,$o,ord_less(real,zero_zero(real)),cosh(real,X)) ).

% cosh_real_pos
tff(fact_4800_cosh__real__nonneg,axiom,
    ! [X: real] : aa(real,$o,ord_less_eq(real,zero_zero(real)),cosh(real,X)) ).

% cosh_real_nonneg
tff(fact_4801_cosh__real__nonneg__le__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
       => ( aa(real,$o,ord_less_eq(real,cosh(real,X)),cosh(real,Y))
        <=> aa(real,$o,ord_less_eq(real,X),Y) ) ) ) ).

% cosh_real_nonneg_le_iff
tff(fact_4802_cosh__real__nonpos__le__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,X),zero_zero(real))
     => ( aa(real,$o,ord_less_eq(real,Y),zero_zero(real))
       => ( aa(real,$o,ord_less_eq(real,cosh(real,X)),cosh(real,Y))
        <=> aa(real,$o,ord_less_eq(real,Y),X) ) ) ) ).

% cosh_real_nonpos_le_iff
tff(fact_4803_atMost__Suc,axiom,
    ! [K: nat] : set_ord_atMost(nat,aa(nat,nat,suc,K)) = aa(set(nat),set(nat),insert(nat,aa(nat,nat,suc,K)),set_ord_atMost(nat,K)) ).

% atMost_Suc
tff(fact_4804_sinh__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : sinh(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),X)) = 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))),sinh(A,X))),cosh(A,X)) ) ).

% sinh_double
tff(fact_4805_cosh__real__ge__1,axiom,
    ! [X: real] : aa(real,$o,ord_less_eq(real,one_one(real)),cosh(real,X)) ).

% cosh_real_ge_1
tff(fact_4806_sum__gp__basic,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,Na: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,one_one(A)),X)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,power_power(A,X)),set_ord_atMost(nat,Na))) = aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,power_power(A,X),aa(nat,nat,suc,Na))) ) ).

% sum_gp_basic
tff(fact_4807_suminf__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [I5: set(A),F2: fun(A,fun(nat,B))] :
          ( ! [I3: A] :
              ( member(A,I3,I5)
             => summable(B,aa(A,fun(nat,B),F2,I3)) )
         => ( suminf(B,aa(fun(A,fun(nat,B)),fun(nat,B),aTP_Lamp_el(set(A),fun(fun(A,fun(nat,B)),fun(nat,B)),I5),F2)) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,aTP_Lamp_ex(fun(A,fun(nat,B)),fun(A,B),F2)),I5) ) ) ) ).

% suminf_sum
tff(fact_4808_choose__row__sum,axiom,
    ! [Na: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,binomial(Na)),set_ord_atMost(nat,Na)) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) ).

% choose_row_sum
tff(fact_4809_binomial,axiom,
    ! [A3: nat,B3: nat,Na: nat] : aa(nat,nat,power_power(nat,aa(nat,nat,plus_plus(nat,A3),B3)),Na) = aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_ey(nat,fun(nat,fun(nat,fun(nat,nat))),A3),B3),Na)),set_ord_atMost(nat,Na)) ).

% binomial
tff(fact_4810_summable__Cauchy__product,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [A3: fun(nat,A),B3: fun(nat,A)] :
          ( summable(real,aTP_Lamp_ez(fun(nat,A),fun(nat,real),A3))
         => ( summable(real,aTP_Lamp_ez(fun(nat,A),fun(nat,real),B3))
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_fb(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A3),B3)) ) ) ) ).

% summable_Cauchy_product
tff(fact_4811_Cauchy__product,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [A3: fun(nat,A),B3: fun(nat,A)] :
          ( summable(real,aTP_Lamp_ez(fun(nat,A),fun(nat,real),A3))
         => ( summable(real,aTP_Lamp_ez(fun(nat,A),fun(nat,real),B3))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,A3)),suminf(A,B3)) = suminf(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_fb(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A3),B3)) ) ) ) ) ).

% Cauchy_product
tff(fact_4812_sum_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Na: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),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,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_fc(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,Na)) ) ).

% sum.in_pairs_0
tff(fact_4813_polynomial__product,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [M: nat,A3: fun(nat,A),Na: nat,B3: fun(nat,A),X: A] :
          ( ! [I3: nat] :
              ( aa(nat,$o,ord_less(nat,M),I3)
             => ( aa(nat,A,A3,I3) = zero_zero(A) ) )
         => ( ! [J2: nat] :
                ( aa(nat,$o,ord_less(nat,Na),J2)
               => ( aa(nat,A,B3,J2) = zero_zero(A) ) )
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_fd(fun(nat,A),fun(A,fun(nat,A)),A3),X)),set_ord_atMost(nat,M))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_fd(fun(nat,A),fun(A,fun(nat,A)),B3),X)),set_ord_atMost(nat,Na))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(fun(nat,A),fun(A,fun(nat,A)),aTP_Lamp_ff(fun(nat,A),fun(fun(nat,A),fun(A,fun(nat,A))),A3),B3),X)),set_ord_atMost(nat,aa(nat,nat,plus_plus(nat,M),Na))) ) ) ) ) ).

% polynomial_product
tff(fact_4814_gbinomial__sum__lower__neg,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_fg(A,fun(nat,A),A3)),set_ord_atMost(nat,M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),M)),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A3),one_one(A))),M)) ) ).

% gbinomial_sum_lower_neg
tff(fact_4815_polynomial__product__nat,axiom,
    ! [M: nat,A3: fun(nat,nat),Na: nat,B3: fun(nat,nat),X: nat] :
      ( ! [I3: nat] :
          ( aa(nat,$o,ord_less(nat,M),I3)
         => ( aa(nat,nat,A3,I3) = zero_zero(nat) ) )
     => ( ! [J2: nat] :
            ( aa(nat,$o,ord_less(nat,Na),J2)
           => ( aa(nat,nat,B3,J2) = zero_zero(nat) ) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_fh(fun(nat,nat),fun(nat,fun(nat,nat)),A3),X)),set_ord_atMost(nat,M))),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_fh(fun(nat,nat),fun(nat,fun(nat,nat)),B3),X)),set_ord_atMost(nat,Na))) = aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aa(fun(nat,nat),fun(nat,fun(nat,nat)),aTP_Lamp_fj(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),A3),B3),X)),set_ord_atMost(nat,aa(nat,nat,plus_plus(nat,M),Na))) ) ) ) ).

% polynomial_product_nat
tff(fact_4816_binomial__ring,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,B3: A,Na: nat] : aa(nat,A,power_power(A,aa(A,A,plus_plus(A,A3),B3)),Na) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_fk(A,fun(A,fun(nat,fun(nat,A))),A3),B3),Na)),set_ord_atMost(nat,Na)) ) ).

% binomial_ring
tff(fact_4817_choose__square__sum,axiom,
    ! [Na: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_fl(nat,fun(nat,nat),Na)),set_ord_atMost(nat,Na)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),Na) ).

% choose_square_sum
tff(fact_4818_pochhammer__binomial__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [A3: A,B3: A,Na: nat] : comm_s3205402744901411588hammer(A,aa(A,A,plus_plus(A,A3),B3),Na) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_fm(A,fun(A,fun(nat,fun(nat,A))),A3),B3),Na)),set_ord_atMost(nat,Na)) ) ).

% pochhammer_binomial_sum
tff(fact_4819_cosh__real__nonpos__less__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,X),zero_zero(real))
     => ( aa(real,$o,ord_less_eq(real,Y),zero_zero(real))
       => ( aa(real,$o,ord_less(real,cosh(real,X)),cosh(real,Y))
        <=> aa(real,$o,ord_less(real,Y),X) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_4820_cosh__real__nonneg__less__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
       => ( aa(real,$o,ord_less(real,cosh(real,X)),cosh(real,Y))
        <=> aa(real,$o,ord_less(real,X),Y) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_4821_cosh__real__strict__mono,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less(real,X),Y)
       => aa(real,$o,ord_less(real,cosh(real,X)),cosh(real,Y)) ) ) ).

% cosh_real_strict_mono
tff(fact_4822_Cauchy__product__sums,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [A3: fun(nat,A),B3: fun(nat,A)] :
          ( summable(real,aTP_Lamp_ez(fun(nat,A),fun(nat,real),A3))
         => ( summable(real,aTP_Lamp_ez(fun(nat,A),fun(nat,real),B3))
           => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_fb(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A3),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,A3)),suminf(A,B3))) ) ) ) ).

% Cauchy_product_sums
tff(fact_4823_atMost__nat__numeral,axiom,
    ! [K: num] : set_ord_atMost(nat,aa(num,nat,numeral_numeral(nat),K)) = aa(set(nat),set(nat),insert(nat,aa(num,nat,numeral_numeral(nat),K)),set_ord_atMost(nat,pred_numeral(K))) ).

% atMost_nat_numeral
tff(fact_4824_cosh__square__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(nat,A,power_power(A,cosh(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,plus_plus(A,aa(nat,A,power_power(A,sinh(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)) ) ).

% cosh_square_eq
tff(fact_4825_hyperbolic__pythagoras,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,minus_minus(A,aa(nat,A,power_power(A,cosh(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,sinh(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(A) ) ).

% hyperbolic_pythagoras
tff(fact_4826_sinh__square__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(nat,A,power_power(A,sinh(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,minus_minus(A,aa(nat,A,power_power(A,cosh(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)) ) ).

% sinh_square_eq
tff(fact_4827_sum__power__add,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,M: nat,I5: set(nat)] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_fn(A,fun(nat,fun(nat,A)),X),M)),I5) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,X),M)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,power_power(A,X)),I5)) ) ).

% sum_power_add
tff(fact_4828_sum_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [P2: nat,K: nat,G: fun(nat,A),H: fun(nat,A)] :
          ( aa(nat,$o,ord_less_eq(nat,one_one(nat)),P2)
         => ( aa(nat,$o,ord_less_eq(nat,K),P2)
           => ( aa(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_fo(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),set_ord_atMost(nat,P2)) = aa(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_fp(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),set_ord_atMost(nat,aa(nat,nat,minus_minus(nat,P2),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% sum.zero_middle
tff(fact_4829_arcosh__cosh__real,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => ( aa(real,real,arcosh(real),cosh(real,X)) = X ) ) ).

% arcosh_cosh_real
tff(fact_4830_gbinomial__partial__sum__poly,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat,A3: A,X: A,Y: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_fq(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A3),X),Y)),set_ord_atMost(nat,M)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_fr(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A3),X),Y)),set_ord_atMost(nat,M)) ) ).

% gbinomial_partial_sum_poly
tff(fact_4831_exp__series__add__commuting,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A,Y: A,Na: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),X),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),X) )
         => ( aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Na))),aa(nat,A,power_power(A,aa(A,A,plus_plus(A,X),Y)),Na)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_fs(A,fun(A,fun(nat,fun(nat,A))),X),Y),Na)),set_ord_atMost(nat,Na)) ) ) ) ).

% exp_series_add_commuting
tff(fact_4832_cosh__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : cosh(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),X)) = aa(A,A,plus_plus(A,aa(nat,A,power_power(A,cosh(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,sinh(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% cosh_double
tff(fact_4833_root__polyfun,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Na: nat,Z: A,A3: A] :
          ( aa(nat,$o,ord_less_eq(nat,one_one(nat)),Na)
         => ( ( aa(nat,A,power_power(A,Z),Na) = A3 )
          <=> ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_ft(nat,fun(A,fun(A,fun(nat,A))),Na),Z),A3)),set_ord_atMost(nat,Na)) = zero_zero(A) ) ) ) ) ).

% root_polyfun
tff(fact_4834_sum__gp0,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [X: A,Na: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,power_power(A,X)),set_ord_atMost(nat,Na)) = $ite(X = one_one(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,plus_plus(nat,Na),one_one(nat))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,power_power(A,X),aa(nat,nat,suc,Na)))),aa(A,A,minus_minus(A,one_one(A)),X))) ) ).

% sum_gp0
tff(fact_4835_choose__alternating__linear__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Na: nat] :
          ( ( Na != one_one(nat) )
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_fu(nat,fun(nat,A),Na)),set_ord_atMost(nat,Na)) = zero_zero(A) ) ) ) ).

% choose_alternating_linear_sum
tff(fact_4836_gbinomial__sum__nat__pow2,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_fv(nat,fun(nat,A),M)),set_ord_atMost(nat,M)) = aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M) ) ).

% gbinomial_sum_nat_pow2
tff(fact_4837_gbinomial__partial__sum__poly__xpos,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat,A3: A,X: A,Y: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_fq(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A3),X),Y)),set_ord_atMost(nat,M)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_fw(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A3),X),Y)),set_ord_atMost(nat,M)) ) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_4838_binomial__r__part__sum,axiom,
    ! [M: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,binomial(aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)),one_one(nat)))),set_ord_atMost(nat,M)) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) ).

% binomial_r_part_sum
tff(fact_4839_choose__linear__sum,axiom,
    ! [Na: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_fx(nat,fun(nat,nat),Na)),set_ord_atMost(nat,Na)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,Na),one_one(nat)))) ).

% choose_linear_sum
tff(fact_4840_choose__alternating__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Na: nat] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_fy(nat,fun(nat,A),Na)),set_ord_atMost(nat,Na)) = zero_zero(A) ) ) ) ).

% choose_alternating_sum
tff(fact_4841_polyfun__extremal__lemma,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [E2: real,C2: fun(nat,A),Na: nat] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),E2)
         => ? [M8: real] :
            ! [Z3: A] :
              ( aa(real,$o,ord_less_eq(real,M8),real_V7770717601297561774m_norm(A,Z3))
             => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_bp(fun(nat,A),fun(A,fun(nat,A)),C2),Z3)),set_ord_atMost(nat,Na)))),aa(real,real,aa(real,fun(real,real),times_times(real),E2),aa(nat,real,power_power(real,real_V7770717601297561774m_norm(A,Z3)),aa(nat,nat,suc,Na)))) ) ) ) ).

% polyfun_extremal_lemma
tff(fact_4842_gbinomial__r__part__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,gbinomial(A,aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,semiring_1_of_nat(A),M))),one_one(A)))),set_ord_atMost(nat,M)) = aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) ) ).

% gbinomial_r_part_sum
tff(fact_4843_gbinomial__partial__row__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_fz(A,fun(nat,A),A3)),set_ord_atMost(nat,M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,plus_plus(A,aa(nat,A,semiring_1_of_nat(A),M)),one_one(A))),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,A,gbinomial(A,A3),aa(nat,nat,plus_plus(nat,M),one_one(nat)))) ) ).

% gbinomial_partial_row_sum
tff(fact_4844_choose__even__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Na: nat] :
          ( aa(nat,$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,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ga(nat,fun(nat,A),Na)),set_ord_atMost(nat,Na))) = aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na) ) ) ) ).

% choose_even_sum
tff(fact_4845_choose__odd__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Na: nat] :
          ( aa(nat,$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,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_gb(nat,fun(nat,A),Na)),set_ord_atMost(nat,Na))) = aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na) ) ) ) ).

% choose_odd_sum
tff(fact_4846_tanh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] :
          ( ( cosh(A,X) != zero_zero(A) )
         => ( ( cosh(A,Y) != zero_zero(A) )
           => ( aa(A,A,tanh(A),aa(A,A,plus_plus(A,X),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,plus_plus(A,aa(A,A,tanh(A),X)),aa(A,A,tanh(A),Y))),aa(A,A,plus_plus(A,one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tanh(A),X)),aa(A,A,tanh(A),Y)))) ) ) ) ) ).

% tanh_add
tff(fact_4847_mask__eq__sum__exp,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Na: nat] : aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)),one_one(A)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2)))),collect(nat,aTP_Lamp_gc(nat,fun(nat,$o),Na))) ) ).

% mask_eq_sum_exp
tff(fact_4848_sinh__zero__iff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( sinh(A,X) = zero_zero(A) )
        <=> member(A,aa(A,A,exp(A),X),aa(set(A),set(A),insert(A,one_one(A)),aa(set(A),set(A),insert(A,aa(A,A,uminus_uminus(A),one_one(A))),bot_bot(set(A))))) ) ) ).

% sinh_zero_iff
tff(fact_4849_cosh__field__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z: A] : cosh(A,Z) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,plus_plus(A,aa(A,A,exp(A),Z)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),Z)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% cosh_field_def
tff(fact_4850_mask__eq__sum__exp__nat,axiom,
    ! [Na: nat] : aa(nat,nat,minus_minus(nat,aa(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,groups7311177749621191930dd_sum(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)))),collect(nat,aTP_Lamp_gc(nat,fun(nat,$o),Na))) ).

% mask_eq_sum_exp_nat
tff(fact_4851_sinh__field__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z: A] : sinh(A,Z) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,exp(A),Z)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),Z)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% sinh_field_def
tff(fact_4852_cosh__zero__iff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cosh(A,X) = zero_zero(A) )
        <=> ( aa(nat,A,power_power(A,aa(A,A,exp(A),X)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% cosh_zero_iff
tff(fact_4853_cosh__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : cosh(A,X) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(A,A,plus_plus(A,aa(A,A,exp(A),X)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),X)))) ) ).

% cosh_def
tff(fact_4854_cosh__ln__real,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ( cosh(real,aa(real,real,ln_ln(real),X)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,plus_plus(real,X),aa(real,real,inverse_inverse(real),X))),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% cosh_ln_real
tff(fact_4855_sinh__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : sinh(A,X) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(A,A,minus_minus(A,aa(A,A,exp(A),X)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),X)))) ) ).

% sinh_def
tff(fact_4856_sinh__ln__real,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ( sinh(real,aa(real,real,ln_ln(real),X)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,X),aa(real,real,inverse_inverse(real),X))),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% sinh_ln_real
tff(fact_4857_cos__x__cos__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_ge(A,fun(A,fun(nat,A)),X),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),cos(A,Y))) ) ).

% cos_x_cos_y
tff(fact_4858_sums__cos__x__plus__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_gg(A,fun(A,fun(nat,A)),X),Y)),cos(A,aa(A,A,plus_plus(A,X),Y))) ) ).

% sums_cos_x_plus_y
tff(fact_4859_sum__abs__ge__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [F2: fun(B,A),A2: set(B)] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_gh(fun(B,A),fun(B,A),F2)),A2)) ) ).

% sum_abs_ge_zero
tff(fact_4860_sum__abs,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [F2: fun(B,A),A2: set(B)] : aa(A,$o,ord_less_eq(A,aa(A,A,abs_abs(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A2))),aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_gh(fun(B,A),fun(B,A),F2)),A2)) ) ).

% sum_abs
tff(fact_4861_sum_Oempty,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),bot_bot(set(B))) = zero_zero(A) ) ).

% sum.empty
tff(fact_4862_sum_Oneutral__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_gi(B,A)),A2) = zero_zero(A) ) ).

% sum.neutral_const
tff(fact_4863_abs__sum__abs,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [F2: fun(B,A),A2: set(B)] : aa(A,A,abs_abs(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_gh(fun(B,A),fun(B,A),F2)),A2)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_gh(fun(B,A),fun(B,A),F2)),A2) ) ).

% abs_sum_abs
tff(fact_4864_int__sum,axiom,
    ! [A: $tType,F2: fun(A,nat),A2: set(A)] : aa(nat,int,semiring_1_of_nat(int),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A2)) = aa(set(A),int,groups7311177749621191930dd_sum(A,int,aTP_Lamp_gj(fun(A,nat),fun(A,int),F2)),A2) ).

% int_sum
tff(fact_4865_sum__subtractf__nat,axiom,
    ! [A: $tType,A2: set(A),G: fun(A,nat),F2: fun(A,nat)] :
      ( ! [X4: A] :
          ( member(A,X4,A2)
         => aa(nat,$o,ord_less_eq(nat,aa(A,nat,G,X4)),aa(A,nat,F2,X4)) )
     => ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_gk(fun(A,nat),fun(fun(A,nat),fun(A,nat)),G),F2)),A2) = aa(nat,nat,minus_minus(nat,aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A2)),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,G),A2)) ) ) ).

% sum_subtractf_nat
tff(fact_4866_Complex__sum_H,axiom,
    ! [A: $tType,F2: fun(A,real),Sa: set(A)] : aa(set(A),complex,groups7311177749621191930dd_sum(A,complex,aTP_Lamp_gl(fun(A,real),fun(A,complex),F2)),Sa) = complex2(aa(set(A),real,groups7311177749621191930dd_sum(A,real,F2),Sa),zero_zero(real)) ).

% Complex_sum'
tff(fact_4867_sum__SucD,axiom,
    ! [A: $tType,F2: fun(A,nat),A2: set(A),Na: nat] :
      ( ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A2) = aa(nat,nat,suc,Na) )
     => ? [X4: A] :
          ( member(A,X4,A2)
          & aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(A,nat,F2,X4)) ) ) ).

% sum_SucD
tff(fact_4868_sum__diff1__nat,axiom,
    ! [A: $tType,F2: fun(A,nat),A2: set(A),A3: A] :
      aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),aa(set(A),set(A),minus_minus(set(A),A2),aa(set(A),set(A),insert(A,A3),bot_bot(set(A))))) = $ite(member(A,A3,A2),aa(nat,nat,minus_minus(nat,aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A2)),aa(A,nat,F2,A3)),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A2)) ).

% sum_diff1_nat
tff(fact_4869_sum__nth__roots,axiom,
    ! [Na: nat,C2: complex] :
      ( aa(nat,$o,ord_less(nat,one_one(nat)),Na)
     => ( aa(set(complex),complex,groups7311177749621191930dd_sum(complex,complex,aTP_Lamp_gm(complex,complex)),collect(complex,aa(complex,fun(complex,$o),aTP_Lamp_gn(nat,fun(complex,fun(complex,$o)),Na),C2))) = zero_zero(complex) ) ) ).

% sum_nth_roots
tff(fact_4870_sum_Oswap,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,fun(C,A)),B2: set(C),A2: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(set(C),fun(B,A),aTP_Lamp_go(fun(B,fun(C,A)),fun(set(C),fun(B,A)),G),B2)),A2) = aa(set(C),A,groups7311177749621191930dd_sum(C,A,aa(set(B),fun(C,A),aTP_Lamp_gq(fun(B,fun(C,A)),fun(set(B),fun(C,A)),G),A2)),B2) ) ).

% sum.swap
tff(fact_4871_sum__roots__unity,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,ord_less(nat,one_one(nat)),Na)
     => ( aa(set(complex),complex,groups7311177749621191930dd_sum(complex,complex,aTP_Lamp_gm(complex,complex)),collect(complex,aTP_Lamp_gr(nat,fun(complex,$o),Na))) = zero_zero(complex) ) ) ).

% sum_roots_unity
tff(fact_4872_sum_Oneutral,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [A2: set(A),G: fun(A,B)] :
          ( ! [X4: A] :
              ( member(A,X4,A2)
             => ( aa(A,B,G,X4) = zero_zero(B) ) )
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A2) = zero_zero(B) ) ) ) ).

% sum.neutral
tff(fact_4873_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),A2: set(B)] :
          ( ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A2) != zero_zero(A) )
         => ~ ! [A4: B] :
                ( member(B,A4,A2)
               => ( aa(B,A,G,A4) = zero_zero(A) ) ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_4874_sum__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [K4: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( ! [I3: A] :
              ( member(A,I3,K4)
             => aa(B,$o,ord_less_eq(B,aa(A,B,F2,I3)),aa(A,B,G,I3)) )
         => aa(B,$o,ord_less_eq(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),K4)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),K4)) ) ) ).

% sum_mono
tff(fact_4875_sum_Odistrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),H: fun(B,A),A2: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_gs(fun(B,A),fun(fun(B,A),fun(B,A)),G),H)),A2) = aa(A,A,plus_plus(A,aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A2)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,H),A2)) ) ).

% sum.distrib
tff(fact_4876_sum__product,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [F2: fun(B,A),A2: set(B),G: fun(C,A),B2: set(C)] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A2)),aa(set(C),A,groups7311177749621191930dd_sum(C,A,G),B2)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_gu(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),F2),G),B2)),A2) ) ).

% sum_product
tff(fact_4877_sum__distrib__right,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [F2: fun(B,A),A2: set(B),R3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A2)),R3) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_gv(fun(B,A),fun(A,fun(B,A)),F2),R3)),A2) ) ).

% sum_distrib_right
tff(fact_4878_sum__distrib__left,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [R3: A,F2: fun(B,A),A2: set(B)] : aa(A,A,aa(A,fun(A,A),times_times(A),R3),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A2)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_gw(A,fun(fun(B,A),fun(B,A)),R3),F2)),A2) ) ).

% sum_distrib_left
tff(fact_4879_sum__subtractf,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(B,A),G: fun(B,A),A2: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_gx(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G)),A2) = aa(A,A,minus_minus(A,aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A2)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A2)) ) ).

% sum_subtractf
tff(fact_4880_sum__divide__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [F2: fun(B,A),A2: set(B),R3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A2)),R3) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_gy(fun(B,A),fun(A,fun(B,A)),F2),R3)),A2) ) ).

% sum_divide_distrib
tff(fact_4881_sum__negf,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(B,A),A2: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_gz(fun(B,A),fun(B,A),F2)),A2) = aa(A,A,uminus_uminus(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A2)) ) ).

% sum_negf
tff(fact_4882_sum__nonpos,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A2: set(A),F2: fun(A,B)] :
          ( ! [X4: A] :
              ( member(A,X4,A2)
             => aa(B,$o,ord_less_eq(B,aa(A,B,F2,X4)),zero_zero(B)) )
         => aa(B,$o,ord_less_eq(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A2)),zero_zero(B)) ) ) ).

% sum_nonpos
tff(fact_4883_sum__nonneg,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A2: set(A),F2: fun(A,B)] :
          ( ! [X4: A] :
              ( member(A,X4,A2)
             => aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,F2,X4)) )
         => aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A2)) ) ) ).

% sum_nonneg
tff(fact_4884_sum__cong__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: set(nat),F2: fun(nat,A),G: fun(nat,A)] :
          ( ~ member(nat,zero_zero(nat),A2)
         => ( ! [X4: nat] :
                ( member(nat,aa(nat,nat,suc,X4),A2)
               => ( aa(nat,A,F2,aa(nat,nat,suc,X4)) = aa(nat,A,G,aa(nat,nat,suc,X4)) ) )
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),A2) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),A2) ) ) ) ) ).

% sum_cong_Suc
tff(fact_4885_convex__sum__bound__le,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),X: fun(A,B),A3: fun(A,B),B3: B,Delta: B] :
          ( ! [I3: A] :
              ( member(A,I3,I5)
             => aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,X,I3)) )
         => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,X),I5) = one_one(B) )
           => ( ! [I3: A] :
                  ( member(A,I3,I5)
                 => aa(B,$o,ord_less_eq(B,aa(B,B,abs_abs(B),aa(B,B,minus_minus(B,aa(A,B,A3,I3)),B3))),Delta) )
             => aa(B,$o,ord_less_eq(B,aa(B,B,abs_abs(B),aa(B,B,minus_minus(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ha(fun(A,B),fun(fun(A,B),fun(A,B)),X),A3)),I5)),B3))),Delta) ) ) ) ) ).

% convex_sum_bound_le
tff(fact_4886_Maclaurin__cos__expansion2,axiom,
    ! [X: real,Na: nat] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
       => ? [T2: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),T2)
            & aa(real,$o,ord_less(real,T2),X)
            & ( cos(real,X) = aa(real,real,plus_plus(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_hb(real,fun(nat,real),X)),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,plus_plus(real,T2),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,power_power(real,X),Na))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_4887_Maclaurin__minus__cos__expansion,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(real,$o,ord_less(real,X),zero_zero(real))
       => ? [T2: real] :
            ( aa(real,$o,ord_less(real,X),T2)
            & aa(real,$o,ord_less(real,T2),zero_zero(real))
            & ( cos(real,X) = aa(real,real,plus_plus(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_hb(real,fun(nat,real),X)),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,plus_plus(real,T2),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,power_power(real,X),Na))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_4888_Maclaurin__sin__expansion3,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),X)
       => ? [T2: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),T2)
            & aa(real,$o,ord_less(real,T2),X)
            & ( sin(real,X) = aa(real,real,plus_plus(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_hc(real,fun(nat,real),X)),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,plus_plus(real,T2),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,power_power(real,X),Na))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_4889_lessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,K: A] :
          ( member(A,I,set_ord_lessThan(A,K))
        <=> aa(A,$o,ord_less(A,I),K) ) ) ).

% lessThan_iff
tff(fact_4890_lessThan__0,axiom,
    set_ord_lessThan(nat,zero_zero(nat)) = bot_bot(set(nat)) ).

% lessThan_0
tff(fact_4891_single__Diff__lessThan,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A] : aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),insert(A,K),bot_bot(set(A)))),set_ord_lessThan(A,K)) = aa(set(A),set(A),insert(A,K),bot_bot(set(A))) ) ).

% single_Diff_lessThan
tff(fact_4892_sumr__cos__zero__one,axiom,
    ! [Na: nat] : aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_hd(nat,real)),set_ord_lessThan(nat,aa(nat,nat,suc,Na))) = one_one(real) ).

% sumr_cos_zero_one
tff(fact_4893_lessThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : set_ord_lessThan(A,U) = collect(A,aTP_Lamp_he(A,fun(A,$o),U)) ) ).

% lessThan_def
tff(fact_4894_lessThan__non__empty,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [X: A] : set_ord_lessThan(A,X) != bot_bot(set(A)) ) ).

% lessThan_non_empty
tff(fact_4895_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_4896_lessThan__strict__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,Na: A] :
          ( aa(set(A),$o,ord_less(set(A),set_ord_lessThan(A,M)),set_ord_lessThan(A,Na))
        <=> aa(A,$o,ord_less(A,M),Na) ) ) ).

% lessThan_strict_subset_iff
tff(fact_4897_lessThan__empty__iff,axiom,
    ! [Na: nat] :
      ( ( set_ord_lessThan(nat,Na) = bot_bot(set(nat)) )
    <=> ( Na = zero_zero(nat) ) ) ).

% lessThan_empty_iff
tff(fact_4898_lessThan__Suc,axiom,
    ! [K: nat] : set_ord_lessThan(nat,aa(nat,nat,suc,K)) = aa(set(nat),set(nat),insert(nat,K),set_ord_lessThan(nat,K)) ).

% lessThan_Suc
tff(fact_4899_Iic__subset__Iio__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(set(A),$o,ord_less_eq(set(A),set_ord_atMost(A,A3)),set_ord_lessThan(A,B3))
        <=> aa(A,$o,ord_less(A,A3),B3) ) ) ).

% Iic_subset_Iio_iff
tff(fact_4900_lessThan__nat__numeral,axiom,
    ! [K: num] : set_ord_lessThan(nat,aa(num,nat,numeral_numeral(nat),K)) = aa(set(nat),set(nat),insert(nat,pred_numeral(K)),set_ord_lessThan(nat,pred_numeral(K))) ).

% lessThan_nat_numeral
tff(fact_4901_sum_Onat__diff__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Na: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_hf(fun(nat,A),fun(nat,fun(nat,A)),G),Na)),set_ord_lessThan(nat,Na)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_ord_lessThan(nat,Na)) ) ).

% sum.nat_diff_reindex
tff(fact_4902_sum__diff__distrib,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Q: fun(A,nat),P: fun(A,nat),Na: A] :
          ( ! [X4: A] : aa(nat,$o,ord_less_eq(nat,aa(A,nat,Q,X4)),aa(A,nat,P,X4))
         => ( aa(nat,nat,minus_minus(nat,aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,P),set_ord_lessThan(A,Na))),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,Q),set_ord_lessThan(A,Na))) = aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_hg(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Q),P)),set_ord_lessThan(A,Na)) ) ) ) ).

% sum_diff_distrib
tff(fact_4903_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),insert(A,U),bot_bot(set(A)))) = set_ord_atMost(A,U) ) ).

% ivl_disj_un_singleton(2)
tff(fact_4904_sum_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Na: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_ord_lessThan(nat,aa(nat,nat,suc,Na))) = aa(A,A,plus_plus(A,aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ep(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,Na))) ) ).

% sum.lessThan_Suc_shift
tff(fact_4905_sum__lessThan__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(nat,A),M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_hh(fun(nat,A),fun(nat,A),F2)),set_ord_lessThan(nat,M)) = aa(A,A,minus_minus(A,aa(nat,A,F2,M)),aa(nat,A,F2,zero_zero(nat))) ) ).

% sum_lessThan_telescope
tff(fact_4906_sum__lessThan__telescope_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(nat,A),M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_eo(fun(nat,A),fun(nat,A),F2)),set_ord_lessThan(nat,M)) = aa(A,A,minus_minus(A,aa(nat,A,F2,zero_zero(nat))),aa(nat,A,F2,M)) ) ).

% sum_lessThan_telescope'
tff(fact_4907_sumr__diff__mult__const2,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [F2: fun(nat,A),Na: nat,R3: A] : aa(A,A,minus_minus(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,Na))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),R3)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_hi(fun(nat,A),fun(A,fun(nat,A)),F2),R3)),set_ord_lessThan(nat,Na)) ) ).

% sumr_diff_mult_const2
tff(fact_4908_summableI__nonneg__bounded,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),X: A] :
          ( ! [N: nat] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,F2,N))
         => ( ! [N: nat] : aa(A,$o,ord_less_eq(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,N))),X)
           => summable(A,F2) ) ) ) ).

% summableI_nonneg_bounded
tff(fact_4909_sums__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),Na: nat,Sa: A] :
          ( aa(A,$o,sums(A,aa(nat,fun(nat,A),aTP_Lamp_ba(fun(nat,A),fun(nat,fun(nat,A)),F2),Na)),Sa)
        <=> aa(A,$o,sums(A,F2),aa(A,A,plus_plus(A,Sa),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,Na)))) ) ) ).

% sums_iff_shift
tff(fact_4910_sums__iff__shift_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),Na: nat,Sa: A] :
          ( aa(A,$o,sums(A,aa(nat,fun(nat,A),aTP_Lamp_ba(fun(nat,A),fun(nat,fun(nat,A)),F2),Na)),aa(A,A,minus_minus(A,Sa),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,Na))))
        <=> aa(A,$o,sums(A,F2),Sa) ) ) ).

% sums_iff_shift'
tff(fact_4911_sums__split__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),Sa: A,Na: nat] :
          ( aa(A,$o,sums(A,F2),Sa)
         => aa(A,$o,sums(A,aa(nat,fun(nat,A),aTP_Lamp_ba(fun(nat,A),fun(nat,fun(nat,A)),F2),Na)),aa(A,A,minus_minus(A,Sa),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,Na)))) ) ) ).

% sums_split_initial_segment
tff(fact_4912_power__diff__1__eq,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,Na: nat] : aa(A,A,minus_minus(A,aa(nat,A,power_power(A,X),Na)),one_one(A)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,X),one_one(A))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,power_power(A,X)),set_ord_lessThan(nat,Na))) ) ).

% power_diff_1_eq
tff(fact_4913_one__diff__power__eq,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,Na: nat] : aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,power_power(A,X),Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,one_one(A)),X)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,power_power(A,X)),set_ord_lessThan(nat,Na))) ) ).

% one_diff_power_eq
tff(fact_4914_geometric__sum,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,Na: nat] :
          ( ( X != one_one(A) )
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,power_power(A,X)),set_ord_lessThan(nat,Na)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(nat,A,power_power(A,X),Na)),one_one(A))),aa(A,A,minus_minus(A,X),one_one(A))) ) ) ) ).

% geometric_sum
tff(fact_4915_sum_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Na: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_ord_atMost(nat,Na)) = aa(A,A,plus_plus(A,aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ep(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,Na))) ) ).

% sum.atMost_shift
tff(fact_4916_suminf__split__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),K: nat] :
          ( summable(A,F2)
         => ( suminf(A,F2) = aa(A,A,plus_plus(A,suminf(A,aa(nat,fun(nat,A),aTP_Lamp_ba(fun(nat,A),fun(nat,fun(nat,A)),F2),K))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,K))) ) ) ) ).

% suminf_split_initial_segment
tff(fact_4917_suminf__minus__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),K: nat] :
          ( summable(A,F2)
         => ( suminf(A,aa(nat,fun(nat,A),aTP_Lamp_ba(fun(nat,A),fun(nat,fun(nat,A)),F2),K)) = aa(A,A,minus_minus(A,suminf(A,F2)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,K))) ) ) ) ).

% suminf_minus_initial_segment
tff(fact_4918_sum__less__suminf,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),Na: nat] :
          ( summable(A,F2)
         => ( ! [M4: nat] :
                ( aa(nat,$o,ord_less_eq(nat,Na),M4)
               => aa(A,$o,ord_less(A,zero_zero(A)),aa(nat,A,F2,M4)) )
           => aa(A,$o,ord_less(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,Na))),suminf(A,F2)) ) ) ) ).

% sum_less_suminf
tff(fact_4919_lemma__termdiff1,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Z: A,H: A,M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_hj(A,fun(A,fun(nat,fun(nat,A))),Z),H),M)),set_ord_lessThan(nat,M)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_hk(A,fun(A,fun(nat,fun(nat,A))),Z),H),M)),set_ord_lessThan(nat,M)) ) ).

% lemma_termdiff1
tff(fact_4920_sum__gp__strict,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [X: A,Na: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,power_power(A,X)),set_ord_lessThan(nat,Na)) = $ite(X = one_one(A),aa(nat,A,semiring_1_of_nat(A),Na),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,power_power(A,X),Na))),aa(A,A,minus_minus(A,one_one(A)),X))) ) ).

% sum_gp_strict
tff(fact_4921_diff__power__eq__sum,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,Na: nat,Y: A] : aa(A,A,minus_minus(A,aa(nat,A,power_power(A,X),aa(nat,nat,suc,Na))),aa(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,minus_minus(A,X),Y)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_hl(A,fun(nat,fun(A,fun(nat,A))),X),Na),Y)),set_ord_lessThan(nat,aa(nat,nat,suc,Na)))) ) ).

% diff_power_eq_sum
tff(fact_4922_power__diff__sumr2,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,Na: nat,Y: A] : aa(A,A,minus_minus(A,aa(nat,A,power_power(A,X),Na)),aa(nat,A,power_power(A,Y),Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,X),Y)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_hm(A,fun(nat,fun(A,fun(nat,A))),X),Na),Y)),set_ord_lessThan(nat,Na))) ) ).

% power_diff_sumr2
tff(fact_4923_polyfun__linear__factor__root,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: fun(nat,A),A3: A,Na: nat] :
          ( ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_fd(fun(nat,A),fun(A,fun(nat,A)),C2),A3)),set_ord_atMost(nat,Na)) = zero_zero(A) )
         => ~ ! [B5: fun(nat,A)] :
                ~ ! [Z3: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_fd(fun(nat,A),fun(A,fun(nat,A)),C2),Z3)),set_ord_atMost(nat,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Z3),A3)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_fd(fun(nat,A),fun(A,fun(nat,A)),B5),Z3)),set_ord_lessThan(nat,Na))) ) ) ).

% polyfun_linear_factor_root
tff(fact_4924_polyfun__linear__factor,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: fun(nat,A),Na: nat,A3: A] :
        ? [B5: fun(nat,A)] :
        ! [Z3: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_fd(fun(nat,A),fun(A,fun(nat,A)),C2),Z3)),set_ord_atMost(nat,Na)) = aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Z3),A3)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_fd(fun(nat,A),fun(A,fun(nat,A)),B5),Z3)),set_ord_lessThan(nat,Na)))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_fd(fun(nat,A),fun(A,fun(nat,A)),C2),A3)),set_ord_atMost(nat,Na))) ) ).

% polyfun_linear_factor
tff(fact_4925_real__sum__nat__ivl__bounded2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat,F2: fun(nat,A),K4: A,K: nat] :
          ( ! [P4: nat] :
              ( aa(nat,$o,ord_less(nat,P4),Na)
             => aa(A,$o,ord_less_eq(A,aa(nat,A,F2,P4)),K4) )
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),K4)
           => aa(A,$o,ord_less_eq(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,aa(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_4926_sum__less__suminf2,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),Na: nat,I: nat] :
          ( summable(A,F2)
         => ( ! [M4: nat] :
                ( aa(nat,$o,ord_less_eq(nat,Na),M4)
               => aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,F2,M4)) )
           => ( aa(nat,$o,ord_less_eq(nat,Na),I)
             => ( aa(A,$o,ord_less(A,zero_zero(A)),aa(nat,A,F2,I))
               => aa(A,$o,ord_less(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,Na))),suminf(A,F2)) ) ) ) ) ) ).

% sum_less_suminf2
tff(fact_4927_one__diff__power__eq_H,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,Na: nat] : aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,power_power(A,X),Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,one_one(A)),X)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_hn(A,fun(nat,fun(nat,A)),X),Na)),set_ord_lessThan(nat,Na))) ) ).

% one_diff_power_eq'
tff(fact_4928_Maclaurin__zero,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [X: real,Na: nat,Diff: fun(nat,fun(A,real))] :
          ( ( X = zero_zero(real) )
         => ( ( Na != zero_zero(nat) )
           => ( aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,fun(A,real)),fun(nat,real),aTP_Lamp_ho(real,fun(fun(nat,fun(A,real)),fun(nat,real)),X),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_4929_Maclaurin__lemma,axiom,
    ! [H: real,F2: fun(real,real),J: fun(nat,real),Na: nat] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),H)
     => ? [B4: real] : aa(real,real,F2,H) = aa(real,real,plus_plus(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_hp(real,fun(fun(nat,real),fun(nat,real)),H),J)),set_ord_lessThan(nat,Na))),aa(real,real,aa(real,fun(real,real),times_times(real),B4),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,power_power(real,H),Na)),semiring_char_0_fact(real,Na)))) ) ).

% Maclaurin_lemma
tff(fact_4930_sum__split__even__odd,axiom,
    ! [F2: fun(nat,real),G: fun(nat,real),Na: nat] : aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_hq(fun(nat,real),fun(fun(nat,real),fun(nat,real)),F2),G)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) = aa(real,real,plus_plus(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_hr(fun(nat,real),fun(nat,real),F2)),set_ord_lessThan(nat,Na))),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_hs(fun(nat,real),fun(nat,real),G)),set_ord_lessThan(nat,Na))) ).

% sum_split_even_odd
tff(fact_4931_Maclaurin__exp__le,axiom,
    ! [X: real,Na: nat] :
    ? [T2: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),T2)),aa(real,real,abs_abs(real),X))
      & ( aa(real,real,exp(real),X) = aa(real,real,plus_plus(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_ht(real,fun(nat,real),X)),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,exp(real),T2)),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,X),Na))) ) ) ).

% Maclaurin_exp_le
tff(fact_4932_polyfun__diff__alt,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Na: nat,A3: fun(nat,A),X: A,Y: A] :
          ( aa(nat,$o,ord_less_eq(nat,one_one(nat)),Na)
         => ( aa(A,A,minus_minus(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_fd(fun(nat,A),fun(A,fun(nat,A)),A3),X)),set_ord_atMost(nat,Na))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_fd(fun(nat,A),fun(A,fun(nat,A)),A3),Y)),set_ord_atMost(nat,Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,X),Y)),aa(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_hv(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Na),A3),X),Y)),set_ord_lessThan(nat,Na))) ) ) ) ).

% polyfun_diff_alt
tff(fact_4933_exp__first__terms,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A,K: nat] : aa(A,A,exp(A),X) = aa(A,A,plus_plus(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_dq(A,fun(nat,A),X)),set_ord_lessThan(nat,K))),suminf(A,aa(nat,fun(nat,A),aTP_Lamp_hw(A,fun(nat,fun(nat,A)),X),K))) ) ).

% exp_first_terms
tff(fact_4934_Maclaurin__sin__bound,axiom,
    ! [X: real,Na: nat] : aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,sin(real,X)),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_hc(real,fun(nat,real),X)),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,power_power(real,aa(real,real,abs_abs(real),X)),Na))) ).

% Maclaurin_sin_bound
tff(fact_4935_sum__pos__lt__pair,axiom,
    ! [F2: fun(nat,real),K: nat] :
      ( summable(real,F2)
     => ( ! [D6: nat] : aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,plus_plus(real,aa(nat,real,F2,aa(nat,nat,plus_plus(nat,K),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))),D6)))),aa(nat,real,F2,aa(nat,nat,plus_plus(nat,K),aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))),D6)),one_one(nat))))))
       => aa(real,$o,ord_less(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,F2),set_ord_lessThan(nat,K))),suminf(real,F2)) ) ) ).

% sum_pos_lt_pair
tff(fact_4936_Maclaurin__exp__lt,axiom,
    ! [X: real,Na: nat] :
      ( ( X != zero_zero(real) )
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
       => ? [T2: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,abs_abs(real),T2))
            & aa(real,$o,ord_less(real,aa(real,real,abs_abs(real),T2)),aa(real,real,abs_abs(real),X))
            & ( aa(real,real,exp(real),X) = aa(real,real,plus_plus(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_ht(real,fun(nat,real),X)),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,exp(real),T2)),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,X),Na))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_4937_lemma__termdiff2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [H: A,Z: A,Na: nat] :
          ( ( H != zero_zero(A) )
         => ( aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(A,A,plus_plus(A,Z),H)),Na)),aa(nat,A,power_power(A,Z),Na))),H)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),aa(nat,A,power_power(A,Z),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),times_times(A),H),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_hy(A,fun(A,fun(nat,fun(nat,A))),H),Z),Na)),set_ord_lessThan(nat,aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat)))))) ) ) ) ).

% lemma_termdiff2
tff(fact_4938_Maclaurin__sin__expansion,axiom,
    ! [X: real,Na: nat] :
    ? [T2: real] : sin(real,X) = aa(real,real,plus_plus(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_hc(real,fun(nat,real),X)),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,plus_plus(real,T2),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,power_power(real,X),Na))) ).

% Maclaurin_sin_expansion
tff(fact_4939_Maclaurin__sin__expansion2,axiom,
    ! [X: real,Na: nat] :
    ? [T2: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),T2)),aa(real,real,abs_abs(real),X))
      & ( sin(real,X) = aa(real,real,plus_plus(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_hc(real,fun(nat,real),X)),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,plus_plus(real,T2),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,power_power(real,X),Na))) ) ) ).

% Maclaurin_sin_expansion2
tff(fact_4940_Maclaurin__cos__expansion,axiom,
    ! [X: real,Na: nat] :
    ? [T2: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),T2)),aa(real,real,abs_abs(real),X))
      & ( cos(real,X) = aa(real,real,plus_plus(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_hb(real,fun(nat,real),X)),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,plus_plus(real,T2),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,power_power(real,X),Na))) ) ) ).

% Maclaurin_cos_expansion
tff(fact_4941_Maclaurin__sin__expansion4,axiom,
    ! [X: real,Na: nat] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => ? [T2: real] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),T2)
          & aa(real,$o,ord_less_eq(real,T2),X)
          & ( sin(real,X) = aa(real,real,plus_plus(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_hc(real,fun(nat,real),X)),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,plus_plus(real,T2),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,power_power(real,X),Na))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_4942_bij__betw__roots__unity,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => bij_betw(nat,complex,aTP_Lamp_hz(nat,fun(nat,complex),Na),set_ord_lessThan(nat,Na),collect(complex,aTP_Lamp_gr(nat,fun(complex,$o),Na))) ) ).

% bij_betw_roots_unity
tff(fact_4943_sum__gp,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [X: A,M: nat,Na: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,power_power(A,X)),set_or1337092689740270186AtMost(nat,M,Na)) = $ite(
            aa(nat,$o,ord_less(nat,Na),M),
            zero_zero(A),
            $ite(X = one_one(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,aa(nat,nat,plus_plus(nat,Na),one_one(nat))),M)),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(nat,A,power_power(A,X),M)),aa(nat,A,power_power(A,X),aa(nat,nat,suc,Na)))),aa(A,A,minus_minus(A,one_one(A)),X))) ) ) ).

% sum_gp
tff(fact_4944_gchoose__row__sum__weighted,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [R3: A,M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_fz(A,fun(nat,A),R3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,M))),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,A,gbinomial(A,R3),aa(nat,nat,suc,M))) ) ).

% gchoose_row_sum_weighted
tff(fact_4945_atLeastatMost__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A] :
          ( ( bot_bot(set(A)) = set_or1337092689740270186AtMost(A,A3,B3) )
        <=> ~ aa(A,$o,ord_less_eq(A,A3),B3) ) ) ).

% atLeastatMost_empty_iff2
tff(fact_4946_atLeastatMost__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A] :
          ( ( set_or1337092689740270186AtMost(A,A3,B3) = bot_bot(set(A)) )
        <=> ~ aa(A,$o,ord_less_eq(A,A3),B3) ) ) ).

% atLeastatMost_empty_iff
tff(fact_4947_atLeastatMost__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,B3),A3)
         => ( set_or1337092689740270186AtMost(A,A3,B3) = bot_bot(set(A)) ) ) ) ).

% atLeastatMost_empty
tff(fact_4948_atLeastAtMost__singleton__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A,C2: A] :
          ( ( set_or1337092689740270186AtMost(A,A3,B3) = aa(set(A),set(A),insert(A,C2),bot_bot(set(A))) )
        <=> ( ( A3 = B3 )
            & ( B3 = C2 ) ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_4949_atLeastAtMost__singleton,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A] : set_or1337092689740270186AtMost(A,A3,A3) = aa(set(A),set(A),insert(A,A3),bot_bot(set(A))) ) ).

% atLeastAtMost_singleton
tff(fact_4950_sum_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,Na))) = $ite(aa(nat,$o,ord_less(nat,aa(nat,nat,suc,Na)),M),zero_zero(A),aa(A,A,plus_plus(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,Na))),aa(nat,A,G,aa(nat,nat,suc,Na)))) ) ).

% sum.cl_ivl_Suc
tff(fact_4951_sum_Oreindex__bij__betw,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [H: fun(A,B),S: set(A),T3: set(B),G: fun(B,C)] :
          ( bij_betw(A,B,H,S,T3)
         => ( aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ia(fun(A,B),fun(fun(B,C),fun(A,C)),H),G)),S) = aa(set(B),C,groups7311177749621191930dd_sum(B,C,G),T3) ) ) ) ).

% sum.reindex_bij_betw
tff(fact_4952_ivl__disj__un__two__touch_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,ord_less_eq(A,L),M)
         => ( aa(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_4953_atLeastAtMost__singleton_H,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = B3 )
         => ( set_or1337092689740270186AtMost(A,A3,B3) = aa(set(A),set(A),insert(A,A3),bot_bot(set(A))) ) ) ) ).

% atLeastAtMost_singleton'
tff(fact_4954_ex__nat__less,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( ? [M3: nat] :
          ( aa(nat,$o,ord_less_eq(nat,M3),Na)
          & aa(nat,$o,P,M3) )
    <=> ? [X3: nat] :
          ( member(nat,X3,set_or1337092689740270186AtMost(nat,zero_zero(nat),Na))
          & aa(nat,$o,P,X3) ) ) ).

% ex_nat_less
tff(fact_4955_all__nat__less,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( ! [M3: nat] :
          ( aa(nat,$o,ord_less_eq(nat,M3),Na)
         => aa(nat,$o,P,M3) )
    <=> ! [X3: nat] :
          ( member(nat,X3,set_or1337092689740270186AtMost(nat,zero_zero(nat),Na))
         => aa(nat,$o,P,X3) ) ) ).

% all_nat_less
tff(fact_4956_atMost__atLeast0,axiom,
    ! [Na: nat] : set_ord_atMost(nat,Na) = set_or1337092689740270186AtMost(nat,zero_zero(nat),Na) ).

% atMost_atLeast0
tff(fact_4957_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Na))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ep(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_4958_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,K: nat,Na: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,plus_plus(nat,M),K),aa(nat,nat,plus_plus(nat,Na),K))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ib(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_4959_atLeastatMost__psubset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A,C2: A,D2: A] :
          ( aa(set(A),$o,ord_less(set(A),set_or1337092689740270186AtMost(A,A3,B3)),set_or1337092689740270186AtMost(A,C2,D2))
        <=> ( ( ~ aa(A,$o,ord_less_eq(A,A3),B3)
              | ( aa(A,$o,ord_less_eq(A,C2),A3)
                & aa(A,$o,ord_less_eq(A,B3),D2)
                & ( aa(A,$o,ord_less(A,C2),A3)
                  | aa(A,$o,ord_less(A,B3),D2) ) ) )
            & aa(A,$o,ord_less_eq(A,C2),D2) ) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4960_atLeast0__atMost__Suc,axiom,
    ! [Na: nat] : set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Na)) = aa(set(nat),set(nat),insert(nat,aa(nat,nat,suc,Na)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) ).

% atLeast0_atMost_Suc
tff(fact_4961_atLeastAtMost__insertL,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),Na)
     => ( aa(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_4962_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,Na: nat] :
      ( aa(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),insert(nat,aa(nat,nat,suc,Na)),set_or1337092689740270186AtMost(nat,M,Na)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_4963_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),Na)
     => ( set_or1337092689740270186AtMost(nat,M,Na) = aa(set(nat),set(nat),insert(nat,M),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),Na)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_4964_word__lessThan__Suc__atMost,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A)] :
          ( ( K != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
         => ( set_ord_lessThan(word(A),aa(word(A),word(A),plus_plus(word(A),K),one_one(word(A)))) = set_ord_atMost(word(A),K) ) ) ) ).

% word_lessThan_Suc_atMost
tff(fact_4965_sum_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Na: nat,M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,Na,M)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_ic(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Na),M)),set_or1337092689740270186AtMost(nat,Na,M)) ) ).

% sum.atLeastAtMost_rev
tff(fact_4966_ivl__disj__un__one_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$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_4967_sum__shift__lb__Suc0__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),K: nat] :
          ( ( aa(nat,A,F2,zero_zero(nat)) = zero_zero(A) )
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),K)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or1337092689740270186AtMost(nat,zero_zero(nat),K)) ) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_4968_sum_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Na: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Na))) = aa(A,A,plus_plus(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na))),aa(nat,A,G,aa(nat,nat,suc,Na))) ) ).

% sum.atLeast0_atMost_Suc
tff(fact_4969_sum_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,ord_less_eq(nat,M),Na)
         => ( aa(A,A,plus_plus(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,Na))),aa(nat,A,G,aa(nat,nat,suc,Na))) = aa(A,A,plus_plus(A,aa(nat,A,G,M)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ep(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,Na))) ) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_4970_sum__Suc__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,Na: nat,F2: fun(nat,A)] :
          ( aa(nat,$o,ord_less_eq(nat,M),aa(nat,nat,suc,Na))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_hh(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,M,Na)) = aa(A,A,minus_minus(A,aa(nat,A,F2,aa(nat,nat,suc,Na))),aa(nat,A,F2,M)) ) ) ) ).

% sum_Suc_diff
tff(fact_4971_sum_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Na: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),Na)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ep(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,Na)) ) ).

% sum.atLeast1_atMost_eq
tff(fact_4972_sum__bounds__lt__plus1,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),Mm: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ep(fun(nat,A),fun(nat,A),F2)),set_ord_lessThan(nat,Mm)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or1337092689740270186AtMost(nat,one_one(nat),Mm)) ) ).

% sum_bounds_lt_plus1
tff(fact_4973_sum_Onested__swap_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: fun(nat,fun(nat,A)),Na: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_id(fun(nat,fun(nat,A)),fun(nat,A),A3)),set_ord_atMost(nat,Na)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_if(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A3),Na)),set_ord_lessThan(nat,Na)) ) ).

% sum.nested_swap'
tff(fact_4974_sum__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),A3: nat,B3: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or1337092689740270186AtMost(nat,A3,B3)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_ig(fun(nat,A),fun(nat,fun(A,A)),F2),A3,B3,zero_zero(A)) ) ).

% sum_atLeastAtMost_code
tff(fact_4975_sum_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Na: nat,G: fun(nat,A),P2: nat] :
          ( aa(nat,$o,ord_less_eq(nat,M),aa(nat,nat,plus_plus(nat,Na),one_one(nat)))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,plus_plus(nat,Na),P2))) = aa(A,A,plus_plus(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,Na))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,plus_plus(nat,Na),one_one(nat)),aa(nat,nat,plus_plus(nat,Na),P2)))) ) ) ) ).

% sum.ub_add_nat
tff(fact_4976_sum__up__index__split,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),M: nat,Na: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_atMost(nat,aa(nat,nat,plus_plus(nat,M),Na))) = aa(A,A,plus_plus(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_atMost(nat,M))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,plus_plus(nat,M),Na)))) ) ).

% sum_up_index_split
tff(fact_4977_sum__natinterval__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(nat,A),M: nat,Na: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ih(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,M,Na)) = $ite(aa(nat,$o,ord_less_eq(nat,M),Na),aa(A,A,minus_minus(A,aa(nat,A,F2,M)),aa(nat,A,F2,aa(nat,nat,plus_plus(nat,Na),one_one(nat)))),zero_zero(A)) ) ).

% sum_natinterval_diff
tff(fact_4978_sum__telescope_H_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,Na: nat,F2: fun(nat,A)] :
          ( aa(nat,$o,ord_less_eq(nat,M),Na)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ii(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),Na)) = aa(A,A,minus_minus(A,aa(nat,A,F2,Na)),aa(nat,A,F2,M)) ) ) ) ).

% sum_telescope''
tff(fact_4979_sum__power__shift,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M: nat,Na: nat,X: A] :
          ( aa(nat,$o,ord_less_eq(nat,M),Na)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,power_power(A,X)),set_or1337092689740270186AtMost(nat,M,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,X),M)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,power_power(A,X)),set_ord_atMost(nat,aa(nat,nat,minus_minus(nat,Na),M)))) ) ) ) ).

% sum_power_shift
tff(fact_4980_atLeast1__atMost__eq__remove0,axiom,
    ! [Na: nat] : set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),Na) = aa(set(nat),set(nat),minus_minus(set(nat),set_ord_atMost(nat,Na)),aa(set(nat),set(nat),insert(nat,zero_zero(nat)),bot_bot(set(nat)))) ).

% atLeast1_atMost_eq_remove0
tff(fact_4981_summable__partial__sum__bound,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),E2: real] :
          ( summable(A,F2)
         => ( aa(real,$o,ord_less(real,zero_zero(real)),E2)
           => ~ ! [N9: nat] :
                  ~ ! [M2: nat] :
                      ( aa(nat,$o,ord_less_eq(nat,N9),M2)
                     => ! [N10: nat] : aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or1337092689740270186AtMost(nat,M2,N10)))),E2) ) ) ) ) ).

% summable_partial_sum_bound
tff(fact_4982_sum__gp__multiplied,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M: nat,Na: nat,X: A] :
          ( aa(nat,$o,ord_less_eq(nat,M),Na)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,one_one(A)),X)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,power_power(A,X)),set_or1337092689740270186AtMost(nat,M,Na))) = aa(A,A,minus_minus(A,aa(nat,A,power_power(A,X),M)),aa(nat,A,power_power(A,X),aa(nat,nat,suc,Na))) ) ) ) ).

% sum_gp_multiplied
tff(fact_4983_sum_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_fc(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% sum.in_pairs
tff(fact_4984_polyfun__eq__const,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Na: nat,K: A] :
          ( ! [X3: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_es(fun(nat,A),fun(A,fun(nat,A)),C2),X3)),set_ord_atMost(nat,Na)) = K
        <=> ( ( aa(nat,A,C2,zero_zero(nat)) = K )
            & ! [X3: nat] :
                ( member(nat,X3,set_or1337092689740270186AtMost(nat,one_one(nat),Na))
               => ( aa(nat,A,C2,X3) = zero_zero(A) ) ) ) ) ) ).

% polyfun_eq_const
tff(fact_4985_gbinomial__sum__up__index,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Na: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ij(nat,fun(nat,A),K)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) = aa(nat,A,gbinomial(A,aa(A,A,plus_plus(A,aa(nat,A,semiring_1_of_nat(A),Na)),one_one(A))),aa(nat,nat,plus_plus(nat,K),one_one(nat))) ) ).

% gbinomial_sum_up_index
tff(fact_4986_gauss__sum__nat,axiom,
    ! [Na: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_ik(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_4987_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,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,plus_plus(A,aa(nat,A,semiring_1_of_nat(A),Na)),one_one(A))) ) ).

% double_gauss_sum
tff(fact_4988_double__arith__series,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: 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,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_il(A,fun(A,fun(nat,A)),A3),D2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,plus_plus(A,aa(nat,A,semiring_1_of_nat(A),Na)),one_one(A))),aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),D2))) ) ).

% double_arith_series
tff(fact_4989_arith__series__nat,axiom,
    ! [A3: nat,D2: nat,Na: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_im(nat,fun(nat,fun(nat,nat)),A3),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,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),A3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),D2)))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% arith_series_nat
tff(fact_4990_Sum__Icc__nat,axiom,
    ! [M: nat,Na: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_ik(nat,nat)),set_or1337092689740270186AtMost(nat,M,Na)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(nat,nat,plus_plus(nat,Na),one_one(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,minus_minus(nat,M),one_one(nat))))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% Sum_Icc_nat
tff(fact_4991_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,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,plus_plus(A,aa(nat,A,semiring_1_of_nat(A),Na)),one_one(A))) ) ).

% double_gauss_sum_from_Suc_0
tff(fact_4992_gauss__sum,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Na: nat] : aa(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,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_4993_arith__series,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A,D2: A,Na: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_in(A,fun(A,fun(nat,A)),A3),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,plus_plus(A,aa(nat,A,semiring_1_of_nat(A),Na)),one_one(A))),aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),D2)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% arith_series
tff(fact_4994_sum__gp__offset,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [X: A,M: nat,Na: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,power_power(A,X)),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,plus_plus(nat,M),Na))) = $ite(X = one_one(A),aa(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,power_power(A,X),M)),aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,power_power(A,X),aa(nat,nat,suc,Na))))),aa(A,A,minus_minus(A,one_one(A)),X))) ) ).

% sum_gp_offset
tff(fact_4995_polyfun__diff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Na: nat,A3: fun(nat,A),X: A,Y: A] :
          ( aa(nat,$o,ord_less_eq(nat,one_one(nat)),Na)
         => ( aa(A,A,minus_minus(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_fd(fun(nat,A),fun(A,fun(nat,A)),A3),X)),set_ord_atMost(nat,Na))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_fd(fun(nat,A),fun(A,fun(nat,A)),A3),Y)),set_ord_atMost(nat,Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,X),Y)),aa(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_ip(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Na),A3),X),Y)),set_ord_lessThan(nat,Na))) ) ) ) ).

% polyfun_diff
tff(fact_4996_gauss__sum__from__Suc__0,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Na: nat] : aa(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,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_4997_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,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))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_iq(A,fun(nat,A),Z),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(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_4998_notIn__Un__bij__betw,axiom,
    ! [A: $tType,B: $tType,B3: A,A2: set(A),F2: fun(A,B),A7: set(B)] :
      ( ~ member(A,B3,A2)
     => ( ~ member(B,aa(A,B,F2,B3),A7)
       => ( bij_betw(A,B,F2,A2,A7)
         => bij_betw(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),aa(set(A),set(A),insert(A,B3),bot_bot(set(A)))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A7),aa(set(B),set(B),insert(B,aa(A,B,F2,B3)),bot_bot(set(B))))) ) ) ) ).

% notIn_Un_bij_betw
tff(fact_4999_notIn__Un__bij__betw3,axiom,
    ! [A: $tType,B: $tType,B3: A,A2: set(A),F2: fun(A,B),A7: set(B)] :
      ( ~ member(A,B3,A2)
     => ( ~ member(B,aa(A,B,F2,B3),A7)
       => ( bij_betw(A,B,F2,A2,A7)
        <=> bij_betw(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),aa(set(A),set(A),insert(A,B3),bot_bot(set(A)))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A7),aa(set(B),set(B),insert(B,aa(A,B,F2,B3)),bot_bot(set(B))))) ) ) ) ).

% notIn_Un_bij_betw3
tff(fact_5000_prod_Oneutral__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: set(B)] : groups7121269368397514597t_prod(B,A,aTP_Lamp_ir(B,A),A2) = one_one(A) ) ).

% prod.neutral_const
tff(fact_5001_of__nat__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F2: fun(B,nat),A2: set(B)] : aa(nat,A,semiring_1_of_nat(A),groups7121269368397514597t_prod(B,nat,F2,A2)) = groups7121269368397514597t_prod(B,A,aTP_Lamp_is(fun(B,nat),fun(B,A),F2),A2) ) ).

% of_nat_prod
tff(fact_5002_of__int__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_ring_1(A)
     => ! [F2: fun(B,int),A2: set(B)] : aa(int,A,ring_1_of_int(A),groups7121269368397514597t_prod(B,int,F2,A2)) = groups7121269368397514597t_prod(B,A,aTP_Lamp_it(fun(B,int),fun(B,A),F2),A2) ) ).

% of_int_prod
tff(fact_5003_of__real__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2191834092415804123ebra_1(A) )
     => ! [F2: fun(B,real),Sa: set(B)] : aa(real,A,real_Vector_of_real(A),groups7121269368397514597t_prod(B,real,F2,Sa)) = groups7121269368397514597t_prod(B,A,aTP_Lamp_iu(fun(B,real),fun(B,A),F2),Sa) ) ).

% of_real_prod
tff(fact_5004_prod_Oempty,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A)] : groups7121269368397514597t_prod(B,A,G,bot_bot(set(B))) = one_one(A) ) ).

% prod.empty
tff(fact_5005_prod_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] :
          groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,Na))) = $ite(aa(nat,$o,ord_less(nat,aa(nat,nat,suc,Na)),M),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,Na))),aa(nat,A,G,aa(nat,nat,suc,Na)))) ) ).

% prod.cl_ivl_Suc
tff(fact_5006_prod_Oreindex__bij__betw,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [H: fun(A,B),S: set(A),T3: set(B),G: fun(B,C)] :
          ( bij_betw(A,B,H,S,T3)
         => ( groups7121269368397514597t_prod(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_iv(fun(A,B),fun(fun(B,C),fun(A,C)),H),G),S) = groups7121269368397514597t_prod(B,C,G,T3) ) ) ) ).

% prod.reindex_bij_betw
tff(fact_5007_prod__nonneg,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A2: set(A),F2: fun(A,B)] :
          ( ! [X4: A] :
              ( member(A,X4,A2)
             => aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,F2,X4)) )
         => aa(B,$o,ord_less_eq(B,zero_zero(B)),groups7121269368397514597t_prod(A,B,F2,A2)) ) ) ).

% prod_nonneg
tff(fact_5008_prod__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A2: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( ! [I3: A] :
              ( member(A,I3,A2)
             => ( aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,F2,I3))
                & aa(B,$o,ord_less_eq(B,aa(A,B,F2,I3)),aa(A,B,G,I3)) ) )
         => aa(B,$o,ord_less_eq(B,groups7121269368397514597t_prod(A,B,F2,A2)),groups7121269368397514597t_prod(A,B,G,A2)) ) ) ).

% prod_mono
tff(fact_5009_prod__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A2: set(A),F2: fun(A,B)] :
          ( ! [X4: A] :
              ( member(A,X4,A2)
             => aa(B,$o,ord_less(B,zero_zero(B)),aa(A,B,F2,X4)) )
         => aa(B,$o,ord_less(B,zero_zero(B)),groups7121269368397514597t_prod(A,B,F2,A2)) ) ) ).

% prod_pos
tff(fact_5010_prod__power__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F2: fun(B,A),A2: set(B),Na: nat] : aa(nat,A,power_power(A,groups7121269368397514597t_prod(B,A,F2,A2)),Na) = groups7121269368397514597t_prod(B,A,aa(nat,fun(B,A),aTP_Lamp_iw(fun(B,A),fun(nat,fun(B,A)),F2),Na),A2) ) ).

% prod_power_distrib
tff(fact_5011_prod__dividef,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [F2: fun(B,A),G: fun(B,A),A2: set(B)] : groups7121269368397514597t_prod(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ix(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),groups7121269368397514597t_prod(B,A,F2,A2)),groups7121269368397514597t_prod(B,A,G,A2)) ) ).

% prod_dividef
tff(fact_5012_prod_Oneutral,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [A2: set(A),G: fun(A,B)] :
          ( ! [X4: A] :
              ( member(A,X4,A2)
             => ( aa(A,B,G,X4) = one_one(B) ) )
         => ( groups7121269368397514597t_prod(A,B,G,A2) = one_one(B) ) ) ) ).

% prod.neutral
tff(fact_5013_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),A2: set(B)] :
          ( ( groups7121269368397514597t_prod(B,A,G,A2) != one_one(A) )
         => ~ ! [A4: B] :
                ( member(B,A4,A2)
               => ( aa(B,A,G,A4) = one_one(A) ) ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_5014_abs__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_field(A)
     => ! [F2: fun(B,A),A2: set(B)] : aa(A,A,abs_abs(A),groups7121269368397514597t_prod(B,A,F2,A2)) = groups7121269368397514597t_prod(B,A,aTP_Lamp_iy(fun(B,A),fun(B,A),F2),A2) ) ).

% abs_prod
tff(fact_5015_prod_Oswap,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,fun(C,A)),B2: set(C),A2: set(B)] : groups7121269368397514597t_prod(B,A,aa(set(C),fun(B,A),aTP_Lamp_iz(fun(B,fun(C,A)),fun(set(C),fun(B,A)),G),B2),A2) = groups7121269368397514597t_prod(C,A,aa(set(B),fun(C,A),aTP_Lamp_jb(fun(B,fun(C,A)),fun(set(B),fun(C,A)),G),A2),B2) ) ).

% prod.swap
tff(fact_5016_prod_Odistrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),H: fun(B,A),A2: set(B)] : groups7121269368397514597t_prod(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_jc(fun(B,A),fun(fun(B,A),fun(B,A)),G),H),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(B,A,G,A2)),groups7121269368397514597t_prod(B,A,H,A2)) ) ).

% prod.distrib
tff(fact_5017_norm__prod__le,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [F2: fun(B,A),A2: set(B)] : aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,groups7121269368397514597t_prod(B,A,F2,A2))),groups7121269368397514597t_prod(B,real,aTP_Lamp_jd(fun(B,A),fun(B,real),F2),A2)) ) ).

% norm_prod_le
tff(fact_5018_prod__norm,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & comm_semiring_1(B) )
     => ! [F2: fun(A,B),A2: set(A)] : groups7121269368397514597t_prod(A,real,aTP_Lamp_je(fun(A,B),fun(A,real),F2),A2) = real_V7770717601297561774m_norm(B,groups7121269368397514597t_prod(A,B,F2,A2)) ) ).

% prod_norm
tff(fact_5019_mod__prod__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [F2: fun(B,A),A3: A,A2: set(B)] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),groups7121269368397514597t_prod(B,A,aa(A,fun(B,A),aTP_Lamp_em(fun(B,A),fun(A,fun(B,A)),F2),A3),A2)),A3) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),groups7121269368397514597t_prod(B,A,F2,A2)),A3) ) ).

% mod_prod_eq
tff(fact_5020_prod__ge__1,axiom,
    ! [B: $tType,A: $tType] :
      ( linord181362715937106298miring(B)
     => ! [A2: set(A),F2: fun(A,B)] :
          ( ! [X4: A] :
              ( member(A,X4,A2)
             => aa(B,$o,ord_less_eq(B,one_one(B)),aa(A,B,F2,X4)) )
         => aa(B,$o,ord_less_eq(B,one_one(B)),groups7121269368397514597t_prod(A,B,F2,A2)) ) ) ).

% prod_ge_1
tff(fact_5021_range__subset__card,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A),C2: word(A),D2: word(A)] :
          ( aa(set(word(A)),$o,ord_less_eq(set(word(A)),set_or1337092689740270186AtMost(word(A),A3,B3)),set_or1337092689740270186AtMost(word(A),C2,D2))
         => ( aa(word(A),$o,ord_less_eq(word(A),A3),B3)
           => ( aa(word(A),$o,ord_less_eq(word(A),C2),D2)
              & aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),minus_minus(word(A),B3),A3)),aa(word(A),word(A),minus_minus(word(A),D2),C2)) ) ) ) ) ).

% range_subset_card
tff(fact_5022_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Na))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_jf(fun(nat,A),fun(nat,A),G),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_5023_power__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [C2: A,F2: fun(B,nat),A2: set(B)] : aa(nat,A,power_power(A,C2),aa(set(B),nat,groups7311177749621191930dd_sum(B,nat,F2),A2)) = groups7121269368397514597t_prod(B,A,aa(fun(B,nat),fun(B,A),aTP_Lamp_jg(A,fun(fun(B,nat),fun(B,A)),C2),F2),A2) ) ).

% power_sum
tff(fact_5024_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,K: nat,Na: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,aa(nat,nat,plus_plus(nat,M),K),aa(nat,nat,plus_plus(nat,Na),K))) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jh(fun(nat,A),fun(nat,fun(nat,A)),G),K),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_5025_prod__le__1,axiom,
    ! [A: $tType,B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [A2: set(A),F2: fun(A,B)] :
          ( ! [X4: A] :
              ( member(A,X4,A2)
             => ( aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,F2,X4))
                & aa(B,$o,ord_less_eq(B,aa(A,B,F2,X4)),one_one(B)) ) )
         => aa(B,$o,ord_less_eq(B,groups7121269368397514597t_prod(A,B,F2,A2)),one_one(B)) ) ) ).

% prod_le_1
tff(fact_5026_bset_I1_J,axiom,
    ! [D3: int,B2: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X4: int] :
          ( ! [Xa: int] :
              ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb2: int] :
                  ( member(int,Xb2,B2)
                 => ( X4 != aa(int,int,plus_plus(int,Xb2),Xa) ) ) )
         => ( aa(int,$o,P,X4)
           => aa(int,$o,P,aa(int,int,minus_minus(int,X4),D3)) ) )
     => ( ! [X4: int] :
            ( ! [Xa: int] :
                ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb2: int] :
                    ( member(int,Xb2,B2)
                   => ( X4 != aa(int,int,plus_plus(int,Xb2),Xa) ) ) )
           => ( aa(int,$o,Q,X4)
             => aa(int,$o,Q,aa(int,int,minus_minus(int,X4),D3)) ) )
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B2)
                   => ( X2 != aa(int,int,plus_plus(int,Xb3),Xa3) ) ) )
           => ( ( aa(int,$o,P,X2)
                & aa(int,$o,Q,X2) )
             => ( aa(int,$o,P,aa(int,int,minus_minus(int,X2),D3))
                & aa(int,$o,Q,aa(int,int,minus_minus(int,X2),D3)) ) ) ) ) ) ).

% bset(1)
tff(fact_5027_bset_I2_J,axiom,
    ! [D3: int,B2: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X4: int] :
          ( ! [Xa: int] :
              ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb2: int] :
                  ( member(int,Xb2,B2)
                 => ( X4 != aa(int,int,plus_plus(int,Xb2),Xa) ) ) )
         => ( aa(int,$o,P,X4)
           => aa(int,$o,P,aa(int,int,minus_minus(int,X4),D3)) ) )
     => ( ! [X4: int] :
            ( ! [Xa: int] :
                ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb2: int] :
                    ( member(int,Xb2,B2)
                   => ( X4 != aa(int,int,plus_plus(int,Xb2),Xa) ) ) )
           => ( aa(int,$o,Q,X4)
             => aa(int,$o,Q,aa(int,int,minus_minus(int,X4),D3)) ) )
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B2)
                   => ( X2 != aa(int,int,plus_plus(int,Xb3),Xa3) ) ) )
           => ( ( aa(int,$o,P,X2)
                | aa(int,$o,Q,X2) )
             => ( aa(int,$o,P,aa(int,int,minus_minus(int,X2),D3))
                | aa(int,$o,Q,aa(int,int,minus_minus(int,X2),D3)) ) ) ) ) ) ).

% bset(2)
tff(fact_5028_aset_I1_J,axiom,
    ! [D3: int,A2: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X4: int] :
          ( ! [Xa: int] :
              ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb2: int] :
                  ( member(int,Xb2,A2)
                 => ( X4 != aa(int,int,minus_minus(int,Xb2),Xa) ) ) )
         => ( aa(int,$o,P,X4)
           => aa(int,$o,P,aa(int,int,plus_plus(int,X4),D3)) ) )
     => ( ! [X4: int] :
            ( ! [Xa: int] :
                ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb2: int] :
                    ( member(int,Xb2,A2)
                   => ( X4 != aa(int,int,minus_minus(int,Xb2),Xa) ) ) )
           => ( aa(int,$o,Q,X4)
             => aa(int,$o,Q,aa(int,int,plus_plus(int,X4),D3)) ) )
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A2)
                   => ( X2 != aa(int,int,minus_minus(int,Xb3),Xa3) ) ) )
           => ( ( aa(int,$o,P,X2)
                & aa(int,$o,Q,X2) )
             => ( aa(int,$o,P,aa(int,int,plus_plus(int,X2),D3))
                & aa(int,$o,Q,aa(int,int,plus_plus(int,X2),D3)) ) ) ) ) ) ).

% aset(1)
tff(fact_5029_aset_I2_J,axiom,
    ! [D3: int,A2: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X4: int] :
          ( ! [Xa: int] :
              ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb2: int] :
                  ( member(int,Xb2,A2)
                 => ( X4 != aa(int,int,minus_minus(int,Xb2),Xa) ) ) )
         => ( aa(int,$o,P,X4)
           => aa(int,$o,P,aa(int,int,plus_plus(int,X4),D3)) ) )
     => ( ! [X4: int] :
            ( ! [Xa: int] :
                ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb2: int] :
                    ( member(int,Xb2,A2)
                   => ( X4 != aa(int,int,minus_minus(int,Xb2),Xa) ) ) )
           => ( aa(int,$o,Q,X4)
             => aa(int,$o,Q,aa(int,int,plus_plus(int,X4),D3)) ) )
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A2)
                   => ( X2 != aa(int,int,minus_minus(int,Xb3),Xa3) ) ) )
           => ( ( aa(int,$o,P,X2)
                | aa(int,$o,Q,X2) )
             => ( aa(int,$o,P,aa(int,int,plus_plus(int,X2),D3))
                | aa(int,$o,Q,aa(int,int,plus_plus(int,X2),D3)) ) ) ) ) ) ).

% aset(2)
tff(fact_5030_prod_Onat__diff__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Na: nat] : groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ji(fun(nat,A),fun(nat,fun(nat,A)),G),Na),set_ord_lessThan(nat,Na)) = groups7121269368397514597t_prod(nat,A,G,set_ord_lessThan(nat,Na)) ) ).

% prod.nat_diff_reindex
tff(fact_5031_prod_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Na: nat,M: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,Na,M)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_jj(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Na),M),set_or1337092689740270186AtMost(nat,Na,M)) ) ).

% prod.atLeastAtMost_rev
tff(fact_5032_prod_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Na: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,zero_zero(nat),Na))),aa(nat,A,G,aa(nat,nat,suc,Na))) ) ).

% prod.atLeast0_atMost_Suc
tff(fact_5033_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,ord_less_eq(word(A),L),M)
           => ( aa(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),plus_plus(word(A),M),one_one(word(A))),U)) = set_or1337092689740270186AtMost(word(A),L,U) ) ) ) ) ) ).

% word_atLeastLessThan_Suc_atLeastAtMost_union
tff(fact_5034_aset_I10_J,axiom,
    ! [D2: int,D3: int,A2: set(int),Ta: int] :
      ( aa(int,$o,dvd_dvd(int,D2),D3)
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb3: int] :
                  ( member(int,Xb3,A2)
                 => ( X2 != aa(int,int,minus_minus(int,Xb3),Xa3) ) ) )
         => ( ~ aa(int,$o,dvd_dvd(int,D2),aa(int,int,plus_plus(int,X2),Ta))
           => ~ aa(int,$o,dvd_dvd(int,D2),aa(int,int,plus_plus(int,aa(int,int,plus_plus(int,X2),D3)),Ta)) ) ) ) ).

% aset(10)
tff(fact_5035_aset_I9_J,axiom,
    ! [D2: int,D3: int,A2: set(int),Ta: int] :
      ( aa(int,$o,dvd_dvd(int,D2),D3)
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb3: int] :
                  ( member(int,Xb3,A2)
                 => ( X2 != aa(int,int,minus_minus(int,Xb3),Xa3) ) ) )
         => ( aa(int,$o,dvd_dvd(int,D2),aa(int,int,plus_plus(int,X2),Ta))
           => aa(int,$o,dvd_dvd(int,D2),aa(int,int,plus_plus(int,aa(int,int,plus_plus(int,X2),D3)),Ta)) ) ) ) ).

% aset(9)
tff(fact_5036_bset_I10_J,axiom,
    ! [D2: int,D3: int,B2: set(int),Ta: int] :
      ( aa(int,$o,dvd_dvd(int,D2),D3)
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb3: int] :
                  ( member(int,Xb3,B2)
                 => ( X2 != aa(int,int,plus_plus(int,Xb3),Xa3) ) ) )
         => ( ~ aa(int,$o,dvd_dvd(int,D2),aa(int,int,plus_plus(int,X2),Ta))
           => ~ aa(int,$o,dvd_dvd(int,D2),aa(int,int,plus_plus(int,aa(int,int,minus_minus(int,X2),D3)),Ta)) ) ) ) ).

% bset(10)
tff(fact_5037_bset_I9_J,axiom,
    ! [D2: int,D3: int,B2: set(int),Ta: int] :
      ( aa(int,$o,dvd_dvd(int,D2),D3)
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb3: int] :
                  ( member(int,Xb3,B2)
                 => ( X2 != aa(int,int,plus_plus(int,Xb3),Xa3) ) ) )
         => ( aa(int,$o,dvd_dvd(int,D2),aa(int,int,plus_plus(int,X2),Ta))
           => aa(int,$o,dvd_dvd(int,D2),aa(int,int,plus_plus(int,aa(int,int,minus_minus(int,X2),D3)),Ta)) ) ) ) ).

% bset(9)
tff(fact_5038_prod_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Na: nat] : groups7121269368397514597t_prod(nat,A,G,set_ord_lessThan(nat,aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),groups7121269368397514597t_prod(nat,A,aTP_Lamp_jf(fun(nat,A),fun(nat,A),G),set_ord_lessThan(nat,Na))) ) ).

% prod.lessThan_Suc_shift
tff(fact_5039_prod_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,ord_less_eq(nat,M),Na)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,Na))),aa(nat,A,G,aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),groups7121269368397514597t_prod(nat,A,aTP_Lamp_jf(fun(nat,A),fun(nat,A),G),set_or1337092689740270186AtMost(nat,M,Na))) ) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_5040_prod_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Na: nat] : groups7121269368397514597t_prod(nat,A,G,set_ord_atMost(nat,aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),groups7121269368397514597t_prod(nat,A,aTP_Lamp_jf(fun(nat,A),fun(nat,A),G),set_ord_atMost(nat,Na))) ) ).

% prod.atMost_Suc_shift
tff(fact_5041_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,Na: int] :
      ( aa(int,$o,ord_less_eq(int,M),aa(int,int,plus_plus(int,one_one(int)),Na))
     => ( set_or1337092689740270186AtMost(int,M,aa(int,int,plus_plus(int,one_one(int)),Na)) = aa(set(int),set(int),insert(int,aa(int,int,plus_plus(int,one_one(int)),Na)),set_or1337092689740270186AtMost(int,M,Na)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_5042_prod_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Na: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),Na)) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_jf(fun(nat,A),fun(nat,A),G),set_ord_lessThan(nat,Na)) ) ).

% prod.atLeast1_atMost_eq
tff(fact_5043_fact__prod,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Na: nat] : semiring_char_0_fact(A,Na) = aa(nat,A,semiring_1_of_nat(A),groups7121269368397514597t_prod(nat,nat,aTP_Lamp_ik(nat,nat),set_or1337092689740270186AtMost(nat,one_one(nat),Na))) ) ).

% fact_prod
tff(fact_5044_prod_Onested__swap_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: fun(nat,fun(nat,A)),Na: nat] : groups7121269368397514597t_prod(nat,A,aTP_Lamp_jk(fun(nat,fun(nat,A)),fun(nat,A),A3),set_ord_atMost(nat,Na)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jm(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A3),Na),set_ord_lessThan(nat,Na)) ) ).

% prod.nested_swap'
tff(fact_5045_prod__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [F2: fun(nat,A),A3: nat,B3: nat] : groups7121269368397514597t_prod(nat,A,F2,set_or1337092689740270186AtMost(nat,A3,B3)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_jn(fun(nat,A),fun(nat,fun(A,A)),F2),A3,B3,one_one(A)) ) ).

% prod_atLeastAtMost_code
tff(fact_5046_prod_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Na: nat,G: fun(nat,A),P2: nat] :
          ( aa(nat,$o,ord_less_eq(nat,M),aa(nat,nat,plus_plus(nat,Na),one_one(nat)))
         => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,aa(nat,nat,plus_plus(nat,Na),P2))) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,Na))),groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,aa(nat,nat,plus_plus(nat,Na),one_one(nat)),aa(nat,nat,plus_plus(nat,Na),P2)))) ) ) ) ).

% prod.ub_add_nat
tff(fact_5047_bset_I3_J,axiom,
    ! [D3: int,Ta: int,B2: set(int)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D3)
     => ( member(int,aa(int,int,minus_minus(int,Ta),one_one(int)),B2)
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B2)
                   => ( X2 != aa(int,int,plus_plus(int,Xb3),Xa3) ) ) )
           => ( ( X2 = Ta )
             => ( aa(int,int,minus_minus(int,X2),D3) = Ta ) ) ) ) ) ).

% bset(3)
tff(fact_5048_bset_I4_J,axiom,
    ! [D3: int,Ta: int,B2: set(int)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D3)
     => ( member(int,Ta,B2)
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B2)
                   => ( X2 != aa(int,int,plus_plus(int,Xb3),Xa3) ) ) )
           => ( ( X2 != Ta )
             => ( aa(int,int,minus_minus(int,X2),D3) != Ta ) ) ) ) ) ).

% bset(4)
tff(fact_5049_bset_I5_J,axiom,
    ! [D3: int,B2: set(int),Ta: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D3)
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb3: int] :
                  ( member(int,Xb3,B2)
                 => ( X2 != aa(int,int,plus_plus(int,Xb3),Xa3) ) ) )
         => ( aa(int,$o,ord_less(int,X2),Ta)
           => aa(int,$o,ord_less(int,aa(int,int,minus_minus(int,X2),D3)),Ta) ) ) ) ).

% bset(5)
tff(fact_5050_bset_I7_J,axiom,
    ! [D3: int,Ta: int,B2: set(int)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D3)
     => ( member(int,Ta,B2)
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B2)
                   => ( X2 != aa(int,int,plus_plus(int,Xb3),Xa3) ) ) )
           => ( aa(int,$o,ord_less(int,Ta),X2)
             => aa(int,$o,ord_less(int,Ta),aa(int,int,minus_minus(int,X2),D3)) ) ) ) ) ).

% bset(7)
tff(fact_5051_aset_I3_J,axiom,
    ! [D3: int,Ta: int,A2: set(int)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D3)
     => ( member(int,aa(int,int,plus_plus(int,Ta),one_one(int)),A2)
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A2)
                   => ( X2 != aa(int,int,minus_minus(int,Xb3),Xa3) ) ) )
           => ( ( X2 = Ta )
             => ( aa(int,int,plus_plus(int,X2),D3) = Ta ) ) ) ) ) ).

% aset(3)
tff(fact_5052_aset_I4_J,axiom,
    ! [D3: int,Ta: int,A2: set(int)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D3)
     => ( member(int,Ta,A2)
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A2)
                   => ( X2 != aa(int,int,minus_minus(int,Xb3),Xa3) ) ) )
           => ( ( X2 != Ta )
             => ( aa(int,int,plus_plus(int,X2),D3) != Ta ) ) ) ) ) ).

% aset(4)
tff(fact_5053_aset_I5_J,axiom,
    ! [D3: int,Ta: int,A2: set(int)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D3)
     => ( member(int,Ta,A2)
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A2)
                   => ( X2 != aa(int,int,minus_minus(int,Xb3),Xa3) ) ) )
           => ( aa(int,$o,ord_less(int,X2),Ta)
             => aa(int,$o,ord_less(int,aa(int,int,plus_plus(int,X2),D3)),Ta) ) ) ) ) ).

% aset(5)
tff(fact_5054_aset_I7_J,axiom,
    ! [D3: int,A2: set(int),Ta: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D3)
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb3: int] :
                  ( member(int,Xb3,A2)
                 => ( X2 != aa(int,int,minus_minus(int,Xb3),Xa3) ) ) )
         => ( aa(int,$o,ord_less(int,Ta),X2)
           => aa(int,$o,ord_less(int,Ta),aa(int,int,plus_plus(int,X2),D3)) ) ) ) ).

% aset(7)
tff(fact_5055_periodic__finite__ex,axiom,
    ! [D2: int,P: fun(int,$o)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D2)
     => ( ! [X4: int,K2: int] :
            ( aa(int,$o,P,X4)
          <=> aa(int,$o,P,aa(int,int,minus_minus(int,X4),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D2))) )
       => ( ? [X_1: int] : aa(int,$o,P,X_1)
        <=> ? [X3: int] :
              ( member(int,X3,set_or1337092689740270186AtMost(int,one_one(int),D2))
              & aa(int,$o,P,X3) ) ) ) ) ).

% periodic_finite_ex
tff(fact_5056_word__subset__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),R3: word(A),Y: word(A),Sa: word(A)] :
          ( aa(set(word(A)),$o,ord_less_eq(set(word(A)),set_or1337092689740270186AtMost(word(A),X,aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),plus_plus(word(A),X),R3)),one_one(word(A))))),set_or1337092689740270186AtMost(word(A),Y,aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),plus_plus(word(A),Y),Sa)),one_one(word(A)))))
         => ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),plus_plus(word(A),X),R3)),one_one(word(A))))
           => ( aa(word(A),$o,ord_less_eq(word(A),Y),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),plus_plus(word(A),Y),Sa)),one_one(word(A))))
             => ( ( Sa != zero_zero(word(A)) )
               => aa(word(A),$o,ord_less_eq(word(A),R3),Sa) ) ) ) ) ) ).

% word_subset_less
tff(fact_5057_norm__prod__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [I5: set(A),Z: fun(A,B),W: fun(A,B)] :
          ( ! [I3: A] :
              ( member(A,I3,I5)
             => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(A,B,Z,I3))),one_one(real)) )
         => ( ! [I3: A] :
                ( member(A,I3,I5)
               => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(A,B,W,I3))),one_one(real)) )
           => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,groups7121269368397514597t_prod(A,B,Z,I5)),groups7121269368397514597t_prod(A,B,W,I5)))),aa(set(A),real,groups7311177749621191930dd_sum(A,real,aa(fun(A,B),fun(A,real),aTP_Lamp_jo(fun(A,B),fun(fun(A,B),fun(A,real)),Z),W)),I5)) ) ) ) ).

% norm_prod_diff
tff(fact_5058_prod_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Na: nat] : groups7121269368397514597t_prod(nat,A,G,set_ord_atMost(nat,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),groups7121269368397514597t_prod(nat,A,aTP_Lamp_jf(fun(nat,A),fun(nat,A),G),set_ord_lessThan(nat,Na))) ) ).

% prod.atMost_shift
tff(fact_5059_fact__eq__fact__times,axiom,
    ! [Na: nat,M: nat] :
      ( aa(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)),groups7121269368397514597t_prod(nat,nat,aTP_Lamp_ik(nat,nat),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Na),M))) ) ) ).

% fact_eq_fact_times
tff(fact_5060_simp__from__to,axiom,
    ! [I: int,J: int] :
      set_or1337092689740270186AtMost(int,I,J) = $ite(aa(int,$o,ord_less(int,J),I),bot_bot(set(int)),aa(set(int),set(int),insert(int,I),set_or1337092689740270186AtMost(int,aa(int,int,plus_plus(int,I),one_one(int)),J))) ).

% simp_from_to
tff(fact_5061_bset_I6_J,axiom,
    ! [D3: int,B2: set(int),Ta: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D3)
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb3: int] :
                  ( member(int,Xb3,B2)
                 => ( X2 != aa(int,int,plus_plus(int,Xb3),Xa3) ) ) )
         => ( aa(int,$o,ord_less_eq(int,X2),Ta)
           => aa(int,$o,ord_less_eq(int,aa(int,int,minus_minus(int,X2),D3)),Ta) ) ) ) ).

% bset(6)
tff(fact_5062_bset_I8_J,axiom,
    ! [D3: int,Ta: int,B2: set(int)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D3)
     => ( member(int,aa(int,int,minus_minus(int,Ta),one_one(int)),B2)
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B2)
                   => ( X2 != aa(int,int,plus_plus(int,Xb3),Xa3) ) ) )
           => ( aa(int,$o,ord_less_eq(int,Ta),X2)
             => aa(int,$o,ord_less_eq(int,Ta),aa(int,int,minus_minus(int,X2),D3)) ) ) ) ) ).

% bset(8)
tff(fact_5063_aset_I6_J,axiom,
    ! [D3: int,Ta: int,A2: set(int)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D3)
     => ( member(int,aa(int,int,plus_plus(int,Ta),one_one(int)),A2)
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A2)
                   => ( X2 != aa(int,int,minus_minus(int,Xb3),Xa3) ) ) )
           => ( aa(int,$o,ord_less_eq(int,X2),Ta)
             => aa(int,$o,ord_less_eq(int,aa(int,int,plus_plus(int,X2),D3)),Ta) ) ) ) ) ).

% aset(6)
tff(fact_5064_aset_I8_J,axiom,
    ! [D3: int,A2: set(int),Ta: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D3)
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb3: int] :
                  ( member(int,Xb3,A2)
                 => ( X2 != aa(int,int,minus_minus(int,Xb3),Xa3) ) ) )
         => ( aa(int,$o,ord_less_eq(int,Ta),X2)
           => aa(int,$o,ord_less_eq(int,Ta),aa(int,int,plus_plus(int,X2),D3)) ) ) ) ).

% aset(8)
tff(fact_5065_cppi,axiom,
    ! [D3: int,P: fun(int,$o),P3: fun(int,$o),A2: set(int)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D3)
     => ( ? [Z3: int] :
          ! [X4: int] :
            ( aa(int,$o,ord_less(int,Z3),X4)
           => ( aa(int,$o,P,X4)
            <=> aa(int,$o,P3,X4) ) )
       => ( ! [X4: int] :
              ( ! [Xa: int] :
                  ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D3))
                 => ! [Xb2: int] :
                      ( member(int,Xb2,A2)
                     => ( X4 != aa(int,int,minus_minus(int,Xb2),Xa) ) ) )
             => ( aa(int,$o,P,X4)
               => aa(int,$o,P,aa(int,int,plus_plus(int,X4),D3)) ) )
         => ( ! [X4: int,K2: int] :
                ( aa(int,$o,P3,X4)
              <=> aa(int,$o,P3,aa(int,int,minus_minus(int,X4),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D3))) )
           => ( ? [X_1: int] : aa(int,$o,P,X_1)
            <=> ( ? [X3: int] :
                    ( member(int,X3,set_or1337092689740270186AtMost(int,one_one(int),D3))
                    & aa(int,$o,P3,X3) )
                | ? [X3: int] :
                    ( member(int,X3,set_or1337092689740270186AtMost(int,one_one(int),D3))
                    & ? [Xa4: int] :
                        ( member(int,Xa4,A2)
                        & aa(int,$o,P,aa(int,int,minus_minus(int,Xa4),X3)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_5066_cpmi,axiom,
    ! [D3: int,P: fun(int,$o),P3: fun(int,$o),B2: set(int)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D3)
     => ( ? [Z3: int] :
          ! [X4: int] :
            ( aa(int,$o,ord_less(int,X4),Z3)
           => ( aa(int,$o,P,X4)
            <=> aa(int,$o,P3,X4) ) )
       => ( ! [X4: int] :
              ( ! [Xa: int] :
                  ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D3))
                 => ! [Xb2: int] :
                      ( member(int,Xb2,B2)
                     => ( X4 != aa(int,int,plus_plus(int,Xb2),Xa) ) ) )
             => ( aa(int,$o,P,X4)
               => aa(int,$o,P,aa(int,int,minus_minus(int,X4),D3)) ) )
         => ( ! [X4: int,K2: int] :
                ( aa(int,$o,P3,X4)
              <=> aa(int,$o,P3,aa(int,int,minus_minus(int,X4),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D3))) )
           => ( ? [X_1: int] : aa(int,$o,P,X_1)
            <=> ( ? [X3: int] :
                    ( member(int,X3,set_or1337092689740270186AtMost(int,one_one(int),D3))
                    & aa(int,$o,P3,X3) )
                | ? [X3: int] :
                    ( member(int,X3,set_or1337092689740270186AtMost(int,one_one(int),D3))
                    & ? [Xa4: int] :
                        ( member(int,Xa4,B2)
                        & aa(int,$o,P,aa(int,int,plus_plus(int,Xa4),X3)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_5067_pochhammer__Suc__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,Na: nat] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,Na)) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_jp(A,fun(nat,A),A3),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) ) ).

% pochhammer_Suc_prod
tff(fact_5068_pochhammer__prod__rev,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,Na: nat] : comm_s3205402744901411588hammer(A,A3,Na) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jq(A,fun(nat,fun(nat,A)),A3),Na),set_or1337092689740270186AtMost(nat,one_one(nat),Na)) ) ).

% pochhammer_prod_rev
tff(fact_5069_fact__div__fact,axiom,
    ! [Na: nat,M: nat] :
      ( aa(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)) = groups7121269368397514597t_prod(nat,nat,aTP_Lamp_ik(nat,nat),set_or1337092689740270186AtMost(nat,aa(nat,nat,plus_plus(nat,Na),one_one(nat)),M)) ) ) ).

% fact_div_fact
tff(fact_5070_prod_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_jr(fun(nat,A),fun(nat,A),G),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% prod.in_pairs
tff(fact_5071_prod_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Na: nat] : groups7121269368397514597t_prod(nat,A,G,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)))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_jr(fun(nat,A),fun(nat,A),G),set_ord_atMost(nat,Na)) ) ).

% prod.in_pairs_0
tff(fact_5072_pochhammer__Suc__prod__rev,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,Na: nat] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,Na)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jq(A,fun(nat,fun(nat,A)),A3),Na),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) ) ).

% pochhammer_Suc_prod_rev
tff(fact_5073_bij__betw__empty2,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A2: set(A)] :
      ( bij_betw(A,B,F2,A2,bot_bot(set(B)))
     => ( A2 = bot_bot(set(A)) ) ) ).

% bij_betw_empty2
tff(fact_5074_bij__betw__empty1,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A2: set(B)] :
      ( bij_betw(A,B,F2,bot_bot(set(A)),A2)
     => ( A2 = bot_bot(set(B)) ) ) ).

% bij_betw_empty1
tff(fact_5075_prod_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P2: nat,K: nat,G: fun(nat,A),H: fun(nat,A)] :
          ( aa(nat,$o,ord_less_eq(nat,one_one(nat)),P2)
         => ( aa(nat,$o,ord_less_eq(nat,K),P2)
           => ( groups7121269368397514597t_prod(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_js(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H),set_ord_atMost(nat,P2)) = groups7121269368397514597t_prod(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_jt(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H),set_ord_atMost(nat,aa(nat,nat,minus_minus(nat,P2),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% prod.zero_middle
tff(fact_5076_gbinomial__Suc,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),groups7121269368397514597t_prod(nat,A,aTP_Lamp_ju(A,fun(nat,A),A3),set_or1337092689740270186AtMost(nat,zero_zero(nat),K))),semiring_char_0_fact(A,aa(nat,nat,suc,K))) ) ).

% gbinomial_Suc
tff(fact_5077_Sum__Icc__int,axiom,
    ! [M: int,Na: int] :
      ( aa(int,$o,ord_less_eq(int,M),Na)
     => ( aa(set(int),int,groups7311177749621191930dd_sum(int,int,aTP_Lamp_jv(int,int)),set_or1337092689740270186AtMost(int,M,Na)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),Na),aa(int,int,plus_plus(int,Na),one_one(int)))),aa(int,int,aa(int,fun(int,int),times_times(int),M),aa(int,int,minus_minus(int,M),one_one(int))))),aa(num,int,numeral_numeral(int),bit0(one2))) ) ) ).

% Sum_Icc_int
tff(fact_5078_bij__betw__nth__root__unity,axiom,
    ! [C2: complex,Na: nat] :
      ( ( C2 != zero_zero(complex) )
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
       => bij_betw(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(real,complex,real_Vector_of_real(complex),aa(real,real,root(Na),real_V7770717601297561774m_norm(complex,C2)))),cis(aa(real,real,aa(real,fun(real,real),divide_divide(real),arg(C2)),aa(nat,real,semiring_1_of_nat(real),Na))))),collect(complex,aTP_Lamp_gr(nat,fun(complex,$o),Na)),collect(complex,aa(nat,fun(complex,$o),aTP_Lamp_jw(complex,fun(nat,fun(complex,$o)),C2),Na))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_5079_Arg__def,axiom,
    ! [Z: complex] :
      arg(Z) = $ite(Z = zero_zero(complex),zero_zero(real),fChoice(real,aTP_Lamp_jx(complex,fun(real,$o),Z))) ).

% Arg_def
tff(fact_5080_cis__multiple__2pi,axiom,
    ! [Na: real] :
      ( member(real,Na,ring_1_Ints(real))
     => ( cis(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))),pi)),Na)) = one_one(complex) ) ) ).

% cis_multiple_2pi
tff(fact_5081_real__root__zero,axiom,
    ! [Na: nat] : aa(real,real,root(Na),zero_zero(real)) = zero_zero(real) ).

% real_root_zero
tff(fact_5082_Ints__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(B)
     => ! [A2: set(A),F2: fun(A,B)] :
          ( ! [X4: A] :
              ( member(A,X4,A2)
             => member(B,aa(A,B,F2,X4),ring_1_Ints(B)) )
         => member(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A2),ring_1_Ints(B)) ) ) ).

% Ints_sum
tff(fact_5083_Ints__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(B)
        & ring_1(B) )
     => ! [A2: set(A),F2: fun(A,B)] :
          ( ! [X4: A] :
              ( member(A,X4,A2)
             => member(B,aa(A,B,F2,X4),ring_1_Ints(B)) )
         => member(B,groups7121269368397514597t_prod(A,B,F2,A2),ring_1_Ints(B)) ) ) ).

% Ints_prod
tff(fact_5084_frac__in__Ints__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( member(A,archimedean_frac(A,X),ring_1_Ints(A))
        <=> member(A,X,ring_1_Ints(A)) ) ) ).

% frac_in_Ints_iff
tff(fact_5085_real__root__Suc__0,axiom,
    ! [X: real] : aa(real,real,root(aa(nat,nat,suc,zero_zero(nat))),X) = X ).

% real_root_Suc_0
tff(fact_5086_real__root__eq__iff,axiom,
    ! [Na: nat,X: real,Y: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( ( aa(real,real,root(Na),X) = aa(real,real,root(Na),Y) )
      <=> ( X = Y ) ) ) ).

% real_root_eq_iff
tff(fact_5087_root__0,axiom,
    ! [X: real] : aa(real,real,root(zero_zero(nat)),X) = zero_zero(real) ).

% root_0
tff(fact_5088_frac__eq__0__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( archimedean_frac(A,X) = zero_zero(A) )
        <=> member(A,X,ring_1_Ints(A)) ) ) ).

% frac_eq_0_iff
tff(fact_5089_some__insert__self,axiom,
    ! [A: $tType,S: set(A)] :
      ( ( S != bot_bot(set(A)) )
     => ( aa(set(A),set(A),insert(A,fChoice(A,aTP_Lamp_aa(set(A),fun(A,$o),S))),S) = S ) ) ).

% some_insert_self
tff(fact_5090_real__root__eq__0__iff,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( ( aa(real,real,root(Na),X) = zero_zero(real) )
      <=> ( X = zero_zero(real) ) ) ) ).

% real_root_eq_0_iff
tff(fact_5091_real__root__less__iff,axiom,
    ! [Na: nat,X: real,Y: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(real,$o,ord_less(real,aa(real,real,root(Na),X)),aa(real,real,root(Na),Y))
      <=> aa(real,$o,ord_less(real,X),Y) ) ) ).

% real_root_less_iff
tff(fact_5092_real__root__le__iff,axiom,
    ! [Na: nat,X: real,Y: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(real,$o,ord_less_eq(real,aa(real,real,root(Na),X)),aa(real,real,root(Na),Y))
      <=> aa(real,$o,ord_less_eq(real,X),Y) ) ) ).

% real_root_le_iff
tff(fact_5093_real__root__one,axiom,
    ! [Na: nat] :
      ( aa(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_5094_real__root__eq__1__iff,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( ( aa(real,real,root(Na),X) = one_one(real) )
      <=> ( X = one_one(real) ) ) ) ).

% real_root_eq_1_iff
tff(fact_5095_floor__add2,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( ( member(A,X,ring_1_Ints(A))
            | member(A,Y,ring_1_Ints(A)) )
         => ( archim6421214686448440834_floor(A,aa(A,A,plus_plus(A,X),Y)) = aa(int,int,plus_plus(int,archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y)) ) ) ) ).

% floor_add2
tff(fact_5096_frac__gt__0__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),archimedean_frac(A,X))
        <=> ~ member(A,X,ring_1_Ints(A)) ) ) ).

% frac_gt_0_iff
tff(fact_5097_real__root__gt__0__iff,axiom,
    ! [Na: nat,Y: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,root(Na),Y))
      <=> aa(real,$o,ord_less(real,zero_zero(real)),Y) ) ) ).

% real_root_gt_0_iff
tff(fact_5098_real__root__lt__0__iff,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(real,$o,ord_less(real,aa(real,real,root(Na),X)),zero_zero(real))
      <=> aa(real,$o,ord_less(real,X),zero_zero(real)) ) ) ).

% real_root_lt_0_iff
tff(fact_5099_real__root__le__0__iff,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(real,$o,ord_less_eq(real,aa(real,real,root(Na),X)),zero_zero(real))
      <=> aa(real,$o,ord_less_eq(real,X),zero_zero(real)) ) ) ).

% real_root_le_0_iff
tff(fact_5100_real__root__ge__0__iff,axiom,
    ! [Na: nat,Y: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,root(Na),Y))
      <=> aa(real,$o,ord_less_eq(real,zero_zero(real)),Y) ) ) ).

% real_root_ge_0_iff
tff(fact_5101_real__root__gt__1__iff,axiom,
    ! [Na: nat,Y: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(real,$o,ord_less(real,one_one(real)),aa(real,real,root(Na),Y))
      <=> aa(real,$o,ord_less(real,one_one(real)),Y) ) ) ).

% real_root_gt_1_iff
tff(fact_5102_real__root__lt__1__iff,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(real,$o,ord_less(real,aa(real,real,root(Na),X)),one_one(real))
      <=> aa(real,$o,ord_less(real,X),one_one(real)) ) ) ).

% real_root_lt_1_iff
tff(fact_5103_real__root__le__1__iff,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(real,$o,ord_less_eq(real,aa(real,real,root(Na),X)),one_one(real))
      <=> aa(real,$o,ord_less_eq(real,X),one_one(real)) ) ) ).

% real_root_le_1_iff
tff(fact_5104_real__root__ge__1__iff,axiom,
    ! [Na: nat,Y: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(real,$o,ord_less_eq(real,one_one(real)),aa(real,real,root(Na),Y))
      <=> aa(real,$o,ord_less_eq(real,one_one(real)),Y) ) ) ).

% real_root_ge_1_iff
tff(fact_5105_real__root__pow__pos2,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
       => ( aa(nat,real,power_power(real,aa(real,real,root(Na),X)),Na) = X ) ) ) ).

% real_root_pow_pos2
tff(fact_5106_int__prod,axiom,
    ! [A: $tType,F2: fun(A,nat),A2: set(A)] : aa(nat,int,semiring_1_of_nat(int),groups7121269368397514597t_prod(A,nat,F2,A2)) = groups7121269368397514597t_prod(A,int,aTP_Lamp_gj(fun(A,nat),fun(A,int),F2),A2) ).

% int_prod
tff(fact_5107_Ints__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A,Na: nat] :
          ( member(A,A3,ring_1_Ints(A))
         => member(A,aa(nat,A,power_power(A,A3),Na),ring_1_Ints(A)) ) ) ).

% Ints_power
tff(fact_5108_real__root__divide,axiom,
    ! [Na: nat,X: real,Y: real] : aa(real,real,root(Na),aa(real,real,aa(real,fun(real,real),divide_divide(real),X),Y)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,root(Na),X)),aa(real,real,root(Na),Y)) ).

% real_root_divide
tff(fact_5109_Ints__add,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A,B3: A] :
          ( member(A,A3,ring_1_Ints(A))
         => ( member(A,B3,ring_1_Ints(A))
           => member(A,aa(A,A,plus_plus(A,A3),B3),ring_1_Ints(A)) ) ) ) ).

% Ints_add
tff(fact_5110_Ints__0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => member(A,zero_zero(A),ring_1_Ints(A)) ) ).

% Ints_0
tff(fact_5111_real__root__minus,axiom,
    ! [Na: nat,X: real] : aa(real,real,root(Na),aa(real,real,uminus_uminus(real),X)) = aa(real,real,uminus_uminus(real),aa(real,real,root(Na),X)) ).

% real_root_minus
tff(fact_5112_minus__in__Ints__iff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: A] :
          ( member(A,aa(A,A,uminus_uminus(A),X),ring_1_Ints(A))
        <=> member(A,X,ring_1_Ints(A)) ) ) ).

% minus_in_Ints_iff
tff(fact_5113_Ints__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A] :
          ( member(A,A3,ring_1_Ints(A))
         => member(A,aa(A,A,uminus_uminus(A),A3),ring_1_Ints(A)) ) ) ).

% Ints_minus
tff(fact_5114_Ints__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => member(A,one_one(A),ring_1_Ints(A)) ) ).

% Ints_1
tff(fact_5115_Ints__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( member(A,A3,ring_1_Ints(A))
         => member(A,aa(A,A,abs_abs(A),A3),ring_1_Ints(A)) ) ) ).

% Ints_abs
tff(fact_5116_some__elem,axiom,
    ! [A: $tType,S: set(A)] :
      ( ( S != bot_bot(set(A)) )
     => member(A,fChoice(A,aTP_Lamp_aa(set(A),fun(A,$o),S)),S) ) ).

% some_elem
tff(fact_5117_Ints__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A,B3: A] :
          ( member(A,A3,ring_1_Ints(A))
         => ( member(A,B3,ring_1_Ints(A))
           => member(A,aa(A,A,minus_minus(A,A3),B3),ring_1_Ints(A)) ) ) ) ).

% Ints_diff
tff(fact_5118_verit__sko__ex_H,axiom,
    ! [A: $tType,P: fun(A,$o),A2: $o] :
      ( ( aa(A,$o,P,fChoice(A,P))
      <=> (A2) )
     => ( ? [X_1: A] : aa(A,$o,P,X_1)
      <=> (A2) ) ) ).

% verit_sko_ex'
tff(fact_5119_verit__sko__forall,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ! [X_1: A] : aa(A,$o,P,X_1)
    <=> aa(A,$o,P,fChoice(A,aTP_Lamp_al(fun(A,$o),fun(A,$o),P))) ) ).

% verit_sko_forall
tff(fact_5120_verit__sko__forall_H,axiom,
    ! [A: $tType,P: fun(A,$o),A2: $o] :
      ( ( aa(A,$o,P,fChoice(A,aTP_Lamp_al(fun(A,$o),fun(A,$o),P)))
      <=> (A2) )
     => ( ! [X_1: A] : aa(A,$o,P,X_1)
      <=> (A2) ) ) ).

% verit_sko_forall'
tff(fact_5121_verit__sko__forall_H_H,axiom,
    ! [A: $tType,B2: A,A2: A,P: fun(A,$o)] :
      ( ( B2 = A2 )
     => ( ( fChoice(A,P) = A2 )
      <=> ( fChoice(A,P) = B2 ) ) ) ).

% verit_sko_forall''
tff(fact_5122_verit__sko__ex__indirect,axiom,
    ! [A: $tType,X: A,P: fun(A,$o)] :
      ( ( X = fChoice(A,P) )
     => ( ? [X_1: A] : aa(A,$o,P,X_1)
      <=> aa(A,$o,P,X) ) ) ).

% verit_sko_ex_indirect
tff(fact_5123_verit__sko__ex__indirect2,axiom,
    ! [A: $tType,X: A,P: fun(A,$o),P3: fun(A,$o)] :
      ( ( X = fChoice(A,P) )
     => ( ! [X4: A] :
            ( aa(A,$o,P,X4)
          <=> aa(A,$o,P3,X4) )
       => ( ? [X_1: A] : aa(A,$o,P3,X_1)
        <=> aa(A,$o,P,X) ) ) ) ).

% verit_sko_ex_indirect2
tff(fact_5124_verit__sko__forall__indirect,axiom,
    ! [A: $tType,X: A,P: fun(A,$o)] :
      ( ( X = fChoice(A,aTP_Lamp_al(fun(A,$o),fun(A,$o),P)) )
     => ( ! [X_1: A] : aa(A,$o,P,X_1)
      <=> aa(A,$o,P,X) ) ) ).

% verit_sko_forall_indirect
tff(fact_5125_verit__sko__forall__indirect2,axiom,
    ! [A: $tType,X: A,P: fun(A,$o),P3: fun(A,$o)] :
      ( ( X = fChoice(A,aTP_Lamp_al(fun(A,$o),fun(A,$o),P)) )
     => ( ! [X4: A] :
            ( aa(A,$o,P,X4)
          <=> aa(A,$o,P3,X4) )
       => ( ! [X_1: A] : aa(A,$o,P3,X_1)
        <=> aa(A,$o,P,X) ) ) ) ).

% verit_sko_forall_indirect2
tff(fact_5126_real__root__commute,axiom,
    ! [M: nat,Na: nat,X: real] : aa(real,real,root(M),aa(real,real,root(Na),X)) = aa(real,real,root(Na),aa(real,real,root(M),X)) ).

% real_root_commute
tff(fact_5127_Ints__numeral,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Na: num] : member(A,aa(num,A,numeral_numeral(A),Na),ring_1_Ints(A)) ) ).

% Ints_numeral
tff(fact_5128_Ints__cases,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Q3: A] :
          ( member(A,Q3,ring_1_Ints(A))
         => ~ ! [Z2: int] : Q3 != aa(int,A,ring_1_of_int(A),Z2) ) ) ).

% Ints_cases
tff(fact_5129_Ints__induct,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Q3: A,P: fun(A,$o)] :
          ( member(A,Q3,ring_1_Ints(A))
         => ( ! [Z2: int] : aa(A,$o,P,aa(int,A,ring_1_of_int(A),Z2))
           => aa(A,$o,P,Q3) ) ) ) ).

% Ints_induct
tff(fact_5130_Ints__of__int,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: int] : member(A,aa(int,A,ring_1_of_int(A),Z),ring_1_Ints(A)) ) ).

% Ints_of_int
tff(fact_5131_Ints__mult,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A,B3: A] :
          ( member(A,A3,ring_1_Ints(A))
         => ( member(A,B3,ring_1_Ints(A))
           => member(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3),ring_1_Ints(A)) ) ) ) ).

% Ints_mult
tff(fact_5132_real__root__mult__exp,axiom,
    ! [M: nat,Na: nat,X: real] : aa(real,real,root(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na)),X) = aa(real,real,root(M),aa(real,real,root(Na),X)) ).

% real_root_mult_exp
tff(fact_5133_real__root__mult,axiom,
    ! [Na: nat,X: real,Y: real] : aa(real,real,root(Na),aa(real,real,aa(real,fun(real,real),times_times(real),X),Y)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,root(Na),X)),aa(real,real,root(Na),Y)) ).

% real_root_mult
tff(fact_5134_real__root__inverse,axiom,
    ! [Na: nat,X: real] : aa(real,real,root(Na),aa(real,real,inverse_inverse(real),X)) = aa(real,real,inverse_inverse(real),aa(real,real,root(Na),X)) ).

% real_root_inverse
tff(fact_5135_Ints__of__nat,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Na: nat] : member(A,aa(nat,A,semiring_1_of_nat(A),Na),ring_1_Ints(A)) ) ).

% Ints_of_nat
tff(fact_5136_real__root__pos__pos__le,axiom,
    ! [X: real,Na: nat] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,root(Na),X)) ) ).

% real_root_pos_pos_le
tff(fact_5137_Ints__double__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [A3: A] :
          ( member(A,A3,ring_1_Ints(A))
         => ( ( aa(A,A,plus_plus(A,A3),A3) = zero_zero(A) )
          <=> ( A3 = zero_zero(A) ) ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_5138_prod__int__eq,axiom,
    ! [I: nat,J: nat] : groups7121269368397514597t_prod(nat,int,semiring_1_of_nat(int),set_or1337092689740270186AtMost(nat,I,J)) = groups7121269368397514597t_prod(int,int,aTP_Lamp_jv(int,int),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),I),aa(nat,int,semiring_1_of_nat(int),J))) ).

% prod_int_eq
tff(fact_5139_real__root__less__mono,axiom,
    ! [Na: nat,X: real,Y: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(real,$o,ord_less(real,X),Y)
       => aa(real,$o,ord_less(real,aa(real,real,root(Na),X)),aa(real,real,root(Na),Y)) ) ) ).

% real_root_less_mono
tff(fact_5140_real__root__le__mono,axiom,
    ! [Na: nat,X: real,Y: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(real,$o,ord_less_eq(real,X),Y)
       => aa(real,$o,ord_less_eq(real,aa(real,real,root(Na),X)),aa(real,real,root(Na),Y)) ) ) ).

% real_root_le_mono
tff(fact_5141_real__root__power,axiom,
    ! [Na: nat,X: real,K: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(real,real,root(Na),aa(nat,real,power_power(real,X),K)) = aa(nat,real,power_power(real,aa(real,real,root(Na),X)),K) ) ) ).

% real_root_power
tff(fact_5142_real__root__abs,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(real,real,root(Na),aa(real,real,abs_abs(real),X)) = aa(real,real,abs_abs(real),aa(real,real,root(Na),X)) ) ) ).

% real_root_abs
tff(fact_5143_sgn__root,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(real,real,sgn_sgn(real),aa(real,real,root(Na),X)) = aa(real,real,sgn_sgn(real),X) ) ) ).

% sgn_root
tff(fact_5144_Ints__odd__nonzero,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [A3: A] :
          ( member(A,A3,ring_1_Ints(A))
         => ( aa(A,A,plus_plus(A,aa(A,A,plus_plus(A,one_one(A)),A3)),A3) != zero_zero(A) ) ) ) ).

% Ints_odd_nonzero
tff(fact_5145_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] : groups7121269368397514597t_prod(nat,int,semiring_1_of_nat(int),set_or1337092689740270186AtMost(nat,I,aa(nat,nat,plus_plus(nat,I),J))) = groups7121269368397514597t_prod(int,int,aTP_Lamp_jv(int,int),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),I),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,plus_plus(nat,I),J)))) ).

% prod_int_plus_eq
tff(fact_5146_of__int__divide__in__Ints,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B3: int,A3: int] :
          ( aa(int,$o,dvd_dvd(int,B3),A3)
         => member(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(int,A,ring_1_of_int(A),A3)),aa(int,A,ring_1_of_int(A),B3)),ring_1_Ints(A)) ) ) ).

% of_int_divide_in_Ints
tff(fact_5147_real__root__gt__zero,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),X)
       => aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,root(Na),X)) ) ) ).

% real_root_gt_zero
tff(fact_5148_real__root__strict__decreasing,axiom,
    ! [Na: nat,N2: nat,X: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(nat,$o,ord_less(nat,Na),N2)
       => ( aa(real,$o,ord_less(real,one_one(real)),X)
         => aa(real,$o,ord_less(real,aa(real,real,root(N2),X)),aa(real,real,root(Na),X)) ) ) ) ).

% real_root_strict_decreasing
tff(fact_5149_sqrt__def,axiom,
    sqrt = root(aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% sqrt_def
tff(fact_5150_root__abs__power,axiom,
    ! [Na: nat,Y: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(real,real,abs_abs(real),aa(real,real,root(Na),aa(nat,real,power_power(real,Y),Na))) = aa(real,real,abs_abs(real),Y) ) ) ).

% root_abs_power
tff(fact_5151_Ints__odd__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( member(A,A3,ring_1_Ints(A))
         => ( aa(A,$o,ord_less(A,aa(A,A,plus_plus(A,aa(A,A,plus_plus(A,one_one(A)),A3)),A3)),zero_zero(A))
          <=> aa(A,$o,ord_less(A,A3),zero_zero(A)) ) ) ) ).

% Ints_odd_less_0
tff(fact_5152_Ints__nonzero__abs__ge1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( member(A,X,ring_1_Ints(A))
         => ( ( X != zero_zero(A) )
           => aa(A,$o,ord_less_eq(A,one_one(A)),aa(A,A,abs_abs(A),X)) ) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_5153_Ints__nonzero__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( member(A,X,ring_1_Ints(A))
         => ( aa(A,$o,ord_less(A,aa(A,A,abs_abs(A),X)),one_one(A))
           => ( X = zero_zero(A) ) ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_5154_Ints__eq__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( member(A,X,ring_1_Ints(A))
         => ( member(A,Y,ring_1_Ints(A))
           => ( ( X = Y )
            <=> aa(A,$o,ord_less(A,aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,X),Y))),one_one(A)) ) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_5155_sin__times__pi__eq__0,axiom,
    ! [X: real] :
      ( ( sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),X),pi)) = zero_zero(real) )
    <=> member(real,X,ring_1_Ints(real)) ) ).

% sin_times_pi_eq_0
tff(fact_5156_real__root__pos__pos,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),X)
       => aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,root(Na),X)) ) ) ).

% real_root_pos_pos
tff(fact_5157_real__root__strict__increasing,axiom,
    ! [Na: nat,N2: nat,X: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(nat,$o,ord_less(nat,Na),N2)
       => ( aa(real,$o,ord_less(real,zero_zero(real)),X)
         => ( aa(real,$o,ord_less(real,X),one_one(real))
           => aa(real,$o,ord_less(real,aa(real,real,root(Na),X)),aa(real,real,root(N2),X)) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_5158_real__root__decreasing,axiom,
    ! [Na: nat,N2: nat,X: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(nat,$o,ord_less_eq(nat,Na),N2)
       => ( aa(real,$o,ord_less_eq(real,one_one(real)),X)
         => aa(real,$o,ord_less_eq(real,aa(real,real,root(N2),X)),aa(real,real,root(Na),X)) ) ) ) ).

% real_root_decreasing
tff(fact_5159_odd__real__root__pow,axiom,
    ! [Na: nat,X: real] :
      ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => ( aa(nat,real,power_power(real,aa(real,real,root(Na),X)),Na) = X ) ) ).

% odd_real_root_pow
tff(fact_5160_odd__real__root__unique,axiom,
    ! [Na: nat,Y: real,X: real] :
      ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => ( ( aa(nat,real,power_power(real,Y),Na) = X )
       => ( aa(real,real,root(Na),X) = Y ) ) ) ).

% odd_real_root_unique
tff(fact_5161_odd__real__root__power__cancel,axiom,
    ! [Na: nat,X: real] :
      ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => ( aa(real,real,root(Na),aa(nat,real,power_power(real,X),Na)) = X ) ) ).

% odd_real_root_power_cancel
tff(fact_5162_real__root__pow__pos,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),X)
       => ( aa(nat,real,power_power(real,aa(real,real,root(Na),X)),Na) = X ) ) ) ).

% real_root_pow_pos
tff(fact_5163_real__root__power__cancel,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
       => ( aa(real,real,root(Na),aa(nat,real,power_power(real,X),Na)) = X ) ) ) ).

% real_root_power_cancel
tff(fact_5164_real__root__pos__unique,axiom,
    ! [Na: nat,Y: real,X: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
       => ( ( aa(nat,real,power_power(real,Y),Na) = X )
         => ( aa(real,real,root(Na),X) = Y ) ) ) ) ).

% real_root_pos_unique
tff(fact_5165_frac__neg,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          archimedean_frac(A,aa(A,A,uminus_uminus(A),X)) = $ite(member(A,X,ring_1_Ints(A)),zero_zero(A),aa(A,A,minus_minus(A,one_one(A)),archimedean_frac(A,X))) ) ).

% frac_neg
tff(fact_5166_real__root__increasing,axiom,
    ! [Na: nat,N2: nat,X: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(nat,$o,ord_less_eq(nat,Na),N2)
       => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
         => ( aa(real,$o,ord_less_eq(real,X),one_one(real))
           => aa(real,$o,ord_less_eq(real,aa(real,real,root(Na),X)),aa(real,real,root(N2),X)) ) ) ) ) ).

% real_root_increasing
tff(fact_5167_sgn__power__root,axiom,
    ! [Na: nat,X: real] :
      ( aa(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),X))),aa(nat,real,power_power(real,aa(real,real,abs_abs(real),aa(real,real,root(Na),X))),Na)) = X ) ) ).

% sgn_power_root
tff(fact_5168_root__sgn__power,axiom,
    ! [Na: nat,Y: real] :
      ( aa(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,power_power(real,aa(real,real,abs_abs(real),Y)),Na))) = Y ) ) ).

% root_sgn_power
tff(fact_5169_le__mult__floor__Ints,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & linordered_idom(B) )
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => ( member(A,A3,ring_1_Ints(A))
           => aa(B,$o,ord_less_eq(B,aa(int,B,ring_1_of_int(B),aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(A,A3)),archim6421214686448440834_floor(A,B3)))),aa(int,B,ring_1_of_int(B),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)))) ) ) ) ).

% le_mult_floor_Ints
tff(fact_5170_mult__ceiling__le__Ints,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & linordered_idom(B) )
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => ( member(A,A3,ring_1_Ints(A))
           => aa(B,$o,ord_less_eq(B,aa(int,B,ring_1_of_int(B),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)))),aa(int,B,ring_1_of_int(B),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(A,A3)),archimedean_ceiling(A,B3)))) ) ) ) ).

% mult_ceiling_le_Ints
tff(fact_5171_frac__unique__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,A3: A] :
          ( ( archimedean_frac(A,X) = A3 )
        <=> ( member(A,aa(A,A,minus_minus(A,X),A3),ring_1_Ints(A))
            & aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
            & aa(A,$o,ord_less(A,A3),one_one(A)) ) ) ) ).

% frac_unique_iff
tff(fact_5172_log__root,axiom,
    ! [Na: nat,A3: real,B3: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
       => ( aa(real,real,log(B3),aa(real,real,root(Na),A3)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,log(B3),A3)),aa(nat,real,semiring_1_of_nat(real),Na)) ) ) ) ).

% log_root
tff(fact_5173_log__base__root,axiom,
    ! [Na: nat,B3: real,X: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),B3)
       => ( aa(real,real,log(aa(real,real,root(Na),B3)),X) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(real,real,log(B3),X)) ) ) ) ).

% log_base_root
tff(fact_5174_ln__root,axiom,
    ! [Na: nat,B3: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),B3)
       => ( aa(real,real,ln_ln(real),aa(real,real,root(Na),B3)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),B3)),aa(nat,real,semiring_1_of_nat(real),Na)) ) ) ) ).

% ln_root
tff(fact_5175_split__root,axiom,
    ! [P: fun(real,$o),Na: nat,X: real] :
      ( aa(real,$o,P,aa(real,real,root(Na),X))
    <=> ( ( ( Na = zero_zero(nat) )
         => aa(real,$o,P,zero_zero(real)) )
        & ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
         => ! [Y6: real] :
              ( ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),Y6)),aa(nat,real,power_power(real,aa(real,real,abs_abs(real),Y6)),Na)) = X )
             => aa(real,$o,P,Y6) ) ) ) ) ).

% split_root
tff(fact_5176_sin__integer__2pi,axiom,
    ! [Na: real] :
      ( member(real,Na,ring_1_Ints(real))
     => ( sin(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))),pi)),Na)) = zero_zero(real) ) ) ).

% sin_integer_2pi
tff(fact_5177_cos__integer__2pi,axiom,
    ! [Na: real] :
      ( member(real,Na,ring_1_Ints(real))
     => ( cos(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))),pi)),Na)) = one_one(real) ) ) ).

% cos_integer_2pi
tff(fact_5178_root__powr__inverse,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),X)
       => ( aa(real,real,root(Na),X) = powr(real,X,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_5179_some__sym__eq__trivial,axiom,
    ! [A: $tType,X: A] : fChoice(A,aa(A,fun(A,$o),fequal(A),X)) = X ).

% some_sym_eq_trivial
tff(fact_5180_some__eq__trivial,axiom,
    ! [A: $tType,X: A] : fChoice(A,aTP_Lamp_ad(A,fun(A,$o),X)) = X ).

% some_eq_trivial
tff(fact_5181_some__equality,axiom,
    ! [A: $tType,P: fun(A,$o),A3: A] :
      ( aa(A,$o,P,A3)
     => ( ! [X4: A] :
            ( aa(A,$o,P,X4)
           => ( X4 = A3 ) )
       => ( fChoice(A,P) = A3 ) ) ) ).

% some_equality
tff(fact_5182_someI2,axiom,
    ! [A: $tType,P: fun(A,$o),A3: A,Q: fun(A,$o)] :
      ( aa(A,$o,P,A3)
     => ( ! [X4: A] :
            ( aa(A,$o,P,X4)
           => aa(A,$o,Q,X4) )
       => aa(A,$o,Q,fChoice(A,P)) ) ) ).

% someI2
tff(fact_5183_someI__ex,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ? [X_13: A] : aa(A,$o,P,X_13)
     => aa(A,$o,P,fChoice(A,P)) ) ).

% someI_ex
tff(fact_5184_someI2__ex,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ? [X_13: A] : aa(A,$o,P,X_13)
     => ( ! [X4: A] :
            ( aa(A,$o,P,X4)
           => aa(A,$o,Q,X4) )
       => aa(A,$o,Q,fChoice(A,P)) ) ) ).

% someI2_ex
tff(fact_5185_someI2__bex,axiom,
    ! [A: $tType,A2: set(A),P: fun(A,$o),Q: fun(A,$o)] :
      ( ? [X2: A] :
          ( member(A,X2,A2)
          & aa(A,$o,P,X2) )
     => ( ! [X4: A] :
            ( ( member(A,X4,A2)
              & aa(A,$o,P,X4) )
           => aa(A,$o,Q,X4) )
       => aa(A,$o,Q,fChoice(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ao(set(A),fun(fun(A,$o),fun(A,$o)),A2),P))) ) ) ).

% someI2_bex
tff(fact_5186_some__eq__ex,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( aa(A,$o,P,fChoice(A,P))
    <=> ? [X_1: A] : aa(A,$o,P,X_1) ) ).

% some_eq_ex
tff(fact_5187_some1__equality,axiom,
    ! [A: $tType,P: fun(A,$o),A3: A] :
      ( ? [X2: A] :
          ( aa(A,$o,P,X2)
          & ! [Y3: A] :
              ( aa(A,$o,P,Y3)
             => ( Y3 = X2 ) ) )
     => ( aa(A,$o,P,A3)
       => ( fChoice(A,P) = A3 ) ) ) ).

% some1_equality
tff(fact_5188_dependent__nat__choice,axiom,
    ! [A: $tType,P: fun(nat,fun(A,$o)),Q: fun(nat,fun(A,fun(A,$o)))] :
      ( ? [X_13: A] : aa(A,$o,aa(nat,fun(A,$o),P,zero_zero(nat)),X_13)
     => ( ! [X4: A,N: nat] :
            ( aa(A,$o,aa(nat,fun(A,$o),P,N),X4)
           => ? [Y4: A] :
                ( aa(A,$o,aa(nat,fun(A,$o),P,aa(nat,nat,suc,N)),Y4)
                & aa(A,$o,aa(A,fun(A,$o),aa(nat,fun(A,fun(A,$o)),Q,N),X4),Y4) ) )
       => ? [F4: fun(nat,A)] :
          ! [N10: nat] :
            ( aa(A,$o,aa(nat,fun(A,$o),P,N10),aa(nat,A,F4,N10))
            & aa(A,$o,aa(A,fun(A,$o),aa(nat,fun(A,fun(A,$o)),Q,N10),aa(nat,A,F4,N10)),aa(nat,A,F4,aa(nat,nat,suc,N10))) ) ) ) ).

% dependent_nat_choice
tff(fact_5189_some__in__eq,axiom,
    ! [A: $tType,A2: set(A)] :
      ( member(A,fChoice(A,aTP_Lamp_aa(set(A),fun(A,$o),A2)),A2)
    <=> ( A2 != bot_bot(set(A)) ) ) ).

% some_in_eq
tff(fact_5190_frame__rule__left,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),R: assn] :
      ( hoare_hoare_triple(A,P,C2,Q)
     => hoare_hoare_triple(A,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),R),P),C2,aa(assn,fun(A,assn),aTP_Lamp_jy(fun(A,assn),fun(assn,fun(A,assn)),Q),R)) ) ).

% frame_rule_left
tff(fact_5191_div__half__nat,axiom,
    ! [Y: nat,X: 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),X),Y)),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),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),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Y)),
            $let(
              r: nat,
              r:= aa(nat,nat,minus_minus(nat,X),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),q),Y)),
              $ite(aa(nat,$o,ord_less_eq(nat,Y),r),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,plus_plus(nat,q),one_one(nat))),aa(nat,nat,minus_minus(nat,r),Y)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),q),r)) ) ) ) ) ).

% div_half_nat
tff(fact_5192_concat__bit__Suc,axiom,
    ! [Na: nat,K: int,L: int] : bit_concat_bit(aa(nat,nat,suc,Na),K,L) = aa(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_5193_concat__bit__0,axiom,
    ! [K: int,L: int] : bit_concat_bit(zero_zero(nat),K,L) = L ).

% concat_bit_0
tff(fact_5194_concat__bit__of__zero__2,axiom,
    ! [Na: nat,K: int] : bit_concat_bit(Na,K,zero_zero(int)) = bit_se2584673776208193580ke_bit(int,Na,K) ).

% concat_bit_of_zero_2
tff(fact_5195_concat__bit__nonnegative__iff,axiom,
    ! [Na: nat,K: int,L: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),bit_concat_bit(Na,K,L))
    <=> aa(int,$o,ord_less_eq(int,zero_zero(int)),L) ) ).

% concat_bit_nonnegative_iff
tff(fact_5196_concat__bit__negative__iff,axiom,
    ! [Na: nat,K: int,L: int] :
      ( aa(int,$o,ord_less(int,bit_concat_bit(Na,K,L)),zero_zero(int))
    <=> aa(int,$o,ord_less(int,L),zero_zero(int)) ) ).

% concat_bit_negative_iff
tff(fact_5197_pred__equals__eq2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S: set(product_prod(A,B))] :
      ( ! [X3: A,Xa4: B] :
          ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Xa4),R)
        <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Xa4),S) )
    <=> ( R = S ) ) ).

% pred_equals_eq2
tff(fact_5198_bot__empty__eq2,axiom,
    ! [B: $tType,A: $tType,X2: A,Xa: B] :
      ( aa(B,$o,aa(A,fun(B,$o),bot_bot(fun(A,fun(B,$o))),X2),Xa)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Xa),bot_bot(set(product_prod(A,B)))) ) ).

% bot_empty_eq2
tff(fact_5199_sup__Un__eq2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S: set(product_prod(A,B)),X2: A,Xa: B] :
      ( aa(B,$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,$o))),aTP_Lamp_jz(set(product_prod(A,B)),fun(A,fun(B,$o)),R)),aTP_Lamp_jz(set(product_prod(A,B)),fun(A,fun(B,$o)),S)),X2),Xa)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Xa),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),R),S)) ) ).

% sup_Un_eq2
tff(fact_5200_pred__subset__eq2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S: set(product_prod(A,B))] :
      ( aa(fun(A,fun(B,$o)),$o,ord_less_eq(fun(A,fun(B,$o)),aTP_Lamp_jz(set(product_prod(A,B)),fun(A,fun(B,$o)),R)),aTP_Lamp_jz(set(product_prod(A,B)),fun(A,fun(B,$o)),S))
    <=> aa(set(product_prod(A,B)),$o,ord_less_eq(set(product_prod(A,B)),R),S) ) ).

% pred_subset_eq2
tff(fact_5201_subrelI,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,B)),Sa: set(product_prod(A,B))] :
      ( ! [X4: A,Y3: B] :
          ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3),R3)
         => member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3),Sa) )
     => aa(set(product_prod(A,B)),$o,ord_less_eq(set(product_prod(A,B)),R3),Sa) ) ).

% subrelI
tff(fact_5202_is__hoare__triple,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn)] :
      ( hoare_hoare_triple(A,P,C2,Q)
     => hoare_hoare_triple(A,P,C2,Q) ) ).

% is_hoare_triple
tff(fact_5203_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,$o,ord_less(nat,Na),M)
          & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),Na) )
        | ( aa(nat,$o,ord_less_eq(nat,M),Na)
          & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),L),aa(nat,nat,minus_minus(nat,Na),M)) ) ) ) ).

% bit_concat_bit_iff
tff(fact_5204_norm__assertion__simps_I2_J,axiom,
    ! [A3: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),one_one(assn)) = A3 ).

% norm_assertion_simps(2)
tff(fact_5205_norm__assertion__simps_I1_J,axiom,
    ! [A3: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),one_one(assn)),A3) = A3 ).

% norm_assertion_simps(1)
tff(fact_5206_norm__assertion__simps_I5_J,axiom,
    ! [X: assn] : aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),bot_bot(assn)),X) = X ).

% norm_assertion_simps(5)
tff(fact_5207_norm__assertion__simps_I6_J,axiom,
    ! [X: assn] : aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),X),bot_bot(assn)) = X ).

% norm_assertion_simps(6)
tff(fact_5208_divmod__step__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [L: num,Q3: A,R3: A] :
          unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),R3)) = $ite(aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),L)),R3),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(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,minus_minus(A,R3),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)),R3)) ) ).

% divmod_step_eq
tff(fact_5209_divides__aux__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q3: A,R3: A] :
          ( unique5940410009612947441es_aux(A,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),R3))
        <=> ( R3 = zero_zero(A) ) ) ) ).

% divides_aux_eq
tff(fact_5210_fun__of__rel__def,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(B,A)),X: B] : fun_of_rel(B,A,R,X) = fChoice(A,aa(B,fun(A,$o),aTP_Lamp_ka(set(product_prod(B,A)),fun(B,fun(A,$o)),R),X)) ).

% fun_of_rel_def
tff(fact_5211_predicate2I,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o)),Q: fun(A,fun(B,$o))] :
      ( ! [X4: A,Y3: B] :
          ( aa(B,$o,aa(A,fun(B,$o),P,X4),Y3)
         => aa(B,$o,aa(A,fun(B,$o),Q,X4),Y3) )
     => aa(fun(A,fun(B,$o)),$o,ord_less_eq(fun(A,fun(B,$o)),P),Q) ) ).

% predicate2I
tff(fact_5212_predicate2D,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),Q: fun(A,fun(B,$o)),X: A,Y: B] :
      ( aa(fun(A,fun(B,$o)),$o,ord_less_eq(fun(A,fun(B,$o)),P),Q)
     => ( aa(B,$o,aa(A,fun(B,$o),P,X),Y)
       => aa(B,$o,aa(A,fun(B,$o),Q,X),Y) ) ) ).

% predicate2D
tff(fact_5213_rev__predicate2D,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),X: A,Y: B,Q: fun(A,fun(B,$o))] :
      ( aa(B,$o,aa(A,fun(B,$o),P,X),Y)
     => ( aa(fun(A,fun(B,$o)),$o,ord_less_eq(fun(A,fun(B,$o)),P),Q)
       => aa(B,$o,aa(A,fun(B,$o),Q,X),Y) ) ) ).

% rev_predicate2D
tff(fact_5214_bot2E,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B] : ~ aa(B,$o,aa(A,fun(B,$o),bot_bot(fun(A,fun(B,$o))),X),Y) ).

% bot2E
tff(fact_5215_VEBT__internal_Oreplicatei_Ocases,axiom,
    ! [A: $tType,X: product_prod(nat,heap_Time_Heap(A))] :
      ( ! [X4: heap_Time_Heap(A)] : X != 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)),X4)
     => ~ ! [N: nat,X4: heap_Time_Heap(A)] : X != 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)),X4) ) ).

% VEBT_internal.replicatei.cases
tff(fact_5216_xor__num_Ocases,axiom,
    ! [X: product_prod(num,num)] :
      ( ( X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),one2) )
     => ( ! [N: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),bit0(N))
       => ( ! [N: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),bit1(N))
         => ( ! [M4: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit0(M4)),one2)
           => ( ! [M4: num,N: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit0(M4)),bit0(N))
             => ( ! [M4: num,N: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit0(M4)),bit1(N))
               => ( ! [M4: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit1(M4)),one2)
                 => ( ! [M4: num,N: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit1(M4)),bit0(N))
                   => ~ ! [M4: num,N: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit1(M4)),bit1(N)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_5217_divmod__algorithm__code_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Na: num] :
          unique8689654367752047608divmod(A,bit0(M),bit1(Na)) = $ite(aa(num,$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_5218_divmod__algorithm__code_I8_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Na: num] :
          unique8689654367752047608divmod(A,bit1(M),bit1(Na)) = $ite(aa(num,$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_5219_neg__eucl__rel__int__mult__2,axiom,
    ! [B3: int,A3: int,Q3: int,R3: int] :
      ( aa(int,$o,ord_less_eq(int,B3),zero_zero(int))
     => ( eucl_rel_int(aa(int,int,plus_plus(int,A3),one_one(int)),B3,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R3))
       => eucl_rel_int(aa(int,int,plus_plus(int,one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A3)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B3),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),R3)),one_one(int)))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_5220_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_5221_dvd__numeral__simp,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Na: num] :
          ( aa(A,$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_5222_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_5223_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_5224_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_5225_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_5226_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_5227_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_5228_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_5229_zminus1__lemma,axiom,
    ! [A3: int,B3: int,Q3: int,R3: int] :
      ( eucl_rel_int(A3,B3,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R3))
     => ( ( B3 != zero_zero(int) )
       => eucl_rel_int(aa(int,int,uminus_uminus(int),A3),B3,
            aa(int,product_prod(int,int),
              aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),
                $ite(R3 = zero_zero(int),aa(int,int,uminus_uminus(int),Q3),aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),Q3)),one_one(int)))),
              $ite(R3 = zero_zero(int),zero_zero(int),aa(int,int,minus_minus(int,B3),R3)))) ) ) ).

% zminus1_lemma
tff(fact_5230_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q3: int,R3: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R3))
    <=> ( ( K = aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),times_times(int),L),Q3)),R3) )
        & $ite(
            aa(int,$o,ord_less(int,zero_zero(int)),L),
            ( aa(int,$o,ord_less_eq(int,zero_zero(int)),R3)
            & aa(int,$o,ord_less(int,R3),L) ),
            $ite(
              aa(int,$o,ord_less(int,L),zero_zero(int)),
              ( aa(int,$o,ord_less(int,L),R3)
              & aa(int,$o,ord_less_eq(int,R3),zero_zero(int)) ),
              Q3 = zero_zero(int) ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_5231_eucl__rel__int__remainderI,axiom,
    ! [R3: int,L: int,K: int,Q3: int] :
      ( ( aa(int,int,sgn_sgn(int),R3) = aa(int,int,sgn_sgn(int),L) )
     => ( aa(int,$o,ord_less(int,aa(int,int,abs_abs(int),R3)),aa(int,int,abs_abs(int),L))
       => ( ( K = aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),times_times(int),Q3),L)),R3) )
         => eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R3)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_5232_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod(int,int)] :
      ( eucl_rel_int(A1,A22,A32)
    <=> ( ? [K3: int] :
            ( ( A1 = K3 )
            & ( A22 = zero_zero(int) )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),K3) ) )
        | ? [L3: int,K3: int,Q5: int] :
            ( ( A1 = K3 )
            & ( A22 = L3 )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q5),zero_zero(int)) )
            & ( L3 != zero_zero(int) )
            & ( K3 = aa(int,int,aa(int,fun(int,int),times_times(int),Q5),L3) ) )
        | ? [R5: int,L3: int,K3: int,Q5: int] :
            ( ( A1 = K3 )
            & ( A22 = L3 )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q5),R5) )
            & ( aa(int,int,sgn_sgn(int),R5) = aa(int,int,sgn_sgn(int),L3) )
            & aa(int,$o,ord_less(int,aa(int,int,abs_abs(int),R5)),aa(int,int,abs_abs(int),L3))
            & ( K3 = aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),times_times(int),Q5),L3)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_5233_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A22: int,A32: product_prod(int,int)] :
      ( eucl_rel_int(A1,A22,A32)
     => ( ( ( A22 = zero_zero(int) )
         => ( A32 != aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),A1) ) )
       => ( ! [Q6: int] :
              ( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q6),zero_zero(int)) )
             => ( ( A22 != zero_zero(int) )
               => ( A1 != aa(int,int,aa(int,fun(int,int),times_times(int),Q6),A22) ) ) )
         => ~ ! [R2: int,Q6: int] :
                ( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q6),R2) )
               => ( ( aa(int,int,sgn_sgn(int),R2) = aa(int,int,sgn_sgn(int),A22) )
                 => ( aa(int,$o,ord_less(int,aa(int,int,abs_abs(int),R2)),aa(int,int,abs_abs(int),A22))
                   => ( A1 != aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),times_times(int),Q6),A22)),R2) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_5234_divmod__divmod__step,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Na: num] :
          unique8689654367752047608divmod(A,M,Na) = $ite(aa(num,$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_5235_pos__eucl__rel__int__mult__2,axiom,
    ! [B3: int,A3: int,Q3: int,R3: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),B3)
     => ( eucl_rel_int(A3,B3,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R3))
       => eucl_rel_int(aa(int,int,plus_plus(int,one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A3)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B3),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),aa(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))),R3)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_5236_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_5237_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_5238_Divides_Oadjust__div__eq,axiom,
    ! [Q3: int,R3: int] : adjust_div(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R3)) = aa(int,int,plus_plus(int,Q3),aa($o,int,zero_neq_one_of_bool(int),R3 != zero_zero(int))) ).

% Divides.adjust_div_eq
tff(fact_5239_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_5240_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_5241_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,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_5242_and__int_Opelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Xa2) = Y )
     => ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X),Xa2))
       => ~ ( ( Y = $ite(
                  ( member(int,X,aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
                  & member(int,Xa2,aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
                  aa(int,int,uminus_uminus(int),
                    aa($o,int,zero_neq_one_of_bool(int),
                      ( ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),X)
                      & ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Xa2) ))),
                  aa(int,int,
                    plus_plus(int,
                      aa($o,int,zero_neq_one_of_bool(int),
                        ( ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),X)
                        & ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Xa2) ))),
                    aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),X),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa2),aa(num,int,numeral_numeral(int),bit0(one2)))))) ) )
           => ~ accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X),Xa2)) ) ) ) ).

% and_int.pelims
tff(fact_5243_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),K),L))
     => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = $ite(
            ( member(int,K,aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
            & member(int,L,aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
            aa(int,int,uminus_uminus(int),
              aa($o,int,zero_neq_one_of_bool(int),
                ( ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K)
                & ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),L) ))),
            aa(int,int,
              plus_plus(int,
                aa($o,int,zero_neq_one_of_bool(int),
                  ( ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K)
                  & ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),L) ))),
              aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ).

% and_int.psimps
tff(fact_5244_zero__natural_Orsp,axiom,
    zero_zero(nat) = zero_zero(nat) ).

% zero_natural.rsp
tff(fact_5245_one__natural_Orsp,axiom,
    one_one(nat) = one_one(nat) ).

% one_natural.rsp
tff(fact_5246_zero__integer_Orsp,axiom,
    zero_zero(int) = zero_zero(int) ).

% zero_integer.rsp
tff(fact_5247_one__integer_Orsp,axiom,
    one_one(int) = one_one(int) ).

% one_integer.rsp
tff(fact_5248_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_5249_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_5250_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_5251_less__eq__integer__code_I1_J,axiom,
    aa(code_integer,$o,ord_less_eq(code_integer,zero_zero(code_integer)),zero_zero(code_integer)) ).

% less_eq_integer_code(1)
tff(fact_5252_less__integer__code_I1_J,axiom,
    ~ aa(code_integer,$o,ord_less(code_integer,zero_zero(code_integer)),zero_zero(code_integer)) ).

% less_integer_code(1)
tff(fact_5253_abs__integer__code,axiom,
    ! [K: code_integer] :
      aa(code_integer,code_integer,abs_abs(code_integer),K) = $ite(aa(code_integer,$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_5254_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun(int,fun(int,$o))] :
      ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1))
     => ( ! [K2: int,L2: int] :
            ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),K2),L2))
           => ( ( ~ ( member(int,K2,aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
                    & member(int,L2,aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) )
               => aa(int,$o,aa(int,fun(int,$o),P,aa(int,int,aa(int,fun(int,int),divide_divide(int),K2),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L2),aa(num,int,numeral_numeral(int),bit0(one2)))) )
             => aa(int,$o,aa(int,fun(int,$o),P,K2),L2) ) )
       => aa(int,$o,aa(int,fun(int,$o),P,A0),A1) ) ) ).

% and_int.pinduct
tff(fact_5255_plus__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa(code_integer,code_integer,plus_plus(code_integer,zero_zero(code_integer)),L) = L ).

% plus_integer_code(2)
tff(fact_5256_plus__integer__code_I1_J,axiom,
    ! [K: code_integer] : aa(code_integer,code_integer,plus_plus(code_integer,K),zero_zero(code_integer)) = K ).

% plus_integer_code(1)
tff(fact_5257_minus__integer__code_I1_J,axiom,
    ! [K: code_integer] : aa(code_integer,code_integer,minus_minus(code_integer,K),zero_zero(code_integer)) = K ).

% minus_integer_code(1)
tff(fact_5258_minus__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa(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_5259_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun(int,fun(int,$o))] :
      ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1))
     => ( ! [I3: int,J2: int] :
            ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),I3),J2))
           => ( ( aa(int,$o,ord_less_eq(int,I3),J2)
               => aa(int,$o,aa(int,fun(int,$o),P,aa(int,int,plus_plus(int,I3),one_one(int))),J2) )
             => aa(int,$o,aa(int,fun(int,$o),P,I3),J2) ) )
       => aa(int,$o,aa(int,fun(int,$o),P,A0),A1) ) ) ).

% upto.pinduct
tff(fact_5260_bitNOT__integer__code,axiom,
    ! [I: code_integer] : aa(code_integer,code_integer,bit_ri4277139882892585799ns_not(code_integer),I) = aa(code_integer,code_integer,minus_minus(code_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),I)),one_one(code_integer)) ).

% bitNOT_integer_code
tff(fact_5261_integer__of__int__code,axiom,
    ! [K: int] :
      code_integer_of_int(K) = $ite(
        aa(int,$o,ord_less(int,K),zero_zero(int)),
        aa(code_integer,code_integer,uminus_uminus(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))),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,plus_plus(code_integer,l),one_one(code_integer))) ) ) ) ).

% integer_of_int_code
tff(fact_5262_zero__integer__def,axiom,
    zero_zero(code_integer) = code_integer_of_int(zero_zero(int)) ).

% zero_integer_def
tff(fact_5263_less__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( aa(code_integer,$o,ord_less(code_integer,code_integer_of_int(Xa2)),code_integer_of_int(X))
    <=> aa(int,$o,ord_less(int,Xa2),X) ) ).

% less_integer.abs_eq
tff(fact_5264_one__integer__def,axiom,
    one_one(code_integer) = code_integer_of_int(one_one(int)) ).

% one_integer_def
tff(fact_5265_lsb__integer__code,axiom,
    ! [X: code_integer] :
      ( aa(code_integer,$o,least_8051144512741203767sb_lsb(code_integer),X)
    <=> aa(nat,$o,aa(code_integer,fun(nat,$o),bit_se5641148757651400278ts_bit(code_integer),X),zero_zero(nat)) ) ).

% lsb_integer_code
tff(fact_5266_shiftr__integer__conv__div__pow2,axiom,
    ! [Na: nat,X: code_integer] : aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se4197421643247451524op_bit(code_integer),Na),X) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),X),aa(nat,code_integer,power_power(code_integer,aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),Na)) ).

% shiftr_integer_conv_div_pow2
tff(fact_5267_Bit__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: $o] : bits_Bit_integer(code_integer_of_int(Xa2),(X)) = code_integer_of_int(aa(int,int,plus_plus(int,aa($o,int,zero_neq_one_of_bool(int),(X))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Xa2))) ).

% Bit_integer.abs_eq
tff(fact_5268_div__half__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: 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)),X),Y)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),X),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)),X)),Y)),
                $let(
                  r: word(A),
                  r:= aa(word(A),word(A),minus_minus(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),q),Y)),
                  $ite(aa(word(A),$o,ord_less_eq(word(A),Y),r),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),plus_plus(word(A),q),one_one(word(A)))),aa(word(A),word(A),minus_minus(word(A),r),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),r)) ) ) ) ) ) ).

% div_half_word
tff(fact_5269_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,minus_minus(int,aa(num,int,numeral_numeral(int),M)),one_one(int))),one_one(int)) ).

% divmod_BitM_2_eq
tff(fact_5270_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_5271_push__bit__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat,A3: A] :
          ( ( aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% push_bit_eq_0_iff
tff(fact_5272_dbl__dec__simps_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_dbl_dec(A,aa(num,A,numeral_numeral(A),K)) = aa(num,A,numeral_numeral(A),bitM(K)) ) ).

% dbl_dec_simps(5)
tff(fact_5273_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_5274_pred__numeral__simps_I2_J,axiom,
    ! [K: num] : pred_numeral(bit0(K)) = aa(num,nat,numeral_numeral(nat),bitM(K)) ).

% pred_numeral_simps(2)
tff(fact_5275_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_5276_shiftl__numeral__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: num,Na: num] : bit_Sh4282982442137083160shiftl(A,aa(num,A,numeral_numeral(A),M),aa(num,nat,numeral_numeral(nat),Na)) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(num,nat,numeral_numeral(nat),Na)),aa(num,A,numeral_numeral(A),M)) ) ).

% shiftl_numeral_numeral
tff(fact_5277_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_5278_push__bit__minus__one__eq__not__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,Na)) ) ).

% push_bit_minus_one_eq_not_mask
tff(fact_5279_push__bit__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [L: num,K: num] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(num,nat,numeral_numeral(nat),L)),aa(num,A,numeral_numeral(A),K)) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),pred_numeral(L)),aa(num,A,numeral_numeral(A),bit0(K))) ) ).

% push_bit_numeral
tff(fact_5280_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_5281_shiftl__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: num,Na: num] : bit_Sh4282982442137083160shiftl(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)),aa(num,nat,numeral_numeral(nat),Na)) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(num,nat,numeral_numeral(nat),Na)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) ) ).

% shiftl_minus_numeral_numeral
tff(fact_5282_push__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(nat,nat,suc,Na)),A3) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% push_bit_Suc
tff(fact_5283_push__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),one_one(A)) = aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na) ) ).

% push_bit_of_1
tff(fact_5284_even__push__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A3: A] :
          ( aa(A,$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),A3))
        <=> ( ( Na != zero_zero(nat) )
            | aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3) ) ) ) ).

% even_push_bit_iff
tff(fact_5285_push__bit__minus__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [L: num,K: num] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(num,nat,numeral_numeral(nat),L)),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),pred_numeral(L)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(K)))) ) ).

% push_bit_minus_numeral
tff(fact_5286_push__bit__numeral__minus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: num] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(num,nat,numeral_numeral(nat),Na)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(num,nat,numeral_numeral(nat),Na))) ) ).

% push_bit_numeral_minus_1
tff(fact_5287_semiring__norm_I26_J,axiom,
    bitM(one2) = one2 ).

% semiring_norm(26)
tff(fact_5288_of__nat__push__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se4730199178511100633sh_bit(nat),M),Na)) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),M),aa(nat,A,semiring_1_of_nat(A),Na)) ) ).

% of_nat_push_bit
tff(fact_5289_push__bit__of__nat,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,M: nat] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),aa(nat,A,semiring_1_of_nat(A),M)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se4730199178511100633sh_bit(nat),Na),M)) ) ).

% push_bit_of_nat
tff(fact_5290_semiring__norm_I28_J,axiom,
    ! [Na: num] : bitM(bit1(Na)) = bit1(bit0(Na)) ).

% semiring_norm(28)
tff(fact_5291_semiring__norm_I27_J,axiom,
    ! [Na: num] : bitM(bit0(Na)) = bit1(bitM(Na)) ).

% semiring_norm(27)
tff(fact_5292_inc__BitM__eq,axiom,
    ! [Na: num] : inc(bitM(Na)) = bit0(Na) ).

% inc_BitM_eq
tff(fact_5293_BitM__inc__eq,axiom,
    ! [Na: num] : bitM(inc(Na)) = bit1(Na) ).

% BitM_inc_eq
tff(fact_5294_div__push__bit__of__1__eq__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,Na: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),one_one(A))) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),A3) ) ).

% div_push_bit_of_1_eq_drop_bit
tff(fact_5295_set__bit__eq__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A3: A] : bit_se5668285175392031749et_bit(A,Na,A3) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),one_one(A))) ) ).

% set_bit_eq_or
tff(fact_5296_flip__bit__eq__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A3: A] : bit_se8732182000553998342ip_bit(A,Na,A3) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),one_one(A))) ) ).

% flip_bit_eq_xor
tff(fact_5297_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_5298_BitM__plus__one,axiom,
    ! [Na: num] : aa(num,num,plus_plus(num,bitM(Na)),one2) = bit0(Na) ).

% BitM_plus_one
tff(fact_5299_one__plus__BitM,axiom,
    ! [Na: num] : aa(num,num,plus_plus(num,one2),bitM(Na)) = bit0(Na) ).

% one_plus_BitM
tff(fact_5300_push__bit__double,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),A3)),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% push_bit_double
tff(fact_5301_bit__iff__and__push__bit__not__eq__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),Na)
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),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_5302_unset__bit__eq__and__not,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,A3: A] : bit_se2638667681897837118et_bit(A,Na,A3) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),one_one(A)))) ) ).

% unset_bit_eq_and_not
tff(fact_5303_numeral__BitM,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Na: num] : aa(num,A,numeral_numeral(A),bitM(Na)) = aa(A,A,minus_minus(A,aa(num,A,numeral_numeral(A),bit0(Na))),one_one(A)) ) ).

% numeral_BitM
tff(fact_5304_odd__numeral__BitM,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [W: num] : ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(num,A,numeral_numeral(A),bitM(W))) ) ).

% odd_numeral_BitM
tff(fact_5305_not__numeral__BitM__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: num] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),bitM(Na))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(Na))) ) ).

% not_numeral_BitM_eq
tff(fact_5306_push__bit__eq__mult,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) ) ).

% push_bit_eq_mult
tff(fact_5307_exp__dvdE,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)),A3)
         => ~ ! [B5: A] : A3 != aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),B5) ) ) ).

% exp_dvdE
tff(fact_5308_word__and__mask__or__conv__and__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A),Index: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Na),Index)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Na),bit_se2239418461657761734s_mask(word(A),Index))),aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4730199178511100633sh_bit(word(A)),Index),one_one(word(A)))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Na),bit_se2239418461657761734s_mask(word(A),aa(nat,nat,plus_plus(nat,Index),one_one(nat)))) ) ) ) ).

% word_and_mask_or_conv_and_mask
tff(fact_5309_signed__take__bit__code,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,A3: A] :
          bit_ri4674362597316999326ke_bit(A,Na,A3) = $let(
            l: A,
            l:= bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Na),A3),
            $ite(aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),l),Na),aa(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_5310_set__bits__aux__Suc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F2: fun(nat,$o),Na: nat,W: word(A)] :
          code_T2661198915054445665ts_aux(A,F2,aa(nat,nat,suc,Na),W) = code_T2661198915054445665ts_aux(A,F2,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)),W)),
              $ite(aa(nat,$o,F2,Na),one_one(word(A)),zero_zero(word(A))))) ) ).

% set_bits_aux_Suc
tff(fact_5311_set__bits__aux__rec,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F2: fun(nat,$o),Na: nat,W: word(A)] :
          code_T2661198915054445665ts_aux(A,F2,Na,W) = $ite(
            Na = zero_zero(nat),
            W,
            $let(
              n2: nat,
              n2:= aa(nat,nat,minus_minus(nat,Na),one_one(nat)),
              code_T2661198915054445665ts_aux(A,F2,n2,
                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)),W)),
                  $ite(aa(nat,$o,F2,n2),one_one(word(A)),zero_zero(word(A))))) ) ) ) ).

% set_bits_aux_rec
tff(fact_5312_test__bit__split,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(C)
        & type_len(A) )
     => ! [C2: word(C),A3: word(A),B3: word(B)] :
          ( ( word_split(C,A,B,C2) = aa(word(B),product_prod(word(A),word(B)),aa(word(A),fun(word(B),product_prod(word(A),word(B))),product_Pair(word(A),word(B)),A3),B3) )
         => ( ! [N10: nat] :
                ( aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),B3),N10)
              <=> ( aa(nat,$o,ord_less(nat,N10),size_size(word(B),B3))
                  & aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),C2),N10) ) )
            & ! [M2: nat] :
                ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),A3),M2)
              <=> ( aa(nat,$o,ord_less(nat,M2),size_size(word(A),A3))
                  & aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),C2),aa(nat,nat,plus_plus(nat,M2),size_size(word(B),B3))) ) ) ) ) ) ).

% test_bit_split
tff(fact_5313_push__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(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,ord_less_eq(int,zero_zero(int)),K) ) ).

% push_bit_nonnegative_int_iff
tff(fact_5314_push__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(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,ord_less(int,K),zero_zero(int)) ) ).

% push_bit_negative_int_iff
tff(fact_5315_concat__bit__of__zero__1,axiom,
    ! [Na: nat,L: int] : bit_concat_bit(Na,zero_zero(int),L) = aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),Na),L) ).

% concat_bit_of_zero_1
tff(fact_5316_set__bits__aux__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F2: fun(nat,$o),W: word(A)] : code_T2661198915054445665ts_aux(A,F2,zero_zero(nat),W) = W ) ).

% set_bits_aux_0
tff(fact_5317_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,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) ).

% push_bit_of_Suc_0
tff(fact_5318_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_5319_flip__bit__nat__def,axiom,
    ! [M: nat,Na: nat] : bit_se8732182000553998342ip_bit(nat,M,Na) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),bit_se4730199178511100633sh_bit(nat),M),one_one(nat))) ).

% flip_bit_nat_def
tff(fact_5320_set__bit__nat__def,axiom,
    ! [M: nat,Na: nat] : bit_se5668285175392031749et_bit(nat,M,Na) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),bit_se4730199178511100633sh_bit(nat),M),one_one(nat))) ).

% set_bit_nat_def
tff(fact_5321_Bit__integer__code_I1_J,axiom,
    ! [I: code_integer] : bits_Bit_integer(I,$false) = aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se4730199178511100633sh_bit(code_integer),one_one(nat)),I) ).

% Bit_integer_code(1)
tff(fact_5322_Bit__Operations_Oset__bit__int__def,axiom,
    ! [Na: nat,K: int] : bit_se5668285175392031749et_bit(int,Na,K) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),Na),one_one(int))) ).

% Bit_Operations.set_bit_int_def
tff(fact_5323_flip__bit__int__def,axiom,
    ! [Na: nat,K: int] : bit_se8732182000553998342ip_bit(int,Na,K) = aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),Na),one_one(int))) ).

% flip_bit_int_def
tff(fact_5324_unset__bit__int__def,axiom,
    ! [Na: nat,K: int] : bit_se2638667681897837118et_bit(int,Na,K) = aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),Na),one_one(int)))) ).

% unset_bit_int_def
tff(fact_5325_push__bit__int__def,axiom,
    ! [Na: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),Na),K) = aa(int,int,aa(int,fun(int,int),times_times(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ).

% push_bit_int_def
tff(fact_5326_Bit__integer__code_I2_J,axiom,
    ! [I: code_integer] : bits_Bit_integer(I,$true) = aa(code_integer,code_integer,plus_plus(code_integer,aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se4730199178511100633sh_bit(code_integer),one_one(nat)),I)),one_one(code_integer)) ).

% Bit_integer_code(2)
tff(fact_5327_push__bit__nat__def,axiom,
    ! [Na: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se4730199178511100633sh_bit(nat),Na),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ).

% push_bit_nat_def
tff(fact_5328_push__bit__minus__one,axiom,
    ! [Na: nat] : aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),Na),aa(int,int,uminus_uminus(int),one_one(int))) = aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ).

% push_bit_minus_one
tff(fact_5329_shiftl__integer__conv__mult__pow2,axiom,
    ! [Na: nat,X: code_integer] : aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se4730199178511100633sh_bit(code_integer),Na),X) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),X),aa(nat,code_integer,power_power(code_integer,aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),Na)) ).

% shiftl_integer_conv_mult_pow2
tff(fact_5330_test__bit__split__eq,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(C)
        & type_len(A) )
     => ! [C2: word(C),A3: word(A),B3: word(B)] :
          ( ( word_split(C,A,B,C2) = aa(word(B),product_prod(word(A),word(B)),aa(word(A),fun(word(B),product_prod(word(A),word(B))),product_Pair(word(A),word(B)),A3),B3) )
        <=> ( ! [N3: nat] :
                ( aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),B3),N3)
              <=> ( aa(nat,$o,ord_less(nat,N3),size_size(word(B),B3))
                  & aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),C2),N3) ) )
            & ! [M3: nat] :
                ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),A3),M3)
              <=> ( aa(nat,$o,ord_less(nat,M3),size_size(word(A),A3))
                  & aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),C2),aa(nat,nat,plus_plus(nat,M3),size_size(word(B),B3))) ) ) ) ) ) ).

% test_bit_split_eq
tff(fact_5331_test__bit__split_H,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(C)
        & type_len(A) )
     => ! [C2: word(C),A3: word(A),B3: word(B)] :
          ( ( word_split(C,A,B,C2) = aa(word(B),product_prod(word(A),word(B)),aa(word(A),fun(word(B),product_prod(word(A),word(B))),product_Pair(word(A),word(B)),A3),B3) )
         => ! [N10: nat,M2: nat] :
              ( ( aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),B3),N10)
              <=> ( aa(nat,$o,ord_less(nat,N10),size_size(word(B),B3))
                  & aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),C2),N10) ) )
              & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),A3),M2)
              <=> ( aa(nat,$o,ord_less(nat,M2),size_size(word(A),A3))
                  & aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),C2),aa(nat,nat,plus_plus(nat,M2),size_size(word(B),B3))) ) ) ) ) ) ).

% test_bit_split'
tff(fact_5332_or__minus__numerals_I5_J,axiom,
    ! [Na: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(Na)))),one_one(int)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(one2,bitM(Na)))) ).

% or_minus_numerals(5)
tff(fact_5333_or__minus__numerals_I1_J,axiom,
    ! [Na: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(Na)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(one2,bitM(Na)))) ).

% or_minus_numerals(1)
tff(fact_5334_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),product_case_prod(A,A,product_prod(A,A),aTP_Lamp_kb(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M,Na)) ) ).

% divmod_algorithm_code(6)
tff(fact_5335_or__minus__numerals_I8_J,axiom,
    ! [Na: num,M: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit1(Na)))),aa(num,int,numeral_numeral(int),M)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,bit0(Na)))) ).

% or_minus_numerals(8)
tff(fact_5336_or__minus__numerals_I4_J,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit1(Na)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,bit0(Na)))) ).

% or_minus_numerals(4)
tff(fact_5337_or__minus__numerals_I3_J,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(Na)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,bitM(Na)))) ).

% or_minus_numerals(3)
tff(fact_5338_or__minus__numerals_I7_J,axiom,
    ! [Na: num,M: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(Na)))),aa(num,int,numeral_numeral(int),M)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,bitM(Na)))) ).

% or_minus_numerals(7)
tff(fact_5339_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),product_case_prod(A,A,product_prod(A,A),aTP_Lamp_kc(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M,Na)) ) ).

% divmod_algorithm_code(5)
tff(fact_5340_or__not__num__neg_Osimps_I1_J,axiom,
    bit_or_not_num_neg(one2,one2) = one2 ).

% or_not_num_neg.simps(1)
tff(fact_5341_nested__case__prod__simp,axiom,
    ! [D: $tType,A: $tType,C: $tType,B: $tType,F2: fun(B,fun(C,fun(D,A))),X: product_prod(B,C),Y: D] : aa(D,A,aa(product_prod(B,C),fun(D,A),product_case_prod(B,C,fun(D,A),F2),X),Y) = aa(product_prod(B,C),A,product_case_prod(B,C,A,aa(D,fun(B,fun(C,A)),aTP_Lamp_kd(fun(B,fun(C,fun(D,A))),fun(D,fun(B,fun(C,A))),F2),Y)),X) ).

% nested_case_prod_simp
tff(fact_5342_or__not__num__neg_Osimps_I4_J,axiom,
    ! [Na: num] : bit_or_not_num_neg(bit0(Na),one2) = bit0(one2) ).

% or_not_num_neg.simps(4)
tff(fact_5343_or__not__num__neg_Osimps_I6_J,axiom,
    ! [Na: num,M: num] : bit_or_not_num_neg(bit0(Na),bit1(M)) = bit0(bit_or_not_num_neg(Na,M)) ).

% or_not_num_neg.simps(6)
tff(fact_5344_or__not__num__neg_Osimps_I7_J,axiom,
    ! [Na: num] : bit_or_not_num_neg(bit1(Na),one2) = one2 ).

% or_not_num_neg.simps(7)
tff(fact_5345_or__not__num__neg_Osimps_I3_J,axiom,
    ! [M: num] : bit_or_not_num_neg(one2,bit1(M)) = bit1(M) ).

% or_not_num_neg.simps(3)
tff(fact_5346_or__not__num__neg_Osimps_I5_J,axiom,
    ! [Na: num,M: num] : bit_or_not_num_neg(bit0(Na),bit0(M)) = bitM(bit_or_not_num_neg(Na,M)) ).

% or_not_num_neg.simps(5)
tff(fact_5347_or__not__num__neg_Osimps_I2_J,axiom,
    ! [M: num] : bit_or_not_num_neg(one2,bit0(M)) = bit1(M) ).

% or_not_num_neg.simps(2)
tff(fact_5348_or__not__num__neg_Osimps_I8_J,axiom,
    ! [Na: num,M: num] : bit_or_not_num_neg(bit1(Na),bit0(M)) = bitM(bit_or_not_num_neg(Na,M)) ).

% or_not_num_neg.simps(8)
tff(fact_5349_or__not__num__neg_Oelims,axiom,
    ! [X: num,Xa2: num,Y: num] :
      ( ( bit_or_not_num_neg(X,Xa2) = Y )
     => ( ( ( X = one2 )
         => ( ( Xa2 = one2 )
           => ( Y != one2 ) ) )
       => ( ( ( X = one2 )
           => ! [M4: num] :
                ( ( Xa2 = bit0(M4) )
               => ( Y != bit1(M4) ) ) )
         => ( ( ( X = one2 )
             => ! [M4: num] :
                  ( ( Xa2 = bit1(M4) )
                 => ( Y != bit1(M4) ) ) )
           => ( ( ? [N: num] : X = bit0(N)
               => ( ( Xa2 = one2 )
                 => ( Y != bit0(one2) ) ) )
             => ( ! [N: num] :
                    ( ( X = bit0(N) )
                   => ! [M4: num] :
                        ( ( Xa2 = bit0(M4) )
                       => ( Y != bitM(bit_or_not_num_neg(N,M4)) ) ) )
               => ( ! [N: num] :
                      ( ( X = bit0(N) )
                     => ! [M4: num] :
                          ( ( Xa2 = bit1(M4) )
                         => ( Y != bit0(bit_or_not_num_neg(N,M4)) ) ) )
                 => ( ( ? [N: num] : X = bit1(N)
                     => ( ( Xa2 = one2 )
                       => ( Y != one2 ) ) )
                   => ( ! [N: num] :
                          ( ( X = bit1(N) )
                         => ! [M4: num] :
                              ( ( Xa2 = bit0(M4) )
                             => ( Y != bitM(bit_or_not_num_neg(N,M4)) ) ) )
                     => ~ ! [N: num] :
                            ( ( X = bit1(N) )
                           => ! [M4: num] :
                                ( ( Xa2 = bit1(M4) )
                               => ( Y != bitM(bit_or_not_num_neg(N,M4)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
tff(fact_5350_int__numeral__or__not__num__neg,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Na))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,Na))) ).

% int_numeral_or_not_num_neg
tff(fact_5351_int__numeral__not__or__num__neg,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),Na)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(Na,M))) ).

% int_numeral_not_or_num_neg
tff(fact_5352_numeral__or__not__num__eq,axiom,
    ! [M: num,Na: num] : aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,Na)) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Na)))) ).

% numeral_or_not_num_eq
tff(fact_5353_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),product_case_prod(nat,nat,product_prod(nat,nat),aTP_Lamp_ke(num,fun(nat,fun(nat,product_prod(nat,nat))),L)),Qr) ).

% divmod_step_nat_def
tff(fact_5354_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),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_kf(num,fun(int,fun(int,product_prod(int,int))),L)),Qr) ).

% divmod_step_int_def
tff(fact_5355_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),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer),aTP_Lamp_kg(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),Qr) ).

% divmod_step_integer_def
tff(fact_5356_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),product_case_prod(A,A,product_prod(A,A),aTP_Lamp_kh(num,fun(A,fun(A,product_prod(A,A))),L)),Qr) ) ).

% divmod_step_def
tff(fact_5357_case__prod__conv,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(B,fun(C,A)),A3: B,B3: C] : aa(product_prod(B,C),A,product_case_prod(B,C,A,F2),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B3)) = aa(C,A,aa(B,fun(C,A),F2,A3),B3) ).

% case_prod_conv
tff(fact_5358_divmod__nat__if,axiom,
    ! [M: nat,Na: nat] :
      divmod_nat(M,Na) = $ite(
        ( ( Na = zero_zero(nat) )
        | aa(nat,$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),product_case_prod(nat,nat,product_prod(nat,nat),aTP_Lamp_ki(nat,fun(nat,product_prod(nat,nat)))),divmod_nat(aa(nat,nat,minus_minus(nat,M),Na),Na)) ) ).

% divmod_nat_if
tff(fact_5359_hash__code__prod__simps,axiom,
    ! [A: $tType,B: $tType,H_a: fun(A,uint32),H_b: fun(B,uint32),X: A,Xa2: 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),X),Xa2)) = aa(uint32,uint32,plus_plus(uint32,aa(uint32,uint32,aa(uint32,fun(uint32,uint32),times_times(uint32),aa(A,uint32,H_a,X)),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,plus_plus(uint32,aa(uint32,uint32,aa(uint32,fun(uint32,uint32),times_times(uint32),aa(B,uint32,H_b,Xa2)),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_5360_Eps__case__prod__eq,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B] : fChoice(product_prod(A,B),product_case_prod(A,B,$o,aa(B,fun(A,fun(B,$o)),aTP_Lamp_kj(A,fun(B,fun(A,fun(B,$o))),X),Y))) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y) ).

% Eps_case_prod_eq
tff(fact_5361_The__split__eq,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B] : the(product_prod(A,B),product_case_prod(A,B,$o,aa(B,fun(A,fun(B,$o)),aTP_Lamp_kj(A,fun(B,fun(A,fun(B,$o))),X),Y))) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y) ).

% The_split_eq
tff(fact_5362_mem__case__prodI2,axiom,
    ! [C: $tType,B: $tType,A: $tType,P2: product_prod(A,B),Z: C,C2: fun(A,fun(B,set(C)))] :
      ( ! [A4: A,B5: B] :
          ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B5) )
         => member(C,Z,aa(B,set(C),aa(A,fun(B,set(C)),C2,A4),B5)) )
     => member(C,Z,aa(product_prod(A,B),set(C),product_case_prod(A,B,set(C),C2),P2)) ) ).

% mem_case_prodI2
tff(fact_5363_mem__case__prodI,axiom,
    ! [A: $tType,B: $tType,C: $tType,Z: A,C2: fun(B,fun(C,set(A))),A3: B,B3: C] :
      ( member(A,Z,aa(C,set(A),aa(B,fun(C,set(A)),C2,A3),B3))
     => member(A,Z,aa(product_prod(B,C),set(A),product_case_prod(B,C,set(A),C2),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B3))) ) ).

% mem_case_prodI
tff(fact_5364_case__prodI2_H,axiom,
    ! [A: $tType,B: $tType,C: $tType,P2: product_prod(A,B),C2: fun(A,fun(B,fun(C,$o))),X: C] :
      ( ! [A4: A,B5: B] :
          ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B5) = P2 )
         => aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),C2,A4),B5),X) )
     => aa(C,$o,aa(product_prod(A,B),fun(C,$o),product_case_prod(A,B,fun(C,$o),C2),P2),X) ) ).

% case_prodI2'
tff(fact_5365_case__prodI2,axiom,
    ! [B: $tType,A: $tType,P2: product_prod(A,B),C2: fun(A,fun(B,$o))] :
      ( ! [A4: A,B5: B] :
          ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B5) )
         => aa(B,$o,aa(A,fun(B,$o),C2,A4),B5) )
     => aa(product_prod(A,B),$o,product_case_prod(A,B,$o,C2),P2) ) ).

% case_prodI2
tff(fact_5366_case__prodI,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(B,$o)),A3: A,B3: B] :
      ( aa(B,$o,aa(A,fun(B,$o),F2,A3),B3)
     => aa(product_prod(A,B),$o,product_case_prod(A,B,$o,F2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3)) ) ).

% case_prodI
tff(fact_5367_split__paired__Eps,axiom,
    ! [B: $tType,A: $tType,P: fun(product_prod(A,B),$o)] : fChoice(product_prod(A,B),P) = fChoice(product_prod(A,B),product_case_prod(A,B,$o,aTP_Lamp_kk(fun(product_prod(A,B),$o),fun(A,fun(B,$o)),P))) ).

% split_paired_Eps
tff(fact_5368_case__prodE_H,axiom,
    ! [B: $tType,A: $tType,C: $tType,C2: fun(A,fun(B,fun(C,$o))),P2: product_prod(A,B),Z: C] :
      ( aa(C,$o,aa(product_prod(A,B),fun(C,$o),product_case_prod(A,B,fun(C,$o),C2),P2),Z)
     => ~ ! [X4: A,Y3: B] :
            ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3) )
           => ~ aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),C2,X4),Y3),Z) ) ) ).

% case_prodE'
tff(fact_5369_case__prodD_H,axiom,
    ! [B: $tType,A: $tType,C: $tType,R: fun(A,fun(B,fun(C,$o))),A3: A,B3: B,C2: C] :
      ( aa(C,$o,aa(product_prod(A,B),fun(C,$o),product_case_prod(A,B,fun(C,$o),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3)),C2)
     => aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),R,A3),B3),C2) ) ).

% case_prodD'
tff(fact_5370_case__prodE,axiom,
    ! [A: $tType,B: $tType,C2: fun(A,fun(B,$o)),P2: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,product_case_prod(A,B,$o,C2),P2)
     => ~ ! [X4: A,Y3: B] :
            ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3) )
           => ~ aa(B,$o,aa(A,fun(B,$o),C2,X4),Y3) ) ) ).

% case_prodE
tff(fact_5371_case__prodD,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(B,$o)),A3: A,B3: B] :
      ( aa(product_prod(A,B),$o,product_case_prod(A,B,$o,F2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3))
     => aa(B,$o,aa(A,fun(B,$o),F2,A3),B3) ) ).

% case_prodD
tff(fact_5372_TBOUND__prod__case,axiom,
    ! [C: $tType,B: $tType,A: $tType,Ta: product_prod(A,B),F2: fun(A,fun(B,heap_Time_Heap(C))),Bnd: fun(A,fun(B,nat))] :
      ( ! [A4: A,B5: B] :
          ( ( Ta = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B5) )
         => time_TBOUND(C,aa(B,heap_Time_Heap(C),aa(A,fun(B,heap_Time_Heap(C)),F2,A4),B5),aa(B,nat,aa(A,fun(B,nat),Bnd,A4),B5)) )
     => time_TBOUND(C,aa(product_prod(A,B),heap_Time_Heap(C),product_case_prod(A,B,heap_Time_Heap(C),F2),Ta),aa(product_prod(A,B),nat,product_case_prod(A,B,nat,Bnd),Ta)) ) ).

% TBOUND_prod_case
tff(fact_5373_sum_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,fun(nat,A)),Na: nat] : aa(set(product_prod(nat,nat)),A,groups7311177749621191930dd_sum(product_prod(nat,nat),A,product_case_prod(nat,nat,A,G)),collect(product_prod(nat,nat),product_case_prod(nat,nat,$o,aTP_Lamp_kl(nat,fun(nat,fun(nat,$o)),Na)))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_kn(fun(nat,fun(nat,A)),fun(nat,A),G)),set_ord_atMost(nat,Na)) ) ).

% sum.triangle_reindex_eq
tff(fact_5374_case__prod__rule,axiom,
    ! [A: $tType,B: $tType,C: $tType,X: product_prod(A,B),P: assn,F2: fun(A,fun(B,heap_Time_Heap(C))),Q: fun(C,assn)] :
      ( ! [A4: A,B5: B] :
          ( ( X = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B5) )
         => hoare_hoare_triple(C,P,aa(B,heap_Time_Heap(C),aa(A,fun(B,heap_Time_Heap(C)),F2,A4),B5),Q) )
     => hoare_hoare_triple(C,P,aa(product_prod(A,B),heap_Time_Heap(C),product_case_prod(A,B,heap_Time_Heap(C),F2),X),Q) ) ).

% case_prod_rule
tff(fact_5375_prod_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,fun(nat,A)),Na: nat] : groups7121269368397514597t_prod(product_prod(nat,nat),A,product_case_prod(nat,nat,A,G),collect(product_prod(nat,nat),product_case_prod(nat,nat,$o,aTP_Lamp_kl(nat,fun(nat,fun(nat,$o)),Na)))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_kp(fun(nat,fun(nat,A)),fun(nat,A),G),set_ord_atMost(nat,Na)) ) ).

% prod.triangle_reindex_eq
tff(fact_5376_sum_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,fun(nat,A)),Na: nat] : aa(set(product_prod(nat,nat)),A,groups7311177749621191930dd_sum(product_prod(nat,nat),A,product_case_prod(nat,nat,A,G)),collect(product_prod(nat,nat),product_case_prod(nat,nat,$o,aTP_Lamp_kq(nat,fun(nat,fun(nat,$o)),Na)))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_kn(fun(nat,fun(nat,A)),fun(nat,A),G)),set_ord_lessThan(nat,Na)) ) ).

% sum.triangle_reindex
tff(fact_5377_prod_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,fun(nat,A)),Na: nat] : groups7121269368397514597t_prod(product_prod(nat,nat),A,product_case_prod(nat,nat,A,G),collect(product_prod(nat,nat),product_case_prod(nat,nat,$o,aTP_Lamp_kq(nat,fun(nat,fun(nat,$o)),Na)))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_kp(fun(nat,fun(nat,A)),fun(nat,A),G),set_ord_lessThan(nat,Na)) ) ).

% prod.triangle_reindex
tff(fact_5378_prod_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,H: fun(B,A),F2: fun(C,fun(D,B)),Prod: product_prod(C,D)] : aa(B,A,H,aa(product_prod(C,D),B,product_case_prod(C,D,B,F2),Prod)) = aa(product_prod(C,D),A,product_case_prod(C,D,A,aa(fun(C,fun(D,B)),fun(C,fun(D,A)),aTP_Lamp_kr(fun(B,A),fun(fun(C,fun(D,B)),fun(C,fun(D,A))),H),F2)),Prod) ).

% prod.case_distrib
tff(fact_5379_Divides_Oadjust__div__def,axiom,
    ! [Qr: product_prod(int,int)] : adjust_div(Qr) = aa(product_prod(int,int),int,product_case_prod(int,int,int,aTP_Lamp_ks(int,fun(int,int))),Qr) ).

% Divides.adjust_div_def
tff(fact_5380_case__prodE2,axiom,
    ! [B: $tType,A: $tType,C: $tType,Q: fun(A,$o),P: fun(B,fun(C,A)),Z: product_prod(B,C)] :
      ( aa(A,$o,Q,aa(product_prod(B,C),A,product_case_prod(B,C,A,P),Z))
     => ~ ! [X4: B,Y3: C] :
            ( ( Z = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X4),Y3) )
           => ~ aa(A,$o,Q,aa(C,A,aa(B,fun(C,A),P,X4),Y3)) ) ) ).

% case_prodE2
tff(fact_5381_case__prod__eta,axiom,
    ! [C: $tType,B: $tType,A: $tType,F2: fun(product_prod(A,B),C)] : product_case_prod(A,B,C,aTP_Lamp_kt(fun(product_prod(A,B),C),fun(A,fun(B,C)),F2)) = F2 ).

% case_prod_eta
tff(fact_5382_cond__case__prod__eta,axiom,
    ! [C: $tType,B: $tType,A: $tType,F2: fun(A,fun(B,C)),G: fun(product_prod(A,B),C)] :
      ( ! [X4: A,Y3: B] : aa(B,C,aa(A,fun(B,C),F2,X4),Y3) = aa(product_prod(A,B),C,G,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3))
     => ( product_case_prod(A,B,C,F2) = G ) ) ).

% cond_case_prod_eta
tff(fact_5383_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,product_case_prod(nat,nat,A,aTP_Lamp_ku(nat,fun(nat,A))),divmod_nat(Na,aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% of_nat_code_if
tff(fact_5384_int__ge__less__than__def,axiom,
    ! [D2: int] : int_ge_less_than(D2) = collect(product_prod(int,int),product_case_prod(int,int,$o,aTP_Lamp_kv(int,fun(int,fun(int,$o)),D2))) ).

% int_ge_less_than_def
tff(fact_5385_int__ge__less__than2__def,axiom,
    ! [D2: int] : int_ge_less_than2(D2) = collect(product_prod(int,int),product_case_prod(int,int,$o,aTP_Lamp_kw(int,fun(int,fun(int,$o)),D2))) ).

% int_ge_less_than2_def
tff(fact_5386_split__part,axiom,
    ! [B: $tType,A: $tType,P: $o,Q: fun(A,fun(B,$o)),X2: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,product_case_prod(A,B,$o,aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_kx($o,fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),(P)),Q)),X2)
    <=> ( (P)
        & aa(product_prod(A,B),$o,product_case_prod(A,B,$o,Q),X2) ) ) ).

% split_part
tff(fact_5387_prod_Odisc__eq__case,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod(A,B)] : aa(product_prod(A,B),$o,product_case_prod(A,B,$o,aTP_Lamp_ky(A,fun(B,$o))),Prod) ).

% prod.disc_eq_case
tff(fact_5388_dup__1,axiom,
    code_dup(one_one(code_integer)) = aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)) ).

% dup_1
tff(fact_5389_rat__inverse__code,axiom,
    ! [P2: rat] : quotient_of(aa(rat,rat,inverse_inverse(rat),P2)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_kz(int,fun(int,product_prod(int,int)))),quotient_of(P2)) ).

% rat_inverse_code
tff(fact_5390_bin__rest__integer_Oabs__eq,axiom,
    ! [X: int] : bits_b2549910563261871055nteger(code_integer_of_int(X)) = code_integer_of_int(aa(int,int,aa(int,fun(int,int),divide_divide(int),X),aa(num,int,numeral_numeral(int),bit0(one2)))) ).

% bin_rest_integer.abs_eq
tff(fact_5391_quotient__of__number_I3_J,axiom,
    ! [K: num] : quotient_of(aa(num,rat,numeral_numeral(rat),K)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(num,int,numeral_numeral(int),K)),one_one(int)) ).

% quotient_of_number(3)
tff(fact_5392_rat__one__code,axiom,
    quotient_of(one_one(rat)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int)) ).

% rat_one_code
tff(fact_5393_rat__zero__code,axiom,
    quotient_of(zero_zero(rat)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) ).

% rat_zero_code
tff(fact_5394_quotient__of__number_I5_J,axiom,
    ! [K: num] : quotient_of(aa(rat,rat,uminus_uminus(rat),aa(num,rat,numeral_numeral(rat),K))) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))),one_one(int)) ).

% quotient_of_number(5)
tff(fact_5395_quotient__of__number_I4_J,axiom,
    quotient_of(aa(rat,rat,uminus_uminus(rat),one_one(rat))) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),one_one(int))),one_one(int)) ).

% quotient_of_number(4)
tff(fact_5396_Code__Numeral_Odup__code_I1_J,axiom,
    code_dup(zero_zero(code_integer)) = zero_zero(code_integer) ).

% Code_Numeral.dup_code(1)
tff(fact_5397_quotient__of__denom__pos,axiom,
    ! [R3: rat,P2: int,Q3: int] :
      ( ( quotient_of(R3) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q3) )
     => aa(int,$o,ord_less(int,zero_zero(int)),Q3) ) ).

% quotient_of_denom_pos
tff(fact_5398_rat__uminus__code,axiom,
    ! [P2: rat] : quotient_of(aa(rat,rat,uminus_uminus(rat),P2)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_la(int,fun(int,product_prod(int,int)))),quotient_of(P2)) ).

% rat_uminus_code
tff(fact_5399_rat__less__code,axiom,
    ! [P2: rat,Q3: rat] :
      ( aa(rat,$o,ord_less(rat,P2),Q3)
    <=> aa(product_prod(int,int),$o,product_case_prod(int,int,$o,aTP_Lamp_lc(rat,fun(int,fun(int,$o)),Q3)),quotient_of(P2)) ) ).

% rat_less_code
tff(fact_5400_rat__floor__code,axiom,
    ! [P2: rat] : archim6421214686448440834_floor(rat,P2) = aa(product_prod(int,int),int,product_case_prod(int,int,int,divide_divide(int)),quotient_of(P2)) ).

% rat_floor_code
tff(fact_5401_rat__abs__code,axiom,
    ! [P2: rat] : quotient_of(aa(rat,rat,abs_abs(rat),P2)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_ld(int,fun(int,product_prod(int,int)))),quotient_of(P2)) ).

% rat_abs_code
tff(fact_5402_rat__less__eq__code,axiom,
    ! [P2: rat,Q3: rat] :
      ( aa(rat,$o,ord_less_eq(rat,P2),Q3)
    <=> aa(product_prod(int,int),$o,product_case_prod(int,int,$o,aTP_Lamp_lf(rat,fun(int,fun(int,$o)),Q3)),quotient_of(P2)) ) ).

% rat_less_eq_code
tff(fact_5403_bin__rest__integer__code,axiom,
    ! [I: code_integer] : bits_b2549910563261871055nteger(I) = 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))) ).

% bin_rest_integer_code
tff(fact_5404_quotient__of__int,axiom,
    ! [A3: int] : quotient_of(of_int(A3)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A3),one_one(int)) ).

% quotient_of_int
tff(fact_5405_bitXOR__integer__unfold,axiom,
    ! [X: code_integer,Y: code_integer] :
      aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344971392196577ns_xor(code_integer),X),Y) = $ite(
        X = zero_zero(code_integer),
        Y,
        $ite(
          X = aa(code_integer,code_integer,uminus_uminus(code_integer),one_one(code_integer)),
          aa(code_integer,code_integer,bit_ri4277139882892585799ns_not(code_integer),Y),
          bits_Bit_integer(aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344971392196577ns_xor(code_integer),bits_b2549910563261871055nteger(X)),bits_b2549910563261871055nteger(Y)),
            ~ bits_b8758750999018896077nteger(X) = bits_b8758750999018896077nteger(Y)) ) ) ).

% bitXOR_integer_unfold
tff(fact_5406_rat__plus__code,axiom,
    ! [P2: rat,Q3: rat] : quotient_of(aa(rat,rat,plus_plus(rat,P2),Q3)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_lh(rat,fun(int,fun(int,product_prod(int,int))),Q3)),quotient_of(P2)) ).

% rat_plus_code
tff(fact_5407_normalize__denom__zero,axiom,
    ! [P2: int] : normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),zero_zero(int))) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) ).

% normalize_denom_zero
tff(fact_5408_normalize__negative,axiom,
    ! [Q3: int,P2: int] :
      ( aa(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),P2),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),P2)),aa(int,int,uminus_uminus(int),Q3))) ) ) ).

% normalize_negative
tff(fact_5409_bitval__bin__last__integer,axiom,
    ! [I: code_integer] : aa($o,code_integer,zero_neq_one_of_bool(code_integer),bits_b8758750999018896077nteger(I)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344872417868541ns_and(code_integer),I),one_one(code_integer)) ).

% bitval_bin_last_integer
tff(fact_5410_bin__last__integer__code,axiom,
    ! [I: code_integer] :
      ( bits_b8758750999018896077nteger(I)
    <=> ( aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344872417868541ns_and(code_integer),I),one_one(code_integer)) != zero_zero(code_integer) ) ) ).

% bin_last_integer_code
tff(fact_5411_normalize__denom__pos,axiom,
    ! [R3: product_prod(int,int),P2: int,Q3: int] :
      ( ( normalize(R3) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q3) )
     => aa(int,$o,ord_less(int,zero_zero(int)),Q3) ) ).

% normalize_denom_pos
tff(fact_5412_normalize__crossproduct,axiom,
    ! [Q3: int,Sa: int,P2: int,R3: int] :
      ( ( Q3 != zero_zero(int) )
     => ( ( Sa != zero_zero(int) )
       => ( ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q3)) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),R3),Sa)) )
         => ( aa(int,int,aa(int,fun(int,int),times_times(int),P2),Sa) = aa(int,int,aa(int,fun(int,int),times_times(int),R3),Q3) ) ) ) ) ).

% normalize_crossproduct
tff(fact_5413_bin__last__integer__nbe,axiom,
    ! [I: code_integer] :
      ( bits_b8758750999018896077nteger(I)
    <=> ( 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) ) ) ).

% bin_last_integer_nbe
tff(fact_5414_rat__divide__code,axiom,
    ! [P2: rat,Q3: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),P2),Q3)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_lj(rat,fun(int,fun(int,product_prod(int,int))),Q3)),quotient_of(P2)) ).

% rat_divide_code
tff(fact_5415_rat__times__code,axiom,
    ! [P2: rat,Q3: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),P2),Q3)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_ll(rat,fun(int,fun(int,product_prod(int,int))),Q3)),quotient_of(P2)) ).

% rat_times_code
tff(fact_5416_bin__last__integer_Oabs__eq,axiom,
    ! [X: int] :
      ( bits_b8758750999018896077nteger(code_integer_of_int(X))
    <=> ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),X) ) ).

% bin_last_integer.abs_eq
tff(fact_5417_rat__minus__code,axiom,
    ! [P2: rat,Q3: rat] : quotient_of(aa(rat,rat,minus_minus(rat,P2),Q3)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_ln(rat,fun(int,fun(int,product_prod(int,int))),Q3)),quotient_of(P2)) ).

% rat_minus_code
tff(fact_5418_bitOR__integer__unfold,axiom,
    ! [X: code_integer,Y: code_integer] :
      aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se1065995026697491101ons_or(code_integer),X),Y) = $ite(
        X = zero_zero(code_integer),
        Y,
        $ite(
          X = aa(code_integer,code_integer,uminus_uminus(code_integer),one_one(code_integer)),
          aa(code_integer,code_integer,uminus_uminus(code_integer),one_one(code_integer)),
          bits_Bit_integer(aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se1065995026697491101ons_or(code_integer),bits_b2549910563261871055nteger(X)),bits_b2549910563261871055nteger(Y)),
            ( bits_b8758750999018896077nteger(X)
            | bits_b8758750999018896077nteger(Y) )) ) ) ).

% bitOR_integer_unfold
tff(fact_5419_bitAND__integer__unfold,axiom,
    ! [X: code_integer,Y: code_integer] :
      aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344872417868541ns_and(code_integer),X),Y) = $ite(
        X = zero_zero(code_integer),
        zero_zero(code_integer),
        $ite(
          X = aa(code_integer,code_integer,uminus_uminus(code_integer),one_one(code_integer)),
          Y,
          bits_Bit_integer(aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344872417868541ns_and(code_integer),bits_b2549910563261871055nteger(X)),bits_b2549910563261871055nteger(Y)),
            ( bits_b8758750999018896077nteger(X)
            & bits_b8758750999018896077nteger(Y) )) ) ) ).

% bitAND_integer_unfold
tff(fact_5420_Frct__code__post_I5_J,axiom,
    ! [K: num] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),aa(num,int,numeral_numeral(int),K))) = aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),one_one(rat)),aa(num,rat,numeral_numeral(rat),K)) ).

% Frct_code_post(5)
tff(fact_5421_case__prod__app,axiom,
    ! [D: $tType,A: $tType,C: $tType,B: $tType,F2: fun(B,fun(C,fun(D,A))),X: product_prod(B,C),Y: D] : aa(D,A,aa(product_prod(B,C),fun(D,A),product_case_prod(B,C,fun(D,A),F2),X),Y) = aa(product_prod(B,C),A,product_case_prod(B,C,A,aa(D,fun(B,fun(C,A)),aTP_Lamp_kd(fun(B,fun(C,fun(D,A))),fun(D,fun(B,fun(C,A))),F2),Y)),X) ).

% case_prod_app
tff(fact_5422_case__prod__Pair__iden,axiom,
    ! [B: $tType,A: $tType,P2: product_prod(A,B)] : aa(product_prod(A,B),product_prod(A,B),product_case_prod(A,B,product_prod(A,B),product_Pair(A,B)),P2) = P2 ).

% case_prod_Pair_iden
tff(fact_5423_fun__cong__unused__0,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( zero(B)
     => ! [F2: fun(fun(A,B),C),G: C] :
          ( ! [X4: fun(A,B)] : aa(fun(A,B),C,F2,X4) = G
         => ( aa(fun(A,B),C,F2,aTP_Lamp_lo(A,B)) = G ) ) ) ).

% fun_cong_unused_0
tff(fact_5424_eq__subset,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o))] : aa(fun(A,fun(A,$o)),$o,ord_less_eq(fun(A,fun(A,$o)),fequal(A)),aTP_Lamp_lp(fun(A,fun(A,$o)),fun(A,fun(A,$o)),P)) ).

% eq_subset
tff(fact_5425_Frct__code__post_I2_J,axiom,
    ! [A3: int] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A3),zero_zero(int))) = zero_zero(rat) ).

% Frct_code_post(2)
tff(fact_5426_Frct__code__post_I1_J,axiom,
    ! [A3: int] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),A3)) = zero_zero(rat) ).

% Frct_code_post(1)
tff(fact_5427_Frct__code__post_I3_J,axiom,
    frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int))) = one_one(rat) ).

% Frct_code_post(3)
tff(fact_5428_Frct__code__post_I6_J,axiom,
    ! [K: num,L: num] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(num,int,numeral_numeral(int),K)),aa(num,int,numeral_numeral(int),L))) = aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),aa(num,rat,numeral_numeral(rat),K)),aa(num,rat,numeral_numeral(rat),L)) ).

% Frct_code_post(6)
tff(fact_5429_Frct__code__post_I4_J,axiom,
    ! [K: num] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(num,int,numeral_numeral(int),K)),one_one(int))) = aa(num,rat,numeral_numeral(rat),K) ).

% Frct_code_post(4)
tff(fact_5430_set__bit__integer__conv__masks,axiom,
    ! [X: code_integer,I: nat,B3: $o] :
      aa($o,code_integer,aa(nat,fun($o,code_integer),aa(code_integer,fun(nat,fun($o,code_integer)),generi7602027413899671122et_bit(code_integer),X),I),(B3)) = $ite((B3),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se1065995026697491101ons_or(code_integer),X),aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se4730199178511100633sh_bit(code_integer),I),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344872417868541ns_and(code_integer),X),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),I),one_one(code_integer))))) ).

% set_bit_integer_conv_masks
tff(fact_5431_Uint32__code,axiom,
    ! [I: code_integer] :
      aa(code_integer,uint32,uint322,I) = $let(
        i: code_integer,
        i:= aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344872417868541ns_and(code_integer),I),aa(num,code_integer,numeral_numeral(code_integer),bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(one2))))))))))))))))))))))))))))))))),
        $ite(aa(nat,$o,aa(code_integer,fun(nat,$o),bit_se5641148757651400278ts_bit(code_integer),i),aa(num,nat,numeral_numeral(nat),bit1(bit1(bit1(bit1(one2)))))),uint32_signed(aa(code_integer,code_integer,minus_minus(code_integer,i),aa(num,code_integer,numeral_numeral(code_integer),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(bit0(one2))))))))))))))))))))))))))))))))))),uint32_signed(i)) ) ).

% Uint32_code
tff(fact_5432_word__cat__split__alt,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B)
        & type_len(C) )
     => ! [W: word(A),U: word(B),V2: word(C)] :
          ( aa(nat,$o,ord_less_eq(nat,size_size(word(A),W)),aa(nat,nat,plus_plus(nat,size_size(word(B),U)),size_size(word(C),V2)))
         => ( ( word_split(A,B,C,W) = 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),V2) )
           => ( word_cat(B,C,A,U,V2) = W ) ) ) ) ).

% word_cat_split_alt
tff(fact_5433_word__cat__id,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [A3: word(B),B3: word(A)] : word_cat(B,A,A,A3,B3) = B3 ) ).

% word_cat_id
tff(fact_5434_test__bit__cat,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(B)
        & type_len(A) )
     => ! [A3: word(B),B3: word(C),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),word_cat(B,C,A,A3,B3)),Na)
        <=> ( aa(nat,$o,ord_less(nat,Na),size_size(word(A),word_cat(B,C,A,A3,B3)))
            & $ite(aa(nat,$o,ord_less(nat,Na),size_size(word(C),B3)),aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),B3),Na),aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),A3),aa(nat,nat,minus_minus(nat,Na),size_size(word(C),B3)))) ) ) ) ).

% test_bit_cat
tff(fact_5435_word__cat__split__size,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(A)
        & type_len(B) )
     => ! [Ta: word(A),U: word(B),V2: word(C)] :
          ( aa(nat,$o,ord_less_eq(nat,size_size(word(A),Ta)),aa(nat,nat,plus_plus(nat,size_size(word(B),U)),size_size(word(C),V2)))
         => ( ( 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),V2) = word_split(A,B,C,Ta) )
           => ( Ta = word_cat(B,C,A,U,V2) ) ) ) ) ).

% word_cat_split_size
tff(fact_5436_word__split__cat__alt,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(B)
        & type_len(A) )
     => ! [W: word(A),U: word(B),V2: word(C)] :
          ( ( W = word_cat(B,C,A,U,V2) )
         => ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,plus_plus(nat,size_size(word(B),U)),size_size(word(C),V2))),size_size(word(A),W))
           => ( word_split(A,B,C,W) = 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),V2) ) ) ) ) ).

% word_split_cat_alt
tff(fact_5437_bit__set__bit__iff__2n,axiom,
    ! [A: $tType] :
      ( generic_set_set_bit(A)
     => ! [A3: A,M: nat,B3: $o,Na: nat] :
          ( aa(nat,$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),A3),M),(B3))),Na)
        <=> ( $ite(M = Na,(B3),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),Na))
            & ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na) != zero_zero(A) ) ) ) ) ).

% bit_set_bit_iff_2n
tff(fact_5438_Uint32__signed__def,axiom,
    ! [I: code_integer] :
      uint32_signed(I) = $ite(
        ( aa(code_integer,$o,ord_less(code_integer,I),aa(code_integer,code_integer,uminus_uminus(code_integer),aa(num,code_integer,numeral_numeral(code_integer),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))))))))))))))))))))))))))))))))))
        | aa(code_integer,$o,ord_less_eq(code_integer,aa(num,code_integer,numeral_numeral(code_integer),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))))))))))))))))))))))))))))))))),I) ),
        aa(code_integer,uint32,aa(fun(code_integer,uint32),fun(code_integer,uint32),undefined(fun(fun(code_integer,uint32),fun(code_integer,uint32))),uint322),I),
        aa(code_integer,uint32,uint322,I) ) ).

% Uint32_signed_def
tff(fact_5439_cat__slices,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A)
        & type_len(C) )
     => ! [A3: word(A),Na: nat,C2: word(B),B3: word(C)] :
          ( ( A3 = aa(word(B),word(A),slice2(B,A,Na),C2) )
         => ( ( B3 = aa(word(B),word(C),slice2(B,C,zero_zero(nat)),C2) )
           => ( ( Na = size_size(word(C),B3) )
             => ( aa(nat,$o,ord_less_eq(nat,size_size(word(B),C2)),aa(nat,nat,plus_plus(nat,size_size(word(A),A3)),size_size(word(C),B3)))
               => ( word_cat(A,C,B,A3,B3) = C2 ) ) ) ) ) ) ).

% cat_slices
tff(fact_5440_slice__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Na: nat] : aa(word(B),word(A),slice2(B,A,Na),zero_zero(word(B))) = zero_zero(word(A)) ) ).

% slice_0
tff(fact_5441_one__bit__shiftl,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] : aa($o,word(A),aa(nat,fun($o,word(A)),aa(word(A),fun(nat,fun($o,word(A))),generi7602027413899671122et_bit(word(A)),zero_zero(word(A))),Na),$true) = bit_Sh4282982442137083160shiftl(word(A),one_one(word(A)),Na) ) ).

% one_bit_shiftl
tff(fact_5442_slice__id,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Ta: word(A)] : aa(word(A),word(A),slice2(A,A,zero_zero(nat)),Ta) = Ta ) ).

% slice_id
tff(fact_5443_test__bit__set__gen,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Na: nat,X: $o,M: nat] :
          ( aa(nat,$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)),W),Na),(X))),M)
        <=> $ite(
              M = Na,
              ( aa(nat,$o,ord_less(nat,Na),size_size(word(A),W))
              & (X) ),
              aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),M) ) ) ) ).

% test_bit_set_gen
tff(fact_5444_test__bit__set,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Na: nat,X: $o] :
          ( aa(nat,$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)),W),Na),(X))),Na)
        <=> ( aa(nat,$o,ord_less(nat,Na),size_size(word(A),W))
            & (X) ) ) ) ).

% test_bit_set
tff(fact_5445_ucast__slice,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [W: word(B)] : aa(word(B),word(A),semiring_1_unsigned(B,word(A)),W) = aa(word(B),word(A),slice2(B,A,zero_zero(nat)),W) ) ).

% ucast_slice
tff(fact_5446_slice__cat2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [A3: word(B),Ta: word(A)] : aa(word(A),word(A),slice2(A,A,zero_zero(nat)),word_cat(B,A,A,A3,Ta)) = Ta ) ).

% slice_cat2
tff(fact_5447_one__bit__pow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] : aa($o,word(A),aa(nat,fun($o,word(A)),aa(word(A),fun(nat,fun($o,word(A))),generi7602027413899671122et_bit(word(A)),zero_zero(word(A))),Na),$true) = aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na) ) ).

% one_bit_pow
tff(fact_5448_int__set__bit__True__conv__OR,axiom,
    ! [I: int,Na: nat] : aa($o,int,aa(nat,fun($o,int),aa(int,fun(nat,fun($o,int)),generi7602027413899671122et_bit(int),I),Na),$true) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),I),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),Na),one_one(int))) ).

% int_set_bit_True_conv_OR
tff(fact_5449_int__set__bit__False__conv__NAND,axiom,
    ! [I: int,Na: nat] : aa($o,int,aa(nat,fun($o,int),aa(int,fun(nat,fun($o,int)),generi7602027413899671122et_bit(int),I),Na),$false) = aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),I),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),Na),one_one(int)))) ).

% int_set_bit_False_conv_NAND
tff(fact_5450_slice__cat1,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(C)
        & type_len(B) )
     => ! [A3: word(A),B3: word(B)] :
          ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,plus_plus(nat,size_size(word(A),A3)),size_size(word(B),B3))),size_size(word(C),word_cat(A,B,C,A3,B3)))
         => ( aa(word(C),word(A),slice2(C,A,size_size(word(B),B3)),word_cat(A,B,C,A3,B3)) = A3 ) ) ) ).

% slice_cat1
tff(fact_5451_split__slices,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(B)
        & type_len(A) )
     => ! [W: word(C),U: word(A),V2: word(B)] :
          ( ( word_split(C,A,B,W) = 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),V2) )
         => ( ( U = aa(word(C),word(A),slice2(C,A,size_size(word(B),V2)),W) )
            & ( V2 = aa(word(C),word(B),slice2(C,B,zero_zero(nat)),W) ) ) ) ) ).

% split_slices
tff(fact_5452_int__set__bit__conv__ops,axiom,
    ! [I: int,Na: nat,B3: $o] :
      aa($o,int,aa(nat,fun($o,int),aa(int,fun(nat,fun($o,int)),generi7602027413899671122et_bit(int),I),Na),(B3)) = $ite((B3),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),I),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),Na),one_one(int))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),I),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),Na),one_one(int))))) ).

% int_set_bit_conv_ops
tff(fact_5453_size__eq__0__iff__empty,axiom,
    ! [A: $tType,M7: multiset(A)] :
      ( ( size_size(multiset(A),M7) = zero_zero(nat) )
    <=> ( M7 = zero_zero(multiset(A)) ) ) ).

% size_eq_0_iff_empty
tff(fact_5454_size__empty,axiom,
    ! [A: $tType] : size_size(multiset(A),zero_zero(multiset(A))) = zero_zero(nat) ).

% size_empty
tff(fact_5455_set__encode__def,axiom,
    nat_set_encode = groups7311177749621191930dd_sum(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% set_encode_def
tff(fact_5456_union__eq__empty,axiom,
    ! [A: $tType,M7: multiset(A),N2: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),plus_plus(multiset(A),M7),N2) = zero_zero(multiset(A)) )
    <=> ( ( M7 = zero_zero(multiset(A)) )
        & ( N2 = zero_zero(multiset(A)) ) ) ) ).

% union_eq_empty
tff(fact_5457_empty__eq__union,axiom,
    ! [A: $tType,M7: multiset(A),N2: multiset(A)] :
      ( ( zero_zero(multiset(A)) = aa(multiset(A),multiset(A),plus_plus(multiset(A),M7),N2) )
    <=> ( ( M7 = zero_zero(multiset(A)) )
        & ( N2 = zero_zero(multiset(A)) ) ) ) ).

% empty_eq_union
tff(fact_5458_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
    ! [A: $tType,X: multiset(A),Y: multiset(A)] :
      ( ( zero_zero(multiset(A)) = aa(multiset(A),multiset(A),plus_plus(multiset(A),X),Y) )
    <=> ( ( X = zero_zero(multiset(A)) )
        & ( Y = zero_zero(multiset(A)) ) ) ) ).

% subset_mset.zero_eq_add_iff_both_eq_0
tff(fact_5459_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
    ! [A: $tType,X: multiset(A),Y: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),plus_plus(multiset(A),X),Y) = zero_zero(multiset(A)) )
    <=> ( ( X = zero_zero(multiset(A)) )
        & ( Y = zero_zero(multiset(A)) ) ) ) ).

% subset_mset.add_eq_0_iff_both_eq_0
tff(fact_5460_set__encode__empty,axiom,
    aa(set(nat),nat,nat_set_encode,bot_bot(set(nat))) = zero_zero(nat) ).

% set_encode_empty
tff(fact_5461_Multiset_Odiff__cancel,axiom,
    ! [A: $tType,A2: multiset(A)] : aa(multiset(A),multiset(A),minus_minus(multiset(A),A2),A2) = zero_zero(multiset(A)) ).

% Multiset.diff_cancel
tff(fact_5462_diff__empty,axiom,
    ! [A: $tType,M7: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),minus_minus(multiset(A),M7),zero_zero(multiset(A))) = M7 )
      & ( aa(multiset(A),multiset(A),minus_minus(multiset(A),zero_zero(multiset(A))),M7) = zero_zero(multiset(A)) ) ) ).

% diff_empty
tff(fact_5463_union__diff__assoc,axiom,
    ! [A: $tType,C3: multiset(A),B2: multiset(A),A2: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),minus_minus(multiset(A),C3),B2) = zero_zero(multiset(A)) )
     => ( aa(multiset(A),multiset(A),minus_minus(multiset(A),aa(multiset(A),multiset(A),plus_plus(multiset(A),A2),B2)),C3) = aa(multiset(A),multiset(A),plus_plus(multiset(A),A2),aa(multiset(A),multiset(A),minus_minus(multiset(A),B2),C3)) ) ) ).

% union_diff_assoc
tff(fact_5464_empty__neutral_I1_J,axiom,
    ! [A: $tType,X: multiset(A)] : aa(multiset(A),multiset(A),plus_plus(multiset(A),zero_zero(multiset(A))),X) = X ).

% empty_neutral(1)
tff(fact_5465_empty__neutral_I2_J,axiom,
    ! [A: $tType,X: multiset(A)] : aa(multiset(A),multiset(A),plus_plus(multiset(A),X),zero_zero(multiset(A))) = X ).

% empty_neutral(2)
tff(fact_5466_nonempty__has__size,axiom,
    ! [A: $tType,S: multiset(A)] :
      ( ( S != zero_zero(multiset(A)) )
    <=> aa(nat,$o,ord_less(nat,zero_zero(nat)),size_size(multiset(A),S)) ) ).

% nonempty_has_size
tff(fact_5467_set__encode__insert,axiom,
    ! [A2: set(nat),Na: nat] :
      ( finite_finite2(nat,A2)
     => ( ~ member(nat,Na,A2)
       => ( aa(set(nat),nat,nat_set_encode,aa(set(nat),set(nat),insert(nat,Na),A2)) = aa(nat,nat,plus_plus(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),aa(set(nat),nat,nat_set_encode,A2)) ) ) ) ).

% set_encode_insert
tff(fact_5468_sdiv__word__min,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(int,$o,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,size_size(word(A),A3)),one_one(nat))))),signed7115095781618012415divide(int,aa(word(A),int,ring_1_signed(A,int),A3),aa(word(A),int,ring_1_signed(A,int),B3))) ) ).

% sdiv_word_min
tff(fact_5469_Cauchy__iff2,axiom,
    ! [X5: fun(nat,real)] :
      ( topolo3814608138187158403Cauchy(real,X5)
    <=> ! [J3: nat] :
        ? [M9: nat] :
        ! [M3: nat] :
          ( aa(nat,$o,ord_less_eq(nat,M9),M3)
         => ! [N3: nat] :
              ( aa(nat,$o,ord_less_eq(nat,M9),N3)
             => aa(real,$o,ord_less(real,aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,aa(nat,real,X5,M3)),aa(nat,real,X5,N3)))),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,J3)))) ) ) ) ).

% Cauchy_iff2
tff(fact_5470_List_Ofinite__set,axiom,
    ! [A: $tType,Xsa: list(A)] : finite_finite2(A,aa(list(A),set(A),set2(A),Xsa)) ).

% List.finite_set
tff(fact_5471_sdiv__int__div__0,axiom,
    ! [X: int] : signed7115095781618012415divide(int,X,zero_zero(int)) = zero_zero(int) ).

% sdiv_int_div_0
tff(fact_5472_sdiv__int__0__div,axiom,
    ! [X: int] : signed7115095781618012415divide(int,zero_zero(int),X) = zero_zero(int) ).

% sdiv_int_0_div
tff(fact_5473_int__sdiv__simps_I2_J,axiom,
    ! [A3: int] : signed7115095781618012415divide(int,A3,zero_zero(int)) = zero_zero(int) ).

% int_sdiv_simps(2)
tff(fact_5474_int__sdiv__simps_I1_J,axiom,
    ! [A3: int] : signed7115095781618012415divide(int,A3,one_one(int)) = A3 ).

% int_sdiv_simps(1)
tff(fact_5475_sum__eq__0__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( canoni5634975068530333245id_add(B)
     => ! [F3: set(A),F2: fun(A,B)] :
          ( finite_finite2(A,F3)
         => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),F3) = zero_zero(B) )
          <=> ! [X3: A] :
                ( member(A,X3,F3)
               => ( aa(A,B,F2,X3) = zero_zero(B) ) ) ) ) ) ).

% sum_eq_0_iff
tff(fact_5476_sum_Oinfinite,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [A2: set(A),G: fun(A,B)] :
          ( ~ finite_finite2(A,A2)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A2) = zero_zero(B) ) ) ) ).

% sum.infinite
tff(fact_5477_infinite__Icc__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A] :
          ( ~ finite_finite2(A,set_or1337092689740270186AtMost(A,A3,B3))
        <=> aa(A,$o,ord_less(A,A3),B3) ) ) ).

% infinite_Icc_iff
tff(fact_5478_prod__zero__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( semidom(B)
     => ! [A2: set(A),F2: fun(A,B)] :
          ( finite_finite2(A,A2)
         => ( ( groups7121269368397514597t_prod(A,B,F2,A2) = zero_zero(B) )
          <=> ? [X3: A] :
                ( member(A,X3,A2)
                & ( aa(A,B,F2,X3) = zero_zero(B) ) ) ) ) ) ).

% prod_zero_iff
tff(fact_5479_prod_Oinfinite,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [A2: set(A),G: fun(A,B)] :
          ( ~ finite_finite2(A,A2)
         => ( groups7121269368397514597t_prod(A,B,G,A2) = one_one(B) ) ) ) ).

% prod.infinite
tff(fact_5480_int__sdiv__same__is__1,axiom,
    ! [A3: int,B3: int] :
      ( ( A3 != zero_zero(int) )
     => ( ( signed7115095781618012415divide(int,A3,B3) = A3 )
      <=> ( B3 = one_one(int) ) ) ) ).

% int_sdiv_same_is_1
tff(fact_5481_int__sdiv__simps_I3_J,axiom,
    ! [A3: int] : signed7115095781618012415divide(int,A3,aa(int,int,uminus_uminus(int),one_one(int))) = aa(int,int,uminus_uminus(int),A3) ).

% int_sdiv_simps(3)
tff(fact_5482_sdiv__int__numeral__numeral,axiom,
    ! [M: num,Na: num] : signed7115095781618012415divide(int,aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),Na)) = 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)) ).

% sdiv_int_numeral_numeral
tff(fact_5483_prod__eq__1__iff,axiom,
    ! [A: $tType,A2: set(A),F2: fun(A,nat)] :
      ( finite_finite2(A,A2)
     => ( ( groups7121269368397514597t_prod(A,nat,F2,A2) = one_one(nat) )
      <=> ! [X3: A] :
            ( member(A,X3,A2)
           => ( aa(A,nat,F2,X3) = one_one(nat) ) ) ) ) ).

% prod_eq_1_iff
tff(fact_5484_sum_Odelta,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [S: set(A),A3: A,B3: fun(A,B)] :
          ( finite_finite2(A,S)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_lq(A,fun(fun(A,B),fun(A,B)),A3),B3)),S) = $ite(member(A,A3,S),aa(A,B,B3,A3),zero_zero(B)) ) ) ) ).

% sum.delta
tff(fact_5485_sum_Odelta_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [S: set(A),A3: A,B3: fun(A,B)] :
          ( finite_finite2(A,S)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_lr(A,fun(fun(A,B),fun(A,B)),A3),B3)),S) = $ite(member(A,A3,S),aa(A,B,B3,A3),zero_zero(B)) ) ) ) ).

% sum.delta'
tff(fact_5486_prod_Odelta_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),A3: A,B3: fun(A,B)] :
          ( finite_finite2(A,S)
         => ( groups7121269368397514597t_prod(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ls(A,fun(fun(A,B),fun(A,B)),A3),B3),S) = $ite(member(A,A3,S),aa(A,B,B3,A3),one_one(B)) ) ) ) ).

% prod.delta'
tff(fact_5487_prod_Odelta,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),A3: A,B3: fun(A,B)] :
          ( finite_finite2(A,S)
         => ( groups7121269368397514597t_prod(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_lt(A,fun(fun(A,B),fun(A,B)),A3),B3),S) = $ite(member(A,A3,S),aa(A,B,B3,A3),one_one(B)) ) ) ) ).

% prod.delta
tff(fact_5488_summable__If__finite__set,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [A2: set(nat),F2: fun(nat,A)] :
          ( finite_finite2(nat,A2)
         => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_lu(set(nat),fun(fun(nat,A),fun(nat,A)),A2),F2)) ) ) ).

% summable_If_finite_set
tff(fact_5489_summable__If__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [P: fun(nat,$o),F2: fun(nat,A)] :
          ( finite_finite2(nat,collect(nat,P))
         => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_lv(fun(nat,$o),fun(fun(nat,A),fun(nat,A)),P),F2)) ) ) ).

% summable_If_finite
tff(fact_5490_sum_Oinsert,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A2: set(A),X: A,G: fun(A,B)] :
          ( finite_finite2(A,A2)
         => ( ~ member(A,X,A2)
           => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),insert(A,X),A2)) = aa(B,B,plus_plus(B,aa(A,B,G,X)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A2)) ) ) ) ) ).

% sum.insert
tff(fact_5491_prod_Oinsert,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A2: set(A),X: A,G: fun(A,B)] :
          ( finite_finite2(A,A2)
         => ( ~ member(A,X,A2)
           => ( groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),insert(A,X),A2)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,X)),groups7121269368397514597t_prod(A,B,G,A2)) ) ) ) ) ).

% prod.insert
tff(fact_5492_int__sdiv__negated__is__minus1,axiom,
    ! [A3: int,B3: int] :
      ( ( A3 != zero_zero(int) )
     => ( ( signed7115095781618012415divide(int,A3,B3) = aa(int,int,uminus_uminus(int),A3) )
      <=> ( B3 = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% int_sdiv_negated_is_minus1
tff(fact_5493_prod__pos__nat__iff,axiom,
    ! [A: $tType,A2: set(A),F2: fun(A,nat)] :
      ( finite_finite2(A,A2)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),groups7121269368397514597t_prod(A,nat,F2,A2))
      <=> ! [X3: A] :
            ( member(A,X3,A2)
           => aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(A,nat,F2,X3)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_5494_sum__zero__power,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: fun(nat,A),A2: set(nat)] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_lw(fun(nat,A),fun(nat,A),C2)),A2) = $ite(
            ( finite_finite2(nat,A2)
            & member(nat,zero_zero(nat),A2) ),
            aa(nat,A,C2,zero_zero(nat)),
            zero_zero(A) ) ) ).

% sum_zero_power
tff(fact_5495_sum__zero__power_H,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: fun(nat,A),D2: fun(nat,A),A2: set(nat)] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_lx(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C2),D2)),A2) = $ite(
            ( finite_finite2(nat,A2)
            & member(nat,zero_zero(nat),A2) ),
            aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,C2,zero_zero(nat))),aa(nat,A,D2,zero_zero(nat))),
            zero_zero(A) ) ) ).

% sum_zero_power'
tff(fact_5496_union__le__mono1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: multiset(A),D3: multiset(A),C3: multiset(A)] :
          ( aa(multiset(A),$o,ord_less(multiset(A),B2),D3)
         => aa(multiset(A),$o,ord_less(multiset(A),aa(multiset(A),multiset(A),plus_plus(multiset(A),B2),C3)),aa(multiset(A),multiset(A),plus_plus(multiset(A),D3),C3)) ) ) ).

% union_le_mono1
tff(fact_5497_union__le__mono2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: multiset(A),D3: multiset(A),C3: multiset(A)] :
          ( aa(multiset(A),$o,ord_less(multiset(A),B2),D3)
         => aa(multiset(A),$o,ord_less(multiset(A),aa(multiset(A),multiset(A),plus_plus(multiset(A),C3),B2)),aa(multiset(A),multiset(A),plus_plus(multiset(A),C3),D3)) ) ) ).

% union_le_mono2
tff(fact_5498_union__less__mono,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: multiset(A),C3: multiset(A),B2: multiset(A),D3: multiset(A)] :
          ( aa(multiset(A),$o,ord_less(multiset(A),A2),C3)
         => ( aa(multiset(A),$o,ord_less(multiset(A),B2),D3)
           => aa(multiset(A),$o,ord_less(multiset(A),aa(multiset(A),multiset(A),plus_plus(multiset(A),A2),B2)),aa(multiset(A),multiset(A),plus_plus(multiset(A),C3),D3)) ) ) ) ).

% union_less_mono
tff(fact_5499_finite__lists__length__eq,axiom,
    ! [A: $tType,A2: set(A),Na: nat] :
      ( finite_finite2(A,A2)
     => finite_finite2(list(A),collect(list(A),aa(nat,fun(list(A),$o),aTP_Lamp_ly(set(A),fun(nat,fun(list(A),$o)),A2),Na))) ) ).

% finite_lists_length_eq
tff(fact_5500_sum__eq__empty__iff,axiom,
    ! [A: $tType,B: $tType,A2: set(A),F2: fun(A,multiset(B))] :
      ( finite_finite2(A,A2)
     => ( ( aa(set(A),multiset(B),groups7311177749621191930dd_sum(A,multiset(B),F2),A2) = zero_zero(multiset(B)) )
      <=> ! [X3: A] :
            ( member(A,X3,A2)
           => ( aa(A,multiset(B),F2,X3) = zero_zero(multiset(B)) ) ) ) ) ).

% sum_eq_empty_iff
tff(fact_5501_prod_Oswap__restrict,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [A2: set(A),B2: set(B),G: fun(A,fun(B,C)),R: fun(A,fun(B,$o))] :
          ( finite_finite2(A,A2)
         => ( finite_finite2(B,B2)
           => ( 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_ma(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),B2),G),R),A2) = groups7121269368397514597t_prod(B,C,aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_md(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),A2),G),R),B2) ) ) ) ) ).

% prod.swap_restrict
tff(fact_5502_sum_Oswap__restrict,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [A2: set(A),B2: set(B),G: fun(A,fun(B,C)),R: fun(A,fun(B,$o))] :
          ( finite_finite2(A,A2)
         => ( finite_finite2(B,B2)
           => ( aa(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_me(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),B2),G),R)),A2) = aa(set(B),C,groups7311177749621191930dd_sum(B,C,aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_mg(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),A2),G),R)),B2) ) ) ) ) ).

% sum.swap_restrict
tff(fact_5503_finite__bit__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : finite_finite2(nat,collect(nat,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W))) ) ).

% finite_bit_word
tff(fact_5504_finite__list,axiom,
    ! [A: $tType,A2: set(A)] :
      ( finite_finite2(A,A2)
     => ? [Xs: list(A)] : aa(list(A),set(A),set2(A),Xs) = A2 ) ).

% finite_list
tff(fact_5505_finite__M__bounded__by__nat,axiom,
    ! [P: fun(nat,$o),I: nat] : finite_finite2(nat,collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_mh(fun(nat,$o),fun(nat,fun(nat,$o)),P),I))) ).

% finite_M_bounded_by_nat
tff(fact_5506_bounded__nat__set__is__finite,axiom,
    ! [N2: set(nat),Na: nat] :
      ( ! [X4: nat] :
          ( member(nat,X4,N2)
         => aa(nat,$o,ord_less(nat,X4),Na) )
     => finite_finite2(nat,N2) ) ).

% bounded_nat_set_is_finite
tff(fact_5507_finite__nat__set__iff__bounded,axiom,
    ! [N2: set(nat)] :
      ( finite_finite2(nat,N2)
    <=> ? [M3: nat] :
        ! [X3: nat] :
          ( member(nat,X3,N2)
         => aa(nat,$o,ord_less(nat,X3),M3) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_5508_finite__less__ub,axiom,
    ! [F2: fun(nat,nat),U: nat] :
      ( ! [N: nat] : aa(nat,$o,ord_less_eq(nat,N),aa(nat,nat,F2,N))
     => finite_finite2(nat,collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_mi(fun(nat,nat),fun(nat,fun(nat,$o)),F2),U))) ) ).

% finite_less_ub
tff(fact_5509_finite__if__eq__beyond__finite,axiom,
    ! [A: $tType,S: set(A),S3: set(A)] :
      ( finite_finite2(A,S)
     => finite_finite2(set(A),collect(set(A),aa(set(A),fun(set(A),$o),aTP_Lamp_mj(set(A),fun(set(A),fun(set(A),$o)),S),S3))) ) ).

% finite_if_eq_beyond_finite
tff(fact_5510_ex__min__if__finite,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [S: set(A)] :
          ( finite_finite2(A,S)
         => ( ( S != bot_bot(set(A)) )
           => ? [X4: A] :
                ( member(A,X4,S)
                & ~ ? [Xa: A] :
                      ( member(A,Xa,S)
                      & aa(A,$o,ord_less(A,Xa),X4) ) ) ) ) ) ).

% ex_min_if_finite
tff(fact_5511_infinite__growing,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X5: set(A)] :
          ( ( X5 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( member(A,X4,X5)
               => ? [Xa: A] :
                    ( member(A,Xa,X5)
                    & aa(A,$o,ord_less(A,X4),Xa) ) )
           => ~ finite_finite2(A,X5) ) ) ) ).

% infinite_growing
tff(fact_5512_infinite__Icc,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ~ finite_finite2(A,set_or1337092689740270186AtMost(A,A3,B3)) ) ) ).

% infinite_Icc
tff(fact_5513_prod__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(B)
     => ! [A2: set(A),F2: fun(A,B)] :
          ( finite_finite2(A,A2)
         => ( ? [X2: A] :
                ( member(A,X2,A2)
                & ( aa(A,B,F2,X2) = zero_zero(B) ) )
           => ( groups7121269368397514597t_prod(A,B,F2,A2) = zero_zero(B) ) ) ) ) ).

% prod_zero
tff(fact_5514_finite__lists__length__le,axiom,
    ! [A: $tType,A2: set(A),Na: nat] :
      ( finite_finite2(A,A2)
     => finite_finite2(list(A),collect(list(A),aa(nat,fun(list(A),$o),aTP_Lamp_mk(set(A),fun(nat,fun(list(A),$o)),A2),Na))) ) ).

% finite_lists_length_le
tff(fact_5515_summable__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [N2: set(nat),F2: fun(nat,A)] :
          ( finite_finite2(nat,N2)
         => ( ! [N: nat] :
                ( ~ member(nat,N,N2)
               => ( aa(nat,A,F2,N) = zero_zero(A) ) )
           => summable(A,F2) ) ) ) ).

% summable_finite
tff(fact_5516_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N2: set(nat),Na: nat] :
      ( aa(set(nat),$o,ord_less_eq(set(nat),N2),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na))
     => finite_finite2(nat,N2) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_5517_sum_Ofinite__Collect__op,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),X: fun(A,B),Y: fun(A,B)] :
          ( finite_finite2(A,collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_ml(set(A),fun(fun(A,B),fun(A,$o)),I5),X)))
         => ( finite_finite2(A,collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_ml(set(A),fun(fun(A,B),fun(A,$o)),I5),Y)))
           => finite_finite2(A,collect(A,aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_mm(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),I5),X),Y))) ) ) ) ).

% sum.finite_Collect_op
tff(fact_5518_prod_Ofinite__Collect__op,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I5: set(A),X: fun(A,B),Y: fun(A,B)] :
          ( finite_finite2(A,collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_mn(set(A),fun(fun(A,B),fun(A,$o)),I5),X)))
         => ( finite_finite2(A,collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_mn(set(A),fun(fun(A,B),fun(A,$o)),I5),Y)))
           => finite_finite2(A,collect(A,aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_mo(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),I5),X),Y))) ) ) ) ).

% prod.finite_Collect_op
tff(fact_5519_set__encode__inf,axiom,
    ! [A2: set(nat)] :
      ( ~ finite_finite2(nat,A2)
     => ( aa(set(nat),nat,nat_set_encode,A2) = zero_zero(nat) ) ) ).

% set_encode_inf
tff(fact_5520_filter__preserves__multiset,axiom,
    ! [A: $tType,M7: fun(A,nat),P: fun(A,$o)] :
      ( finite_finite2(A,collect(A,aTP_Lamp_mp(fun(A,nat),fun(A,$o),M7)))
     => finite_finite2(A,collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_mq(fun(A,nat),fun(fun(A,$o),fun(A,$o)),M7),P))) ) ).

% filter_preserves_multiset
tff(fact_5521_sum_Ointer__filter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A2: set(A),G: fun(A,B),P: fun(A,$o)] :
          ( finite_finite2(A,A2)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ao(set(A),fun(fun(A,$o),fun(A,$o)),A2),P))) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_mr(fun(A,B),fun(fun(A,$o),fun(A,B)),G),P)),A2) ) ) ) ).

% sum.inter_filter
tff(fact_5522_prod_Ointer__filter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A2: set(A),G: fun(A,B),P: fun(A,$o)] :
          ( finite_finite2(A,A2)
         => ( groups7121269368397514597t_prod(A,B,G,collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ao(set(A),fun(fun(A,$o),fun(A,$o)),A2),P))) = groups7121269368397514597t_prod(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_ms(fun(A,B),fun(fun(A,$o),fun(A,B)),G),P),A2) ) ) ) ).

% prod.inter_filter
tff(fact_5523_finite__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A,B3: A] : finite_finite2(A,collect(A,aa(A,fun(A,$o),aTP_Lamp_mt(A,fun(A,fun(A,$o)),A3),B3))) ) ).

% finite_int_segment
tff(fact_5524_sum__le__included,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(C)
     => ! [Sa: set(A),Ta: set(B),G: fun(B,C),I: fun(B,A),F2: fun(A,C)] :
          ( finite_finite2(A,Sa)
         => ( finite_finite2(B,Ta)
           => ( ! [X4: B] :
                  ( member(B,X4,Ta)
                 => aa(C,$o,ord_less_eq(C,zero_zero(C)),aa(B,C,G,X4)) )
             => ( ! [X4: A] :
                    ( member(A,X4,Sa)
                   => ? [Xa: B] :
                        ( member(B,Xa,Ta)
                        & ( aa(B,A,I,Xa) = X4 )
                        & aa(C,$o,ord_less_eq(C,aa(A,C,F2,X4)),aa(B,C,G,Xa)) ) )
               => aa(C,$o,ord_less_eq(C,aa(set(A),C,groups7311177749621191930dd_sum(A,C,F2),Sa)),aa(set(B),C,groups7311177749621191930dd_sum(B,C,G),Ta)) ) ) ) ) ) ).

% sum_le_included
tff(fact_5525_sum__nonneg__eq__0__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A2: set(A),F2: fun(A,B)] :
          ( finite_finite2(A,A2)
         => ( ! [X4: A] :
                ( member(A,X4,A2)
               => aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,F2,X4)) )
           => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A2) = zero_zero(B) )
            <=> ! [X3: A] :
                  ( member(A,X3,A2)
                 => ( aa(A,B,F2,X3) = zero_zero(B) ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5526_sum__strict__mono__ex1,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [A2: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( finite_finite2(A,A2)
         => ( ! [X4: A] :
                ( member(A,X4,A2)
               => aa(B,$o,ord_less_eq(B,aa(A,B,F2,X4)),aa(A,B,G,X4)) )
           => ( ? [X2: A] :
                  ( member(A,X2,A2)
                  & aa(B,$o,ord_less(B,aa(A,B,F2,X2)),aa(A,B,G,X2)) )
             => aa(B,$o,ord_less(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A2)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A2)) ) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5527_sum_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [R: fun(A,fun(A,$o)),S: set(B),H: fun(B,A),G: fun(B,A)] :
          ( aa(A,$o,aa(A,fun(A,$o),R,zero_zero(A)),zero_zero(A))
         => ( ! [X1: A,Y1: A,X23: A,Y22: A] :
                ( ( aa(A,$o,aa(A,fun(A,$o),R,X1),X23)
                  & aa(A,$o,aa(A,fun(A,$o),R,Y1),Y22) )
               => aa(A,$o,aa(A,fun(A,$o),R,aa(A,A,plus_plus(A,X1),Y1)),aa(A,A,plus_plus(A,X23),Y22)) )
           => ( finite_finite2(B,S)
             => ( ! [X4: B] :
                    ( member(B,X4,S)
                   => aa(A,$o,aa(A,fun(A,$o),R,aa(B,A,H,X4)),aa(B,A,G,X4)) )
               => aa(A,$o,aa(A,fun(A,$o),R,aa(set(B),A,groups7311177749621191930dd_sum(B,A,H),S)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),S)) ) ) ) ) ) ).

% sum.related
tff(fact_5528_finite__ranking__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [S: set(A),P: fun(set(A),$o),F2: fun(A,B)] :
          ( finite_finite2(A,S)
         => ( aa(set(A),$o,P,bot_bot(set(A)))
           => ( ! [X4: A,S4: set(A)] :
                  ( finite_finite2(A,S4)
                 => ( ! [Y4: A] :
                        ( member(A,Y4,S4)
                       => aa(B,$o,ord_less_eq(B,aa(A,B,F2,Y4)),aa(A,B,F2,X4)) )
                   => ( aa(set(A),$o,P,S4)
                     => aa(set(A),$o,P,aa(set(A),set(A),insert(A,X4),S4)) ) ) )
             => aa(set(A),$o,P,S) ) ) ) ) ).

% finite_ranking_induct
tff(fact_5529_finite__linorder__min__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: set(A),P: fun(set(A),$o)] :
          ( finite_finite2(A,A2)
         => ( aa(set(A),$o,P,bot_bot(set(A)))
           => ( ! [B5: A,A8: set(A)] :
                  ( finite_finite2(A,A8)
                 => ( ! [X2: A] :
                        ( member(A,X2,A8)
                       => aa(A,$o,ord_less(A,B5),X2) )
                   => ( aa(set(A),$o,P,A8)
                     => aa(set(A),$o,P,aa(set(A),set(A),insert(A,B5),A8)) ) ) )
             => aa(set(A),$o,P,A2) ) ) ) ) ).

% finite_linorder_min_induct
tff(fact_5530_finite__linorder__max__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: set(A),P: fun(set(A),$o)] :
          ( finite_finite2(A,A2)
         => ( aa(set(A),$o,P,bot_bot(set(A)))
           => ( ! [B5: A,A8: set(A)] :
                  ( finite_finite2(A,A8)
                 => ( ! [X2: A] :
                        ( member(A,X2,A8)
                       => aa(A,$o,ord_less(A,X2),B5) )
                   => ( aa(set(A),$o,P,A8)
                     => aa(set(A),$o,P,aa(set(A),set(A),insert(A,B5),A8)) ) ) )
             => aa(set(A),$o,P,A2) ) ) ) ) ).

% finite_linorder_max_induct
tff(fact_5531_sum__strict__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( strict7427464778891057005id_add(B)
     => ! [A2: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( finite_finite2(A,A2)
         => ( ( A2 != bot_bot(set(A)) )
           => ( ! [X4: A] :
                  ( member(A,X4,A2)
                 => aa(B,$o,ord_less(B,aa(A,B,F2,X4)),aa(A,B,G,X4)) )
             => aa(B,$o,ord_less(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A2)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A2)) ) ) ) ) ).

% sum_strict_mono
tff(fact_5532_prod_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [R: fun(A,fun(A,$o)),S: set(B),H: fun(B,A),G: fun(B,A)] :
          ( aa(A,$o,aa(A,fun(A,$o),R,one_one(A)),one_one(A))
         => ( ! [X1: A,Y1: A,X23: A,Y22: A] :
                ( ( aa(A,$o,aa(A,fun(A,$o),R,X1),X23)
                  & aa(A,$o,aa(A,fun(A,$o),R,Y1),Y22) )
               => aa(A,$o,aa(A,fun(A,$o),R,aa(A,A,aa(A,fun(A,A),times_times(A),X1),Y1)),aa(A,A,aa(A,fun(A,A),times_times(A),X23),Y22)) )
           => ( finite_finite2(B,S)
             => ( ! [X4: B] :
                    ( member(B,X4,S)
                   => aa(A,$o,aa(A,fun(A,$o),R,aa(B,A,H,X4)),aa(B,A,G,X4)) )
               => aa(A,$o,aa(A,fun(A,$o),R,groups7121269368397514597t_prod(B,A,H,S)),groups7121269368397514597t_prod(B,A,G,S)) ) ) ) ) ) ).

% prod.related
tff(fact_5533_sum_Oinsert__if,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A2: set(A),G: fun(A,B),X: A] :
          ( finite_finite2(A,A2)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),insert(A,X),A2)) = $ite(member(A,X,A2),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A2),aa(B,B,plus_plus(B,aa(A,B,G,X)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A2))) ) ) ) ).

% sum.insert_if
tff(fact_5534_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [S5: set(A),T4: set(B),S: set(A),I: fun(B,A),J: fun(A,B),T3: set(B),G: fun(A,C),H: fun(B,C)] :
          ( finite_finite2(A,S5)
         => ( finite_finite2(B,T4)
           => ( ! [A4: A] :
                  ( member(A,A4,aa(set(A),set(A),minus_minus(set(A),S),S5))
                 => ( aa(B,A,I,aa(A,B,J,A4)) = A4 ) )
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),minus_minus(set(A),S),S5))
                   => member(B,aa(A,B,J,A4),aa(set(B),set(B),minus_minus(set(B),T3),T4)) )
               => ( ! [B5: B] :
                      ( member(B,B5,aa(set(B),set(B),minus_minus(set(B),T3),T4))
                     => ( aa(A,B,J,aa(B,A,I,B5)) = B5 ) )
                 => ( ! [B5: B] :
                        ( member(B,B5,aa(set(B),set(B),minus_minus(set(B),T3),T4))
                       => member(A,aa(B,A,I,B5),aa(set(A),set(A),minus_minus(set(A),S),S5)) )
                   => ( ! [A4: A] :
                          ( member(A,A4,S5)
                         => ( aa(A,C,G,A4) = zero_zero(C) ) )
                     => ( ! [B5: B] :
                            ( member(B,B5,T4)
                           => ( aa(B,C,H,B5) = zero_zero(C) ) )
                       => ( ! [A4: A] :
                              ( member(A,A4,S)
                             => ( aa(B,C,H,aa(A,B,J,A4)) = aa(A,C,G,A4) ) )
                         => ( aa(set(A),C,groups7311177749621191930dd_sum(A,C,G),S) = aa(set(B),C,groups7311177749621191930dd_sum(B,C,H),T3) ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_5535_prod_Oinsert__if,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A2: set(A),G: fun(A,B),X: A] :
          ( finite_finite2(A,A2)
         => ( groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),insert(A,X),A2)) = $ite(member(A,X,A2),groups7121269368397514597t_prod(A,B,G,A2),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,X)),groups7121269368397514597t_prod(A,B,G,A2))) ) ) ) ).

% prod.insert_if
tff(fact_5536_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [S5: set(A),T4: set(B),S: set(A),I: fun(B,A),J: fun(A,B),T3: set(B),G: fun(A,C),H: fun(B,C)] :
          ( finite_finite2(A,S5)
         => ( finite_finite2(B,T4)
           => ( ! [A4: A] :
                  ( member(A,A4,aa(set(A),set(A),minus_minus(set(A),S),S5))
                 => ( aa(B,A,I,aa(A,B,J,A4)) = A4 ) )
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),minus_minus(set(A),S),S5))
                   => member(B,aa(A,B,J,A4),aa(set(B),set(B),minus_minus(set(B),T3),T4)) )
               => ( ! [B5: B] :
                      ( member(B,B5,aa(set(B),set(B),minus_minus(set(B),T3),T4))
                     => ( aa(A,B,J,aa(B,A,I,B5)) = B5 ) )
                 => ( ! [B5: B] :
                        ( member(B,B5,aa(set(B),set(B),minus_minus(set(B),T3),T4))
                       => member(A,aa(B,A,I,B5),aa(set(A),set(A),minus_minus(set(A),S),S5)) )
                   => ( ! [A4: A] :
                          ( member(A,A4,S5)
                         => ( aa(A,C,G,A4) = one_one(C) ) )
                     => ( ! [B5: B] :
                            ( member(B,B5,T4)
                           => ( aa(B,C,H,B5) = one_one(C) ) )
                       => ( ! [A4: A] :
                              ( member(A,A4,S)
                             => ( aa(B,C,H,aa(A,B,J,A4)) = aa(A,C,G,A4) ) )
                         => ( groups7121269368397514597t_prod(A,C,G,S) = groups7121269368397514597t_prod(B,C,H,T3) ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_5537_sum__eq__Suc0__iff,axiom,
    ! [A: $tType,A2: set(A),F2: fun(A,nat)] :
      ( finite_finite2(A,A2)
     => ( ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A2) = aa(nat,nat,suc,zero_zero(nat)) )
      <=> ? [X3: A] :
            ( member(A,X3,A2)
            & ( aa(A,nat,F2,X3) = aa(nat,nat,suc,zero_zero(nat)) )
            & ! [Xa4: A] :
                ( member(A,Xa4,A2)
               => ( ( X3 != Xa4 )
                 => ( aa(A,nat,F2,Xa4) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_5538_sum__eq__1__iff,axiom,
    ! [A: $tType,A2: set(A),F2: fun(A,nat)] :
      ( finite_finite2(A,A2)
     => ( ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A2) = one_one(nat) )
      <=> ? [X3: A] :
            ( member(A,X3,A2)
            & ( aa(A,nat,F2,X3) = one_one(nat) )
            & ! [Xa4: A] :
                ( member(A,Xa4,A2)
               => ( ( X3 != Xa4 )
                 => ( aa(A,nat,F2,Xa4) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_5539_sum__nonneg__0,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [Sa: set(A),F2: fun(A,B),I: A] :
          ( finite_finite2(A,Sa)
         => ( ! [I3: A] :
                ( member(A,I3,Sa)
               => aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,F2,I3)) )
           => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),Sa) = zero_zero(B) )
             => ( member(A,I,Sa)
               => ( aa(A,B,F2,I) = zero_zero(B) ) ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5540_sum__nonneg__leq__bound,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [Sa: set(A),F2: fun(A,B),B2: B,I: A] :
          ( finite_finite2(A,Sa)
         => ( ! [I3: A] :
                ( member(A,I3,Sa)
               => aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,F2,I3)) )
           => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),Sa) = B2 )
             => ( member(A,I,Sa)
               => aa(B,$o,ord_less_eq(B,aa(A,B,F2,I)),B2) ) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5541_add__mset__in__multiset,axiom,
    ! [A: $tType,M7: fun(A,nat),A3: A] :
      ( finite_finite2(A,collect(A,aTP_Lamp_mp(fun(A,nat),fun(A,$o),M7)))
     => finite_finite2(A,collect(A,aa(A,fun(A,$o),aTP_Lamp_mu(fun(A,nat),fun(A,fun(A,$o)),M7),A3))) ) ).

% add_mset_in_multiset
tff(fact_5542_diff__preserves__multiset,axiom,
    ! [A: $tType,M7: fun(A,nat),N2: fun(A,nat)] :
      ( finite_finite2(A,collect(A,aTP_Lamp_mp(fun(A,nat),fun(A,$o),M7)))
     => finite_finite2(A,collect(A,aa(fun(A,nat),fun(A,$o),aTP_Lamp_mv(fun(A,nat),fun(fun(A,nat),fun(A,$o)),M7),N2))) ) ).

% diff_preserves_multiset
tff(fact_5543_sum_Osetdiff__irrelevant,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A2: set(A),G: fun(A,B)] :
          ( finite_finite2(A,A2)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),minus_minus(set(A),A2),collect(A,aTP_Lamp_mw(fun(A,B),fun(A,$o),G)))) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A2) ) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5544_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
     => finite_finite2(nat,collect(nat,aTP_Lamp_mx(nat,fun(nat,$o),M))) ) ).

% finite_divisors_nat
tff(fact_5545_prod_Osetdiff__irrelevant,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A2: set(A),G: fun(A,B)] :
          ( finite_finite2(A,A2)
         => ( groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),minus_minus(set(A),A2),collect(A,aTP_Lamp_my(fun(A,B),fun(A,$o),G)))) = groups7121269368397514597t_prod(A,B,G,A2) ) ) ) ).

% prod.setdiff_irrelevant
tff(fact_5546_sums__If__finite__set,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [A2: set(nat),F2: fun(nat,A)] :
          ( finite_finite2(nat,A2)
         => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_lu(set(nat),fun(fun(nat,A),fun(nat,A)),A2),F2)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),A2)) ) ) ).

% sums_If_finite_set
tff(fact_5547_sums__If__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [P: fun(nat,$o),F2: fun(nat,A)] :
          ( finite_finite2(nat,collect(nat,P))
         => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_lv(fun(nat,$o),fun(fun(nat,A),fun(nat,A)),P),F2)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),collect(nat,P))) ) ) ).

% sums_If_finite
tff(fact_5548_sums__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [N2: set(nat),F2: fun(nat,A)] :
          ( finite_finite2(nat,N2)
         => ( ! [N: nat] :
                ( ~ member(nat,N,N2)
               => ( aa(nat,A,F2,N) = zero_zero(A) ) )
           => aa(A,$o,sums(A,F2),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),N2)) ) ) ) ).

% sums_finite
tff(fact_5549_suminf__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [N2: set(nat),F2: fun(nat,A)] :
          ( finite_finite2(nat,N2)
         => ( ! [N: nat] :
                ( ~ member(nat,N,N2)
               => ( aa(nat,A,F2,N) = zero_zero(A) ) )
           => ( suminf(A,F2) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),N2) ) ) ) ) ).

% suminf_finite
tff(fact_5550_finite__abs__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A] : finite_finite2(A,collect(A,aTP_Lamp_mz(A,fun(A,$o),A3))) ) ).

% finite_abs_int_segment
tff(fact_5551_exp__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_Vector_banach(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [I5: set(A),F2: fun(A,B)] :
          ( finite_finite2(A,I5)
         => ( aa(B,B,exp(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),I5)) = groups7121269368397514597t_prod(A,B,aTP_Lamp_na(fun(A,B),fun(A,B),F2),I5) ) ) ) ).

% exp_sum
tff(fact_5552_sgn__sdiv__eq__sgn__mult,axiom,
    ! [A3: int,B3: int] :
      ( ( signed7115095781618012415divide(int,A3,B3) != zero_zero(int) )
     => ( aa(int,int,sgn_sgn(int),signed7115095781618012415divide(int,A3,B3)) = aa(int,int,sgn_sgn(int),aa(int,int,aa(int,fun(int,int),times_times(int),A3),B3)) ) ) ).

% sgn_sdiv_eq_sgn_mult
tff(fact_5553_sum__pos2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(A),I: A,F2: fun(A,B)] :
          ( finite_finite2(A,I5)
         => ( member(A,I,I5)
           => ( aa(B,$o,ord_less(B,zero_zero(B)),aa(A,B,F2,I))
             => ( ! [I3: A] :
                    ( member(A,I3,I5)
                   => aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,F2,I3)) )
               => aa(B,$o,ord_less(B,zero_zero(B)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),I5)) ) ) ) ) ) ).

% sum_pos2
tff(fact_5554_sum__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(A),F2: fun(A,B)] :
          ( finite_finite2(A,I5)
         => ( ( I5 != bot_bot(set(A)) )
           => ( ! [I3: A] :
                  ( member(A,I3,I5)
                 => aa(B,$o,ord_less(B,zero_zero(B)),aa(A,B,F2,I3)) )
             => aa(B,$o,ord_less(B,zero_zero(B)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),I5)) ) ) ) ) ).

% sum_pos
tff(fact_5555_less__1__prod2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),I: A,F2: fun(A,B)] :
          ( finite_finite2(A,I5)
         => ( member(A,I,I5)
           => ( aa(B,$o,ord_less(B,one_one(B)),aa(A,B,F2,I))
             => ( ! [I3: A] :
                    ( member(A,I3,I5)
                   => aa(B,$o,ord_less_eq(B,one_one(B)),aa(A,B,F2,I3)) )
               => aa(B,$o,ord_less(B,one_one(B)),groups7121269368397514597t_prod(A,B,F2,I5)) ) ) ) ) ) ).

% less_1_prod2
tff(fact_5556_less__1__prod,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),F2: fun(A,B)] :
          ( finite_finite2(A,I5)
         => ( ( I5 != bot_bot(set(A)) )
           => ( ! [I3: A] :
                  ( member(A,I3,I5)
                 => aa(B,$o,ord_less(B,one_one(B)),aa(A,B,F2,I3)) )
             => aa(B,$o,ord_less(B,one_one(B)),groups7121269368397514597t_prod(A,B,F2,I5)) ) ) ) ) ).

% less_1_prod
tff(fact_5557_sum_Osame__carrier,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [C3: set(A),A2: set(A),B2: set(A),G: fun(A,B),H: fun(A,B)] :
          ( finite_finite2(A,C3)
         => ( aa(set(A),$o,ord_less_eq(set(A),A2),C3)
           => ( aa(set(A),$o,ord_less_eq(set(A),B2),C3)
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),minus_minus(set(A),C3),A2))
                   => ( aa(A,B,G,A4) = zero_zero(B) ) )
               => ( ! [B5: A] :
                      ( member(A,B5,aa(set(A),set(A),minus_minus(set(A),C3),B2))
                     => ( aa(A,B,H,B5) = zero_zero(B) ) )
                 => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A2) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),B2) )
                  <=> ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),C3) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),C3) ) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_5558_sum_Osame__carrierI,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [C3: set(A),A2: set(A),B2: set(A),G: fun(A,B),H: fun(A,B)] :
          ( finite_finite2(A,C3)
         => ( aa(set(A),$o,ord_less_eq(set(A),A2),C3)
           => ( aa(set(A),$o,ord_less_eq(set(A),B2),C3)
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),minus_minus(set(A),C3),A2))
                   => ( aa(A,B,G,A4) = zero_zero(B) ) )
               => ( ! [B5: A] :
                      ( member(A,B5,aa(set(A),set(A),minus_minus(set(A),C3),B2))
                     => ( aa(A,B,H,B5) = zero_zero(B) ) )
                 => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),C3) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),C3) )
                   => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A2) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),B2) ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_5559_sum_Omono__neutral__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T3: set(A),S: set(A),G: fun(A,B)] :
          ( finite_finite2(A,T3)
         => ( aa(set(A),$o,ord_less_eq(set(A),S),T3)
           => ( ! [X4: A] :
                  ( member(A,X4,aa(set(A),set(A),minus_minus(set(A),T3),S))
                 => ( aa(A,B,G,X4) = zero_zero(B) ) )
             => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),S) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),T3) ) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_5560_sum_Omono__neutral__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T3: set(A),S: set(A),G: fun(A,B)] :
          ( finite_finite2(A,T3)
         => ( aa(set(A),$o,ord_less_eq(set(A),S),T3)
           => ( ! [X4: A] :
                  ( member(A,X4,aa(set(A),set(A),minus_minus(set(A),T3),S))
                 => ( aa(A,B,G,X4) = zero_zero(B) ) )
             => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),T3) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),S) ) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_5561_sum_Omono__neutral__cong__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T3: set(A),S: set(A),H: fun(A,B),G: fun(A,B)] :
          ( finite_finite2(A,T3)
         => ( aa(set(A),$o,ord_less_eq(set(A),S),T3)
           => ( ! [X4: A] :
                  ( member(A,X4,aa(set(A),set(A),minus_minus(set(A),T3),S))
                 => ( aa(A,B,H,X4) = zero_zero(B) ) )
             => ( ! [X4: A] :
                    ( member(A,X4,S)
                   => ( aa(A,B,G,X4) = aa(A,B,H,X4) ) )
               => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),S) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),T3) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_5562_sum_Omono__neutral__cong__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T3: set(A),S: set(A),G: fun(A,B),H: fun(A,B)] :
          ( finite_finite2(A,T3)
         => ( aa(set(A),$o,ord_less_eq(set(A),S),T3)
           => ( ! [X4: A] :
                  ( member(A,X4,aa(set(A),set(A),minus_minus(set(A),T3),S))
                 => ( aa(A,B,G,X4) = zero_zero(B) ) )
             => ( ! [X4: A] :
                    ( member(A,X4,S)
                   => ( aa(A,B,G,X4) = aa(A,B,H,X4) ) )
               => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),T3) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),S) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_5563_prod_Omono__neutral__cong__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T3: set(A),S: set(A),G: fun(A,B),H: fun(A,B)] :
          ( finite_finite2(A,T3)
         => ( aa(set(A),$o,ord_less_eq(set(A),S),T3)
           => ( ! [X4: A] :
                  ( member(A,X4,aa(set(A),set(A),minus_minus(set(A),T3),S))
                 => ( aa(A,B,G,X4) = one_one(B) ) )
             => ( ! [X4: A] :
                    ( member(A,X4,S)
                   => ( aa(A,B,G,X4) = aa(A,B,H,X4) ) )
               => ( groups7121269368397514597t_prod(A,B,G,T3) = groups7121269368397514597t_prod(A,B,H,S) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_5564_prod_Omono__neutral__cong__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T3: set(A),S: set(A),H: fun(A,B),G: fun(A,B)] :
          ( finite_finite2(A,T3)
         => ( aa(set(A),$o,ord_less_eq(set(A),S),T3)
           => ( ! [X4: A] :
                  ( member(A,X4,aa(set(A),set(A),minus_minus(set(A),T3),S))
                 => ( aa(A,B,H,X4) = one_one(B) ) )
             => ( ! [X4: A] :
                    ( member(A,X4,S)
                   => ( aa(A,B,G,X4) = aa(A,B,H,X4) ) )
               => ( groups7121269368397514597t_prod(A,B,G,S) = groups7121269368397514597t_prod(A,B,H,T3) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_5565_prod_Omono__neutral__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T3: set(A),S: set(A),G: fun(A,B)] :
          ( finite_finite2(A,T3)
         => ( aa(set(A),$o,ord_less_eq(set(A),S),T3)
           => ( ! [X4: A] :
                  ( member(A,X4,aa(set(A),set(A),minus_minus(set(A),T3),S))
                 => ( aa(A,B,G,X4) = one_one(B) ) )
             => ( groups7121269368397514597t_prod(A,B,G,T3) = groups7121269368397514597t_prod(A,B,G,S) ) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_5566_prod_Omono__neutral__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T3: set(A),S: set(A),G: fun(A,B)] :
          ( finite_finite2(A,T3)
         => ( aa(set(A),$o,ord_less_eq(set(A),S),T3)
           => ( ! [X4: A] :
                  ( member(A,X4,aa(set(A),set(A),minus_minus(set(A),T3),S))
                 => ( aa(A,B,G,X4) = one_one(B) ) )
             => ( groups7121269368397514597t_prod(A,B,G,S) = groups7121269368397514597t_prod(A,B,G,T3) ) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_5567_prod_Osame__carrierI,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [C3: set(A),A2: set(A),B2: set(A),G: fun(A,B),H: fun(A,B)] :
          ( finite_finite2(A,C3)
         => ( aa(set(A),$o,ord_less_eq(set(A),A2),C3)
           => ( aa(set(A),$o,ord_less_eq(set(A),B2),C3)
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),minus_minus(set(A),C3),A2))
                   => ( aa(A,B,G,A4) = one_one(B) ) )
               => ( ! [B5: A] :
                      ( member(A,B5,aa(set(A),set(A),minus_minus(set(A),C3),B2))
                     => ( aa(A,B,H,B5) = one_one(B) ) )
                 => ( ( groups7121269368397514597t_prod(A,B,G,C3) = groups7121269368397514597t_prod(A,B,H,C3) )
                   => ( groups7121269368397514597t_prod(A,B,G,A2) = groups7121269368397514597t_prod(A,B,H,B2) ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_5568_prod_Osame__carrier,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [C3: set(A),A2: set(A),B2: set(A),G: fun(A,B),H: fun(A,B)] :
          ( finite_finite2(A,C3)
         => ( aa(set(A),$o,ord_less_eq(set(A),A2),C3)
           => ( aa(set(A),$o,ord_less_eq(set(A),B2),C3)
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),minus_minus(set(A),C3),A2))
                   => ( aa(A,B,G,A4) = one_one(B) ) )
               => ( ! [B5: A] :
                      ( member(A,B5,aa(set(A),set(A),minus_minus(set(A),C3),B2))
                     => ( aa(A,B,H,B5) = one_one(B) ) )
                 => ( ( groups7121269368397514597t_prod(A,B,G,A2) = groups7121269368397514597t_prod(A,B,H,B2) )
                  <=> ( groups7121269368397514597t_prod(A,B,G,C3) = groups7121269368397514597t_prod(A,B,H,C3) ) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_5569_infinite__imp__bij__betw,axiom,
    ! [A: $tType,A2: set(A),A3: A] :
      ( ~ finite_finite2(A,A2)
     => ? [H2: fun(A,A)] : bij_betw(A,A,H2,A2,aa(set(A),set(A),minus_minus(set(A),A2),aa(set(A),set(A),insert(A,A3),bot_bot(set(A))))) ) ).

% infinite_imp_bij_betw
tff(fact_5570_infinite__imp__bij__betw2,axiom,
    ! [A: $tType,A2: set(A),A3: A] :
      ( ~ finite_finite2(A,A2)
     => ? [H2: fun(A,A)] : bij_betw(A,A,H2,A2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),aa(set(A),set(A),insert(A,A3),bot_bot(set(A))))) ) ).

% infinite_imp_bij_betw2
tff(fact_5571_finite__roots__unity,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Na: nat] :
          ( aa(nat,$o,ord_less_eq(nat,one_one(nat)),Na)
         => finite_finite2(A,collect(A,aTP_Lamp_nb(nat,fun(A,$o),Na))) ) ) ).

% finite_roots_unity
tff(fact_5572_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [S5: set(A),T4: set(B),H: fun(A,B),S: set(A),T3: set(B),G: fun(B,C)] :
          ( finite_finite2(A,S5)
         => ( finite_finite2(B,T4)
           => ( bij_betw(A,B,H,aa(set(A),set(A),minus_minus(set(A),S),S5),aa(set(B),set(B),minus_minus(set(B),T3),T4))
             => ( ! [A4: A] :
                    ( member(A,A4,S5)
                   => ( aa(B,C,G,aa(A,B,H,A4)) = zero_zero(C) ) )
               => ( ! [B5: B] :
                      ( member(B,B5,T4)
                     => ( aa(B,C,G,B5) = zero_zero(C) ) )
                 => ( aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ia(fun(A,B),fun(fun(B,C),fun(A,C)),H),G)),S) = aa(set(B),C,groups7311177749621191930dd_sum(B,C,G),T3) ) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
tff(fact_5573_sums__If__finite__set_H,axiom,
    ! [A: $tType] :
      ( ( topolo1287966508704411220up_add(A)
        & topological_t2_space(A) )
     => ! [G: fun(nat,A),S: A,A2: set(nat),S5: A,F2: fun(nat,A)] :
          ( aa(A,$o,sums(A,G),S)
         => ( finite_finite2(nat,A2)
           => ( ( S5 = aa(A,A,plus_plus(A,S),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_nc(fun(nat,A),fun(fun(nat,A),fun(nat,A)),G),F2)),A2)) )
             => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aa(set(nat),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_nd(fun(nat,A),fun(set(nat),fun(fun(nat,A),fun(nat,A))),G),A2),F2)),S5) ) ) ) ) ).

% sums_If_finite_set'
tff(fact_5574_prod_Oreindex__bij__betw__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [S5: set(A),T4: set(B),H: fun(A,B),S: set(A),T3: set(B),G: fun(B,C)] :
          ( finite_finite2(A,S5)
         => ( finite_finite2(B,T4)
           => ( bij_betw(A,B,H,aa(set(A),set(A),minus_minus(set(A),S),S5),aa(set(B),set(B),minus_minus(set(B),T3),T4))
             => ( ! [A4: A] :
                    ( member(A,A4,S5)
                   => ( aa(B,C,G,aa(A,B,H,A4)) = one_one(C) ) )
               => ( ! [B5: B] :
                      ( member(B,B5,T4)
                     => ( aa(B,C,G,B5) = one_one(C) ) )
                 => ( groups7121269368397514597t_prod(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_iv(fun(A,B),fun(fun(B,C),fun(A,C)),H),G),S) = groups7121269368397514597t_prod(B,C,G,T3) ) ) ) ) ) ) ) ).

% prod.reindex_bij_betw_not_neutral
tff(fact_5575_prod__mono__strict,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A2: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( finite_finite2(A,A2)
         => ( ! [I3: A] :
                ( member(A,I3,A2)
               => ( aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,F2,I3))
                  & aa(B,$o,ord_less(B,aa(A,B,F2,I3)),aa(A,B,G,I3)) ) )
           => ( ( A2 != bot_bot(set(A)) )
             => aa(B,$o,ord_less(B,groups7121269368397514597t_prod(A,B,F2,A2)),groups7121269368397514597t_prod(A,B,G,A2)) ) ) ) ) ).

% prod_mono_strict
tff(fact_5576_even__prod__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_parity(B)
     => ! [A2: set(A),F2: fun(A,B)] :
          ( finite_finite2(A,A2)
         => ( aa(B,$o,dvd_dvd(B,aa(num,B,numeral_numeral(B),bit0(one2))),groups7121269368397514597t_prod(A,B,F2,A2))
          <=> ? [X3: A] :
                ( member(A,X3,A2)
                & aa(B,$o,dvd_dvd(B,aa(num,B,numeral_numeral(B),bit0(one2))),aa(A,B,F2,X3)) ) ) ) ) ).

% even_prod_iff
tff(fact_5577_sum__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [B2: set(A),A2: set(A),F2: fun(A,B)] :
          ( finite_finite2(A,B2)
         => ( aa(set(A),$o,ord_less_eq(set(A),A2),B2)
           => ( ! [B5: A] :
                  ( member(A,B5,aa(set(A),set(A),minus_minus(set(A),B2),A2))
                 => aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,F2,B5)) )
             => aa(B,$o,ord_less_eq(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A2)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),B2)) ) ) ) ) ).

% sum_mono2
tff(fact_5578_sum_Oremove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A2: set(A),X: A,G: fun(A,B)] :
          ( finite_finite2(A,A2)
         => ( member(A,X,A2)
           => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A2) = aa(B,B,plus_plus(B,aa(A,B,G,X)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),minus_minus(set(A),A2),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))))) ) ) ) ) ).

% sum.remove
tff(fact_5579_sum_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A2: set(A),G: fun(A,B),X: A] :
          ( finite_finite2(A,A2)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),insert(A,X),A2)) = aa(B,B,plus_plus(B,aa(A,B,G,X)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),minus_minus(set(A),A2),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))))) ) ) ) ).

% sum.insert_remove
tff(fact_5580_sum__diff1,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [A2: set(A),F2: fun(A,B),A3: A] :
          ( finite_finite2(A,A2)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),aa(set(A),set(A),minus_minus(set(A),A2),aa(set(A),set(A),insert(A,A3),bot_bot(set(A))))) = $ite(member(A,A3,A2),aa(B,B,minus_minus(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A2)),aa(A,B,F2,A3)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A2)) ) ) ) ).

% sum_diff1
tff(fact_5581_prod_Oremove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A2: set(A),X: A,G: fun(A,B)] :
          ( finite_finite2(A,A2)
         => ( member(A,X,A2)
           => ( groups7121269368397514597t_prod(A,B,G,A2) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,X)),groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),minus_minus(set(A),A2),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))))) ) ) ) ) ).

% prod.remove
tff(fact_5582_prod_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A2: set(A),G: fun(A,B),X: A] :
          ( finite_finite2(A,A2)
         => ( groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),insert(A,X),A2)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,X)),groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),minus_minus(set(A),A2),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))))) ) ) ) ).

% prod.insert_remove
tff(fact_5583_sum__le__suminf,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),I5: set(nat)] :
          ( summable(A,F2)
         => ( finite_finite2(nat,I5)
           => ( ! [N: nat] :
                  ( member(nat,N,aa(set(nat),set(nat),uminus_uminus(set(nat)),I5))
                 => aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,F2,N)) )
             => aa(A,$o,ord_less_eq(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),I5)),suminf(A,F2)) ) ) ) ) ).

% sum_le_suminf
tff(fact_5584_sum_Odelta__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S: set(A),A3: A,B3: fun(A,B),C2: fun(A,B)] :
          ( finite_finite2(A,S)
         => ( aa(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_ne(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A3),B3),C2)),S) = $ite(member(A,A3,S),aa(B,B,plus_plus(B,aa(A,B,B3,A3)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,C2),aa(set(A),set(A),minus_minus(set(A),S),aa(set(A),set(A),insert(A,A3),bot_bot(set(A)))))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,C2),aa(set(A),set(A),minus_minus(set(A),S),aa(set(A),set(A),insert(A,A3),bot_bot(set(A)))))) ) ) ) ).

% sum.delta_remove
tff(fact_5585_prod_Odelta__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),A3: A,B3: fun(A,B),C2: fun(A,B)] :
          ( finite_finite2(A,S)
         => ( groups7121269368397514597t_prod(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_nf(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A3),B3),C2),S) = $ite(member(A,A3,S),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,B3,A3)),groups7121269368397514597t_prod(A,B,C2,aa(set(A),set(A),minus_minus(set(A),S),aa(set(A),set(A),insert(A,A3),bot_bot(set(A)))))),groups7121269368397514597t_prod(A,B,C2,aa(set(A),set(A),minus_minus(set(A),S),aa(set(A),set(A),insert(A,A3),bot_bot(set(A)))))) ) ) ) ).

% prod.delta_remove
tff(fact_5586_CauchyD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),E2: real] :
          ( topolo3814608138187158403Cauchy(A,X5)
         => ( aa(real,$o,ord_less(real,zero_zero(real)),E2)
           => ? [M8: nat] :
              ! [M2: nat] :
                ( aa(nat,$o,ord_less_eq(nat,M8),M2)
               => ! [N10: nat] :
                    ( aa(nat,$o,ord_less_eq(nat,M8),N10)
                   => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X5,M2)),aa(nat,A,X5,N10)))),E2) ) ) ) ) ) ).

% CauchyD
tff(fact_5587_CauchyI,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( ! [E: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),E)
             => ? [M10: nat] :
                ! [M4: nat] :
                  ( aa(nat,$o,ord_less_eq(nat,M10),M4)
                 => ! [N: nat] :
                      ( aa(nat,$o,ord_less_eq(nat,M10),N)
                     => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X5,M4)),aa(nat,A,X5,N)))),E) ) ) )
         => topolo3814608138187158403Cauchy(A,X5) ) ) ).

% CauchyI
tff(fact_5588_Cauchy__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X5)
        <=> ! [E3: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),E3)
             => ? [M9: nat] :
                ! [M3: nat] :
                  ( aa(nat,$o,ord_less_eq(nat,M9),M3)
                 => ! [N3: nat] :
                      ( aa(nat,$o,ord_less_eq(nat,M9),N3)
                     => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X5,M3)),aa(nat,A,X5,N3)))),E3) ) ) ) ) ) ).

% Cauchy_iff
tff(fact_5589_sum__strict__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [B2: set(A),A2: set(A),B3: A,F2: fun(A,B)] :
          ( finite_finite2(A,B2)
         => ( aa(set(A),$o,ord_less_eq(set(A),A2),B2)
           => ( member(A,B3,aa(set(A),set(A),minus_minus(set(A),B2),A2))
             => ( aa(B,$o,ord_less(B,zero_zero(B)),aa(A,B,F2,B3))
               => ( ! [X4: A] :
                      ( member(A,X4,B2)
                     => aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,F2,X4)) )
                 => aa(B,$o,ord_less(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A2)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),B2)) ) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_5590_member__le__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [I: A,A2: set(A),F2: fun(A,B)] :
          ( member(A,I,A2)
         => ( ! [X4: A] :
                ( member(A,X4,aa(set(A),set(A),minus_minus(set(A),A2),aa(set(A),set(A),insert(A,I),bot_bot(set(A)))))
               => aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,F2,X4)) )
           => ( finite_finite2(A,A2)
             => aa(B,$o,ord_less_eq(B,aa(A,B,F2,I)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A2)) ) ) ) ) ).

% member_le_sum
tff(fact_5591_prod__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [B2: set(A),A2: set(A),F2: fun(A,B)] :
          ( finite_finite2(A,B2)
         => ( aa(set(A),$o,ord_less_eq(set(A),A2),B2)
           => ( ! [B5: A] :
                  ( member(A,B5,aa(set(A),set(A),minus_minus(set(A),B2),A2))
                 => aa(B,$o,ord_less_eq(B,one_one(B)),aa(A,B,F2,B5)) )
             => ( ! [A4: A] :
                    ( member(A,A4,A2)
                   => aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,F2,A4)) )
               => aa(B,$o,ord_less_eq(B,groups7121269368397514597t_prod(A,B,F2,A2)),groups7121269368397514597t_prod(A,B,F2,B2)) ) ) ) ) ) ).

% prod_mono2
tff(fact_5592_prod__diff1,axiom,
    ! [B: $tType,A: $tType] :
      ( semidom_divide(B)
     => ! [A2: set(A),F2: fun(A,B),A3: A] :
          ( finite_finite2(A,A2)
         => ( ( aa(A,B,F2,A3) != zero_zero(B) )
           => ( groups7121269368397514597t_prod(A,B,F2,aa(set(A),set(A),minus_minus(set(A),A2),aa(set(A),set(A),insert(A,A3),bot_bot(set(A))))) = $ite(member(A,A3,A2),aa(B,B,aa(B,fun(B,B),divide_divide(B),groups7121269368397514597t_prod(A,B,F2,A2)),aa(A,B,F2,A3)),groups7121269368397514597t_prod(A,B,F2,A2)) ) ) ) ) ).

% prod_diff1
tff(fact_5593_even__set__encode__iff,axiom,
    ! [A2: set(nat)] :
      ( finite_finite2(nat,A2)
     => ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(set(nat),nat,nat_set_encode,A2))
      <=> ~ member(nat,zero_zero(nat),A2) ) ) ).

% even_set_encode_iff
tff(fact_5594_ln__prod,axiom,
    ! [A: $tType,I5: set(A),F2: fun(A,real)] :
      ( finite_finite2(A,I5)
     => ( ! [I3: A] :
            ( member(A,I3,I5)
           => aa(real,$o,ord_less(real,zero_zero(real)),aa(A,real,F2,I3)) )
       => ( aa(real,real,ln_ln(real),groups7121269368397514597t_prod(A,real,F2,I5)) = aa(set(A),real,groups7311177749621191930dd_sum(A,real,aTP_Lamp_ng(fun(A,real),fun(A,real),F2)),I5) ) ) ) ).

% ln_prod
tff(fact_5595_polyfun__roots__finite,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),K: nat,Na: nat] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( aa(nat,$o,ord_less_eq(nat,K),Na)
           => finite_finite2(A,collect(A,aa(nat,fun(A,$o),aTP_Lamp_nh(fun(nat,A),fun(nat,fun(A,$o)),C2),Na))) ) ) ) ).

% polyfun_roots_finite
tff(fact_5596_polyfun__finite__roots,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Na: nat] :
          ( finite_finite2(A,collect(A,aa(nat,fun(A,$o),aTP_Lamp_nh(fun(nat,A),fun(nat,fun(A,$o)),C2),Na)))
        <=> ? [I4: nat] :
              ( aa(nat,$o,ord_less_eq(nat,I4),Na)
              & ( aa(nat,A,C2,I4) != zero_zero(A) ) ) ) ) ).

% polyfun_finite_roots
tff(fact_5597_sdiv__word__max,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(int,$o,ord_less_eq(int,signed7115095781618012415divide(int,aa(word(A),int,ring_1_signed(A,int),A3),aa(word(A),int,ring_1_signed(A,int),B3))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,size_size(word(A),A3)),aa(nat,nat,suc,zero_zero(nat))))) ) ).

% sdiv_word_max
tff(fact_5598_finite__Diff__insert,axiom,
    ! [A: $tType,A2: set(A),A3: A,B2: set(A)] :
      ( finite_finite2(A,aa(set(A),set(A),minus_minus(set(A),A2),aa(set(A),set(A),insert(A,A3),B2)))
    <=> finite_finite2(A,aa(set(A),set(A),minus_minus(set(A),A2),B2)) ) ).

% finite_Diff_insert
tff(fact_5599_finite__Collect__le__nat,axiom,
    ! [K: nat] : finite_finite2(nat,collect(nat,aTP_Lamp_ni(nat,fun(nat,$o),K))) ).

% finite_Collect_le_nat
tff(fact_5600_finite__Collect__less__nat,axiom,
    ! [K: nat] : finite_finite2(nat,collect(nat,aTP_Lamp_gc(nat,fun(nat,$o),K))) ).

% finite_Collect_less_nat
tff(fact_5601_finite__Collect__disjI,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( finite_finite2(A,collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ab(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)))
    <=> ( finite_finite2(A,collect(A,P))
        & finite_finite2(A,collect(A,Q)) ) ) ).

% finite_Collect_disjI
tff(fact_5602_finite__Collect__conjI,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ( finite_finite2(A,collect(A,P))
        | finite_finite2(A,collect(A,Q)) )
     => finite_finite2(A,collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_nj(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q))) ) ).

% finite_Collect_conjI
tff(fact_5603_finite__interval__int1,axiom,
    ! [A3: int,B3: int] : finite_finite2(int,collect(int,aa(int,fun(int,$o),aTP_Lamp_nk(int,fun(int,fun(int,$o)),A3),B3))) ).

% finite_interval_int1
tff(fact_5604_finite__interval__int4,axiom,
    ! [A3: int,B3: int] : finite_finite2(int,collect(int,aa(int,fun(int,$o),aTP_Lamp_nl(int,fun(int,fun(int,$o)),A3),B3))) ).

% finite_interval_int4
tff(fact_5605_finite__insert,axiom,
    ! [A: $tType,A3: A,A2: set(A)] :
      ( finite_finite2(A,aa(set(A),set(A),insert(A,A3),A2))
    <=> finite_finite2(A,A2) ) ).

% finite_insert
tff(fact_5606_finite__Un,axiom,
    ! [A: $tType,F3: set(A),G2: set(A)] :
      ( finite_finite2(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),F3),G2))
    <=> ( finite_finite2(A,F3)
        & finite_finite2(A,G2) ) ) ).

% finite_Un
tff(fact_5607_finite__interval__int3,axiom,
    ! [A3: int,B3: int] : finite_finite2(int,collect(int,aa(int,fun(int,$o),aTP_Lamp_nm(int,fun(int,fun(int,$o)),A3),B3))) ).

% finite_interval_int3
tff(fact_5608_finite__interval__int2,axiom,
    ! [A3: int,B3: int] : finite_finite2(int,collect(int,aa(int,fun(int,$o),aTP_Lamp_nn(int,fun(int,fun(int,$o)),A3),B3))) ).

% finite_interval_int2
tff(fact_5609_finite__nth__roots,axiom,
    ! [Na: nat,C2: complex] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => finite_finite2(complex,collect(complex,aa(complex,fun(complex,$o),aTP_Lamp_gn(nat,fun(complex,fun(complex,$o)),Na),C2))) ) ).

% finite_nth_roots
tff(fact_5610_finite__Collect__subsets,axiom,
    ! [A: $tType,A2: set(A)] :
      ( finite_finite2(A,A2)
     => finite_finite2(set(A),collect(set(A),aTP_Lamp_no(set(A),fun(set(A),$o),A2))) ) ).

% finite_Collect_subsets
tff(fact_5611_mset__le__asym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M7: multiset(A),N2: multiset(A)] :
          ( aa(multiset(A),$o,ord_less(multiset(A),M7),N2)
         => ~ aa(multiset(A),$o,ord_less(multiset(A),N2),M7) ) ) ).

% mset_le_asym
tff(fact_5612_mset__le__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K4: multiset(A),M7: multiset(A),N2: multiset(A)] :
          ( aa(multiset(A),$o,ord_less(multiset(A),K4),M7)
         => ( aa(multiset(A),$o,ord_less(multiset(A),M7),N2)
           => aa(multiset(A),$o,ord_less(multiset(A),K4),N2) ) ) ) ).

% mset_le_trans
tff(fact_5613_mset__le__irrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M7: multiset(A)] : ~ aa(multiset(A),$o,ord_less(multiset(A),M7),M7) ) ).

% mset_le_irrefl
tff(fact_5614_mset__le__not__sym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M7: multiset(A),N2: multiset(A)] :
          ( aa(multiset(A),$o,ord_less(multiset(A),M7),N2)
         => ~ aa(multiset(A),$o,ord_less(multiset(A),N2),M7) ) ) ).

% mset_le_not_sym
tff(fact_5615_mset__le__not__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M7: multiset(A)] : ~ aa(multiset(A),$o,ord_less(multiset(A),M7),M7) ) ).

% mset_le_not_refl
tff(fact_5616_less__eq__multiset__def,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M7: multiset(A),N2: multiset(A)] :
          ( aa(multiset(A),$o,ord_less_eq(multiset(A),M7),N2)
        <=> ( aa(multiset(A),$o,ord_less(multiset(A),M7),N2)
            | ( M7 = N2 ) ) ) ) ).

% less_eq_multiset_def
tff(fact_5617_finite__maxlen,axiom,
    ! [A: $tType,M7: set(list(A))] :
      ( finite_finite2(list(A),M7)
     => ? [N: nat] :
        ! [X2: list(A)] :
          ( member(list(A),X2,M7)
         => aa(nat,$o,ord_less(nat,size_size(list(A),X2)),N) ) ) ).

% finite_maxlen
tff(fact_5618_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I != zero_zero(int) )
     => finite_finite2(int,collect(int,aTP_Lamp_np(int,fun(int,$o),I))) ) ).

% finite_divisors_int
tff(fact_5619_pigeonhole__infinite__rel,axiom,
    ! [A: $tType,B: $tType,A2: set(A),B2: set(B),R: fun(A,fun(B,$o))] :
      ( ~ finite_finite2(A,A2)
     => ( finite_finite2(B,B2)
       => ( ! [X4: A] :
              ( member(A,X4,A2)
             => ? [Xa: B] :
                  ( member(B,Xa,B2)
                  & aa(B,$o,aa(A,fun(B,$o),R,X4),Xa) ) )
         => ? [X4: B] :
              ( member(B,X4,B2)
              & ~ finite_finite2(A,collect(A,aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_mc(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),A2),R),X4))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_5620_not__finite__existsD,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ~ finite_finite2(A,collect(A,P))
     => ? [X_12: A] : aa(A,$o,P,X_12) ) ).

% not_finite_existsD
tff(fact_5621_infinite__imp__nonempty,axiom,
    ! [A: $tType,S: set(A)] :
      ( ~ finite_finite2(A,S)
     => ( S != bot_bot(set(A)) ) ) ).

% infinite_imp_nonempty
tff(fact_5622_finite_OemptyI,axiom,
    ! [A: $tType] : finite_finite2(A,bot_bot(set(A))) ).

% finite.emptyI
tff(fact_5623_finite_OinsertI,axiom,
    ! [A: $tType,A2: set(A),A3: A] :
      ( finite_finite2(A,A2)
     => finite_finite2(A,aa(set(A),set(A),insert(A,A3),A2)) ) ).

% finite.insertI
tff(fact_5624_finite__UnI,axiom,
    ! [A: $tType,F3: set(A),G2: set(A)] :
      ( finite_finite2(A,F3)
     => ( finite_finite2(A,G2)
       => finite_finite2(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),F3),G2)) ) ) ).

% finite_UnI
tff(fact_5625_Un__infinite,axiom,
    ! [A: $tType,S: set(A),T3: set(A)] :
      ( ~ finite_finite2(A,S)
     => ~ finite_finite2(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T3)) ) ).

% Un_infinite
tff(fact_5626_infinite__Un,axiom,
    ! [A: $tType,S: set(A),T3: set(A)] :
      ( ~ finite_finite2(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T3))
    <=> ( ~ finite_finite2(A,S)
        | ~ finite_finite2(A,T3) ) ) ).

% infinite_Un
tff(fact_5627_finite__psubset__induct,axiom,
    ! [A: $tType,A2: set(A),P: fun(set(A),$o)] :
      ( finite_finite2(A,A2)
     => ( ! [A8: set(A)] :
            ( finite_finite2(A,A8)
           => ( ! [B8: set(A)] :
                  ( aa(set(A),$o,ord_less(set(A),B8),A8)
                 => aa(set(A),$o,P,B8) )
             => aa(set(A),$o,P,A8) ) )
       => aa(set(A),$o,P,A2) ) ) ).

% finite_psubset_induct
tff(fact_5628_finite__has__maximal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: set(A)] :
          ( finite_finite2(A,A2)
         => ( ( A2 != bot_bot(set(A)) )
           => ? [X4: A] :
                ( member(A,X4,A2)
                & ! [Xa: A] :
                    ( member(A,Xa,A2)
                   => ( aa(A,$o,ord_less_eq(A,X4),Xa)
                     => ( X4 = Xa ) ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_5629_finite__has__minimal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: set(A)] :
          ( finite_finite2(A,A2)
         => ( ( A2 != bot_bot(set(A)) )
           => ? [X4: A] :
                ( member(A,X4,A2)
                & ! [Xa: A] :
                    ( member(A,Xa,A2)
                   => ( aa(A,$o,ord_less_eq(A,Xa),X4)
                     => ( X4 = Xa ) ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_5630_finite_Ocases,axiom,
    ! [A: $tType,A3: set(A)] :
      ( finite_finite2(A,A3)
     => ( ( A3 != bot_bot(set(A)) )
       => ~ ! [A8: set(A)] :
              ( ? [A4: A] : A3 = aa(set(A),set(A),insert(A,A4),A8)
             => ~ finite_finite2(A,A8) ) ) ) ).

% finite.cases
tff(fact_5631_finite_Osimps,axiom,
    ! [A: $tType,A3: set(A)] :
      ( finite_finite2(A,A3)
    <=> ( ( A3 = bot_bot(set(A)) )
        | ? [A9: set(A),A10: A] :
            ( ( A3 = aa(set(A),set(A),insert(A,A10),A9) )
            & finite_finite2(A,A9) ) ) ) ).

% finite.simps
tff(fact_5632_finite__induct,axiom,
    ! [A: $tType,F3: set(A),P: fun(set(A),$o)] :
      ( finite_finite2(A,F3)
     => ( aa(set(A),$o,P,bot_bot(set(A)))
       => ( ! [X4: A,F5: set(A)] :
              ( finite_finite2(A,F5)
             => ( ~ member(A,X4,F5)
               => ( aa(set(A),$o,P,F5)
                 => aa(set(A),$o,P,aa(set(A),set(A),insert(A,X4),F5)) ) ) )
         => aa(set(A),$o,P,F3) ) ) ) ).

% finite_induct
tff(fact_5633_finite__ne__induct,axiom,
    ! [A: $tType,F3: set(A),P: fun(set(A),$o)] :
      ( finite_finite2(A,F3)
     => ( ( F3 != bot_bot(set(A)) )
       => ( ! [X4: A] : aa(set(A),$o,P,aa(set(A),set(A),insert(A,X4),bot_bot(set(A))))
         => ( ! [X4: A,F5: set(A)] :
                ( finite_finite2(A,F5)
               => ( ( F5 != bot_bot(set(A)) )
                 => ( ~ member(A,X4,F5)
                   => ( aa(set(A),$o,P,F5)
                     => aa(set(A),$o,P,aa(set(A),set(A),insert(A,X4),F5)) ) ) ) )
           => aa(set(A),$o,P,F3) ) ) ) ) ).

% finite_ne_induct
tff(fact_5634_infinite__finite__induct,axiom,
    ! [A: $tType,P: fun(set(A),$o),A2: set(A)] :
      ( ! [A8: set(A)] :
          ( ~ finite_finite2(A,A8)
         => aa(set(A),$o,P,A8) )
     => ( aa(set(A),$o,P,bot_bot(set(A)))
       => ( ! [X4: A,F5: set(A)] :
              ( finite_finite2(A,F5)
             => ( ~ member(A,X4,F5)
               => ( aa(set(A),$o,P,F5)
                 => aa(set(A),$o,P,aa(set(A),set(A),insert(A,X4),F5)) ) ) )
         => aa(set(A),$o,P,A2) ) ) ) ).

% infinite_finite_induct
tff(fact_5635_finite__subset__induct_H,axiom,
    ! [A: $tType,F3: set(A),A2: set(A),P: fun(set(A),$o)] :
      ( finite_finite2(A,F3)
     => ( aa(set(A),$o,ord_less_eq(set(A),F3),A2)
       => ( aa(set(A),$o,P,bot_bot(set(A)))
         => ( ! [A4: A,F5: set(A)] :
                ( finite_finite2(A,F5)
               => ( member(A,A4,A2)
                 => ( aa(set(A),$o,ord_less_eq(set(A),F5),A2)
                   => ( ~ member(A,A4,F5)
                     => ( aa(set(A),$o,P,F5)
                       => aa(set(A),$o,P,aa(set(A),set(A),insert(A,A4),F5)) ) ) ) ) )
           => aa(set(A),$o,P,F3) ) ) ) ) ).

% finite_subset_induct'
tff(fact_5636_finite__subset__induct,axiom,
    ! [A: $tType,F3: set(A),A2: set(A),P: fun(set(A),$o)] :
      ( finite_finite2(A,F3)
     => ( aa(set(A),$o,ord_less_eq(set(A),F3),A2)
       => ( aa(set(A),$o,P,bot_bot(set(A)))
         => ( ! [A4: A,F5: set(A)] :
                ( finite_finite2(A,F5)
               => ( member(A,A4,A2)
                 => ( ~ member(A,A4,F5)
                   => ( aa(set(A),$o,P,F5)
                     => aa(set(A),$o,P,aa(set(A),set(A),insert(A,A4),F5)) ) ) ) )
           => aa(set(A),$o,P,F3) ) ) ) ) ).

% finite_subset_induct
tff(fact_5637_finite__empty__induct,axiom,
    ! [A: $tType,A2: set(A),P: fun(set(A),$o)] :
      ( finite_finite2(A,A2)
     => ( aa(set(A),$o,P,A2)
       => ( ! [A4: A,A8: set(A)] :
              ( finite_finite2(A,A8)
             => ( member(A,A4,A8)
               => ( aa(set(A),$o,P,A8)
                 => aa(set(A),$o,P,aa(set(A),set(A),minus_minus(set(A),A8),aa(set(A),set(A),insert(A,A4),bot_bot(set(A))))) ) ) )
         => aa(set(A),$o,P,bot_bot(set(A))) ) ) ) ).

% finite_empty_induct
tff(fact_5638_infinite__coinduct,axiom,
    ! [A: $tType,X5: fun(set(A),$o),A2: set(A)] :
      ( aa(set(A),$o,X5,A2)
     => ( ! [A8: set(A)] :
            ( aa(set(A),$o,X5,A8)
           => ? [X2: A] :
                ( member(A,X2,A8)
                & ( aa(set(A),$o,X5,aa(set(A),set(A),minus_minus(set(A),A8),aa(set(A),set(A),insert(A,X2),bot_bot(set(A)))))
                  | ~ finite_finite2(A,aa(set(A),set(A),minus_minus(set(A),A8),aa(set(A),set(A),insert(A,X2),bot_bot(set(A))))) ) ) )
       => ~ finite_finite2(A,A2) ) ) ).

% infinite_coinduct
tff(fact_5639_infinite__remove,axiom,
    ! [A: $tType,S: set(A),A3: A] :
      ( ~ finite_finite2(A,S)
     => ~ finite_finite2(A,aa(set(A),set(A),minus_minus(set(A),S),aa(set(A),set(A),insert(A,A3),bot_bot(set(A))))) ) ).

% infinite_remove
tff(fact_5640_finite__remove__induct,axiom,
    ! [A: $tType,B2: set(A),P: fun(set(A),$o)] :
      ( finite_finite2(A,B2)
     => ( aa(set(A),$o,P,bot_bot(set(A)))
       => ( ! [A8: set(A)] :
              ( finite_finite2(A,A8)
             => ( ( A8 != bot_bot(set(A)) )
               => ( aa(set(A),$o,ord_less_eq(set(A),A8),B2)
                 => ( ! [X2: A] :
                        ( member(A,X2,A8)
                       => aa(set(A),$o,P,aa(set(A),set(A),minus_minus(set(A),A8),aa(set(A),set(A),insert(A,X2),bot_bot(set(A))))) )
                   => aa(set(A),$o,P,A8) ) ) ) )
         => aa(set(A),$o,P,B2) ) ) ) ).

% finite_remove_induct
tff(fact_5641_remove__induct,axiom,
    ! [A: $tType,P: fun(set(A),$o),B2: set(A)] :
      ( aa(set(A),$o,P,bot_bot(set(A)))
     => ( ( ~ finite_finite2(A,B2)
         => aa(set(A),$o,P,B2) )
       => ( ! [A8: set(A)] :
              ( finite_finite2(A,A8)
             => ( ( A8 != bot_bot(set(A)) )
               => ( aa(set(A),$o,ord_less_eq(set(A),A8),B2)
                 => ( ! [X2: A] :
                        ( member(A,X2,A8)
                       => aa(set(A),$o,P,aa(set(A),set(A),minus_minus(set(A),A8),aa(set(A),set(A),insert(A,X2),bot_bot(set(A))))) )
                   => aa(set(A),$o,P,A8) ) ) ) )
         => aa(set(A),$o,P,B2) ) ) ) ).

% remove_induct
tff(fact_5642_finite__induct__select,axiom,
    ! [A: $tType,S: set(A),P: fun(set(A),$o)] :
      ( finite_finite2(A,S)
     => ( aa(set(A),$o,P,bot_bot(set(A)))
       => ( ! [T5: set(A)] :
              ( aa(set(A),$o,ord_less(set(A),T5),S)
             => ( aa(set(A),$o,P,T5)
               => ? [X2: A] :
                    ( member(A,X2,aa(set(A),set(A),minus_minus(set(A),S),T5))
                    & aa(set(A),$o,P,aa(set(A),set(A),insert(A,X2),T5)) ) ) )
         => aa(set(A),$o,P,S) ) ) ) ).

% finite_induct_select
tff(fact_5643_sum__count__set,axiom,
    ! [A: $tType,Xsa: list(A),X5: set(A)] :
      ( aa(set(A),$o,ord_less_eq(set(A),aa(list(A),set(A),set2(A),Xsa)),X5)
     => ( finite_finite2(A,X5)
       => ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,count_list(A,Xsa)),X5) = size_size(list(A),Xsa) ) ) ) ).

% sum_count_set
tff(fact_5644_sum__diff1_H__aux,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [F3: set(A),I5: set(A),F2: fun(A,B),I: A] :
          ( finite_finite2(A,F3)
         => ( aa(set(A),$o,ord_less_eq(set(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_nq(set(A),fun(fun(A,B),fun(A,$o)),I5),F2))),F3)
           => ( groups1027152243600224163dd_sum(A,B,F2,aa(set(A),set(A),minus_minus(set(A),I5),aa(set(A),set(A),insert(A,I),bot_bot(set(A))))) = $ite(member(A,I,I5),aa(B,B,minus_minus(B,groups1027152243600224163dd_sum(A,B,F2,I5)),aa(A,B,F2,I)),groups1027152243600224163dd_sum(A,B,F2,I5)) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_5645_int__of__nat__def,axiom,
    code_T6385005292777649522of_nat = semiring_1_of_nat(int) ).

% int_of_nat_def
tff(fact_5646_sum_Oempty_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [P2: fun(B,A)] : groups1027152243600224163dd_sum(B,A,P2,bot_bot(set(B))) = zero_zero(A) ) ).

% sum.empty'
tff(fact_5647_count__notin,axiom,
    ! [A: $tType,X: A,Xsa: list(A)] :
      ( ~ member(A,X,aa(list(A),set(A),set2(A),Xsa))
     => ( aa(A,nat,count_list(A,Xsa),X) = zero_zero(nat) ) ) ).

% count_notin
tff(fact_5648_sum_Oinsert_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),P2: fun(A,B),I: A] :
          ( finite_finite2(A,collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_ml(set(A),fun(fun(A,B),fun(A,$o)),I5),P2)))
         => ( groups1027152243600224163dd_sum(A,B,P2,aa(set(A),set(A),insert(A,I),I5)) = $ite(member(A,I,I5),groups1027152243600224163dd_sum(A,B,P2,I5),aa(B,B,plus_plus(B,aa(A,B,P2,I)),groups1027152243600224163dd_sum(A,B,P2,I5))) ) ) ) ).

% sum.insert'
tff(fact_5649_sum_Onon__neutral_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),I5: set(B)] : groups1027152243600224163dd_sum(B,A,G,collect(B,aa(set(B),fun(B,$o),aTP_Lamp_nr(fun(B,A),fun(set(B),fun(B,$o)),G),I5))) = groups1027152243600224163dd_sum(B,A,G,I5) ) ).

% sum.non_neutral'
tff(fact_5650_sum_Odistrib__triv_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),G: fun(A,B),H: fun(A,B)] :
          ( finite_finite2(A,I5)
         => ( groups1027152243600224163dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ns(fun(A,B),fun(fun(A,B),fun(A,B)),G),H),I5) = aa(B,B,plus_plus(B,groups1027152243600224163dd_sum(A,B,G,I5)),groups1027152243600224163dd_sum(A,B,H,I5)) ) ) ) ).

% sum.distrib_triv'
tff(fact_5651_sum_Omono__neutral__cong__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S: set(A),T3: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,ord_less_eq(set(A),S),T3)
         => ( ! [X4: A] :
                ( member(A,X4,aa(set(A),set(A),minus_minus(set(A),T3),S))
               => ( aa(A,B,G,X4) = zero_zero(B) ) )
           => ( ! [X4: A] :
                  ( member(A,X4,S)
                 => ( aa(A,B,G,X4) = aa(A,B,H,X4) ) )
             => ( groups1027152243600224163dd_sum(A,B,G,T3) = groups1027152243600224163dd_sum(A,B,H,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_5652_sum_Omono__neutral__cong__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S: set(A),T3: set(A),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,ord_less_eq(set(A),S),T3)
         => ( ! [I3: A] :
                ( member(A,I3,aa(set(A),set(A),minus_minus(set(A),T3),S))
               => ( aa(A,B,H,I3) = zero_zero(B) ) )
           => ( ! [X4: A] :
                  ( member(A,X4,S)
                 => ( aa(A,B,G,X4) = aa(A,B,H,X4) ) )
             => ( groups1027152243600224163dd_sum(A,B,G,S) = groups1027152243600224163dd_sum(A,B,H,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_5653_sum_Omono__neutral__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S: set(A),T3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,ord_less_eq(set(A),S),T3)
         => ( ! [X4: A] :
                ( member(A,X4,aa(set(A),set(A),minus_minus(set(A),T3),S))
               => ( aa(A,B,G,X4) = zero_zero(B) ) )
           => ( groups1027152243600224163dd_sum(A,B,G,T3) = groups1027152243600224163dd_sum(A,B,G,S) ) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_5654_sum_Omono__neutral__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S: set(A),T3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,ord_less_eq(set(A),S),T3)
         => ( ! [X4: A] :
                ( member(A,X4,aa(set(A),set(A),minus_minus(set(A),T3),S))
               => ( aa(A,B,G,X4) = zero_zero(B) ) )
           => ( groups1027152243600224163dd_sum(A,B,G,S) = groups1027152243600224163dd_sum(A,B,G,T3) ) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_5655_sum_Odistrib_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),G: fun(A,B),H: fun(A,B)] :
          ( finite_finite2(A,collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_ml(set(A),fun(fun(A,B),fun(A,$o)),I5),G)))
         => ( finite_finite2(A,collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_ml(set(A),fun(fun(A,B),fun(A,$o)),I5),H)))
           => ( groups1027152243600224163dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ns(fun(A,B),fun(fun(A,B),fun(A,B)),G),H),I5) = aa(B,B,plus_plus(B,groups1027152243600224163dd_sum(A,B,G,I5)),groups1027152243600224163dd_sum(A,B,H,I5)) ) ) ) ) ).

% sum.distrib'
tff(fact_5656_sum_OG__def,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [P2: fun(B,A),I5: set(B)] :
          groups1027152243600224163dd_sum(B,A,P2,I5) = $ite(finite_finite2(B,collect(B,aa(set(B),fun(B,$o),aTP_Lamp_nr(fun(B,A),fun(set(B),fun(B,$o)),P2),I5))),aa(set(B),A,groups7311177749621191930dd_sum(B,A,P2),collect(B,aa(set(B),fun(B,$o),aTP_Lamp_nr(fun(B,A),fun(set(B),fun(B,$o)),P2),I5))),zero_zero(A)) ) ).

% sum.G_def
tff(fact_5657_count__le__length,axiom,
    ! [A: $tType,Xsa: list(A),X: A] : aa(nat,$o,ord_less_eq(nat,aa(A,nat,count_list(A,Xsa),X)),size_size(list(A),Xsa)) ).

% count_le_length
tff(fact_5658_sum__diff1_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [I5: set(A),F2: fun(A,B),I: A] :
          ( finite_finite2(A,collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_nq(set(A),fun(fun(A,B),fun(A,$o)),I5),F2)))
         => ( groups1027152243600224163dd_sum(A,B,F2,aa(set(A),set(A),minus_minus(set(A),I5),aa(set(A),set(A),insert(A,I),bot_bot(set(A))))) = $ite(member(A,I,I5),aa(B,B,minus_minus(B,groups1027152243600224163dd_sum(A,B,F2,I5)),aa(A,B,F2,I)),groups1027152243600224163dd_sum(A,B,F2,I5)) ) ) ) ).

% sum_diff1'
tff(fact_5659_infinite__int__iff__unbounded,axiom,
    ! [S: set(int)] :
      ( ~ finite_finite2(int,S)
    <=> ! [M3: int] :
        ? [N3: int] :
          ( aa(int,$o,ord_less(int,M3),aa(int,int,abs_abs(int),N3))
          & member(int,N3,S) ) ) ).

% infinite_int_iff_unbounded
tff(fact_5660_Sum__Ico__nat,axiom,
    ! [M: nat,Na: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_ik(nat,nat)),set_or7035219750837199246ssThan(nat,M,Na)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(nat,nat,minus_minus(nat,Na),one_one(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,minus_minus(nat,M),one_one(nat))))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% Sum_Ico_nat
tff(fact_5661_sum__power2,axiom,
    ! [K: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)))),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),K)),one_one(nat)) ).

% sum_power2
tff(fact_5662_atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( member(A,I,set_or7035219750837199246ssThan(A,L,U))
        <=> ( aa(A,$o,ord_less_eq(A,L),I)
            & aa(A,$o,ord_less(A,I),U) ) ) ) ).

% atLeastLessThan_iff
tff(fact_5663_atLeastLessThan__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
         => ( set_or7035219750837199246ssThan(A,A3,B3) = bot_bot(set(A)) ) ) ) ).

% atLeastLessThan_empty
tff(fact_5664_atLeastLessThan__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A] :
          ( ( set_or7035219750837199246ssThan(A,A3,B3) = bot_bot(set(A)) )
        <=> ~ aa(A,$o,ord_less(A,A3),B3) ) ) ).

% atLeastLessThan_empty_iff
tff(fact_5665_atLeastLessThan__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A] :
          ( ( bot_bot(set(A)) = set_or7035219750837199246ssThan(A,A3,B3) )
        <=> ~ aa(A,$o,ord_less(A,A3),B3) ) ) ).

% atLeastLessThan_empty_iff2
tff(fact_5666_infinite__Ico__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A] :
          ( ~ finite_finite2(A,set_or7035219750837199246ssThan(A,A3,B3))
        <=> aa(A,$o,ord_less(A,A3),B3) ) ) ).

% infinite_Ico_iff
tff(fact_5667_atLeastLessThan__singleton,axiom,
    ! [M: nat] : set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,M)) = aa(set(nat),set(nat),insert(nat,M),bot_bot(set(nat))) ).

% atLeastLessThan_singleton
tff(fact_5668_sum_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,Na))) = $ite(aa(nat,$o,ord_less(nat,Na),M),zero_zero(A),aa(A,A,plus_plus(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,Na))),aa(nat,A,G,Na))) ) ).

% sum.op_ivl_Suc
tff(fact_5669_prod_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] :
          groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,Na))) = $ite(aa(nat,$o,ord_less(nat,Na),M),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,Na))),aa(nat,A,G,Na))) ) ).

% prod.op_ivl_Suc
tff(fact_5670_infinite__Ico,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ~ finite_finite2(A,set_or7035219750837199246ssThan(A,A3,B3)) ) ) ).

% infinite_Ico
tff(fact_5671_ivl__disj__un__two_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,ord_less_eq(A,L),M)
         => ( aa(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_5672_atLeastLessThan0,axiom,
    ! [M: nat] : set_or7035219750837199246ssThan(nat,M,zero_zero(nat)) = bot_bot(set(nat)) ).

% atLeastLessThan0
tff(fact_5673_lessThan__atLeast0,axiom,
    ! [Na: nat] : set_ord_lessThan(nat,Na) = set_or7035219750837199246ssThan(nat,zero_zero(nat),Na) ).

% lessThan_atLeast0
tff(fact_5674_all__nat__less__eq,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( ! [M3: nat] :
          ( aa(nat,$o,ord_less(nat,M3),Na)
         => aa(nat,$o,P,M3) )
    <=> ! [X3: nat] :
          ( member(nat,X3,set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))
         => aa(nat,$o,P,X3) ) ) ).

% all_nat_less_eq
tff(fact_5675_ex__nat__less__eq,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( ? [M3: nat] :
          ( aa(nat,$o,ord_less(nat,M3),Na)
          & aa(nat,$o,P,M3) )
    <=> ? [X3: nat] :
          ( member(nat,X3,set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))
          & aa(nat,$o,P,X3) ) ) ).

% ex_nat_less_eq
tff(fact_5676_atLeastLessThan__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C2: A,D2: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(A,$o,ord_less(A,C2),D2)
           => ( ( set_or7035219750837199246ssThan(A,A3,B3) = set_or7035219750837199246ssThan(A,C2,D2) )
            <=> ( ( A3 = C2 )
                & ( B3 = D2 ) ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_5677_atLeastLessThan__inj_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C2: A,D2: A] :
          ( ( set_or7035219750837199246ssThan(A,A3,B3) = set_or7035219750837199246ssThan(A,C2,D2) )
         => ( aa(A,$o,ord_less(A,A3),B3)
           => ( aa(A,$o,ord_less(A,C2),D2)
             => ( A3 = C2 ) ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_5678_atLeastLessThan__inj_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C2: A,D2: A] :
          ( ( set_or7035219750837199246ssThan(A,A3,B3) = set_or7035219750837199246ssThan(A,C2,D2) )
         => ( aa(A,$o,ord_less(A,A3),B3)
           => ( aa(A,$o,ord_less(A,C2),D2)
             => ( B3 = D2 ) ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_5679_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Na))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ep(fun(nat,A),fun(nat,A),G)),set_or7035219750837199246ssThan(nat,M,Na)) ) ).

% sum.shift_bounds_Suc_ivl
tff(fact_5680_sum_Oshift__bounds__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,K: nat,Na: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,aa(nat,nat,plus_plus(nat,M),K),aa(nat,nat,plus_plus(nat,Na),K))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ib(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or7035219750837199246ssThan(nat,M,Na)) ) ).

% sum.shift_bounds_nat_ivl
tff(fact_5681_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Na))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_jf(fun(nat,A),fun(nat,A),G),set_or7035219750837199246ssThan(nat,M,Na)) ) ).

% prod.shift_bounds_Suc_ivl
tff(fact_5682_prod_Oshift__bounds__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,K: nat,Na: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,aa(nat,nat,plus_plus(nat,M),K),aa(nat,nat,plus_plus(nat,Na),K))) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jh(fun(nat,A),fun(nat,fun(nat,A)),G),K),set_or7035219750837199246ssThan(nat,M,Na)) ) ).

% prod.shift_bounds_nat_ivl
tff(fact_5683_sum_Oivl__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & comm_monoid_add(B) )
     => ! [A3: A,C2: A,B3: A,D2: A,G: fun(A,B),H: fun(A,B)] :
          ( ( A3 = C2 )
         => ( ( B3 = D2 )
           => ( ! [X4: A] :
                  ( aa(A,$o,ord_less_eq(A,C2),X4)
                 => ( aa(A,$o,ord_less(A,X4),D2)
                   => ( aa(A,B,G,X4) = aa(A,B,H,X4) ) ) )
             => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),set_or7035219750837199246ssThan(A,A3,B3)) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),set_or7035219750837199246ssThan(A,C2,D2)) ) ) ) ) ) ).

% sum.ivl_cong
tff(fact_5684_prod_Oivl__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & comm_monoid_mult(B) )
     => ! [A3: A,C2: A,B3: A,D2: A,G: fun(A,B),H: fun(A,B)] :
          ( ( A3 = C2 )
         => ( ( B3 = D2 )
           => ( ! [X4: A] :
                  ( aa(A,$o,ord_less_eq(A,C2),X4)
                 => ( aa(A,$o,ord_less(A,X4),D2)
                   => ( aa(A,B,G,X4) = aa(A,B,H,X4) ) ) )
             => ( groups7121269368397514597t_prod(A,B,G,set_or7035219750837199246ssThan(A,A3,B3)) = groups7121269368397514597t_prod(A,B,H,set_or7035219750837199246ssThan(A,C2,D2)) ) ) ) ) ) ).

% prod.ivl_cong
tff(fact_5685_ivl__disj__un__two_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,ord_less_eq(A,L),M)
         => ( aa(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_5686_ivl__disj__un__one_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$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_5687_atLeast0__lessThan__Suc,axiom,
    ! [Na: nat] : set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Na)) = aa(set(nat),set(nat),insert(nat,Na),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)) ).

% atLeast0_lessThan_Suc
tff(fact_5688_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => ( set_or7035219750837199246ssThan(nat,I,aa(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,plus_plus(nat,J),K))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_5689_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N2: set(nat),Na: nat] :
      ( aa(set(nat),$o,ord_less_eq(set(nat),N2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))
     => finite_finite2(nat,N2) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_5690_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A,C2: A,D2: A] :
          ( aa(set(A),$o,ord_less_eq(set(A),set_or1337092689740270186AtMost(A,A3,B3)),set_or7035219750837199246ssThan(A,C2,D2))
        <=> ( aa(A,$o,ord_less_eq(A,A3),B3)
           => ( aa(A,$o,ord_less_eq(A,C2),A3)
              & aa(A,$o,ord_less(A,B3),D2) ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_5691_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A,C2: A,D2: A] :
          ( aa(set(A),$o,ord_less_eq(set(A),set_or7035219750837199246ssThan(A,A3,B3)),set_or1337092689740270186AtMost(A,C2,D2))
        <=> ( aa(A,$o,ord_less(A,A3),B3)
           => ( aa(A,$o,ord_less_eq(A,C2),A3)
              & aa(A,$o,ord_less_eq(A,B3),D2) ) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
tff(fact_5692_ivl__disj__un__two__touch_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,ord_less_eq(A,L),M)
         => ( aa(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_5693_sum__shift__lb__Suc0__0__upt,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),K: nat] :
          ( ( aa(nat,A,F2,zero_zero(nat)) = zero_zero(A) )
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),K)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_5694_sum_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Na: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Na))) = aa(A,A,plus_plus(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))),aa(nat,A,G,Na)) ) ).

% sum.atLeast0_lessThan_Suc
tff(fact_5695_sum_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,ord_less(nat,M),Na)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,Na)) = aa(A,A,plus_plus(A,aa(nat,A,G,M)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),Na))) ) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_5696_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] : set_or7035219750837199246ssThan(A,A3,B3) = aa(set(A),set(A),minus_minus(set(A),set_or1337092689740270186AtMost(A,A3,B3)),aa(set(A),set(A),insert(A,B3),bot_bot(set(A)))) ) ).

% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_5697_prod_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Na: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))),aa(nat,A,G,Na)) ) ).

% prod.atLeast0_lessThan_Suc
tff(fact_5698_prod_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,ord_less(nat,M),Na)
         => ( groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),Na))) ) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_5699_atLeastLessThanSuc,axiom,
    ! [M: nat,Na: nat] :
      set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,Na)) = $ite(aa(nat,$o,ord_less_eq(nat,M),Na),aa(set(nat),set(nat),insert(nat,Na),set_or7035219750837199246ssThan(nat,M,Na)),bot_bot(set(nat))) ).

% atLeastLessThanSuc
tff(fact_5700_le__mask__high__bits,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Na: nat] :
          ( aa(word(A),$o,ord_less_eq(word(A),W),bit_se2239418461657761734s_mask(word(A),Na))
        <=> ! [X3: nat] :
              ( member(nat,X3,set_or7035219750837199246ssThan(nat,Na,size_size(word(A),W)))
             => ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),X3) ) ) ) ).

% le_mask_high_bits
tff(fact_5701_sum__Suc__diff_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,Na: nat,F2: fun(nat,A)] :
          ( aa(nat,$o,ord_less_eq(nat,M),Na)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_hh(fun(nat,A),fun(nat,A),F2)),set_or7035219750837199246ssThan(nat,M,Na)) = aa(A,A,minus_minus(A,aa(nat,A,F2,Na)),aa(nat,A,F2,M)) ) ) ) ).

% sum_Suc_diff'
tff(fact_5702_sum_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Na: nat,M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,Na,M)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_nt(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Na),M)),set_or7035219750837199246ssThan(nat,Na,M)) ) ).

% sum.atLeastLessThan_rev
tff(fact_5703_sum_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: fun(nat,fun(nat,A)),Na: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_nu(fun(nat,fun(nat,A)),fun(nat,A),A3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_if(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A3),Na)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)) ) ).

% sum.nested_swap
tff(fact_5704_prod_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Na: nat,M: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,Na,M)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_nv(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Na),M),set_or7035219750837199246ssThan(nat,Na,M)) ) ).

% prod.atLeastLessThan_rev
tff(fact_5705_prod_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: fun(nat,fun(nat,A)),Na: nat] : groups7121269368397514597t_prod(nat,A,aTP_Lamp_nw(fun(nat,fun(nat,A)),fun(nat,A),A3),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jm(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A3),Na),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)) ) ).

% prod.nested_swap
tff(fact_5706_sum_Onat__group,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),K: nat,Na: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_nx(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_ord_lessThan(nat,Na)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),K))) ) ).

% sum.nat_group
tff(fact_5707_prod_Onat__group,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),K: nat,Na: nat] : groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ny(fun(nat,A),fun(nat,fun(nat,A)),G),K),set_ord_lessThan(nat,Na)) = groups7121269368397514597t_prod(nat,A,G,set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),K))) ) ).

% prod.nat_group
tff(fact_5708_prod__Suc__fact,axiom,
    ! [Na: nat] : groups7121269368397514597t_prod(nat,nat,suc,set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)) = semiring_char_0_fact(nat,Na) ).

% prod_Suc_fact
tff(fact_5709_prod__Suc__Suc__fact,axiom,
    ! [Na: 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_5710_ivl__disj__un__singleton_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$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),insert(A,U),bot_bot(set(A)))) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(6)
tff(fact_5711_sum_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,Na)) = $ite(aa(nat,$o,ord_less(nat,Na),M),zero_zero(A),aa(A,A,plus_plus(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,Na))),aa(nat,A,G,Na))) ) ).

% sum.head_if
tff(fact_5712_prod_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] :
          groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,Na)) = $ite(aa(nat,$o,ord_less(nat,Na),M),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,Na))),aa(nat,A,G,Na))) ) ).

% prod.head_if
tff(fact_5713_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      set_or7035219750837199246ssThan(nat,M,aa(num,nat,numeral_numeral(nat),K)) = $ite(aa(nat,$o,ord_less_eq(nat,M),pred_numeral(K)),aa(set(nat),set(nat),insert(nat,pred_numeral(K)),set_or7035219750837199246ssThan(nat,M,pred_numeral(K))),bot_bot(set(nat))) ).

% atLeastLessThan_nat_numeral
tff(fact_5714_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),groups7121269368397514597t_prod(nat,nat,suc,set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))) ) ).

% fact_prod_Suc
tff(fact_5715_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Na: nat,M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,Na,M)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_ic(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Na),M)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Na),M)) ) ).

% sum.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_5716_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Na: nat,M: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,Na,M)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_jj(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Na),M),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Na),M)) ) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_5717_pochhammer__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,Na: nat] : comm_s3205402744901411588hammer(A,A3,Na) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_jp(A,fun(nat,A),A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)) ) ).

% pochhammer_prod
tff(fact_5718_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),groups7121269368397514597t_prod(nat,nat,minus_minus(nat,Na),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))) ) ).

% fact_prod_rev
tff(fact_5719_summable__Cauchy,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
        <=> ! [E3: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),E3)
             => ? [N8: nat] :
                ! [M3: nat] :
                  ( aa(nat,$o,ord_less_eq(nat,N8),M3)
                 => ! [N3: nat] : aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or7035219750837199246ssThan(nat,M3,N3)))),E3) ) ) ) ) ).

% summable_Cauchy
tff(fact_5720_sums__group,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A),Sa: A,K: nat] :
          ( aa(A,$o,sums(A,F2),Sa)
         => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
           => aa(A,$o,sums(A,aa(nat,fun(nat,A),aTP_Lamp_nz(fun(nat,A),fun(nat,fun(nat,A)),F2),K)),Sa) ) ) ) ).

% sums_group
tff(fact_5721_atLeast1__lessThan__eq__remove0,axiom,
    ! [Na: nat] : set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),Na) = aa(set(nat),set(nat),minus_minus(set(nat),set_ord_lessThan(nat,Na)),aa(set(nat),set(nat),insert(nat,zero_zero(nat)),bot_bot(set(nat)))) ).

% atLeast1_lessThan_eq_remove0
tff(fact_5722_take__bit__sum,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat,A3: A] : bit_se2584673776208193580ke_bit(A,Na,A3) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_oa(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)) ) ).

% take_bit_sum
tff(fact_5723_finite__transitivity__chain,axiom,
    ! [A: $tType,A2: set(A),R: fun(A,fun(A,$o))] :
      ( finite_finite2(A,A2)
     => ( ! [X4: A] : ~ aa(A,$o,aa(A,fun(A,$o),R,X4),X4)
       => ( ! [X4: A,Y3: A,Z2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),R,X4),Y3)
             => ( aa(A,$o,aa(A,fun(A,$o),R,Y3),Z2)
               => aa(A,$o,aa(A,fun(A,$o),R,X4),Z2) ) )
         => ( ! [X4: A] :
                ( member(A,X4,A2)
               => ? [Y4: A] :
                    ( member(A,Y4,A2)
                    & aa(A,$o,aa(A,fun(A,$o),R,X4),Y4) ) )
           => ( A2 = bot_bot(set(A)) ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_5724_fact__split,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [K: nat,Na: nat] :
          ( aa(nat,$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),groups7121269368397514597t_prod(nat,nat,suc,set_or7035219750837199246ssThan(nat,aa(nat,nat,minus_minus(nat,Na),K),Na)))),semiring_char_0_fact(A,aa(nat,nat,minus_minus(nat,Na),K))) ) ) ) ).

% fact_split
tff(fact_5725_binomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,ord_less_eq(nat,K),Na)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Na),K)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ob(nat,fun(nat,fun(nat,A)),K),Na),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ) ) ).

% binomial_altdef_of_nat
tff(fact_5726_gbinomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),K) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_oc(A,fun(nat,fun(nat,A)),A3),K),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_altdef_of_nat
tff(fact_5727_infinite__nat__iff__unbounded,axiom,
    ! [S: set(nat)] :
      ( ~ finite_finite2(nat,S)
    <=> ! [M3: nat] :
        ? [N3: nat] :
          ( aa(nat,$o,ord_less(nat,M3),N3)
          & member(nat,N3,S) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_5728_unbounded__k__infinite,axiom,
    ! [K: nat,S: set(nat)] :
      ( ! [M4: nat] :
          ( aa(nat,$o,ord_less(nat,K),M4)
         => ? [N10: nat] :
              ( aa(nat,$o,ord_less(nat,M4),N10)
              & member(nat,N10,S) ) )
     => ~ finite_finite2(nat,S) ) ).

% unbounded_k_infinite
tff(fact_5729_gbinomial__mult__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),aa(nat,A,gbinomial(A,A3),K)) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_od(A,fun(nat,A),A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact
tff(fact_5730_gbinomial__mult__fact_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),K)),semiring_char_0_fact(A,K)) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_od(A,fun(nat,A),A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact'
tff(fact_5731_gbinomial__prod__rev,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),K) = aa(A,A,aa(A,fun(A,A),divide_divide(A),groups7121269368397514597t_prod(nat,A,aTP_Lamp_ju(A,fun(nat,A),A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),K))),semiring_char_0_fact(A,K)) ) ).

% gbinomial_prod_rev
tff(fact_5732_Chebyshev__sum__upper,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat,A3: fun(nat,A),B3: fun(nat,A)] :
          ( ! [I3: nat,J2: nat] :
              ( aa(nat,$o,ord_less_eq(nat,I3),J2)
             => ( aa(nat,$o,ord_less(nat,J2),Na)
               => aa(A,$o,ord_less_eq(A,aa(nat,A,A3,I3)),aa(nat,A,A3,J2)) ) )
         => ( ! [I3: nat,J2: nat] :
                ( aa(nat,$o,ord_less_eq(nat,I3),J2)
               => ( aa(nat,$o,ord_less(nat,J2),Na)
                 => aa(A,$o,ord_less_eq(A,aa(nat,A,B3,J2)),aa(nat,A,B3,I3)) ) )
           => aa(A,$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,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_oe(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A3),B3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,B3),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)))) ) ) ) ).

% Chebyshev_sum_upper
tff(fact_5733_Chebyshev__sum__upper__nat,axiom,
    ! [Na: nat,A3: fun(nat,nat),B3: fun(nat,nat)] :
      ( ! [I3: nat,J2: nat] :
          ( aa(nat,$o,ord_less_eq(nat,I3),J2)
         => ( aa(nat,$o,ord_less(nat,J2),Na)
           => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,A3,I3)),aa(nat,nat,A3,J2)) ) )
     => ( ! [I3: nat,J2: nat] :
            ( aa(nat,$o,ord_less_eq(nat,I3),J2)
           => ( aa(nat,$o,ord_less(nat,J2),Na)
             => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,B3,J2)),aa(nat,nat,B3,I3)) ) )
       => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(fun(nat,nat),fun(nat,nat),aTP_Lamp_of(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),A3),B3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,B3),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_5734_smod__int__range,axiom,
    ! [B3: int,A3: int] :
      ( ( B3 != zero_zero(int) )
     => member(int,signed6721504322012087516modulo(int,A3,B3),set_or1337092689740270186AtMost(int,aa(int,int,plus_plus(int,aa(int,int,uminus_uminus(int),aa(int,int,abs_abs(int),B3))),one_one(int)),aa(int,int,minus_minus(int,aa(int,int,abs_abs(int),B3)),one_one(int)))) ) ).

% smod_int_range
tff(fact_5735_smod__int__mod__0,axiom,
    ! [X: int] : signed6721504322012087516modulo(int,X,zero_zero(int)) = X ).

% smod_int_mod_0
tff(fact_5736_smod__int__0__mod,axiom,
    ! [X: int] : signed6721504322012087516modulo(int,zero_zero(int),X) = zero_zero(int) ).

% smod_int_0_mod
tff(fact_5737_smod__int__numeral__numeral,axiom,
    ! [M: num,Na: num] : signed6721504322012087516modulo(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)) ).

% smod_int_numeral_numeral
tff(fact_5738_finite__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] : finite_finite2(code_integer,set_or7035219750837199246ssThan(code_integer,zero_zero(code_integer),U)) ).

% finite_atLeastZeroLessThan_integer
tff(fact_5739_word__to__1__set,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( set_or7035219750837199246ssThan(word(A),zero_zero(word(A)),one_one(word(A))) = aa(set(word(A)),set(word(A)),insert(word(A),zero_zero(word(A))),bot_bot(set(word(A)))) ) ) ).

% word_to_1_set
tff(fact_5740_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : finite_finite2(int,set_or7035219750837199246ssThan(int,zero_zero(int),U)) ).

% finite_atLeastZeroLessThan_int
tff(fact_5741_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or7035219750837199246ssThan(code_integer,L,aa(code_integer,code_integer,plus_plus(code_integer,U),one_one(code_integer))) = set_or1337092689740270186AtMost(code_integer,L,U) ).

% atLeastLessThanPlusOne_atLeastAtMost_integer
tff(fact_5742_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : set_or7035219750837199246ssThan(int,L,aa(int,int,plus_plus(int,U),one_one(int))) = set_or1337092689740270186AtMost(int,L,U) ).

% atLeastLessThanPlusOne_atLeastAtMost_int
tff(fact_5743_smod__int__compares_I1_J,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),A3)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),B3)
       => aa(int,$o,ord_less(int,signed6721504322012087516modulo(int,A3,B3)),B3) ) ) ).

% smod_int_compares(1)
tff(fact_5744_smod__int__compares_I2_J,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),A3)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),B3)
       => aa(int,$o,ord_less_eq(int,zero_zero(int)),signed6721504322012087516modulo(int,A3,B3)) ) ) ).

% smod_int_compares(2)
tff(fact_5745_smod__int__compares_I4_J,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,A3),zero_zero(int))
     => ( aa(int,$o,ord_less(int,zero_zero(int)),B3)
       => aa(int,$o,ord_less_eq(int,signed6721504322012087516modulo(int,A3,B3)),zero_zero(int)) ) ) ).

% smod_int_compares(4)
tff(fact_5746_smod__int__compares_I6_J,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),A3)
     => ( aa(int,$o,ord_less(int,B3),zero_zero(int))
       => aa(int,$o,ord_less_eq(int,zero_zero(int)),signed6721504322012087516modulo(int,A3,B3)) ) ) ).

% smod_int_compares(6)
tff(fact_5747_smod__int__compares_I7_J,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,A3),zero_zero(int))
     => ( aa(int,$o,ord_less(int,B3),zero_zero(int))
       => aa(int,$o,ord_less_eq(int,signed6721504322012087516modulo(int,A3,B3)),zero_zero(int)) ) ) ).

% smod_int_compares(7)
tff(fact_5748_smod__int__compares_I8_J,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,A3),zero_zero(int))
     => ( aa(int,$o,ord_less(int,B3),zero_zero(int))
       => aa(int,$o,ord_less_eq(int,B3),signed6721504322012087516modulo(int,A3,B3)) ) ) ).

% smod_int_compares(8)
tff(fact_5749_smod__mod__positive,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),A3)
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),B3)
       => ( signed6721504322012087516modulo(int,A3,B3) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B3) ) ) ) ).

% smod_mod_positive
tff(fact_5750_word__atLeastLessThan__Suc__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [U: word(A),L: word(A)] :
          ( ( U != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
         => ( set_or7035219750837199246ssThan(word(A),L,aa(word(A),word(A),plus_plus(word(A),U),one_one(word(A)))) = set_or1337092689740270186AtMost(word(A),L,U) ) ) ) ).

% word_atLeastLessThan_Suc_atLeastAtMost
tff(fact_5751_word__range__minus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: word(A),A3: word(A)] :
          ( ( B3 != zero_zero(word(A)) )
         => ( set_or1337092689740270186AtMost(word(A),A3,aa(word(A),word(A),minus_minus(word(A),B3),one_one(word(A)))) = set_or7035219750837199246ssThan(word(A),A3,B3) ) ) ) ).

% word_range_minus_1
tff(fact_5752_smod__int__compares_I3_J,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,A3),zero_zero(int))
     => ( aa(int,$o,ord_less(int,zero_zero(int)),B3)
       => aa(int,$o,ord_less(int,aa(int,int,uminus_uminus(int),B3)),signed6721504322012087516modulo(int,A3,B3)) ) ) ).

% smod_int_compares(3)
tff(fact_5753_smod__int__compares_I5_J,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),A3)
     => ( aa(int,$o,ord_less(int,B3),zero_zero(int))
       => aa(int,$o,ord_less(int,signed6721504322012087516modulo(int,A3,B3)),aa(int,int,uminus_uminus(int),B3)) ) ) ).

% smod_int_compares(5)
tff(fact_5754_word__count__from__top,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A)] :
          ( ( Na != zero_zero(word(A)) )
         => ( set_or7035219750837199246ssThan(word(A),zero_zero(word(A)),Na) = aa(set(word(A)),set(word(A)),aa(set(word(A)),fun(set(word(A)),set(word(A))),sup_sup(set(word(A))),set_or7035219750837199246ssThan(word(A),zero_zero(word(A)),aa(word(A),word(A),minus_minus(word(A),Na),one_one(word(A))))),aa(set(word(A)),set(word(A)),insert(word(A),aa(word(A),word(A),minus_minus(word(A),Na),one_one(word(A)))),bot_bot(set(word(A))))) ) ) ) ).

% word_count_from_top
tff(fact_5755_Multiset_Ois__empty__def,axiom,
    ! [A: $tType,A2: multiset(A)] :
      ( is_empty(A,A2)
    <=> ( A2 = zero_zero(multiset(A)) ) ) ).

% Multiset.is_empty_def
tff(fact_5756_mod__word__minus__1__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,minus_minus(int,aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(int))),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3))))) ) ).

% mod_word_minus_1_minus_numeral
tff(fact_5757_horner__sum__of__bool__2__less,axiom,
    ! [Bs: list($o)] : aa(int,$o,ord_less(int,groups4207007520872428315er_sum($o,int,zero_neq_one_of_bool(int),aa(num,int,numeral_numeral(int),bit0(one2)),Bs)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),size_size(list($o),Bs))) ).

% horner_sum_of_bool_2_less
tff(fact_5758_take__bit__length__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : bit_se2584673776208193580ke_bit(word(A),type_len0_len_of(A,type2(A)),W) = W ) ).

% take_bit_length_eq
tff(fact_5759_drop__bit__word__beyond,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] :
          ( aa(nat,$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),W) = zero_zero(word(A)) ) ) ) ).

% drop_bit_word_beyond
tff(fact_5760_push__bit__word__beyond,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] :
          ( aa(nat,$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),W) = zero_zero(word(A)) ) ) ) ).

% push_bit_word_beyond
tff(fact_5761_uint__bintrunc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: num] : aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),Bin)) = bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(num,int,numeral_numeral(int),Bin)) ) ).

% uint_bintrunc
tff(fact_5762_signed__1,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ring_1(A)
        & type_len(B) )
     => ( aa(word(B),A,ring_1_signed(B,A),one_one(word(B))) = $ite(type_len0_len_of(B,type2(B)) = one_one(nat),aa(A,A,uminus_uminus(A),one_one(A)),one_one(A)) ) ) ).

% signed_1
tff(fact_5763_word__exp__length__eq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),type_len0_len_of(A,type2(A))) = zero_zero(word(A)) ) ) ).

% word_exp_length_eq_0
tff(fact_5764_less__eq__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(num,word(A),numeral_numeral(word(A)),B3))
        <=> aa(int,$o,ord_less_eq(int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(num,int,numeral_numeral(int),A3))),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(num,int,numeral_numeral(int),B3))) ) ) ).

% less_eq_word_numeral_numeral
tff(fact_5765_less__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( aa(word(A),$o,ord_less(word(A),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(num,word(A),numeral_numeral(word(A)),B3))
        <=> aa(int,$o,ord_less(int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(num,int,numeral_numeral(int),A3))),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(num,int,numeral_numeral(int),B3))) ) ) ).

% less_word_numeral_numeral
tff(fact_5766_unat__bintrunc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: num] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(num,word(A),numeral_numeral(word(A)),Bin)) = nat2(bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(num,int,numeral_numeral(int),Bin))) ) ).

% unat_bintrunc
tff(fact_5767_bit__numeral__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: num,Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(num,word(A),numeral_numeral(word(A)),W)),Na)
        <=> ( aa(nat,$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),W)),Na) ) ) ) ).

% bit_numeral_word_iff
tff(fact_5768_ucast__bintr,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [W: num] : aa(word(B),word(A),semiring_1_unsigned(B,word(A)),aa(num,word(B),numeral_numeral(word(B)),W)) = aa(int,word(A),ring_1_of_int(word(A)),bit_se2584673776208193580ke_bit(int,type_len0_len_of(B,type2(B)),aa(num,int,numeral_numeral(int),W))) ) ).

% ucast_bintr
tff(fact_5769_unsigned__numeral,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & semiring_1(A) )
     => ! [Na: num] : aa(word(B),A,semiring_1_unsigned(B,A),aa(num,word(B),numeral_numeral(word(B)),Na)) = aa(nat,A,semiring_1_of_nat(A),bit_se2584673776208193580ke_bit(nat,type_len0_len_of(B,type2(B)),aa(num,nat,numeral_numeral(nat),Na))) ) ).

% unsigned_numeral
tff(fact_5770_unat__lt2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(nat,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% unat_lt2p
tff(fact_5771_uint__lt2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(int,$o,ord_less(int,aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_lt2p
tff(fact_5772_uint__bounded,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(int,$o,ord_less(int,aa(word(A),int,semiring_1_unsigned(A,int),W)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_bounded
tff(fact_5773_of__nat__2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) = zero_zero(word(A)) ) ) ).

% of_nat_2p
tff(fact_5774_exp__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( ( aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na) = zero_zero(word(A)) )
        <=> aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),Na) ) ) ).

% exp_eq_zero_iff
tff(fact_5775_signed__take__bit__word__Suc__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,K: num] : 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)),bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,Na),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_5776_signed__take__bit__word__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num,K: num] : bit_ri4674362597316999326ke_bit(word(A),aa(num,nat,numeral_numeral(nat),Na),aa(num,word(A),numeral_numeral(word(A)),K)) = aa(int,word(A),ring_1_of_int(word(A)),bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),Na),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(num,int,numeral_numeral(int),K)))) ) ).

% signed_take_bit_word_numeral
tff(fact_5777_sint__sbintrunc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: num] : aa(word(A),int,ring_1_signed(A,int),aa(num,word(A),numeral_numeral(word(A)),Bin)) = bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),aa(num,int,numeral_numeral(int),Bin)) ) ).

% sint_sbintrunc
tff(fact_5778_uint__bintrunc__neg,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: num] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Bin))) = bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Bin))) ) ).

% uint_bintrunc_neg
tff(fact_5779_div__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(num,int,numeral_numeral(int),A3))),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(num,int,numeral_numeral(int),B3)))) ) ).

% div_word_numeral_numeral
tff(fact_5780_mod__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(num,int,numeral_numeral(int),A3))),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(num,int,numeral_numeral(int),B3)))) ) ).

% mod_word_numeral_numeral
tff(fact_5781_signed__numeral,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_1(A) )
     => ! [Na: num] : aa(word(B),A,ring_1_signed(B,A),aa(num,word(B),numeral_numeral(word(B)),Na)) = aa(int,A,ring_1_of_int(A),bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(B,type2(B))),one_one(nat)),aa(num,int,numeral_numeral(int),Na))) ) ).

% signed_numeral
tff(fact_5782_less__eq__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3)))
        <=> aa(int,$o,ord_less_eq(int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3)))),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3)))) ) ) ).

% less_eq_word_minus_numeral_minus_numeral
tff(fact_5783_less__eq__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3)))
        <=> aa(int,$o,ord_less_eq(int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(num,int,numeral_numeral(int),A3))),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3)))) ) ) ).

% less_eq_word_numeral_minus_numeral
tff(fact_5784_less__eq__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3))),aa(num,word(A),numeral_numeral(word(A)),B3))
        <=> aa(int,$o,ord_less_eq(int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3)))),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(num,int,numeral_numeral(int),B3))) ) ) ).

% less_eq_word_minus_numeral_numeral
tff(fact_5785_less__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3))),aa(num,word(A),numeral_numeral(word(A)),B3))
        <=> aa(int,$o,ord_less(int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3)))),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(num,int,numeral_numeral(int),B3))) ) ) ).

% less_word_minus_numeral_numeral
tff(fact_5786_less__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( aa(word(A),$o,ord_less(word(A),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3)))
        <=> aa(int,$o,ord_less(int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(num,int,numeral_numeral(int),A3))),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3)))) ) ) ).

% less_word_numeral_minus_numeral
tff(fact_5787_less__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3)))
        <=> aa(int,$o,ord_less(int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3)))),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3)))) ) ) ).

% less_word_minus_numeral_minus_numeral
tff(fact_5788_unat__bintrunc__neg,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: num] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Bin))) = nat2(bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Bin)))) ) ).

% unat_bintrunc_neg
tff(fact_5789_bit__neg__numeral__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: num,Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),W))),Na)
        <=> ( aa(nat,$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),W))),Na) ) ) ) ).

% bit_neg_numeral_word_iff
tff(fact_5790_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)),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_5791_drop__bit__word__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num,K: num] : aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),aa(num,nat,numeral_numeral(nat),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(num,nat,numeral_numeral(nat),Na)),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(num,int,numeral_numeral(int),K)))) ) ).

% drop_bit_word_numeral
tff(fact_5792_unat__power__lower,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$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),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) ) ) ) ).

% unat_power_lower
tff(fact_5793_unsigned__neg__numeral,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & semiring_1(A) )
     => ! [Na: num] : aa(word(B),A,semiring_1_unsigned(B,A),aa(word(B),word(B),uminus_uminus(word(B)),aa(num,word(B),numeral_numeral(word(B)),Na))) = aa(nat,A,semiring_1_of_nat(A),nat2(bit_se2584673776208193580ke_bit(int,type_len0_len_of(B,type2(B)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na))))) ) ).

% unsigned_neg_numeral
tff(fact_5794_signed__take__bit__word__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,K: num] : 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)),bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,Na),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_5795_signed__take__bit__word__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num,K: num] : bit_ri4674362597316999326ke_bit(word(A),aa(num,nat,numeral_numeral(nat),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)),bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),Na),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_minus_numeral
tff(fact_5796_sint__sbintrunc__neg,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: num] : aa(word(A),int,ring_1_signed(A,int),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Bin))) = bit_ri4674362597316999326ke_bit(int,aa(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),Bin))) ) ).

% sint_sbintrunc_neg
tff(fact_5797_scast__sbintr,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [W: num] : aa(word(B),word(A),ring_1_signed(B,word(A)),aa(num,word(B),numeral_numeral(word(B)),W)) = aa(int,word(A),ring_1_of_int(word(A)),bit_ri4674362597316999326ke_bit(int,aa(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),W))) ) ).

% scast_sbintr
tff(fact_5798_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))),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_5799_div__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3)))),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3))))) ) ).

% div_word_minus_numeral_minus_numeral
tff(fact_5800_div__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(num,int,numeral_numeral(int),A3))),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3))))) ) ).

% div_word_numeral_minus_numeral
tff(fact_5801_div__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3))),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3)))),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(num,int,numeral_numeral(int),B3)))) ) ).

% div_word_minus_numeral_numeral
tff(fact_5802_mod__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3))),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3)))),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(num,int,numeral_numeral(int),B3)))) ) ).

% mod_word_minus_numeral_numeral
tff(fact_5803_mod__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(num,int,numeral_numeral(int),A3))),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3))))) ) ).

% mod_word_numeral_minus_numeral
tff(fact_5804_mod__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3)))),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3))))) ) ).

% mod_word_minus_numeral_minus_numeral
tff(fact_5805_word__less__sub__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,X: word(A)] :
          ( aa(nat,$o,ord_less(nat,Na),type_len0_len_of(A,type2(A)))
         => ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),one_one(word(A))))
          <=> aa(word(A),$o,ord_less(word(A),X),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) ) ) ) ).

% word_less_sub_le
tff(fact_5806_signed__neg__numeral,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_1(A) )
     => ! [Na: num] : aa(word(B),A,ring_1_signed(B,A),aa(word(B),word(B),uminus_uminus(word(B)),aa(num,word(B),numeral_numeral(word(B)),Na))) = aa(int,A,ring_1_of_int(A),bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(B,type2(B))),one_one(nat)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na)))) ) ).

% signed_neg_numeral
tff(fact_5807_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)),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_5808_drop__bit__word__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num,K: num] : aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),aa(num,nat,numeral_numeral(nat),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(num,nat,numeral_numeral(nat),Na)),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_minus_numeral
tff(fact_5809_less__word__numeral__minus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num] :
          ( aa(word(A),$o,ord_less(word(A),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))))
        <=> aa(int,$o,ord_less(int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(num,int,numeral_numeral(int),A3))),aa(int,int,minus_minus(int,aa(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_5810_less__word__minus__numeral__minus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3))),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))))
        <=> aa(int,$o,ord_less(int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3)))),aa(int,int,minus_minus(int,aa(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_5811_div__word__minus__1__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,minus_minus(int,aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(int))),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(num,int,numeral_numeral(int),B3)))) ) ).

% div_word_minus_1_numeral
tff(fact_5812_mod__word__minus__1__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,minus_minus(int,aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(int))),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(num,int,numeral_numeral(int),B3)))) ) ).

% mod_word_minus_1_numeral
tff(fact_5813_div__word__minus__1__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,minus_minus(int,aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(int))),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3))))) ) ).

% div_word_minus_1_minus_numeral
tff(fact_5814_ucast__mask__drop,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Na: nat,X: word(B)] :
          ( aa(nat,$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)),X),bit_se2239418461657761734s_mask(word(B),Na))) = aa(word(B),word(A),semiring_1_unsigned(B,word(A)),X) ) ) ) ).

% ucast_mask_drop
tff(fact_5815_uint__word__arith__bintrs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(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)),A3),B3)) = bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,semiring_1_unsigned(A,int),A3)),aa(word(A),int,semiring_1_unsigned(A,int),B3))) ) ).

% uint_word_arith_bintrs(3)
tff(fact_5816_signed__take__bit__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_ri3973907225187159222ations(A) )
     => ! [Na: nat,W: word(B)] :
          aa(word(B),A,ring_1_signed(B,A),bit_se2584673776208193580ke_bit(word(B),Na,W)) = $ite(aa(nat,$o,ord_less(nat,Na),type_len0_len_of(B,type2(B))),bit_se2584673776208193580ke_bit(A,Na,aa(word(B),A,ring_1_signed(B,A),W)),aa(word(B),A,ring_1_signed(B,A),W)) ) ).

% signed_take_bit_eq
tff(fact_5817_word__cat__inj,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(C)
        & type_len(A) )
     => ! [A3: word(A),B3: word(B),C2: word(A),D2: word(B)] :
          ( aa(nat,$o,ord_less_eq(nat,aa(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,A3,B3) = word_cat(A,B,C,C2,D2) )
          <=> ( ( A3 = C2 )
              & ( B3 = D2 ) ) ) ) ) ).

% word_cat_inj
tff(fact_5818_ucast__ucast__add,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [X: word(A),Y: word(B)] :
          ( aa(nat,$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),plus_plus(word(B),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),X)),Y)) = aa(word(A),word(A),plus_plus(word(A),X),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Y)) ) ) ) ).

% ucast_ucast_add
tff(fact_5819_bintr__uint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),Na)
         => ( bit_se2584673776208193580ke_bit(int,Na,aa(word(A),int,semiring_1_unsigned(A,int),W)) = aa(word(A),int,semiring_1_unsigned(A,int),W) ) ) ) ).

% bintr_uint
tff(fact_5820_wi__bintr,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: int] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),Na)
         => ( aa(int,word(A),ring_1_of_int(word(A)),bit_se2584673776208193580ke_bit(int,Na,W)) = aa(int,word(A),ring_1_of_int(word(A)),W) ) ) ) ).

% wi_bintr
tff(fact_5821_eq__ucast__ucast__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(B),Y: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( ( X = aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y) )
           => ( aa(word(B),word(A),semiring_1_unsigned(B,word(A)),X) = Y ) ) ) ) ).

% eq_ucast_ucast_eq
tff(fact_5822_up__ucast__inj__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(A),Y: word(A)] :
          ( aa(nat,$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)),X) = aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y) )
          <=> ( X = Y ) ) ) ) ).

% up_ucast_inj_eq
tff(fact_5823_ucast__ucast__eq,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(B)
        & type_len(A) )
     => ! [X: word(B),Y: word(C)] :
          ( ( aa(word(B),word(A),semiring_1_unsigned(B,word(A)),X) = aa(word(B),word(A),semiring_1_unsigned(B,word(A)),aa(word(C),word(B),semiring_1_unsigned(C,word(B)),Y)) )
         => ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(B,type2(B))),type_len0_len_of(C,type2(C)))
           => ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(C,type2(C))),type_len0_len_of(A,type2(A)))
             => ( X = aa(word(C),word(B),semiring_1_unsigned(C,word(B)),Y) ) ) ) ) ) ).

% ucast_ucast_eq
tff(fact_5824_up__ucast__inj,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [X: word(B),Y: word(B)] :
          ( ( aa(word(B),word(A),semiring_1_unsigned(B,word(A)),X) = aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Y) )
         => ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(B,type2(B))),type_len0_len_of(A,type2(A)))
           => ( X = Y ) ) ) ) ).

% up_ucast_inj
tff(fact_5825_up__scast__inj__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(A),Y: word(A)] :
          ( aa(nat,$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)),X) = aa(word(A),word(B),ring_1_signed(A,word(B)),Y) )
          <=> ( X = Y ) ) ) ) ).

% up_scast_inj_eq
tff(fact_5826_take__bit__word__beyond__length__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),Na)
         => ( bit_se2584673776208193580ke_bit(word(A),Na,W) = W ) ) ) ).

% take_bit_word_beyond_length_eq
tff(fact_5827_unat__ucast__up__simp,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(A)] :
          ( aa(nat,$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)),X)) = aa(word(A),nat,semiring_1_unsigned(A,nat),X) ) ) ) ).

% unat_ucast_up_simp
tff(fact_5828_less__ucast__ucast__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(B),Y: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(B),$o,ord_less(word(B),X),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y))
           => aa(word(A),$o,ord_less(word(A),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),X)),Y) ) ) ) ).

% less_ucast_ucast_less
tff(fact_5829_ucast__less__ucast,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(A),Y: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(B),$o,ord_less(word(B),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),X)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y))
          <=> aa(word(A),$o,ord_less(word(A),X),Y) ) ) ) ).

% ucast_less_ucast
tff(fact_5830_ucast__up__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(A),Y: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(A),$o,ord_less(word(A),X),Y)
           => aa(word(B),$o,ord_less(word(B),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),X)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y)) ) ) ) ).

% ucast_up_mono
tff(fact_5831_ucast__up__mono__le,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(A),Y: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(A),$o,ord_less_eq(word(A),X),Y)
           => aa(word(B),$o,ord_less_eq(word(B),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),X)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y)) ) ) ) ).

% ucast_up_mono_le
tff(fact_5832_ucast__le__ucast,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(A),Y: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(B),$o,ord_less_eq(word(B),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),X)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y))
          <=> aa(word(A),$o,ord_less_eq(word(A),X),Y) ) ) ) ).

% ucast_le_ucast
tff(fact_5833_ucast__drop__bit__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Na: nat,W: word(A)] :
          ( aa(nat,$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),W)) = 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)),W)) ) ) ) ).

% ucast_drop_bit_eq
tff(fact_5834_up__scast__inj,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [X: word(B),Y: word(B)] :
          ( ( aa(word(B),word(A),ring_1_signed(B,word(A)),X) = aa(word(B),word(A),ring_1_signed(B,word(A)),Y) )
         => ( aa(nat,$o,ord_less_eq(nat,size_size(word(B),X)),type_len0_len_of(A,type2(A)))
           => ( X = Y ) ) ) ) ).

% up_scast_inj
tff(fact_5835_word__of__int__less__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: int,L: int] :
          ( aa(word(A),$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,ord_less_eq(int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),K)),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),L)) ) ) ).

% word_of_int_less_eq_iff
tff(fact_5836_uint__word__arith__bintrs_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),uminus_uminus(word(A)),A3)) = bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(int,int,uminus_uminus(int),aa(word(A),int,semiring_1_unsigned(A,int),A3))) ) ).

% uint_word_arith_bintrs(4)
tff(fact_5837_unsigned__minus__1__eq__mask,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_se359711467146920520ations(A) )
     => ( aa(word(B),A,semiring_1_unsigned(B,A),aa(word(B),word(B),uminus_uminus(word(B)),one_one(word(B)))) = bit_se2239418461657761734s_mask(A,type_len0_len_of(B,type2(B))) ) ) ).

% unsigned_minus_1_eq_mask
tff(fact_5838_mask__over__length,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),Na)
         => ( bit_se2239418461657761734s_mask(word(A),Na) = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% mask_over_length
tff(fact_5839_unsigned__not__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_ri3973907225187159222ations(A) )
     => ! [W: word(B)] : aa(word(B),A,semiring_1_unsigned(B,A),aa(word(B),word(B),bit_ri4277139882892585799ns_not(word(B)),W)) = bit_se2584673776208193580ke_bit(A,type_len0_len_of(B,type2(B)),aa(A,A,bit_ri4277139882892585799ns_not(A),aa(word(B),A,semiring_1_unsigned(B,A),W))) ) ).

% unsigned_not_eq
tff(fact_5840_uint__word__arith__bintrs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),minus_minus(word(A),A3),B3)) = bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),A3)),aa(word(A),int,semiring_1_unsigned(A,int),B3))) ) ).

% uint_word_arith_bintrs(2)
tff(fact_5841_test__bit__bin,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),Na)
        <=> ( aa(nat,$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),W)),Na) ) ) ) ).

% test_bit_bin
tff(fact_5842_bit__uint__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Na: nat] :
          ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(word(A),int,semiring_1_unsigned(A,int),W)),Na)
        <=> ( aa(nat,$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)),W),Na) ) ) ) ).

% bit_uint_iff
tff(fact_5843_ucast__sub__ucast,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Y: word(A),X: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Y),X)
         => ( aa(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),minus_minus(word(A),X),Y)) = aa(word(B),word(B),minus_minus(word(B),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),X)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y)) ) ) ) ) ).

% ucast_sub_ucast
tff(fact_5844_word__and__full__mask__simp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),bit_se2239418461657761734s_mask(word(A),type_len0_len_of(A,type2(A)))) = X ) ).

% word_and_full_mask_simp
tff(fact_5845_ucast__ucast__mask,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [X: word(A)] : aa(word(B),word(A),semiring_1_unsigned(B,word(A)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),X)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),bit_se2239418461657761734s_mask(word(A),type_len0_len_of(B,type2(B)))) ) ).

% ucast_ucast_mask
tff(fact_5846_one__word_Orsp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),one_one(int)) = bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),one_one(int)) ) ) ).

% one_word.rsp
tff(fact_5847_uint__sint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),W) = bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(word(A),int,ring_1_signed(A,int),W)) ) ).

% uint_sint
tff(fact_5848_word__of__int__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: int,L: int] :
          ( ( aa(int,word(A),ring_1_of_int(word(A)),K) = aa(int,word(A),ring_1_of_int(word(A)),L) )
        <=> ( bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),K) = bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),L) ) ) ) ).

% word_of_int_eq_iff
tff(fact_5849_uint__word__of__int__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: int] : aa(word(A),int,semiring_1_unsigned(A,int),aa(int,word(A),ring_1_of_int(word(A)),K)) = bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),K) ) ).

% uint_word_of_int_eq
tff(fact_5850_unsigned__ucast__eq,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( type_len(C)
        & bit_se359711467146920520ations(A)
        & type_len(B) )
     => ! [W: word(C)] : aa(word(B),A,semiring_1_unsigned(B,A),aa(word(C),word(B),semiring_1_unsigned(C,word(B)),W)) = bit_se2584673776208193580ke_bit(A,type_len0_len_of(B,type2(B)),aa(word(C),A,semiring_1_unsigned(C,A),W)) ) ).

% unsigned_ucast_eq
tff(fact_5851_word__of__nat__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,Na: nat] :
          ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),M) = aa(nat,word(A),semiring_1_of_nat(word(A)),Na) )
        <=> ( bit_se2584673776208193580ke_bit(nat,type_len0_len_of(A,type2(A)),M) = bit_se2584673776208193580ke_bit(nat,type_len0_len_of(A,type2(A)),Na) ) ) ) ).

% word_of_nat_eq_iff
tff(fact_5852_not__bit__length,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),type_len0_len_of(A,type2(A))) ) ).

% not_bit_length
tff(fact_5853_uint32_Osize__eq__length,axiom,
    aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(one2)))))) = type_len0_len_of(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))),type2(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))) ).

% uint32.size_eq_length
tff(fact_5854_num__of__bintr_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( ( bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(num,int,numeral_numeral(int),A3)) = aa(num,int,numeral_numeral(int),B3) )
         => ( aa(num,word(A),numeral_numeral(word(A)),A3) = aa(num,word(A),numeral_numeral(word(A)),B3) ) ) ) ).

% num_of_bintr'
tff(fact_5855_word__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : size_size(word(A),W) = type_len0_len_of(A,type2(A)) ) ).

% word_size
tff(fact_5856_size__word_Orep__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : size_size(word(A),X) = type_len0_len_of(A,type2(A)) ) ).

% size_word.rep_eq
tff(fact_5857_zero__word_Orsp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),zero_zero(int)) = bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),zero_zero(int)) ) ) ).

% zero_word.rsp
tff(fact_5858_size__0__same,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),V2: word(A)] :
          ( ( type_len0_len_of(A,type2(A)) = zero_zero(nat) )
         => ( W = V2 ) ) ) ).

% size_0_same
tff(fact_5859_degenerate__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( ( type_len0_len_of(A,type2(A)) = one_one(nat) )
         => ( ( X = zero_zero(word(A)) )
            | ( X = one_one(word(A)) ) ) ) ) ).

% degenerate_word
tff(fact_5860_word__of__int__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: int,L: int] :
          ( aa(word(A),$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,ord_less(int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),K)),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),L)) ) ) ).

% word_of_int_less_iff
tff(fact_5861_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,ord_less(nat,Na),type_len0_len_of(A,type2(A))) ) ) ).

% max_test_bit
tff(fact_5862_uint__word__arith__bintrs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),plus_plus(word(A),A3),B3)) = bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(int,int,plus_plus(int,aa(word(A),int,semiring_1_unsigned(A,int),A3)),aa(word(A),int,semiring_1_unsigned(A,int),B3))) ) ).

% uint_word_arith_bintrs(1)
tff(fact_5863_uint__word__arith__bintrs_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),int,semiring_1_unsigned(A,int),one_one(word(A))) = bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),one_one(int)) ) ) ).

% uint_word_arith_bintrs(8)
tff(fact_5864_uint__word__arith__bintrs_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),int,semiring_1_unsigned(A,int),zero_zero(word(A))) = bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),zero_zero(int)) ) ) ).

% uint_word_arith_bintrs(7)
tff(fact_5865_num__abs__bintr,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: num] : aa(num,word(A),numeral_numeral(word(A)),X) = aa(int,word(A),ring_1_of_int(word(A)),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(num,int,numeral_numeral(int),X))) ) ).

% num_abs_bintr
tff(fact_5866_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,ord_less(nat,zero_zero(nat)),type_len0_len_of(A,type2(A)))
            & ( Na = zero_zero(nat) ) ) ) ) ).

% test_bit_1'
tff(fact_5867_bit__set__bit__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),M: nat,B3: $o,Na: nat] :
          ( aa(nat,$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)),W),M),(B3))),Na)
        <=> $ite(
              M = Na,
              ( aa(nat,$o,ord_less(nat,Na),type_len0_len_of(A,type2(A)))
              & (B3) ),
              aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),Na) ) ) ) ).

% bit_set_bit_word_iff
tff(fact_5868_neg__test__bit,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),X)),Na)
        <=> ( ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),X),Na)
            & aa(nat,$o,ord_less(nat,Na),type_len0_len_of(A,type2(A))) ) ) ) ).

% neg_test_bit
tff(fact_5869_test__bit__conj__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),M: nat] :
          ( ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),X),M)
            & aa(nat,$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)),X),M) ) ) ).

% test_bit_conj_lt
tff(fact_5870_bit__imp__le__length,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),Na)
         => aa(nat,$o,ord_less(nat,Na),type_len0_len_of(A,type2(A))) ) ) ).

% bit_imp_le_length
tff(fact_5871_word__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ( X = Y )
        <=> ! [N3: nat] :
              ( aa(nat,$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)),X),N3)
              <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),N3) ) ) ) ) ).

% word_eq_iff
tff(fact_5872_bit__word__eqI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          ( ! [N: nat] :
              ( aa(nat,$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)),A3),N)
              <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),B3),N) ) )
         => ( A3 = B3 ) ) ) ).

% bit_word_eqI
tff(fact_5873_bit__word__ucast__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [W: word(B),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),W)),Na)
        <=> ( aa(nat,$o,ord_less(nat,Na),type_len0_len_of(B,type2(B)))
            & aa(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)),W),Na) ) ) ) ).

% bit_word_ucast_iff
tff(fact_5874_nth__ucast,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [W: word(B),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),W)),Na)
        <=> ( aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),W),Na)
            & aa(nat,$o,ord_less(nat,Na),type_len0_len_of(A,type2(A))) ) ) ) ).

% nth_ucast
tff(fact_5875_bit__ucast__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [A3: word(B),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),A3)),Na)
        <=> ( aa(nat,$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)),A3),Na) ) ) ) ).

% bit_ucast_iff
tff(fact_5876_bin__nth__uint__imp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Na: nat] :
          ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(word(A),int,semiring_1_unsigned(A,int),W)),Na)
         => aa(nat,$o,ord_less(nat,Na),type_len0_len_of(A,type2(A))) ) ) ).

% bin_nth_uint_imp
tff(fact_5877_max__word__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) = bit_se2239418461657761734s_mask(word(A),type_len0_len_of(A,type2(A))) ) ) ).

% max_word_mask
tff(fact_5878_ucast__less__ucast__weak,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(A),Y: word(A)] :
          ( aa(nat,$o,ord_less(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(B),$o,ord_less(word(B),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),X)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y))
          <=> aa(word(A),$o,ord_less(word(A),X),Y) ) ) ) ).

% ucast_less_ucast_weak
tff(fact_5879_ucast__ucast__id,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(A)] :
          ( aa(nat,$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)),X)) = X ) ) ) ).

% ucast_ucast_id
tff(fact_5880_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,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_5881_test__bit__wi,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: int,Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(int,word(A),ring_1_of_int(word(A)),X)),Na)
        <=> ( aa(nat,$o,ord_less(nat,Na),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),X),Na) ) ) ) ).

% test_bit_wi
tff(fact_5882_signed__not__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_ri3973907225187159222ations(A) )
     => ! [W: word(B)] : aa(word(B),A,ring_1_signed(B,A),aa(word(B),word(B),bit_ri4277139882892585799ns_not(word(B)),W)) = bit_ri4674362597316999326ke_bit(A,type_len0_len_of(B,type2(B)),aa(A,A,bit_ri4277139882892585799ns_not(A),aa(word(B),A,ring_1_signed(B,A),W))) ) ).

% signed_not_eq
tff(fact_5883_unsigned__drop__bit__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_un5681908812861735899ations(A) )
     => ! [Na: nat,W: word(B)] : aa(word(B),A,semiring_1_unsigned(B,A),aa(word(B),word(B),aa(nat,fun(word(B),word(B)),bit_se4197421643247451524op_bit(word(B)),Na),W)) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),bit_se2584673776208193580ke_bit(A,type_len0_len_of(B,type2(B)),aa(word(B),A,semiring_1_unsigned(B,A),W))) ) ).

% unsigned_drop_bit_eq
tff(fact_5884_scast__1,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ( aa(word(B),word(A),ring_1_signed(B,word(A)),one_one(word(B))) = $ite(type_len0_len_of(B,type2(B)) = one_one(nat),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))),one_one(word(A))) ) ) ).

% scast_1
tff(fact_5885_unsigned__push__bit__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_se359711467146920520ations(A) )
     => ! [Na: nat,W: word(B)] : aa(word(B),A,semiring_1_unsigned(B,A),aa(word(B),word(B),aa(nat,fun(word(B),word(B)),bit_se4730199178511100633sh_bit(word(B)),Na),W)) = bit_se2584673776208193580ke_bit(A,type_len0_len_of(B,type2(B)),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),aa(word(B),A,semiring_1_unsigned(B,A),W))) ) ).

% unsigned_push_bit_eq
tff(fact_5886_unsigned__of__nat,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & semiring_1(A) )
     => ! [Na: nat] : aa(word(B),A,semiring_1_unsigned(B,A),aa(nat,word(B),semiring_1_of_nat(word(B)),Na)) = aa(nat,A,semiring_1_of_nat(A),bit_se2584673776208193580ke_bit(nat,type_len0_len_of(B,type2(B)),Na)) ) ).

% unsigned_of_nat
tff(fact_5887_bit__word__cat__iff,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(B)
        & type_len(A) )
     => ! [V2: word(B),W: word(C),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),word_cat(B,C,A,V2,W)),Na)
        <=> ( aa(nat,$o,ord_less(nat,Na),type_len0_len_of(A,type2(A)))
            & $ite(aa(nat,$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)),W),Na),aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),V2),aa(nat,nat,minus_minus(nat,Na),type_len0_len_of(C,type2(C))))) ) ) ) ).

% bit_word_cat_iff
tff(fact_5888_bit__set__bit__aux,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F2: fun(nat,$o),Na: nat,W: word(A),M: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),code_T2661198915054445665ts_aux(A,F2,Na,W)),M)
        <=> ( aa(nat,$o,ord_less(nat,M),type_len0_len_of(A,type2(A)))
            & $ite(aa(nat,$o,ord_less(nat,M),Na),aa(nat,$o,F2,M),aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),aa(nat,nat,minus_minus(nat,M),Na))) ) ) ) ).

% bit_set_bit_aux
tff(fact_5889_word__of__nat__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,Na: nat] :
          ( aa(word(A),$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,ord_less(nat,bit_se2584673776208193580ke_bit(nat,type_len0_len_of(A,type2(A)),M)),bit_se2584673776208193580ke_bit(nat,type_len0_len_of(A,type2(A)),Na)) ) ) ).

% word_of_nat_less_iff
tff(fact_5890_word__of__nat__less__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,Na: nat] :
          ( aa(word(A),$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,ord_less_eq(nat,bit_se2584673776208193580ke_bit(nat,type_len0_len_of(A,type2(A)),M)),bit_se2584673776208193580ke_bit(nat,type_len0_len_of(A,type2(A)),Na)) ) ) ).

% word_of_nat_less_eq_iff
tff(fact_5891_nth__slice,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Na: nat,W: word(B),M: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(B),word(A),slice2(B,A,Na),W)),M)
        <=> ( aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),W),aa(nat,nat,plus_plus(nat,M),Na))
            & aa(nat,$o,ord_less(nat,M),type_len0_len_of(A,type2(A))) ) ) ) ).

% nth_slice
tff(fact_5892_word__cat__eq_H,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(C)
        & type_len(A) )
     => ! [A3: word(B),B3: word(C)] : word_cat(B,C,A,A3,B3) = aa(int,word(A),ring_1_of_int(word(A)),bit_concat_bit(type_len0_len_of(C,type2(C)),aa(word(C),int,semiring_1_unsigned(C,int),B3),aa(word(B),int,semiring_1_unsigned(B,int),A3))) ) ).

% word_cat_eq'
tff(fact_5893_word__cat__bin_H,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(A)
        & type_len(B) )
     => ! [V2: word(B),W: word(C)] : word_cat(B,C,A,V2,W) = aa(word(A),word(A),plus_plus(word(A),aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4730199178511100633sh_bit(word(A)),type_len0_len_of(C,type2(C))),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),V2))),aa(word(C),word(A),semiring_1_unsigned(C,word(A)),W)) ) ).

% word_cat_bin'
tff(fact_5894_two__power__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat] :
          ( aa(nat,$o,ord_less(nat,Na),type_len0_len_of(A,type2(A)))
         => ( aa(nat,$o,ord_less(nat,M),type_len0_len_of(A,type2(A)))
           => ( ( aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na) = aa(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_5895_signed__take__bit__decr__length__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( bit_ri3973907225187159222ations(A)
        & type_len(B) )
     => ! [K: A,L: A] :
          ( ( bit_ri4674362597316999326ke_bit(A,aa(nat,nat,minus_minus(nat,type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat))),K) = bit_ri4674362597316999326ke_bit(A,aa(nat,nat,minus_minus(nat,type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat))),L) )
        <=> ( bit_se2584673776208193580ke_bit(A,type_len0_len_of(B,type2(B)),K) = bit_se2584673776208193580ke_bit(A,type_len0_len_of(B,type2(B)),L) ) ) ) ).

% signed_take_bit_decr_length_iff
tff(fact_5896_num__of__sbintr_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( ( bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),aa(num,int,numeral_numeral(int),A3)) = aa(num,int,numeral_numeral(int),B3) )
         => ( aa(num,word(A),numeral_numeral(word(A)),A3) = aa(num,word(A),numeral_numeral(word(A)),B3) ) ) ) ).

% num_of_sbintr'
tff(fact_5897_word__split__bin_H,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(B)
        & type_len(A) )
     => ! [W: word(C)] : word_split(C,A,B,W) = 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)),aa(word(C),word(A),semiring_1_unsigned(C,word(A)),aa(word(C),word(C),aa(nat,fun(word(C),word(C)),bit_se4197421643247451524op_bit(word(C)),type_len0_len_of(B,type2(B))),W))),aa(word(C),word(B),semiring_1_unsigned(C,word(B)),W)) ) ).

% word_split_bin'
tff(fact_5898_unsigned__of__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & semiring_1(A) )
     => ! [K: int] : aa(word(B),A,semiring_1_unsigned(B,A),aa(int,word(B),ring_1_of_int(word(B)),K)) = aa(nat,A,semiring_1_of_nat(A),nat2(bit_se2584673776208193580ke_bit(int,type_len0_len_of(B,type2(B)),K))) ) ).

% unsigned_of_int
tff(fact_5899_word__sint__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),int,ring_1_signed(A,int),one_one(word(A))) = $ite(type_len0_len_of(A,type2(A)) = one_one(nat),aa(int,int,uminus_uminus(int),one_one(int)),one_one(int)) ) ) ).

% word_sint_1
tff(fact_5900_bin__nth__sint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] :
          ( aa(nat,$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),W)),Na)
          <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(word(A),int,ring_1_signed(A,int),W)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))) ) ) ) ).

% bin_nth_sint
tff(fact_5901_sint__sbintrunc_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: int] : aa(word(A),int,ring_1_signed(A,int),aa(int,word(A),ring_1_of_int(word(A)),Bin)) = bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),Bin) ) ).

% sint_sbintrunc'
tff(fact_5902_neg__mask__test__bit,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),Na))),M)
        <=> ( aa(nat,$o,ord_less_eq(nat,Na),M)
            & aa(nat,$o,ord_less(nat,M),type_len0_len_of(A,type2(A))) ) ) ) ).

% neg_mask_test_bit
tff(fact_5903_word__of__int__bin__cat__eq__iff,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(C)
        & type_len(B) )
     => ! [B3: word(B),A3: word(A),D2: word(B),C2: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,aa(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),B3),aa(word(A),int,semiring_1_unsigned(A,int),A3))) = 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),C2))) )
          <=> ( ( B3 = D2 )
              & ( A3 = C2 ) ) ) ) ) ).

% word_of_int_bin_cat_eq_iff
tff(fact_5904_mask__exceed,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,X: word(A)] :
          ( aa(nat,$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)),X),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),Na))) = zero_zero(word(A)) ) ) ) ).

% mask_exceed
tff(fact_5905_unsigned__less,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & unique1627219031080169319umeral(A) )
     => ! [W: word(B)] : aa(A,$o,ord_less(A,aa(word(B),A,semiring_1_unsigned(B,A),W)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),type_len0_len_of(B,type2(B)))) ) ).

% unsigned_less
tff(fact_5906_not__degenerate__imp__2__neq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(nat,$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_5907_word__nchotomy,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W4: word(A)] :
        ? [N: nat] :
          ( ( W4 = aa(nat,word(A),semiring_1_of_nat(word(A)),N) )
          & aa(nat,$o,ord_less(nat,N),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% word_nchotomy
tff(fact_5908_word__nat__cases,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ~ ! [N: nat] :
              ( ( X = aa(nat,word(A),semiring_1_of_nat(word(A)),N) )
             => ~ aa(nat,$o,ord_less(nat,N),aa(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_5909_of__nat__inj,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: nat,Y: nat] :
          ( aa(nat,$o,ord_less(nat,X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,ord_less(nat,Y),aa(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)),X) = aa(nat,word(A),semiring_1_of_nat(word(A)),Y) )
            <=> ( X = Y ) ) ) ) ) ).

% of_nat_inj
tff(fact_5910_word__of__nat__inj,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: nat,Y: nat] :
          ( aa(nat,$o,ord_less(nat,X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,ord_less(nat,Y),aa(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)),X) = aa(nat,word(A),semiring_1_of_nat(word(A)),Y) )
             => ( X = Y ) ) ) ) ) ).

% word_of_nat_inj
tff(fact_5911_More__Word_Opower__not__zero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,ord_less(nat,Na),type_len0_len_of(A,type2(A)))
         => ( aa(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_5912_word__power__increasing,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: nat,Y: nat] :
          ( aa(word(A),$o,ord_less(word(A),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),X)),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Y))
         => ( aa(nat,$o,ord_less(nat,X),type_len0_len_of(A,type2(A)))
           => ( aa(nat,$o,ord_less(nat,Y),type_len0_len_of(A,type2(A)))
             => aa(nat,$o,ord_less(nat,X),Y) ) ) ) ) ).

% word_power_increasing
tff(fact_5913_power__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),Na)
         => ( aa(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_5914_nth__w2p__same,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),Na)
        <=> aa(nat,$o,ord_less(nat,Na),type_len0_len_of(A,type2(A))) ) ) ).

% nth_w2p_same
tff(fact_5915_nth__w2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),M)
        <=> ( ( M = Na )
            & aa(nat,$o,ord_less(nat,M),type_len0_len_of(A,type2(A))) ) ) ) ).

% nth_w2p
tff(fact_5916_uint__idem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(word(A),int,semiring_1_unsigned(A,int),W)),aa(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),W) ) ).

% uint_idem
tff(fact_5917_word__of__int__2p__len,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(int,word(A),ring_1_of_int(word(A)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) = zero_zero(word(A)) ) ) ).

% word_of_int_2p_len
tff(fact_5918_of__nat__neq__iff__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: nat,Y: nat] :
          ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),aa(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),modulo_modulo(nat),Y),aa(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)),X) != aa(nat,word(A),semiring_1_of_nat(word(A)),Y) )
          <=> ( X != Y ) ) ) ) ).

% of_nat_neq_iff_word
tff(fact_5919_sint__uint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(word(A),int,ring_1_signed(A,int),W) = bit_ri4674362597316999326ke_bit(int,aa(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),W)) ) ).

% sint_uint
tff(fact_5920_num__abs__sbintr,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: num] : aa(num,word(A),numeral_numeral(word(A)),X) = aa(int,word(A),ring_1_of_int(word(A)),bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),aa(num,int,numeral_numeral(int),X))) ) ).

% num_abs_sbintr
tff(fact_5921_ucast__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),X),Y)
         => ( aa(word(A),$o,ord_less(word(A),Y),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),type_len0_len_of(B,type2(B))))
           => aa(word(B),$o,ord_less(word(B),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),X)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y)) ) ) ) ).

% ucast_mono
tff(fact_5922_ucast__ucast__len,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),X),aa(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)),X)) = X ) ) ) ).

% ucast_ucast_len
tff(fact_5923_sint__word__ariths_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),int,ring_1_signed(A,int),zero_zero(word(A))) = bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),zero_zero(int)) ) ) ).

% sint_word_ariths(7)
tff(fact_5924_sint__word__ariths_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),int,ring_1_signed(A,int),one_one(word(A))) = bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),one_one(int)) ) ) ).

% sint_word_ariths(8)
tff(fact_5925_sint__word__ariths_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(word(A),int,ring_1_signed(A,int),aa(word(A),word(A),plus_plus(word(A),A3),B3)) = bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),aa(int,int,plus_plus(int,aa(word(A),int,ring_1_signed(A,int),A3)),aa(word(A),int,ring_1_signed(A,int),B3))) ) ).

% sint_word_ariths(1)
tff(fact_5926_nth__sint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Na: nat] :
          ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(word(A),int,ring_1_signed(A,int),W)),Na)
        <=> $ite(aa(nat,$o,ord_less(nat,Na),aa(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)),W),Na),aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)))) ) ) ).

% nth_sint
tff(fact_5927_bit__sint__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Na: nat] :
          ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(word(A),int,ring_1_signed(A,int),W)),Na)
        <=> ( ( aa(nat,$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)),W),aa(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)),W),Na) ) ) ) ).

% bit_sint_iff
tff(fact_5928_sint__word__ariths_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : aa(word(A),int,ring_1_signed(A,int),aa(word(A),word(A),uminus_uminus(word(A)),A3)) = bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),aa(int,int,uminus_uminus(int),aa(word(A),int,ring_1_signed(A,int),A3))) ) ).

% sint_word_ariths(4)
tff(fact_5929_drop__bit__eq__zero__iff__not__bit__last,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( ( aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),W) = zero_zero(word(A)) )
        <=> ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),aa(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_5930_sint__word__ariths_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(word(A),int,ring_1_signed(A,int),aa(word(A),word(A),minus_minus(word(A),A3),B3)) = bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),aa(int,int,minus_minus(int,aa(word(A),int,ring_1_signed(A,int),A3)),aa(word(A),int,ring_1_signed(A,int),B3))) ) ).

% sint_word_ariths(2)
tff(fact_5931_signed__scast__eq,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( type_len(C)
        & bit_ri3973907225187159222ations(A)
        & type_len(B) )
     => ! [W: word(C)] : aa(word(B),A,ring_1_signed(B,A),aa(word(C),word(B),ring_1_signed(C,word(B)),W)) = bit_ri4674362597316999326ke_bit(A,aa(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),W)) ) ).

% signed_scast_eq
tff(fact_5932_sint__word__ariths_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(word(A),int,ring_1_signed(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A3),B3)) = bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,ring_1_signed(A,int),A3)),aa(word(A),int,ring_1_signed(A,int),B3))) ) ).

% sint_word_ariths(3)
tff(fact_5933_less__Suc__unat__less__bound,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,X: word(A)] :
          ( aa(nat,$o,ord_less(nat,Na),aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),X)))
         => aa(nat,$o,ord_less(nat,Na),aa(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_5934_uint__2__id,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(nat,$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_5935_lt2p__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),Na)
         => aa(int,$o,ord_less(int,aa(word(A),int,semiring_1_unsigned(A,int),W)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ) ).

% lt2p_lem
tff(fact_5936_power__le__mono,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))
         => ( aa(nat,$o,ord_less(nat,Na),type_len0_len_of(A,type2(A)))
           => ( aa(nat,$o,ord_less(nat,M),type_len0_len_of(A,type2(A)))
             => aa(nat,$o,ord_less_eq(nat,Na),M) ) ) ) ) ).

% power_le_mono
tff(fact_5937_two__power__increasing,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat] :
          ( aa(nat,$o,ord_less_eq(nat,Na),M)
         => ( aa(nat,$o,ord_less(nat,M),type_len0_len_of(A,type2(A)))
           => aa(word(A),$o,ord_less_eq(word(A),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)) ) ) ) ).

% two_power_increasing
tff(fact_5938_unat__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: num] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(num,nat,numeral_numeral(nat),B3)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% unat_numeral
tff(fact_5939_of__nat__mono__maybe,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: nat,Y: nat] :
          ( aa(nat,$o,ord_less(nat,X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,ord_less(nat,Y),X)
           => aa(word(A),$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)),X)) ) ) ) ).

% of_nat_mono_maybe
tff(fact_5940_of__nat__mono__maybe_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: nat,Y: nat] :
          ( aa(nat,$o,ord_less(nat,X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,ord_less(nat,Y),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(nat,$o,ord_less(nat,Y),X)
            <=> aa(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)),X)) ) ) ) ) ).

% of_nat_mono_maybe'
tff(fact_5941_unat__split,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(nat,$o),X: word(A)] :
          ( aa(nat,$o,P,aa(word(A),nat,semiring_1_unsigned(A,nat),X))
        <=> ! [N3: nat] :
              ( ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),N3) = X )
                & aa(nat,$o,ord_less(nat,N3),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) )
             => aa(nat,$o,P,N3) ) ) ) ).

% unat_split
tff(fact_5942_of__nat__inverse,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [R3: nat,A3: word(A)] :
          ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),R3) = A3 )
         => ( aa(nat,$o,ord_less(nat,R3),aa(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),A3) = R3 ) ) ) ) ).

% of_nat_inverse
tff(fact_5943_unat__split__asm,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(nat,$o),X: word(A)] :
          ( aa(nat,$o,P,aa(word(A),nat,semiring_1_unsigned(A,nat),X))
        <=> ~ ? [N3: nat] :
                ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),N3) = X )
                & aa(nat,$o,ord_less(nat,N3),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
                & ~ aa(nat,$o,P,N3) ) ) ) ).

% unat_split_asm
tff(fact_5944_unat__eq__of__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,X: word(A)] :
          ( aa(nat,$o,ord_less(nat,Na),aa(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),X) = Na )
          <=> ( X = aa(nat,word(A),semiring_1_of_nat(word(A)),Na) ) ) ) ) ).

% unat_eq_of_nat
tff(fact_5945_unat__of__nat__len,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: nat] :
          ( aa(nat,$o,ord_less(nat,X),aa(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)),X)) = X ) ) ) ).

% unat_of_nat_len
tff(fact_5946_x__less__2__0__1_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( ( type_len0_len_of(A,type2(A)) != one_one(nat) )
         => ( aa(word(A),$o,ord_less(word(A),X),aa(num,word(A),numeral_numeral(word(A)),bit0(one2)))
           => ( ( X = zero_zero(word(A)) )
              | ( X = one_one(word(A)) ) ) ) ) ) ).

% x_less_2_0_1'
tff(fact_5947_test__bit__2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(int,word(A),ring_1_of_int(word(A)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))),M)
        <=> ( ( M = Na )
            & aa(nat,$o,ord_less(nat,M),type_len0_len_of(A,type2(A))) ) ) ) ).

% test_bit_2p
tff(fact_5948_Word_Oof__nat__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat] :
          ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),M) = zero_zero(word(A)) )
        <=> ? [Q5: nat] : M = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q5),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% Word.of_nat_0
tff(fact_5949_word__1__le__power,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,ord_less(nat,Na),type_len0_len_of(A,type2(A)))
         => aa(word(A),$o,ord_less_eq(word(A),one_one(word(A))),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) ) ) ).

% word_1_le_power
tff(fact_5950_ucast__of__nat__small,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [X: nat] :
          ( aa(nat,$o,ord_less(nat,X),aa(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)),X)) = aa(nat,word(B),semiring_1_of_nat(word(B)),X) ) ) ) ).

% ucast_of_nat_small
tff(fact_5951_uint__sub__lt2p,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(A),Y: word(B)] : aa(int,$o,ord_less(int,aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(B),int,semiring_1_unsigned(B,int),Y))),aa(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_5952_uint__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: num] : aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(num,int,numeral_numeral(int),B3)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_numeral
tff(fact_5953_p2__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))
        <=> aa(nat,$o,ord_less(nat,Na),type_len0_len_of(A,type2(A))) ) ) ).

% p2_gt_0
tff(fact_5954_word__of__nat__eq__0__iff,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,dvd_dvd(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))),Na) ) ) ).

% word_of_nat_eq_0_iff
tff(fact_5955_unat__ucast,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(B)] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),X)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(word(B),nat,semiring_1_unsigned(B,nat),X)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% unat_ucast
tff(fact_5956_word__of__int__minus,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: int] : aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,minus_minus(int,aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))),I)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,uminus_uminus(int),I)) ) ).

% word_of_int_minus
tff(fact_5957_bit__last__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
        <=> aa(int,$o,ord_less(int,aa(word(A),int,ring_1_signed(A,int),W)),zero_zero(int)) ) ) ).

% bit_last_iff
tff(fact_5958_mask__lt__2pn,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,ord_less(nat,Na),type_len0_len_of(A,type2(A)))
         => aa(word(A),$o,ord_less(word(A),bit_se2239418461657761734s_mask(word(A),Na)),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) ) ) ).

% mask_lt_2pn
tff(fact_5959_unat__of__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: nat] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(nat,word(A),semiring_1_of_nat(word(A)),X)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% unat_of_nat
tff(fact_5960_uint__word__of__int,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: int] : aa(word(A),int,semiring_1_unsigned(A,int),aa(int,word(A),ring_1_of_int(word(A)),K)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_word_of_int
tff(fact_5961_ucast__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(A)] :
          ( aa(nat,$o,ord_less(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => aa(word(B),$o,ord_less(word(B),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),X)),aa(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_5962_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),bit_ri4674362597316999326ke_bit(int,aa(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_5963_word__of__int__eq__0__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: int] :
          ( ( aa(int,word(A),ring_1_of_int(word(A)),K) = zero_zero(word(A)) )
        <=> aa(int,$o,dvd_dvd(int,aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))),K) ) ) ).

% word_of_int_eq_0_iff
tff(fact_5964_of__nat__n__less__equal__power__2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,ord_less(nat,Na),type_len0_len_of(A,type2(A)))
         => aa(word(A),$o,ord_less(word(A),aa(nat,word(A),semiring_1_of_nat(word(A)),Na)),aa(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_5965_signed__ucast__eq,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( type_len(C)
        & bit_ri3973907225187159222ations(A)
        & type_len(B) )
     => ! [W: word(C)] : aa(word(B),A,ring_1_signed(B,A),aa(word(C),word(B),semiring_1_unsigned(C,word(B)),W)) = bit_ri4674362597316999326ke_bit(A,aa(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),W)) ) ).

% signed_ucast_eq
tff(fact_5966_complement__nth__w2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N6: nat,Na: nat] :
          ( aa(nat,$o,ord_less(nat,N6),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),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))),N6)
          <=> ( N6 != Na ) ) ) ) ).

% complement_nth_w2p
tff(fact_5967_upper__trivial,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( ( X != aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(word(A))) )
         => aa(word(A),$o,ord_less(word(A),X),aa(word(A),word(A),minus_minus(word(A),aa(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_5968_bit__word__scast__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [W: word(B),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(B),word(A),ring_1_signed(B,word(A)),W)),Na)
        <=> ( aa(nat,$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)),W),Na)
              | ( aa(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)),W),aa(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_5969_minus__one__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) = aa(word(A),word(A),minus_minus(word(A),aa(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))) ) ) ).

% minus_one_word
tff(fact_5970_word__power__less__diff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,Q3: word(A),M: nat] :
          ( aa(word(A),$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),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),Q3)),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))
         => ( aa(word(A),$o,ord_less(word(A),Q3),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),Na)))
           => aa(word(A),$o,ord_less(word(A),Q3),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,M),Na))) ) ) ) ).

% word_power_less_diff
tff(fact_5971_unat__word__ariths_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),A3),B3)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A3)),aa(word(A),nat,semiring_1_unsigned(A,nat),B3))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% unat_word_ariths(7)
tff(fact_5972_ucast__mono__le,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),X),Y)
         => ( aa(word(A),$o,ord_less(word(A),Y),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),type_len0_len_of(B,type2(B))))
           => aa(word(B),$o,ord_less_eq(word(B),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),X)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y)) ) ) ) ).

% ucast_mono_le
tff(fact_5973_take__bit__word__eq__self__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] :
          ( ( bit_se2584673776208193580ke_bit(word(A),Na,W) = W )
        <=> ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),Na)
            | aa(word(A),$o,ord_less(word(A),W),aa(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_5974_signed__push__bit__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_ri3973907225187159222ations(A) )
     => ! [Na: nat,W: 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),W)) = bit_ri4674362597316999326ke_bit(A,aa(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),W))) ) ).

% signed_push_bit_eq
tff(fact_5975_msb0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A),I: nat] :
          ( ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),X),Y)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))
          <=> ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),X),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))
              | aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I))) ) )
          & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),Y)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))
          <=> ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),X),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))
              & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I))) ) )
          & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),X),Y)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))
          <=> ~ ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),X),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))
              <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I))) ) )
          & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),X)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))
          <=> ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),X),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I))) ) ) ) ).

% msb0
tff(fact_5976_unat__add__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(nat,$o,ord_less(nat,aa(nat,nat,plus_plus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(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),plus_plus(word(A),X),Y)) = aa(nat,nat,plus_plus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) ) ) ).

% unat_add_lem
tff(fact_5977_unat__add__lem_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(nat,$o,ord_less(nat,aa(nat,nat,plus_plus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(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),plus_plus(word(A),X),Y)) = aa(nat,nat,plus_plus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) ) ) ).

% unat_add_lem'
tff(fact_5978_Word_Oof__nat__neq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: nat] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
         => ( aa(nat,$o,ord_less(nat,K),aa(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_5979_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,ord_less(nat,Na),aa(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_5980_of__nat__mono__maybe__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: nat,Y: nat] :
          ( aa(nat,$o,ord_less(nat,X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,ord_less(nat,Y),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(nat,$o,ord_less_eq(nat,Y),X)
            <=> aa(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)),X)) ) ) ) ) ).

% of_nat_mono_maybe_le
tff(fact_5981_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,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% unat_word_ariths(4)
tff(fact_5982_unat__word__ariths_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),plus_plus(word(A),A3),B3)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,plus_plus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),A3)),aa(word(A),nat,semiring_1_unsigned(A,nat),B3))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% unat_word_ariths(1)
tff(fact_5983_bool__mask_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( aa(nat,$o,ord_less(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))
         => ( aa(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)),X),one_one(word(A))))
          <=> ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),one_one(word(A))) = one_one(word(A)) ) ) ) ) ).

% bool_mask'
tff(fact_5984_uint__range_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(word(A),int,semiring_1_unsigned(A,int),X))
          & aa(int,$o,ord_less(int,aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% uint_range'
tff(fact_5985_sint__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(int,$o,ord_less(int,aa(word(A),int,ring_1_signed(A,int),X)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)))) ) ).

% sint_lt
tff(fact_5986_ucast__mono__le_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Y: word(A),X: word(A)] :
          ( aa(nat,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(B,type2(B))))
         => ( aa(nat,$o,ord_less(nat,type_len0_len_of(B,type2(B))),type_len0_len_of(A,type2(A)))
           => ( aa(word(A),$o,ord_less_eq(word(A),X),Y)
             => aa(word(B),$o,ord_less_eq(word(B),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),X)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y)) ) ) ) ) ).

% ucast_mono_le'
tff(fact_5987_of__nat__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] :
          ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),Na) = W )
        <=> ? [Q5: nat] : Na = aa(nat,nat,plus_plus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),W)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q5),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ) ).

% of_nat_eq
tff(fact_5988_unat__mult__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(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)),X),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) ) ) ).

% unat_mult_lem
tff(fact_5989_word__of__int__inj,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: int,Y: int] :
          ( ( aa(int,$o,ord_less_eq(int,zero_zero(int)),X)
            & aa(int,$o,ord_less(int,X),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) )
         => ( ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Y)
              & aa(int,$o,ord_less(int,Y),aa(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)),X) = aa(int,word(A),ring_1_of_int(word(A)),Y) )
            <=> ( X = Y ) ) ) ) ) ).

% word_of_int_inj
tff(fact_5990_word__int__cases,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ~ ! [N: int] :
              ( ( X = aa(int,word(A),ring_1_of_int(word(A)),N) )
             => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),N)
               => ~ aa(int,$o,ord_less(int,N),aa(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_5991_unat__ucast__no__overflow__le,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [B3: word(A),F2: word(B)] :
          ( aa(nat,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),B3)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(B,type2(B))))
         => ( aa(nat,$o,ord_less(nat,type_len0_len_of(B,type2(B))),type_len0_len_of(A,type2(A)))
           => ( aa(word(A),$o,ord_less(word(A),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),F2)),B3)
            <=> aa(nat,$o,ord_less(nat,aa(word(B),nat,semiring_1_unsigned(B,nat),F2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B3)) ) ) ) ) ).

% unat_ucast_no_overflow_le
tff(fact_5992_uint__m2p__not__non__neg,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : ~ aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(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_5993_uint__m2p__neg,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(int,$o,ord_less(int,aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(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_5994_unat__ucast__less__no__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,F2: word(A)] :
          ( aa(nat,$o,ord_less(nat,Na),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),F2)),Na)
           => aa(word(A),$o,ord_less(word(A),F2),aa(nat,word(A),semiring_1_of_nat(word(A)),Na)) ) ) ) ).

% unat_ucast_less_no_overflow
tff(fact_5995_unat__ucast__less__no__overflow__simp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,F2: word(A)] :
          ( aa(nat,$o,ord_less(nat,Na),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),F2)),Na)
          <=> aa(word(A),$o,ord_less(word(A),F2),aa(nat,word(A),semiring_1_of_nat(word(A)),Na)) ) ) ) ).

% unat_ucast_less_no_overflow_simp
tff(fact_5996_uint__power__lower,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$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),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) = aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na) ) ) ) ).

% uint_power_lower
tff(fact_5997_nth__bounded,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Na: nat,M: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),X),Na)
         => ( aa(word(A),$o,ord_less(word(A),X),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))
           => ( aa(nat,$o,ord_less_eq(nat,M),type_len0_len_of(A,type2(A)))
             => aa(nat,$o,ord_less(nat,Na),M) ) ) ) ) ).

% nth_bounded
tff(fact_5998_upper__bits__unset__is__l2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,P2: word(A)] :
          ( aa(nat,$o,ord_less(nat,Na),type_len0_len_of(A,type2(A)))
         => ( ! [N11: nat] :
                ( aa(nat,$o,ord_less_eq(nat,Na),N11)
               => ( aa(nat,$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)),P2),N11) ) )
          <=> aa(word(A),$o,ord_less(word(A),P2),aa(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_5999_less__2p__is__upper__bits__unset,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),Na: nat] :
          ( aa(word(A),$o,ord_less(word(A),P2),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))
        <=> ( aa(nat,$o,ord_less(nat,Na),type_len0_len_of(A,type2(A)))
            & ! [N11: nat] :
                ( aa(nat,$o,ord_less_eq(nat,Na),N11)
               => ( aa(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)),P2),N11) ) ) ) ) ) ).

% less_2p_is_upper_bits_unset
tff(fact_6000_uint__add__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(int,$o,ord_less(int,aa(int,int,plus_plus(int,aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Y))),aa(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),plus_plus(word(A),X),Y)) = aa(int,int,plus_plus(int,aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ) ) ).

% uint_add_lem
tff(fact_6001_uint__word__ariths_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),int,semiring_1_unsigned(A,int),zero_zero(word(A))) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),zero_zero(int)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% uint_word_ariths(7)
tff(fact_6002_uint__word__ariths_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),int,semiring_1_unsigned(A,int),one_one(word(A))) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),one_one(int)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% uint_word_ariths(8)
tff(fact_6003_wi__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: int,M: int] :
          ( aa(word(A),$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,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Na),aa(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,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ) ).

% wi_le
tff(fact_6004_uint__word__ariths_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),plus_plus(word(A),A3),B3)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,plus_plus(int,aa(word(A),int,semiring_1_unsigned(A,int),A3)),aa(word(A),int,semiring_1_unsigned(A,int),B3))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_word_ariths(1)
tff(fact_6005_word__2p__mult__inc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat] :
          ( aa(word(A),$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),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),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)))
         => ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,Na)),type_len0_len_of(A,type2(A)))
           => aa(word(A),$o,ord_less(word(A),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)) ) ) ) ).

% word_2p_mult_inc
tff(fact_6006_unat__word__ariths_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(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)),A3),B3)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A3)),aa(word(A),nat,semiring_1_unsigned(A,nat),B3))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% unat_word_ariths(2)
tff(fact_6007_wi__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: int,M: int] :
          ( aa(word(A),$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,ord_less(int,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Na),aa(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,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ) ).

% wi_less
tff(fact_6008_unat__word__ariths_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : 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)),A3),B3)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A3)),aa(word(A),nat,semiring_1_unsigned(A,nat),B3))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% unat_word_ariths(6)
tff(fact_6009_power__2__ge__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),aa(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),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))
        <=> aa(nat,$o,ord_less(nat,Na),type_len0_len_of(A,type2(A))) ) ) ).

% power_2_ge_iff
tff(fact_6010_word__power__less__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Sz: nat] :
          ( aa(nat,$o,ord_less(nat,Sz),type_len0_len_of(A,type2(A)))
         => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),aa(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),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Sz)) ) ) ).

% word_power_less_1
tff(fact_6011_uint__word__ariths_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),uminus_uminus(word(A)),A3)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),aa(word(A),int,semiring_1_unsigned(A,int),A3))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_word_ariths(4)
tff(fact_6012_le__mask__iff__lt__2n,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] :
          ( aa(nat,$o,ord_less(nat,Na),type_len0_len_of(A,type2(A)))
        <=> ( aa(word(A),$o,ord_less_eq(word(A),W),bit_se2239418461657761734s_mask(word(A),Na))
          <=> aa(word(A),$o,ord_less(word(A),W),aa(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_6013_eq__mask__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Na: nat] :
          ( ( W = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),Na)) )
         => ( aa(nat,$o,ord_less(nat,Na),type_len0_len_of(A,type2(A)))
           => aa(word(A),$o,ord_less(word(A),W),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) ) ) ) ).

% eq_mask_less
tff(fact_6014_and__mask__less_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] :
          ( aa(nat,$o,ord_less(nat,Na),type_len0_len_of(A,type2(A)))
         => aa(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)),W),bit_se2239418461657761734s_mask(word(A),Na))),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) ) ) ).

% and_mask_less'
tff(fact_6015_sint__1__cases,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] :
          ( ( ( type_len0_len_of(A,type2(A)) = one_one(nat) )
           => ( ( A3 = zero_zero(word(A)) )
             => ( aa(word(A),int,ring_1_signed(A,int),A3) != zero_zero(int) ) ) )
         => ( ( ( type_len0_len_of(A,type2(A)) = one_one(nat) )
             => ( ( A3 = one_one(word(A)) )
               => ( aa(word(A),int,ring_1_signed(A,int),one_one(word(A))) != aa(int,int,uminus_uminus(int),one_one(int)) ) ) )
           => ~ ( aa(nat,$o,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_6016_uint__word__ariths_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),minus_minus(word(A),A3),B3)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),A3)),aa(word(A),int,semiring_1_unsigned(A,int),B3))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_word_ariths(2)
tff(fact_6017_uint__mult__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Y))),aa(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)),X),Y)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ) ) ).

% uint_mult_lem
tff(fact_6018_uint__word__ariths_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(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)),A3),B3)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,semiring_1_unsigned(A,int),A3)),aa(word(A),int,semiring_1_unsigned(A,int),B3))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_word_ariths(3)
tff(fact_6019_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),bit_ri4674362597316999326ke_bit(int,aa(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_6020_word__power__mod__div,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat,X: word(A)] :
          ( aa(nat,$o,ord_less(nat,Na),type_len0_len_of(A,type2(A)))
         => ( aa(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)),X),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))),aa(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)),X),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,Na),M))) ) ) ) ) ).

% word_power_mod_div
tff(fact_6021_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)),bit_ri4674362597316999326ke_bit(int,aa(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_6022_msb1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),X),Y)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
          <=> ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),X),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
              | aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))) ) )
          & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),Y)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
          <=> ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),X),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
              & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))) ) )
          & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),X),Y)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
          <=> ~ ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),X),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
              <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))) ) )
          & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),X)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
          <=> ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),X),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))) ) ) ) ).

% msb1
tff(fact_6023_unat__plus__if_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),plus_plus(word(A),A3),B3)) = $ite(aa(nat,$o,ord_less(nat,aa(nat,nat,plus_plus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),A3)),aa(word(A),nat,semiring_1_unsigned(A,nat),B3))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))),aa(nat,nat,plus_plus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),A3)),aa(word(A),nat,semiring_1_unsigned(A,nat),B3)),aa(nat,nat,minus_minus(nat,aa(nat,nat,plus_plus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),A3)),aa(word(A),nat,semiring_1_unsigned(A,nat),B3))),aa(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_6024_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,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% unat_word_ariths(5)
tff(fact_6025_unat__sub__if_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),minus_minus(word(A),X),Y)) = $ite(aa(nat,$o,ord_less_eq(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(nat,nat,minus_minus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),aa(nat,nat,minus_minus(nat,aa(nat,nat,plus_plus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(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_6026_no__olen__add__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),plus_plus(word(A),X),Y))
        <=> aa(nat,$o,ord_less(nat,aa(nat,nat,plus_plus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(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_6027_word__add__le__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( aa(nat,$o,ord_less(nat,aa(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,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,ord_less(nat,aa(nat,nat,plus_plus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),plus_plus(word(A),I),K)),aa(word(A),word(A),plus_plus(word(A),J),K))
            <=> aa(word(A),$o,ord_less_eq(word(A),I),J) ) ) ) ) ).

% word_add_le_iff
tff(fact_6028_word__add__le__dest,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),plus_plus(word(A),I),K)),aa(word(A),word(A),plus_plus(word(A),J),K))
         => ( aa(nat,$o,ord_less(nat,aa(nat,nat,plus_plus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(nat,$o,ord_less(nat,aa(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,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => aa(word(A),$o,ord_less_eq(word(A),I),J) ) ) ) ) ).

% word_add_le_dest
tff(fact_6029_word__add__le__mono1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),J: word(A),K: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),I),J)
         => ( aa(nat,$o,ord_less(nat,aa(nat,nat,plus_plus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),plus_plus(word(A),I),K)),aa(word(A),word(A),plus_plus(word(A),J),K)) ) ) ) ).

% word_add_le_mono1
tff(fact_6030_word__add__le__mono2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),J: word(A),K: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),I),J)
         => ( aa(nat,$o,ord_less(nat,aa(nat,nat,plus_plus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),plus_plus(word(A),K),I)),aa(word(A),word(A),plus_plus(word(A),K),J)) ) ) ) ).

% word_add_le_mono2
tff(fact_6031_sint__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(int,$o,ord_less(int,aa(word(A),int,ring_1_signed(A,int),W)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))) ) ).

% sint_less
tff(fact_6032_word__add__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( aa(nat,$o,ord_less(nat,aa(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,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,ord_less(nat,aa(nat,nat,plus_plus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),plus_plus(word(A),I),K)),aa(word(A),word(A),plus_plus(word(A),J),K))
            <=> aa(word(A),$o,ord_less(word(A),I),J) ) ) ) ) ).

% word_add_less_iff
tff(fact_6033_word__add__less__dest,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),plus_plus(word(A),I),K)),aa(word(A),word(A),plus_plus(word(A),J),K))
         => ( aa(nat,$o,ord_less(nat,aa(nat,nat,plus_plus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(nat,$o,ord_less(nat,aa(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,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => aa(word(A),$o,ord_less(word(A),I),J) ) ) ) ) ).

% word_add_less_dest
tff(fact_6034_word__add__less__mono1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),J: word(A),K: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),I),J)
         => ( aa(nat,$o,ord_less(nat,aa(nat,nat,plus_plus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),plus_plus(word(A),I),K)),aa(word(A),word(A),plus_plus(word(A),J),K)) ) ) ) ).

% word_add_less_mono1
tff(fact_6035_unat__minus__one__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) = aa(nat,nat,minus_minus(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(nat)) ) ) ).

% unat_minus_one_word
tff(fact_6036_sint__ge,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(int,$o,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))))),aa(word(A),int,ring_1_signed(A,int),X)) ) ).

% sint_ge
tff(fact_6037_unat__less__power,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Sz: nat,K: word(A)] :
          ( aa(nat,$o,ord_less(nat,Sz),type_len0_len_of(A,type2(A)))
         => ( aa(word(A),$o,ord_less(word(A),K),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Sz))
           => aa(nat,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),K)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Sz)) ) ) ) ).

% unat_less_power
tff(fact_6038_word__of__int__inverse,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [R3: int,A3: word(A)] :
          ( ( aa(int,word(A),ring_1_of_int(word(A)),R3) = A3 )
         => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),R3)
           => ( aa(int,$o,ord_less(int,R3),aa(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),A3) = R3 ) ) ) ) ) ).

% word_of_int_inverse
tff(fact_6039_uint__split__asm,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(int,$o),X: word(A)] :
          ( aa(int,$o,P,aa(word(A),int,semiring_1_unsigned(A,int),X))
        <=> ~ ? [I4: int] :
                ( ( aa(int,word(A),ring_1_of_int(word(A)),I4) = X )
                & aa(int,$o,ord_less_eq(int,zero_zero(int)),I4)
                & aa(int,$o,ord_less(int,I4),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))
                & ~ aa(int,$o,P,I4) ) ) ) ).

% uint_split_asm
tff(fact_6040_uint__split,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(int,$o),X: word(A)] :
          ( aa(int,$o,P,aa(word(A),int,semiring_1_unsigned(A,int),X))
        <=> ! [I4: int] :
              ( ( ( aa(int,word(A),ring_1_of_int(word(A)),I4) = X )
                & aa(int,$o,ord_less_eq(int,zero_zero(int)),I4)
                & aa(int,$o,ord_less(int,I4),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) )
             => aa(int,$o,P,I4) ) ) ) ).

% uint_split
tff(fact_6041_word__mult__less__dest,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( aa(word(A),$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,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(nat,$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,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => aa(word(A),$o,ord_less(word(A),I),J) ) ) ) ) ).

% word_mult_less_dest
tff(fact_6042_div__lt_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),X: word(A)] :
          ( aa(word(A),$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),X))
         => aa(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),X))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% div_lt'
tff(fact_6043_div__lt_H_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),X: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),K),X))
         => aa(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),X))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% div_lt''
tff(fact_6044_double__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: 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))),A3) = zero_zero(word(A)) )
        <=> ( ( A3 = zero_zero(word(A)) )
            | ( A3 = aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(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_6045_word__le__exists_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),X),Y)
         => ? [Z2: word(A)] :
              ( ( Y = aa(word(A),word(A),plus_plus(word(A),X),Z2) )
              & aa(int,$o,ord_less(int,aa(int,int,plus_plus(int,aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Z2))),aa(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_6046_no__olen__add_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),plus_plus(word(A),Y),X))
        <=> aa(int,$o,ord_less(int,aa(int,int,plus_plus(int,aa(word(A),int,semiring_1_unsigned(A,int),Y)),aa(word(A),int,semiring_1_unsigned(A,int),X))),aa(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_6047_no__olen__add,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),X),aa(word(A),word(A),plus_plus(word(A),X),Y))
        <=> aa(int,$o,ord_less(int,aa(int,int,plus_plus(int,aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Y))),aa(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_6048_More__Word_Oof__nat__power,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: nat,X: nat] :
          ( aa(nat,$o,ord_less(nat,P2),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),X))
         => ( aa(nat,$o,ord_less(nat,X),type_len0_len_of(A,type2(A)))
           => aa(word(A),$o,ord_less(word(A),aa(nat,word(A),semiring_1_of_nat(word(A)),P2)),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),X)) ) ) ) ).

% More_Word.of_nat_power
tff(fact_6049_uint__plus__if_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),plus_plus(word(A),A3),B3)) = $ite(aa(int,$o,ord_less(int,aa(int,int,plus_plus(int,aa(word(A),int,semiring_1_unsigned(A,int),A3)),aa(word(A),int,semiring_1_unsigned(A,int),B3))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))),aa(int,int,plus_plus(int,aa(word(A),int,semiring_1_unsigned(A,int),A3)),aa(word(A),int,semiring_1_unsigned(A,int),B3)),aa(int,int,minus_minus(int,aa(int,int,plus_plus(int,aa(word(A),int,semiring_1_unsigned(A,int),A3)),aa(word(A),int,semiring_1_unsigned(A,int),B3))),aa(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_6050_word__less__power__trans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A),M: nat,K: nat] :
          ( aa(word(A),$o,ord_less(word(A),Na),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,M),K)))
         => ( aa(nat,$o,ord_less_eq(nat,K),M)
           => ( aa(nat,$o,ord_less(nat,M),type_len0_len_of(A,type2(A)))
             => aa(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),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),K)),Na)),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)) ) ) ) ) ).

% word_less_power_trans
tff(fact_6051_word__less__power__trans2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A),M: nat,K: nat] :
          ( aa(word(A),$o,ord_less(word(A),Na),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,M),K)))
         => ( aa(nat,$o,ord_less_eq(nat,K),M)
           => ( aa(nat,$o,ord_less(nat,M),type_len0_len_of(A,type2(A)))
             => aa(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),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),K))),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)) ) ) ) ) ).

% word_less_power_trans2
tff(fact_6052_uint__sub__if_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),minus_minus(word(A),A3),B3)) = $ite(aa(int,$o,ord_less_eq(int,aa(word(A),int,semiring_1_unsigned(A,int),B3)),aa(word(A),int,semiring_1_unsigned(A,int),A3)),aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),A3)),aa(word(A),int,semiring_1_unsigned(A,int),B3)),aa(int,int,plus_plus(int,aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),A3)),aa(word(A),int,semiring_1_unsigned(A,int),B3))),aa(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_6053_word__less__two__pow__divI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Na: nat,M: nat] :
          ( aa(word(A),$o,ord_less(word(A),X),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,Na),M)))
         => ( aa(nat,$o,ord_less_eq(nat,M),Na)
           => ( aa(nat,$o,ord_less(nat,Na),type_len0_len_of(A,type2(A)))
             => aa(word(A),$o,ord_less(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),aa(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_6054_uint__neg__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: num] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3))) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_neg_numeral
tff(fact_6055_word__power__nonzero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Na: nat] :
          ( aa(word(A),$o,ord_less(word(A),X),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),Na)))
         => ( aa(nat,$o,ord_less(nat,Na),type_len0_len_of(A,type2(A)))
           => ( ( X != zero_zero(word(A)) )
             => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),X),aa(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_6056_mult__pow2__inj,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,Na: nat,X: word(A),Y: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,plus_plus(nat,M),Na)),type_len0_len_of(A,type2(A)))
         => ( aa(word(A),$o,ord_less_eq(word(A),X),bit_se2239418461657761734s_mask(word(A),M))
           => ( aa(word(A),$o,ord_less_eq(word(A),Y),bit_se2239418461657761734s_mask(word(A),M))
             => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),X),aa(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),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) )
               => ( X = Y ) ) ) ) ) ) ).

% mult_pow2_inj
tff(fact_6057_div__lt__uint_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),X: word(A)] :
          ( aa(word(A),$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),X))
         => aa(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),X))),aa(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_6058_div__lt__uint_H_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),X: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),K),X))
         => aa(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),X))),aa(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_6059_push__bit__word__eq__nonzero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),M: nat,Na: nat] :
          ( aa(word(A),$o,ord_less(word(A),W),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))
         => ( aa(nat,$o,ord_less(nat,aa(nat,nat,plus_plus(nat,M),Na)),type_len0_len_of(A,type2(A)))
           => ( ( W != zero_zero(word(A)) )
             => ( aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4730199178511100633sh_bit(word(A)),Na),W) != zero_zero(word(A)) ) ) ) ) ) ).

% push_bit_word_eq_nonzero
tff(fact_6060_uint__and__mask__or__full,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A),Mask1: word(A),Mask2: int] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Na),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)))
         => ( ( Mask1 = bit_se2239418461657761734s_mask(word(A),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))) )
           => ( ( Mask2 = aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),one_one(int)) )
             => ( aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(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)),Na),Mask1))),Mask2) = aa(word(A),int,semiring_1_unsigned(A,int),Na) ) ) ) ) ) ).

% uint_and_mask_or_full
tff(fact_6061_sint__greater__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(int,$o,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(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),W)) ) ).

% sint_greater_eq
tff(fact_6062_int__eq__sint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: nat] :
          ( aa(nat,$o,ord_less(nat,X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(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)),X)) = aa(nat,int,semiring_1_of_nat(int),X) ) ) ) ).

% int_eq_sint
tff(fact_6063_word__mult__less__mono1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),J: word(A),K: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),I),J)
         => ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),K)
           => ( aa(nat,$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,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => aa(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_6064_word__mult__less__cancel,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A),I: word(A),J: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),K)
         => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(nat,$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,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => ( aa(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,ord_less(word(A),I),J) ) ) ) ) ) ).

% word_mult_less_cancel
tff(fact_6065_smod__word__max,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(int,$o,ord_less(int,signed6721504322012087516modulo(int,aa(word(A),int,ring_1_signed(A,int),A3),aa(word(A),int,ring_1_signed(A,int),B3))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))) ) ).

% smod_word_max
tff(fact_6066_le2p__bits__unset,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),Na: nat] :
          ( aa(word(A),$o,ord_less_eq(word(A),P2),aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),one_one(word(A))))
         => ! [N4: nat] :
              ( aa(nat,$o,ord_less_eq(nat,Na),N4)
             => ( aa(nat,$o,ord_less(nat,N4),type_len0_len_of(A,type2(A)))
               => ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),P2),N4) ) ) ) ) ).

% le2p_bits_unset
tff(fact_6067_le__2p__upper__bits,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),Na: nat] :
          ( aa(word(A),$o,ord_less_eq(word(A),P2),aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),one_one(word(A))))
         => ( aa(nat,$o,ord_less(nat,Na),type_len0_len_of(A,type2(A)))
           => ! [N4: nat] :
                ( aa(nat,$o,ord_less_eq(nat,Na),N4)
               => ( aa(nat,$o,ord_less(nat,N4),type_len0_len_of(A,type2(A)))
                 => ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),P2),N4) ) ) ) ) ) ).

% le_2p_upper_bits
tff(fact_6068_word__add__offset__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Na: nat,X: word(A),M: nat,Sz: nat] :
          ( aa(word(A),$o,ord_less(word(A),Y),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))
         => ( aa(word(A),$o,ord_less(word(A),X),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))
           => ( aa(nat,$o,ord_less(nat,Sz),type_len0_len_of(A,type2(A)))
             => ( aa(word(A),$o,ord_less(word(A),X),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),Na)))
               => ( ( Sz = aa(nat,nat,plus_plus(nat,M),Na) )
                 => aa(word(A),$o,ord_less(word(A),aa(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)),X),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))),Y)),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Sz)) ) ) ) ) ) ) ).

% word_add_offset_less
tff(fact_6069_div__power__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: nat,Y: nat] :
          ( aa(nat,$o,ord_less_eq(nat,X),Y)
         => ( aa(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),minus_minus(word(A),aa(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),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),X)) = aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,Y),X))),one_one(word(A))) ) ) ) ) ).

% div_power_helper
tff(fact_6070_even__mult__exp__div__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),M: nat,Na: nat] :
          ( aa(word(A),$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)),A3),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)))
        <=> ~ ( aa(nat,$o,ord_less_eq(nat,M),Na)
              & aa(nat,$o,ord_less(nat,Na),type_len0_len_of(A,type2(A)))
              & ~ aa(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)),A3),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,Na),M)))) ) ) ) ).

% even_mult_exp_div_word_iff
tff(fact_6071_Suc__2p__unat__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,K: nat] :
          ( aa(nat,$o,ord_less(nat,Na),type_len0_len_of(A,type2(A)))
         => ( aa(nat,nat,suc,aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),K)),aa(word(A),nat,semiring_1_unsigned(A,nat),bit_se2239418461657761734s_mask(word(A),Na)))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),aa(nat,nat,plus_plus(nat,K),one_one(nat))) ) ) ) ).

% Suc_2p_unat_mask
tff(fact_6072_sint__of__nat__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: nat,A3: nat] :
          ( aa(nat,$o,ord_less(nat,B3),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))))
         => ( aa(nat,$o,ord_less_eq(nat,A3),B3)
           => aa(int,$o,ord_less_eq(int,aa(word(A),int,ring_1_signed(A,int),aa(nat,word(A),semiring_1_of_nat(word(A)),A3))),aa(word(A),int,ring_1_signed(A,int),aa(nat,word(A),semiring_1_of_nat(word(A)),B3))) ) ) ) ).

% sint_of_nat_le
tff(fact_6073_sint__of__nat__ge__zero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: nat] :
          ( aa(nat,$o,ord_less(nat,X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))))
         => aa(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)),X))) ) ) ).

% sint_of_nat_ge_zero
tff(fact_6074_sint__int__max__plus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),int,ring_1_signed(A,int),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(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,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(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_6075_sint__of__int__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: int] :
          ( aa(int,$o,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))))),X)
         => ( aa(int,$o,ord_less(int,X),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(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)),X)) = X ) ) ) ) ).

% sint_of_int_eq
tff(fact_6076_word__mult__le__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A),I: word(A),J: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),K)
         => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(nat,$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,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => ( aa(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,ord_less_eq(word(A),I),J) ) ) ) ) ) ).

% word_mult_le_iff
tff(fact_6077_word__mult__le__mono1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),J: word(A),K: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),I),J)
         => ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),K)
           => ( aa(nat,$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,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => aa(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_6078_sint__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: num] : aa(word(A),int,ring_1_signed(A,int),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,plus_plus(int,aa(num,int,numeral_numeral(int),B3)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)))) ) ).

% sint_numeral
tff(fact_6079_smod__word__min,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(int,$o,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))),signed6721504322012087516modulo(int,aa(word(A),int,ring_1_signed(A,int),A3),aa(word(A),int,ring_1_signed(A,int),B3))) ) ).

% smod_word_min
tff(fact_6080_int__word__sint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: int] : aa(word(A),int,ring_1_signed(A,int),aa(int,word(A),ring_1_of_int(word(A)),X)) = aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,plus_plus(int,X),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)))) ) ).

% int_word_sint
tff(fact_6081_Word_Oword__div__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),Y)
         => ( aa(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),X)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(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)),X),Y)),Y) = X ) ) ) ) ).

% Word.word_div_mult
tff(fact_6082_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),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(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,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).

% sint_int_min
tff(fact_6083_word__less__power__trans__ofnat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat,K: nat] :
          ( aa(nat,$o,ord_less(nat,Na),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,M),K)))
         => ( aa(nat,$o,ord_less_eq(nat,K),M)
           => ( aa(nat,$o,ord_less(nat,M),type_len0_len_of(A,type2(A)))
             => aa(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),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),K))),aa(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_6084_word__bit__induct,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(word(A),$o),A3: word(A)] :
          ( aa(word(A),$o,P,zero_zero(word(A)))
         => ( ! [A4: word(A)] :
                ( aa(word(A),$o,P,A4)
               => ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),A4)
                 => ( aa(word(A),$o,ord_less(word(A),A4),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))
                   => aa(word(A),$o,P,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),A4)) ) ) )
           => ( ! [A4: word(A)] :
                  ( aa(word(A),$o,P,A4)
                 => ( aa(word(A),$o,ord_less(word(A),A4),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))
                   => aa(word(A),$o,P,aa(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,P,A3) ) ) ) ) ).

% word_bit_induct
tff(fact_6085_unat__mult__power__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: nat,Sz: nat] :
          ( aa(nat,$o,ord_less(nat,K),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(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),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,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Sz)),K) ) ) ) ).

% unat_mult_power_lem
tff(fact_6086_bit__word__half__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: $o] :
          ( aa(word(A),$o,ord_less(word(A),A3),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(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),plus_plus(word(A),aa($o,word(A),zero_neq_one_of_bool(word(A)),(B3))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A3),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))))),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))) = A3 ) ) ) ).

% bit_word_half_eq
tff(fact_6087_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 = 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,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,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,ord_less(int,aa(int,int,minus_minus(int,i),Shift)),Least)
                            | aa(int,$o,ord_less_eq(int,Overflow),aa(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,minus_minus(int,i),Shift)) ),
                          $ite(
                            ( aa(int,$o,ord_less(int,i),Least)
                            | aa(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_6088_Suc__div__unat__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Sz: nat,Us: nat] :
          ( aa(nat,$o,ord_less(nat,Sz),type_len0_len_of(A,type2(A)))
         => ( aa(nat,$o,ord_less_eq(nat,Us),Sz)
           => ( aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,Sz),Us)) = 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),minus_minus(word(A),aa(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),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Us)))) ) ) ) ) ).

% Suc_div_unat_helper
tff(fact_6089_alignUp__div__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: nat,Na: nat,X: word(A),A3: word(A)] :
          ( aa(nat,$o,ord_less(nat,K),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),Na)))
         => ( ( X = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(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),$o,ord_less_eq(word(A),A3),X)
             => ( aa(nat,$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)),A3),aa(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,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),A3),aa(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_6090_decr__length__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,ord_less(nat,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Na)
        <=> aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),Na) ) ) ).

% decr_length_less_iff
tff(fact_6091_less__eq__decr__length__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,ord_less_eq(nat,Na),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
        <=> aa(nat,$o,ord_less(nat,Na),type_len0_len_of(A,type2(A))) ) ) ).

% less_eq_decr_length_iff
tff(fact_6092_length__not__greater__eq__2__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( ~ aa(nat,$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_6093_len__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(nat,$o,ord_less(nat,zero_zero(nat)),type_len0_len_of(A,type2(A))) ) ).

% len_gt_0
tff(fact_6094_len__num0,axiom,
    ! [Uu: itself(numeral_num0)] : type_len0_len_of(numeral_num0,Uu) = zero_zero(nat) ).

% len_num0
tff(fact_6095_len__of__finite__1__def,axiom,
    ! [X: itself(finite_1)] : type_len0_len_of(finite_1,X) = one_one(nat) ).

% len_of_finite_1_def
tff(fact_6096_len__num1,axiom,
    ! [Uu: itself(numeral_num1)] : type_len0_len_of(numeral_num1,Uu) = one_one(nat) ).

% len_num1
tff(fact_6097_len__of__finite__2__def,axiom,
    ! [X: itself(finite_2)] : type_len0_len_of(finite_2,X) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% len_of_finite_2_def
tff(fact_6098_len__of__finite__3__def,axiom,
    ! [X: itself(finite_3)] : type_len0_len_of(finite_3,X) = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ).

% len_of_finite_3_def
tff(fact_6099_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_6100_len__bit0,axiom,
    ! [A: $tType] :
      ( type_len0(A)
     => ! [Uu: itself(numeral_bit0(A))] : type_len0_len_of(numeral_bit0(A),Uu) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))) ) ).

% len_bit0
tff(fact_6101_two__less__eq__exp__length,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & linordered_idom(A) )
     => aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(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_6102_len__bit1,axiom,
    ! [A: $tType] :
      ( type_len0(A)
     => ! [Uu: itself(numeral_bit1(A))] : type_len0_len_of(numeral_bit1(A),Uu) = aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(nat)) ) ).

% len_bit1
tff(fact_6103_divmod__via__sdivmod,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: 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)),X),Y)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),X),Y)) = $ite(
                aa(word(A),$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,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),one_one(word(A)))),Y),
                $ite(aa(word(A),$o,ord_less(word(A),X),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))),X),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),minus_minus(word(A),X),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)),signed7115095781618012415divide(word(A),aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),one_one(nat)),X),Y)),
                  $let(
                    r: word(A),
                    r:= aa(word(A),word(A),minus_minus(word(A),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),q),Y)),
                    $ite(aa(word(A),$o,ord_less_eq(word(A),Y),r),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),plus_plus(word(A),q),one_one(word(A)))),aa(word(A),word(A),minus_minus(word(A),r),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),r)) ) ) ) ) ) ) ).

% divmod_via_sdivmod
tff(fact_6104_signed__drop__bit__word__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num,K: num] : signed_drop_bit(A,aa(num,nat,numeral_numeral(nat),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(num,nat,numeral_numeral(nat),Na)),bit_ri4674362597316999326ke_bit(int,aa(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_minus_numeral
tff(fact_6105_signed__drop__bit__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : signed_drop_bit(A,zero_zero(nat),W) = W ) ).

% signed_drop_bit_0
tff(fact_6106_signed__drop__bit__signed__drop__bit,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,Na: nat,W: word(A)] : signed_drop_bit(A,M,signed_drop_bit(A,Na,W)) = signed_drop_bit(A,aa(nat,nat,plus_plus(nat,M),Na),W) ) ).

% signed_drop_bit_signed_drop_bit
tff(fact_6107_signed__drop__bit__of__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] : signed_drop_bit(A,Na,zero_zero(word(A))) = zero_zero(word(A)) ) ).

% signed_drop_bit_of_0
tff(fact_6108_word__sdiv__div1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : signed7115095781618012415divide(word(A),A3,one_one(word(A))) = A3 ) ).

% word_sdiv_div1
tff(fact_6109_word__sdiv__div0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : signed7115095781618012415divide(word(A),A3,zero_zero(word(A))) = zero_zero(word(A)) ) ).

% word_sdiv_div0
tff(fact_6110_signed__drop__bit__of__minus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] : signed_drop_bit(A,Na,aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ).

% signed_drop_bit_of_minus_1
tff(fact_6111_word__sdiv__div__minus1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : signed7115095781618012415divide(word(A),A3,aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) = aa(word(A),word(A),uminus_uminus(word(A)),A3) ) ).

% word_sdiv_div_minus1
tff(fact_6112_minus__one__sdiv__word__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : signed7115095781618012415divide(word(A),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))),W) = aa(word(A),word(A),uminus_uminus(word(A)),signed7115095781618012415divide(word(A),one_one(word(A)),W)) ) ).

% minus_one_sdiv_word_eq
tff(fact_6113_one__sdiv__word__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          signed7115095781618012415divide(word(A),one_one(word(A)),W) = aa(word(A),word(A),
            aa(word(A),fun(word(A),word(A)),times_times(word(A)),
              aa($o,word(A),zero_neq_one_of_bool(word(A)),
                ( ( W = one_one(word(A)) )
                | ( W = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ))),
            W) ) ).

% one_sdiv_word_eq
tff(fact_6114_signed__drop__bit__of__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          signed_drop_bit(A,Na,one_one(word(A))) = aa($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_6115_sdiv__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : signed7115095781618012415divide(word(A),aa(num,word(A),numeral_numeral(word(A)),A3),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),aa(num,int,numeral_numeral(int),A3)),bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),aa(num,int,numeral_numeral(int),B3)))) ) ).

% sdiv_word_numeral_numeral
tff(fact_6116_signed__drop__bit__word__Suc__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,K: num] : 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)),bit_ri4674362597316999326ke_bit(int,aa(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_6117_signed__drop__bit__word__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num,K: num] : signed_drop_bit(A,aa(num,nat,numeral_numeral(nat),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(num,nat,numeral_numeral(nat),Na)),bit_ri4674362597316999326ke_bit(int,aa(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_numeral
tff(fact_6118_sdiv__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : signed7115095781618012415divide(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(int,bit_ri4674362597316999326ke_bit(int,aa(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),A3))),bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),aa(num,int,numeral_numeral(int),B3)))) ) ).

% sdiv_word_minus_numeral_numeral
tff(fact_6119_sdiv__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : signed7115095781618012415divide(word(A),aa(num,word(A),numeral_numeral(word(A)),A3),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3))) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),aa(num,int,numeral_numeral(int),A3)),bit_ri4674362597316999326ke_bit(int,aa(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),B3))))) ) ).

% sdiv_word_numeral_minus_numeral
tff(fact_6120_sdiv__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : signed7115095781618012415divide(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3))) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(int,bit_ri4674362597316999326ke_bit(int,aa(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),A3))),bit_ri4674362597316999326ke_bit(int,aa(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),B3))))) ) ).

% sdiv_word_minus_numeral_minus_numeral
tff(fact_6121_signed__drop__bit__word__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,K: num] : 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)),bit_ri4674362597316999326ke_bit(int,aa(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_6122_sint__signed__drop__bit__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] : aa(word(A),int,ring_1_signed(A,int),signed_drop_bit(A,Na,W)) = aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),Na),aa(word(A),int,ring_1_signed(A,int),W)) ) ).

% sint_signed_drop_bit_eq
tff(fact_6123_word__sdiv__numerals_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: num,Y: num] : signed7115095781618012415divide(word(A),aa(num,word(A),numeral_numeral(word(A)),X),aa(num,word(A),numeral_numeral(word(A)),Y)) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(int,aa(word(A),int,ring_1_signed(A,int),aa(num,word(A),numeral_numeral(word(A)),X)),aa(word(A),int,ring_1_signed(A,int),aa(num,word(A),numeral_numeral(word(A)),Y)))) ) ).

% word_sdiv_numerals(1)
tff(fact_6124_word__sdiv__numerals__lhs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: num,W: word(A)] : signed7115095781618012415divide(word(A),aa(num,word(A),numeral_numeral(word(A)),X),W) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(int,aa(word(A),int,ring_1_signed(A,int),aa(num,word(A),numeral_numeral(word(A)),X)),aa(word(A),int,ring_1_signed(A,int),W))) ) ).

% word_sdiv_numerals_lhs(1)
tff(fact_6125_word__sdiv__numerals__lhs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : signed7115095781618012415divide(word(A),one_one(word(A)),W) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(int,aa(word(A),int,ring_1_signed(A,int),one_one(word(A))),aa(word(A),int,ring_1_signed(A,int),W))) ) ).

% word_sdiv_numerals_lhs(3)
tff(fact_6126_word__sdiv__numerals_I9_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( signed7115095781618012415divide(word(A),one_one(word(A)),one_one(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(int,aa(word(A),int,ring_1_signed(A,int),one_one(word(A))),aa(word(A),int,ring_1_signed(A,int),one_one(word(A))))) ) ) ).

% word_sdiv_numerals(9)
tff(fact_6127_word__sdiv__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( signed7115095781618012415divide(word(A),zero_zero(word(A)),zero_zero(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(int,aa(word(A),int,ring_1_signed(A,int),zero_zero(word(A))),aa(word(A),int,ring_1_signed(A,int),zero_zero(word(A))))) ) ) ).

% word_sdiv_numerals(5)
tff(fact_6128_word__sdiv__numerals__lhs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : signed7115095781618012415divide(word(A),zero_zero(word(A)),W) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(int,aa(word(A),int,ring_1_signed(A,int),zero_zero(word(A))),aa(word(A),int,ring_1_signed(A,int),W))) ) ).

% word_sdiv_numerals_lhs(2)
tff(fact_6129_word__sdiv__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: num] : signed7115095781618012415divide(word(A),one_one(word(A)),aa(num,word(A),numeral_numeral(word(A)),Y)) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(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)),Y)))) ) ).

% word_sdiv_numerals(3)
tff(fact_6130_word__sdiv__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: num] : signed7115095781618012415divide(word(A),aa(num,word(A),numeral_numeral(word(A)),X),one_one(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(int,aa(word(A),int,ring_1_signed(A,int),aa(num,word(A),numeral_numeral(word(A)),X)),aa(word(A),int,ring_1_signed(A,int),one_one(word(A))))) ) ).

% word_sdiv_numerals(7)
tff(fact_6131_word__sdiv__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: num] : signed7115095781618012415divide(word(A),aa(num,word(A),numeral_numeral(word(A)),X),zero_zero(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(int,aa(word(A),int,ring_1_signed(A,int),aa(num,word(A),numeral_numeral(word(A)),X)),aa(word(A),int,ring_1_signed(A,int),zero_zero(word(A))))) ) ).

% word_sdiv_numerals(4)
tff(fact_6132_word__sdiv__numerals_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: num] : signed7115095781618012415divide(word(A),zero_zero(word(A)),aa(num,word(A),numeral_numeral(word(A)),Y)) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(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)),Y)))) ) ).

% word_sdiv_numerals(2)
tff(fact_6133_word__sdiv__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( signed7115095781618012415divide(word(A),zero_zero(word(A)),one_one(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(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))))) ) ) ).

% word_sdiv_numerals(8)
tff(fact_6134_word__sdiv__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( signed7115095781618012415divide(word(A),one_one(word(A)),zero_zero(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(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))))) ) ) ).

% word_sdiv_numerals(6)
tff(fact_6135_signed__div__arith,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(word(A),int,ring_1_signed(A,int),signed7115095781618012415divide(word(A),A3,B3)) = bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),signed7115095781618012415divide(int,aa(word(A),int,ring_1_signed(A,int),A3),aa(word(A),int,ring_1_signed(A,int),B3))) ) ).

% signed_div_arith
tff(fact_6136_bit__signed__drop__bit__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,W: word(A),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),signed_drop_bit(A,M,W)),Na)
        <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),
              $ite(
                ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),M)),Na)
                & aa(nat,$o,ord_less(nat,Na),type_len0_len_of(A,type2(A))) ),
                aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),
                aa(nat,nat,plus_plus(nat,M),Na) )) ) ) ).

% bit_signed_drop_bit_iff
tff(fact_6137_signed__drop__bit__beyond,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),Na)
         => ( signed_drop_bit(A,Na,W) = $ite(aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),aa(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_6138_word__int__split__asm,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [P: fun(A,$o),F2: fun(int,A),X: word(B)] :
          ( aa(A,$o,P,word_int_case(A,B,F2,X))
        <=> ~ ? [N3: int] :
                ( ( X = aa(int,word(B),ring_1_of_int(word(B)),N3) )
                & aa(int,$o,ord_less_eq(int,zero_zero(int)),N3)
                & aa(int,$o,ord_less(int,N3),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(B,type2(B))))
                & ~ aa(A,$o,P,aa(int,A,F2,N3)) ) ) ) ).

% word_int_split_asm
tff(fact_6139_word__int__split,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [P: fun(A,$o),F2: fun(int,A),X: word(B)] :
          ( aa(A,$o,P,word_int_case(A,B,F2,X))
        <=> ! [I4: int] :
              ( ( ( X = aa(int,word(B),ring_1_of_int(word(B)),I4) )
                & aa(int,$o,ord_less_eq(int,zero_zero(int)),I4)
                & aa(int,$o,ord_less(int,I4),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(B,type2(B)))) )
             => aa(A,$o,P,aa(int,A,F2,I4)) ) ) ) ).

% word_int_split
tff(fact_6140_word__int__case__eq__uint,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(B)
     => ! [F2: fun(int,A),W: word(B)] : word_int_case(A,B,F2,W) = aa(int,A,F2,aa(word(B),int,semiring_1_unsigned(B,int),W)) ) ).

% word_int_case_eq_uint
tff(fact_6141_word__int__case__wi,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(B)
     => ! [F2: fun(int,A),I: int] : word_int_case(A,B,F2,aa(int,word(B),ring_1_of_int(word(B)),I)) = aa(int,A,F2,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),I),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(B,type2(B))))) ) ).

% word_int_case_wi
tff(fact_6142_smod__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : signed6721504322012087516modulo(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(int,bit_ri4674362597316999326ke_bit(int,aa(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),A3))),bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),aa(num,int,numeral_numeral(int),B3)))) ) ).

% smod_word_minus_numeral_numeral
tff(fact_6143_smod__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : signed6721504322012087516modulo(word(A),aa(num,word(A),numeral_numeral(word(A)),A3),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3))) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),aa(num,int,numeral_numeral(int),A3)),bit_ri4674362597316999326ke_bit(int,aa(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),B3))))) ) ).

% smod_word_numeral_minus_numeral
tff(fact_6144_smod__word__mod__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : signed6721504322012087516modulo(word(A),X,zero_zero(word(A))) = X ) ).

% smod_word_mod_0
tff(fact_6145_smod__word__0__mod,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : signed6721504322012087516modulo(word(A),zero_zero(word(A)),X) = zero_zero(word(A)) ) ).

% smod_word_0_mod
tff(fact_6146_smod__word__zero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : signed6721504322012087516modulo(word(A),W,zero_zero(word(A))) = W ) ).

% smod_word_zero
tff(fact_6147_smod__word__one,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : signed6721504322012087516modulo(word(A),W,one_one(word(A))) = zero_zero(word(A)) ) ).

% smod_word_one
tff(fact_6148_minus__one__smod__word__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : signed6721504322012087516modulo(word(A),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))),W) = aa(word(A),word(A),uminus_uminus(word(A)),signed6721504322012087516modulo(word(A),one_one(word(A)),W)) ) ).

% minus_one_smod_word_eq
tff(fact_6149_smod__word__minus__one,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : signed6721504322012087516modulo(word(A),W,aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) = zero_zero(word(A)) ) ).

% smod_word_minus_one
tff(fact_6150_one__smod__word__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          signed6721504322012087516modulo(word(A),one_one(word(A)),W) = aa(word(A),word(A),minus_minus(word(A),one_one(word(A))),
            aa($o,word(A),zero_neq_one_of_bool(word(A)),
              ( ( W = one_one(word(A)) )
              | ( W = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ))) ) ).

% one_smod_word_eq
tff(fact_6151_smod__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : signed6721504322012087516modulo(word(A),aa(num,word(A),numeral_numeral(word(A)),A3),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),aa(num,int,numeral_numeral(int),A3)),bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),aa(num,int,numeral_numeral(int),B3)))) ) ).

% smod_word_numeral_numeral
tff(fact_6152_smod__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : signed6721504322012087516modulo(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3))) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(int,bit_ri4674362597316999326ke_bit(int,aa(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),A3))),bit_ri4674362597316999326ke_bit(int,aa(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),B3))))) ) ).

% smod_word_minus_numeral_minus_numeral
tff(fact_6153_word__smod__numerals_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: num,Y: num] : signed6721504322012087516modulo(word(A),aa(num,word(A),numeral_numeral(word(A)),X),aa(num,word(A),numeral_numeral(word(A)),Y)) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(int,aa(word(A),int,ring_1_signed(A,int),aa(num,word(A),numeral_numeral(word(A)),X)),aa(word(A),int,ring_1_signed(A,int),aa(num,word(A),numeral_numeral(word(A)),Y)))) ) ).

% word_smod_numerals(1)
tff(fact_6154_word__smod__numerals__lhs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: num,W: word(A)] : signed6721504322012087516modulo(word(A),aa(num,word(A),numeral_numeral(word(A)),X),W) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(int,aa(word(A),int,ring_1_signed(A,int),aa(num,word(A),numeral_numeral(word(A)),X)),aa(word(A),int,ring_1_signed(A,int),W))) ) ).

% word_smod_numerals_lhs(1)
tff(fact_6155_word__smod__numerals_I9_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( signed6721504322012087516modulo(word(A),one_one(word(A)),one_one(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(int,aa(word(A),int,ring_1_signed(A,int),one_one(word(A))),aa(word(A),int,ring_1_signed(A,int),one_one(word(A))))) ) ) ).

% word_smod_numerals(9)
tff(fact_6156_word__smod__numerals__lhs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : signed6721504322012087516modulo(word(A),one_one(word(A)),W) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(int,aa(word(A),int,ring_1_signed(A,int),one_one(word(A))),aa(word(A),int,ring_1_signed(A,int),W))) ) ).

% word_smod_numerals_lhs(3)
tff(fact_6157_word__smod__numerals__lhs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : signed6721504322012087516modulo(word(A),zero_zero(word(A)),W) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(int,aa(word(A),int,ring_1_signed(A,int),zero_zero(word(A))),aa(word(A),int,ring_1_signed(A,int),W))) ) ).

% word_smod_numerals_lhs(2)
tff(fact_6158_word__smod__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( signed6721504322012087516modulo(word(A),zero_zero(word(A)),zero_zero(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(int,aa(word(A),int,ring_1_signed(A,int),zero_zero(word(A))),aa(word(A),int,ring_1_signed(A,int),zero_zero(word(A))))) ) ) ).

% word_smod_numerals(5)
tff(fact_6159_word__smod__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: num] : signed6721504322012087516modulo(word(A),one_one(word(A)),aa(num,word(A),numeral_numeral(word(A)),Y)) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(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)),Y)))) ) ).

% word_smod_numerals(3)
tff(fact_6160_word__smod__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: num] : signed6721504322012087516modulo(word(A),aa(num,word(A),numeral_numeral(word(A)),X),one_one(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(int,aa(word(A),int,ring_1_signed(A,int),aa(num,word(A),numeral_numeral(word(A)),X)),aa(word(A),int,ring_1_signed(A,int),one_one(word(A))))) ) ).

% word_smod_numerals(7)
tff(fact_6161_word__smod__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: num] : signed6721504322012087516modulo(word(A),aa(num,word(A),numeral_numeral(word(A)),X),zero_zero(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(int,aa(word(A),int,ring_1_signed(A,int),aa(num,word(A),numeral_numeral(word(A)),X)),aa(word(A),int,ring_1_signed(A,int),zero_zero(word(A))))) ) ).

% word_smod_numerals(4)
tff(fact_6162_word__smod__numerals_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: num] : signed6721504322012087516modulo(word(A),zero_zero(word(A)),aa(num,word(A),numeral_numeral(word(A)),Y)) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(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)),Y)))) ) ).

% word_smod_numerals(2)
tff(fact_6163_word__smod__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( signed6721504322012087516modulo(word(A),zero_zero(word(A)),one_one(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(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))))) ) ) ).

% word_smod_numerals(8)
tff(fact_6164_word__smod__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( signed6721504322012087516modulo(word(A),one_one(word(A)),zero_zero(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(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))))) ) ) ).

% word_smod_numerals(6)
tff(fact_6165_signed__mod__arith,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(word(A),int,ring_1_signed(A,int),signed6721504322012087516modulo(word(A),A3,B3)) = bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),signed6721504322012087516modulo(int,aa(word(A),int,ring_1_signed(A,int),A3),aa(word(A),int,ring_1_signed(A,int),B3))) ) ).

% signed_mod_arith
tff(fact_6166_sshiftr__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          bit_Sh8784991116023147202shiftr(A,one_one(word(A)),Na) = aa($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_6167_bit__sshiftr__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),M: nat,Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),bit_Sh8784991116023147202shiftr(A,W,M)),Na)
        <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),
              $ite(
                ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),M)),Na)
                & aa(nat,$o,ord_less(nat,Na),type_len0_len_of(A,type2(A))) ),
                aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),
                aa(nat,nat,plus_plus(nat,M),Na) )) ) ) ).

% bit_sshiftr_iff
tff(fact_6168_sshiftr__of__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : bit_Sh8784991116023147202shiftr(A,W,zero_zero(nat)) = W ) ).

% sshiftr_of_0
tff(fact_6169_sshiftr__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] : bit_Sh8784991116023147202shiftr(A,zero_zero(word(A)),Na) = zero_zero(word(A)) ) ).

% sshiftr_0
tff(fact_6170_sshiftr__numeral__Suc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: num,Na: nat] : bit_Sh8784991116023147202shiftr(A,aa(num,word(A),numeral_numeral(word(A)),M),aa(nat,nat,suc,Na)) = signed_drop_bit(A,aa(nat,nat,suc,Na),aa(num,word(A),numeral_numeral(word(A)),M)) ) ).

% sshiftr_numeral_Suc
tff(fact_6171_sshiftr__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: num,Na: num] : bit_Sh8784991116023147202shiftr(A,aa(num,word(A),numeral_numeral(word(A)),M),aa(num,nat,numeral_numeral(nat),Na)) = signed_drop_bit(A,aa(num,nat,numeral_numeral(nat),Na),aa(num,word(A),numeral_numeral(word(A)),M)) ) ).

% sshiftr_numeral_numeral
tff(fact_6172_sshiftr__minus__numeral__Suc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: num,Na: nat] : 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)) = 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_6173_sshiftr__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: num,Na: num] : bit_Sh8784991116023147202shiftr(A,aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),M)),aa(num,nat,numeral_numeral(nat),Na)) = signed_drop_bit(A,aa(num,nat,numeral_numeral(nat),Na),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),M))) ) ).

% sshiftr_minus_numeral_numeral
tff(fact_6174_uint__word__ariths_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),word_pred(A,A3)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),A3)),one_one(int))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_word_ariths(6)
tff(fact_6175_slice1__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Na: nat,W: word(B)] :
          aa(word(B),word(A),slice1(B,A,Na),W) = $ite(aa(nat,$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,minus_minus(nat,type_len0_len_of(B,type2(B))),Na)),W)),aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4730199178511100633sh_bit(word(A)),aa(nat,nat,minus_minus(nat,Na),type_len0_len_of(B,type2(B)))),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),W))) ) ).

% slice1_def
tff(fact_6176_slice1__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Na: nat] : aa(word(B),word(A),slice1(B,A,Na),zero_zero(word(B))) = zero_zero(word(A)) ) ).

% slice1_0
tff(fact_6177_succ__pred__no_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: num] : word_pred(A,aa(num,word(A),numeral_numeral(word(A)),W)) = aa(word(A),word(A),minus_minus(word(A),aa(num,word(A),numeral_numeral(word(A)),W)),one_one(word(A))) ) ).

% succ_pred_no(2)
tff(fact_6178_succ__pred__no_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: num] : word_pred(A,aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),W))) = aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),W))),one_one(word(A))) ) ).

% succ_pred_no(4)
tff(fact_6179_word__m1__ge,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A)] : aa(word(A),$o,ord_less_eq(word(A),Y),word_pred(A,zero_zero(word(A)))) ) ).

% word_m1_ge
tff(fact_6180_word__not__simps_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A)] : ~ aa(word(A),$o,ord_less(word(A),word_pred(A,zero_zero(word(A)))),Y) ) ).

% word_not_simps(2)
tff(fact_6181_word__pred__m1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : word_pred(A,A3) = aa(word(A),word(A),minus_minus(word(A),A3),one_one(word(A))) ) ).

% word_pred_m1
tff(fact_6182_mask__eqs_I12_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),Na: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),word_pred(A,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),A3),bit_se2239418461657761734s_mask(word(A),Na)))),bit_se2239418461657761734s_mask(word(A),Na)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),word_pred(A,A3)),bit_se2239418461657761734s_mask(word(A),Na)) ) ).

% mask_eqs(12)
tff(fact_6183_ucast__slice1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [W: word(B)] : aa(word(B),word(A),semiring_1_unsigned(B,word(A)),W) = aa(word(B),word(A),slice1(B,A,size_size(word(B),W)),W) ) ).

% ucast_slice1
tff(fact_6184_wi__hom__pred,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: int] : word_pred(A,aa(int,word(A),ring_1_of_int(word(A)),A3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,minus_minus(int,A3),one_one(int))) ) ).

% wi_hom_pred
tff(fact_6185_word__pred__0__n1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( word_pred(A,zero_zero(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,uminus_uminus(int),one_one(int))) ) ) ).

% word_pred_0_n1
tff(fact_6186_word__pred__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : word_pred(A,A3) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),A3)),one_one(int))) ) ).

% word_pred_alt
tff(fact_6187_Word_Oslice__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Na: nat] : slice2(A,B,Na) = slice1(A,B,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),Na)) ) ).

% Word.slice_def
tff(fact_6188_uint__word__arith__bintrs_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),word_pred(A,A3)) = bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),A3)),one_one(int))) ) ).

% uint_word_arith_bintrs(6)
tff(fact_6189_sint__word__ariths_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : aa(word(A),int,ring_1_signed(A,int),word_pred(A,A3)) = bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),aa(int,int,minus_minus(int,aa(word(A),int,ring_1_signed(A,int),A3)),one_one(int))) ) ).

% sint_word_ariths(6)
tff(fact_6190_uint__word__ariths_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),word_succ(A,A3)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,plus_plus(int,aa(word(A),int,semiring_1_unsigned(A,int),A3)),one_one(int))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_word_ariths(5)
tff(fact_6191_of__nat__less__two__pow__div__set,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat] :
          ( aa(nat,$o,ord_less(nat,Na),type_len0_len_of(A,type2(A)))
         => ( collect(word(A),aa(nat,fun(word(A),$o),aTP_Lamp_og(nat,fun(nat,fun(word(A),$o)),Na),M)) = aa(set(nat),set(word(A)),image(nat,word(A),semiring_1_of_nat(word(A))),collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_oh(nat,fun(nat,fun(nat,$o)),Na),M))) ) ) ) ).

% of_nat_less_two_pow_div_set
tff(fact_6192_image__eqI,axiom,
    ! [A: $tType,B: $tType,B3: A,F2: fun(B,A),X: B,A2: set(B)] :
      ( ( B3 = aa(B,A,F2,X) )
     => ( member(B,X,A2)
       => member(A,B3,aa(set(B),set(A),image(B,A,F2),A2)) ) ) ).

% image_eqI
tff(fact_6193_image__ident,axiom,
    ! [A: $tType,Y7: set(A)] : aa(set(A),set(A),image(A,A,aTP_Lamp_oi(A,A)),Y7) = Y7 ).

% image_ident
tff(fact_6194_image__is__empty,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A2: set(B)] :
      ( ( aa(set(B),set(A),image(B,A,F2),A2) = bot_bot(set(A)) )
    <=> ( A2 = bot_bot(set(B)) ) ) ).

% image_is_empty
tff(fact_6195_empty__is__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A2: set(B)] :
      ( ( bot_bot(set(A)) = aa(set(B),set(A),image(B,A,F2),A2) )
    <=> ( A2 = bot_bot(set(B)) ) ) ).

% empty_is_image
tff(fact_6196_image__empty,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A)] : aa(set(B),set(A),image(B,A,F2),bot_bot(set(B))) = bot_bot(set(A)) ).

% image_empty
tff(fact_6197_insert__image,axiom,
    ! [B: $tType,A: $tType,X: A,A2: set(A),F2: fun(A,B)] :
      ( member(A,X,A2)
     => ( aa(set(B),set(B),insert(B,aa(A,B,F2,X)),aa(set(A),set(B),image(A,B,F2),A2)) = aa(set(A),set(B),image(A,B,F2),A2) ) ) ).

% insert_image
tff(fact_6198_image__insert,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: B,B2: set(B)] : aa(set(B),set(A),image(B,A,F2),aa(set(B),set(B),insert(B,A3),B2)) = aa(set(A),set(A),insert(A,aa(B,A,F2,A3)),aa(set(B),set(A),image(B,A,F2),B2)) ).

% image_insert
tff(fact_6199_image__add__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(A)] : aa(set(A),set(A),image(A,A,plus_plus(A,zero_zero(A))),S) = S ) ).

% image_add_0
tff(fact_6200_bij__betw__of__nat,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N2: set(nat),A2: set(A)] :
          ( bij_betw(nat,A,semiring_1_of_nat(A),N2,A2)
        <=> ( aa(set(nat),set(A),image(nat,A,semiring_1_of_nat(A)),N2) = A2 ) ) ) ).

% bij_betw_of_nat
tff(fact_6201_image__add__atLeastAtMost_H,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I: A,J: A] : aa(set(A),set(A),image(A,A,aTP_Lamp_oj(A,fun(A,A),K)),set_or1337092689740270186AtMost(A,I,J)) = set_or1337092689740270186AtMost(A,aa(A,A,plus_plus(A,I),K),aa(A,A,plus_plus(A,J),K)) ) ).

% image_add_atLeastAtMost'
tff(fact_6202_image__add__atLeastLessThan_H,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I: A,J: A] : aa(set(A),set(A),image(A,A,aTP_Lamp_oj(A,fun(A,A),K)),set_or7035219750837199246ssThan(A,I,J)) = set_or7035219750837199246ssThan(A,aa(A,A,plus_plus(A,I),K),aa(A,A,plus_plus(A,J),K)) ) ).

% image_add_atLeastLessThan'
tff(fact_6203_image__minus__const__atLeastAtMost_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [D2: A,A3: A,B3: A] : aa(set(A),set(A),image(A,A,aTP_Lamp_ok(A,fun(A,A),D2)),set_or1337092689740270186AtMost(A,A3,B3)) = set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,A3),D2),aa(A,A,minus_minus(A,B3),D2)) ) ).

% image_minus_const_atLeastAtMost'
tff(fact_6204_succ__pred__no_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: num] : word_succ(A,aa(num,word(A),numeral_numeral(word(A)),W)) = aa(word(A),word(A),plus_plus(word(A),aa(num,word(A),numeral_numeral(word(A)),W)),one_one(word(A))) ) ).

% succ_pred_no(1)
tff(fact_6205_image__mult__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D2: A,A3: A,B3: A] :
          ( aa(A,$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,A3,B3)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),D2),A3),aa(A,A,aa(A,fun(A,A),times_times(A),D2),B3)) ) ) ) ).

% image_mult_atLeastAtMost
tff(fact_6206_image__divide__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),D2)
         => ( aa(set(A),set(A),image(A,A,aTP_Lamp_ol(A,fun(A,A),D2)),set_or1337092689740270186AtMost(A,A3,B3)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),D2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),D2)) ) ) ) ).

% image_divide_atLeastAtMost
tff(fact_6207_succ__pred__no_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: num] : word_succ(A,aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),W))) = aa(word(A),word(A),plus_plus(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),W))),one_one(word(A))) ) ).

% succ_pred_no(3)
tff(fact_6208_word__pred__succ,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : word_pred(A,word_succ(A,A3)) = A3 ) ).

% word_pred_succ
tff(fact_6209_word__succ__pred,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : word_succ(A,word_pred(A,A3)) = A3 ) ).

% word_succ_pred
tff(fact_6210_finite__conv__nat__seg__image,axiom,
    ! [A: $tType,A2: set(A)] :
      ( finite_finite2(A,A2)
    <=> ? [N3: nat,F6: fun(nat,A)] : A2 = aa(set(nat),set(A),image(nat,A,F6),collect(nat,aTP_Lamp_gc(nat,fun(nat,$o),N3))) ) ).

% finite_conv_nat_seg_image
tff(fact_6211_nat__seg__image__imp__finite,axiom,
    ! [A: $tType,A2: set(A),F2: fun(nat,A),Na: nat] :
      ( ( A2 = aa(set(nat),set(A),image(nat,A,F2),collect(nat,aTP_Lamp_gc(nat,fun(nat,$o),Na))) )
     => finite_finite2(A,A2) ) ).

% nat_seg_image_imp_finite
tff(fact_6212_pigeonhole__infinite,axiom,
    ! [B: $tType,A: $tType,A2: set(A),F2: fun(A,B)] :
      ( ~ finite_finite2(A,A2)
     => ( finite_finite2(B,aa(set(A),set(B),image(A,B,F2),A2))
       => ? [X4: A] :
            ( member(A,X4,A2)
            & ~ finite_finite2(A,collect(A,aa(A,fun(A,$o),aa(fun(A,B),fun(A,fun(A,$o)),aTP_Lamp_om(set(A),fun(fun(A,B),fun(A,fun(A,$o))),A2),F2),X4))) ) ) ) ).

% pigeonhole_infinite
tff(fact_6213_rev__image__eqI,axiom,
    ! [B: $tType,A: $tType,X: A,A2: set(A),B3: B,F2: fun(A,B)] :
      ( member(A,X,A2)
     => ( ( B3 = aa(A,B,F2,X) )
       => member(B,B3,aa(set(A),set(B),image(A,B,F2),A2)) ) ) ).

% rev_image_eqI
tff(fact_6214_ball__imageD,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A2: set(B),P: fun(A,$o)] :
      ( ! [X4: A] :
          ( member(A,X4,aa(set(B),set(A),image(B,A,F2),A2))
         => aa(A,$o,P,X4) )
     => ! [X2: B] :
          ( member(B,X2,A2)
         => aa(A,$o,P,aa(B,A,F2,X2)) ) ) ).

% ball_imageD
tff(fact_6215_image__cong,axiom,
    ! [B: $tType,A: $tType,M7: set(A),N2: set(A),F2: fun(A,B),G: fun(A,B)] :
      ( ( M7 = N2 )
     => ( ! [X4: A] :
            ( member(A,X4,N2)
           => ( aa(A,B,F2,X4) = aa(A,B,G,X4) ) )
       => ( aa(set(A),set(B),image(A,B,F2),M7) = aa(set(A),set(B),image(A,B,G),N2) ) ) ) ).

% image_cong
tff(fact_6216_bex__imageD,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A2: set(B),P: fun(A,$o)] :
      ( ? [X2: A] :
          ( member(A,X2,aa(set(B),set(A),image(B,A,F2),A2))
          & aa(A,$o,P,X2) )
     => ? [X4: B] :
          ( member(B,X4,A2)
          & aa(A,$o,P,aa(B,A,F2,X4)) ) ) ).

% bex_imageD
tff(fact_6217_image__iff,axiom,
    ! [A: $tType,B: $tType,Z: A,F2: fun(B,A),A2: set(B)] :
      ( member(A,Z,aa(set(B),set(A),image(B,A,F2),A2))
    <=> ? [X3: B] :
          ( member(B,X3,A2)
          & ( Z = aa(B,A,F2,X3) ) ) ) ).

% image_iff
tff(fact_6218_imageI,axiom,
    ! [B: $tType,A: $tType,X: A,A2: set(A),F2: fun(A,B)] :
      ( member(A,X,A2)
     => member(B,aa(A,B,F2,X),aa(set(A),set(B),image(A,B,F2),A2)) ) ).

% imageI
tff(fact_6219_Compr__image__eq,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A2: set(B),P: fun(A,$o)] : collect(A,aa(fun(A,$o),fun(A,$o),aa(set(B),fun(fun(A,$o),fun(A,$o)),aTP_Lamp_on(fun(B,A),fun(set(B),fun(fun(A,$o),fun(A,$o))),F2),A2),P)) = aa(set(B),set(A),image(B,A,F2),collect(B,aa(fun(A,$o),fun(B,$o),aa(set(B),fun(fun(A,$o),fun(B,$o)),aTP_Lamp_oo(fun(B,A),fun(set(B),fun(fun(A,$o),fun(B,$o))),F2),A2),P))) ).

% Compr_image_eq
tff(fact_6220_image__image,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(B,A),G: fun(C,B),A2: set(C)] : aa(set(B),set(A),image(B,A,F2),aa(set(C),set(B),image(C,B,G),A2)) = aa(set(C),set(A),image(C,A,aa(fun(C,B),fun(C,A),aTP_Lamp_op(fun(B,A),fun(fun(C,B),fun(C,A)),F2),G)),A2) ).

% image_image
tff(fact_6221_imageE,axiom,
    ! [A: $tType,B: $tType,B3: A,F2: fun(B,A),A2: set(B)] :
      ( member(A,B3,aa(set(B),set(A),image(B,A,F2),A2))
     => ~ ! [X4: B] :
            ( ( B3 = aa(B,A,F2,X4) )
           => ~ member(B,X4,A2) ) ) ).

% imageE
tff(fact_6222_image__Un,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A2: set(B),B2: set(B)] : aa(set(B),set(A),image(B,A,F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A2),B2)) = 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,F2),A2)),aa(set(B),set(A),image(B,A,F2),B2)) ).

% image_Un
tff(fact_6223_image__diff__subset,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A2: set(B),B2: set(B)] : aa(set(A),$o,ord_less_eq(set(A),aa(set(A),set(A),minus_minus(set(A),aa(set(B),set(A),image(B,A,F2),A2)),aa(set(B),set(A),image(B,A,F2),B2))),aa(set(B),set(A),image(B,A,F2),aa(set(B),set(B),minus_minus(set(B),A2),B2))) ).

% image_diff_subset
tff(fact_6224_subset__image__iff,axiom,
    ! [A: $tType,B: $tType,B2: set(A),F2: fun(B,A),A2: set(B)] :
      ( aa(set(A),$o,ord_less_eq(set(A),B2),aa(set(B),set(A),image(B,A,F2),A2))
    <=> ? [AA: set(B)] :
          ( aa(set(B),$o,ord_less_eq(set(B),AA),A2)
          & ( B2 = aa(set(B),set(A),image(B,A,F2),AA) ) ) ) ).

% subset_image_iff
tff(fact_6225_image__subset__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A2: set(B),B2: set(A)] :
      ( aa(set(A),$o,ord_less_eq(set(A),aa(set(B),set(A),image(B,A,F2),A2)),B2)
    <=> ! [X3: B] :
          ( member(B,X3,A2)
         => member(A,aa(B,A,F2,X3),B2) ) ) ).

% image_subset_iff
tff(fact_6226_subset__imageE,axiom,
    ! [A: $tType,B: $tType,B2: set(A),F2: fun(B,A),A2: set(B)] :
      ( aa(set(A),$o,ord_less_eq(set(A),B2),aa(set(B),set(A),image(B,A,F2),A2))
     => ~ ! [C7: set(B)] :
            ( aa(set(B),$o,ord_less_eq(set(B),C7),A2)
           => ( B2 != aa(set(B),set(A),image(B,A,F2),C7) ) ) ) ).

% subset_imageE
tff(fact_6227_image__subsetI,axiom,
    ! [A: $tType,B: $tType,A2: set(A),F2: fun(A,B),B2: set(B)] :
      ( ! [X4: A] :
          ( member(A,X4,A2)
         => member(B,aa(A,B,F2,X4),B2) )
     => aa(set(B),$o,ord_less_eq(set(B),aa(set(A),set(B),image(A,B,F2),A2)),B2) ) ).

% image_subsetI
tff(fact_6228_image__mono,axiom,
    ! [B: $tType,A: $tType,A2: set(A),B2: set(A),F2: fun(A,B)] :
      ( aa(set(A),$o,ord_less_eq(set(A),A2),B2)
     => aa(set(B),$o,ord_less_eq(set(B),aa(set(A),set(B),image(A,B,F2),A2)),aa(set(A),set(B),image(A,B,F2),B2)) ) ).

% image_mono
tff(fact_6229_image__constant,axiom,
    ! [A: $tType,B: $tType,X: A,A2: set(A),C2: B] :
      ( member(A,X,A2)
     => ( aa(set(A),set(B),image(A,B,aTP_Lamp_oq(B,fun(A,B),C2)),A2) = aa(set(B),set(B),insert(B,C2),bot_bot(set(B))) ) ) ).

% image_constant
tff(fact_6230_image__constant__conv,axiom,
    ! [B: $tType,A: $tType,C2: A,A2: set(B)] :
      aa(set(B),set(A),image(B,A,aTP_Lamp_or(A,fun(B,A),C2)),A2) = $ite(A2 = bot_bot(set(B)),bot_bot(set(A)),aa(set(A),set(A),insert(A,C2),bot_bot(set(A)))) ).

% image_constant_conv
tff(fact_6231_sum_Oimage__gen,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S: set(A),H: fun(A,B),G: fun(A,C)] :
          ( finite_finite2(A,S)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),S) = aa(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_ot(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),S),H),G)),aa(set(A),set(C),image(A,C,G),S)) ) ) ) ).

% sum.image_gen
tff(fact_6232_translation__subtract__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,Sa: set(A),Ta: set(A)] : aa(set(A),set(A),image(A,A,aTP_Lamp_ou(A,fun(A,A),A3)),aa(set(A),set(A),minus_minus(set(A),Sa),Ta)) = aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),image(A,A,aTP_Lamp_ou(A,fun(A,A),A3)),Sa)),aa(set(A),set(A),image(A,A,aTP_Lamp_ou(A,fun(A,A),A3)),Ta)) ) ).

% translation_subtract_diff
tff(fact_6233_prod_Oimage__gen,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),H: fun(A,B),G: fun(A,C)] :
          ( finite_finite2(A,S)
         => ( groups7121269368397514597t_prod(A,B,H,S) = groups7121269368397514597t_prod(C,B,aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_ov(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),S),H),G),aa(set(A),set(C),image(A,C,G),S)) ) ) ) ).

% prod.image_gen
tff(fact_6234_translation__subtract__Compl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,Ta: set(A)] : aa(set(A),set(A),image(A,A,aTP_Lamp_ou(A,fun(A,A),A3)),aa(set(A),set(A),uminus_uminus(set(A)),Ta)) = aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),image(A,A,aTP_Lamp_ou(A,fun(A,A),A3)),Ta)) ) ).

% translation_subtract_Compl
tff(fact_6235_the__elem__image__unique,axiom,
    ! [B: $tType,A: $tType,A2: set(A),F2: fun(A,B),X: A] :
      ( ( A2 != bot_bot(set(A)) )
     => ( ! [Y3: A] :
            ( member(A,Y3,A2)
           => ( aa(A,B,F2,Y3) = aa(A,B,F2,X) ) )
       => ( the_elem(B,aa(set(A),set(B),image(A,B,F2),A2)) = aa(A,B,F2,X) ) ) ) ).

% the_elem_image_unique
tff(fact_6236_Abs__fnat__hom__Suc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: nat] : word_succ(A,aa(nat,word(A),semiring_1_of_nat(word(A)),A3)) = aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,suc,A3)) ) ).

% Abs_fnat_hom_Suc
tff(fact_6237_word__succ__p1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : word_succ(A,A3) = aa(word(A),word(A),plus_plus(word(A),A3),one_one(word(A))) ) ).

% word_succ_p1
tff(fact_6238_mask__eqs_I11_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),Na: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),word_succ(A,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),A3),bit_se2239418461657761734s_mask(word(A),Na)))),bit_se2239418461657761734s_mask(word(A),Na)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),word_succ(A,A3)),bit_se2239418461657761734s_mask(word(A),Na)) ) ).

% mask_eqs(11)
tff(fact_6239_word__mult__succ,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),word_succ(A,A3)),B3) = aa(word(A),word(A),plus_plus(word(A),B3),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A3),B3)) ) ).

% word_mult_succ
tff(fact_6240_sum_Ogroup,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [S: set(A),T3: set(B),G: fun(A,B),H: fun(A,C)] :
          ( finite_finite2(A,S)
         => ( finite_finite2(B,T3)
           => ( aa(set(B),$o,ord_less_eq(set(B),aa(set(A),set(B),image(A,B,G),S)),T3)
             => ( aa(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_ox(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),S),G),H)),T3) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,H),S) ) ) ) ) ) ).

% sum.group
tff(fact_6241_prod_Ogroup,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [S: set(A),T3: set(B),G: fun(A,B),H: fun(A,C)] :
          ( finite_finite2(A,S)
         => ( finite_finite2(B,T3)
           => ( aa(set(B),$o,ord_less_eq(set(B),aa(set(A),set(B),image(A,B,G),S)),T3)
             => ( groups7121269368397514597t_prod(B,C,aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_oy(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),S),G),H),T3) = groups7121269368397514597t_prod(A,C,H,S) ) ) ) ) ) ).

% prod.group
tff(fact_6242_scaleR__image__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,X: A,Y: A] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
         => ( aa(set(A),set(A),image(A,A,real_V8093663219630862766scaleR(A,C2)),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),X),aa(A,A,real_V8093663219630862766scaleR(A,C2),Y)) ) ) ) ).

% scaleR_image_atLeastAtMost
tff(fact_6243_wi__hom__succ,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: int] : word_succ(A,aa(int,word(A),ring_1_of_int(word(A)),A3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,plus_plus(int,A3),one_one(int))) ) ).

% wi_hom_succ
tff(fact_6244_word__arith__nat__Suc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : word_succ(A,A3) = aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),A3))) ) ).

% word_arith_nat_Suc
tff(fact_6245_image__mult__atLeastAtMost__if,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,X: A,Y: A] :
          aa(set(A),set(A),image(A,A,aa(A,fun(A,A),times_times(A),C2)),set_or1337092689740270186AtMost(A,X,Y)) = $ite(
            aa(A,$o,ord_less(A,zero_zero(A)),C2),
            set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),X),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Y)),
            $ite(aa(A,$o,ord_less_eq(A,X),Y),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),Y),aa(A,A,aa(A,fun(A,A),times_times(A),C2),X)),bot_bot(set(A))) ) ) ).

% image_mult_atLeastAtMost_if
tff(fact_6246_word__succ__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : word_succ(A,A3) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,plus_plus(int,aa(word(A),int,semiring_1_unsigned(A,int),A3)),one_one(int))) ) ).

% word_succ_alt
tff(fact_6247_image__mult__atLeastAtMost__if_H,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,X: A,Y: A] :
          aa(set(A),set(A),image(A,A,aTP_Lamp_oz(A,fun(A,A),C2)),set_or1337092689740270186AtMost(A,X,Y)) = $ite(
            aa(A,$o,ord_less_eq(A,X),Y),
            $ite(aa(A,$o,ord_less(A,zero_zero(A)),C2),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),C2),aa(A,A,aa(A,fun(A,A),times_times(A),Y),C2)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),Y),C2),aa(A,A,aa(A,fun(A,A),times_times(A),X),C2))),
            bot_bot(set(A)) ) ) ).

% image_mult_atLeastAtMost_if'
tff(fact_6248_image__affinity__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C2: A,A3: A,B3: A] :
          aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_pa(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A3,B3)) = $ite(
            set_or1337092689740270186AtMost(A,A3,B3) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,ord_less_eq(A,zero_zero(A)),M),set_or1337092689740270186AtMost(A,aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),A3)),C2),aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),B3)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),B3)),C2),aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),A3)),C2))) ) ) ).

% image_affinity_atLeastAtMost
tff(fact_6249_image__affinity__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C2: A,A3: A,B3: A] :
          aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_pb(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A3,B3)) = $ite(
            set_or1337092689740270186AtMost(A,A3,B3) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,ord_less_eq(A,zero_zero(A)),M),set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),A3)),C2),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),B3)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),B3)),C2),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),A3)),C2))) ) ) ).

% image_affinity_atLeastAtMost_diff
tff(fact_6250_image__affinity__atLeastAtMost__div,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C2: A,A3: A,B3: A] :
          aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_pc(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A3,B3)) = $ite(
            set_or1337092689740270186AtMost(A,A3,B3) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,ord_less_eq(A,zero_zero(A)),M),set_or1337092689740270186AtMost(A,aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),M)),C2),aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),M)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),M)),C2),aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),M)),C2))) ) ) ).

% image_affinity_atLeastAtMost_div
tff(fact_6251_image__affinity__atLeastAtMost__div__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C2: A,A3: A,B3: A] :
          aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_pd(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A3,B3)) = $ite(
            set_or1337092689740270186AtMost(A,A3,B3) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,ord_less_eq(A,zero_zero(A)),M),set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),M)),C2),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),M)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),M)),C2),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),M)),C2))) ) ) ).

% image_affinity_atLeastAtMost_div_diff
tff(fact_6252_word__sp__01,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType] :
      ( ( type_len(D)
        & type_len(B)
        & type_len(A)
        & type_len(C) )
     => ( ( word_succ(A,aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) = zero_zero(word(A)) )
        & ( word_succ(B,zero_zero(word(B))) = one_one(word(B)) )
        & ( word_pred(C,zero_zero(word(C))) = aa(word(C),word(C),uminus_uminus(word(C)),one_one(word(C))) )
        & ( word_pred(D,one_one(word(D))) = zero_zero(word(D)) ) ) ) ).

% word_sp_01
tff(fact_6253_uint__word__arith__bintrs_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),word_succ(A,A3)) = bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),aa(int,int,plus_plus(int,aa(word(A),int,semiring_1_unsigned(A,int),A3)),one_one(int))) ) ).

% uint_word_arith_bintrs(5)
tff(fact_6254_sint__word__ariths_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : aa(word(A),int,ring_1_signed(A,int),word_succ(A,A3)) = bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),aa(int,int,plus_plus(int,aa(word(A),int,ring_1_signed(A,int),A3)),one_one(int))) ) ).

% sint_word_ariths(5)
tff(fact_6255_unat__word__ariths_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),word_succ(A,A3)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),A3))),aa(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_6256_bit__word__roti__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: int,W: word(A),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),word_roti(A,K,W)),Na)
        <=> ( aa(nat,$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)),W),nat2(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(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_6257_sless__eq__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( word_sle(A,aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3)))
        <=> aa(int,$o,ord_less_eq(int,bit_ri4674362597316999326ke_bit(int,aa(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),A3)))),bit_ri4674362597316999326ke_bit(int,aa(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),B3)))) ) ) ).

% sless_eq_word_minus_numeral_minus_numeral
tff(fact_6258_bij__betw__Suc,axiom,
    ! [M7: set(nat),N2: set(nat)] :
      ( bij_betw(nat,nat,suc,M7,N2)
    <=> ( aa(set(nat),set(nat),image(nat,nat,suc),M7) = N2 ) ) ).

% bij_betw_Suc
tff(fact_6259_signed_Oorder__refl,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : word_sle(A,X,X) ) ).

% signed.order_refl
tff(fact_6260_signed_Odual__order_Orefl,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : word_sle(A,A3,A3) ) ).

% signed.dual_order.refl
tff(fact_6261_word__roti__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : word_roti(A,zero_zero(int),W) = W ) ).

% word_roti_0
tff(fact_6262_pair__imageI,axiom,
    ! [C: $tType,B: $tType,A: $tType,A3: A,B3: B,A2: set(product_prod(A,B)),F2: 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),A3),B3),A2)
     => member(C,aa(B,C,aa(A,fun(B,C),F2,A3),B3),aa(set(product_prod(A,B)),set(C),image(product_prod(A,B),C,product_case_prod(A,B,C,F2)),A2)) ) ).

% pair_imageI
tff(fact_6263_word__roti__0_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: int] : word_roti(A,Na,zero_zero(word(A))) = zero_zero(word(A)) ) ).

% word_roti_0'
tff(fact_6264_word__sle__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( word_sle(A,aa(num,word(A),numeral_numeral(word(A)),A3),aa(num,word(A),numeral_numeral(word(A)),B3))
        <=> aa(int,$o,ord_less_eq(int,aa(word(A),int,ring_1_signed(A,int),aa(num,word(A),numeral_numeral(word(A)),A3))),aa(word(A),int,ring_1_signed(A,int),aa(num,word(A),numeral_numeral(word(A)),B3))) ) ) ).

% word_sle_no
tff(fact_6265_extra__sle__sless__unfolds_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num] :
          ( word_sle(A,aa(num,word(A),numeral_numeral(word(A)),Na),one_one(word(A)))
        <=> aa(int,$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_6266_extra__sle__sless__unfolds_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num] :
          ( word_sle(A,one_one(word(A)),aa(num,word(A),numeral_numeral(word(A)),Na))
        <=> aa(int,$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_6267_extra__sle__sless__unfolds_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num] :
          ( word_sle(A,zero_zero(word(A)),aa(num,word(A),numeral_numeral(word(A)),Na))
        <=> aa(int,$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_6268_extra__sle__sless__unfolds_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num] :
          ( word_sle(A,aa(num,word(A),numeral_numeral(word(A)),Na),zero_zero(word(A)))
        <=> aa(int,$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_6269_extra__sle__sless__unfolds_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( word_sle(A,one_one(word(A)),zero_zero(word(A)))
      <=> aa(int,$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_6270_extra__sle__sless__unfolds_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( word_sle(A,zero_zero(word(A)),one_one(word(A)))
      <=> aa(int,$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_6271_sless__eq__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( word_sle(A,aa(num,word(A),numeral_numeral(word(A)),A3),aa(num,word(A),numeral_numeral(word(A)),B3))
        <=> aa(int,$o,ord_less_eq(int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),aa(num,int,numeral_numeral(int),A3))),bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),aa(num,int,numeral_numeral(int),B3))) ) ) ).

% sless_eq_word_numeral_numeral
tff(fact_6272_sless__eq__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( word_sle(A,aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(num,word(A),numeral_numeral(word(A)),B3))
        <=> aa(int,$o,ord_less_eq(int,bit_ri4674362597316999326ke_bit(int,aa(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),A3)))),bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),aa(num,int,numeral_numeral(int),B3))) ) ) ).

% sless_eq_word_minus_numeral_numeral
tff(fact_6273_sless__eq__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( word_sle(A,aa(num,word(A),numeral_numeral(word(A)),A3),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3)))
        <=> aa(int,$o,ord_less_eq(int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),aa(num,int,numeral_numeral(int),A3))),bit_ri4674362597316999326ke_bit(int,aa(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),B3)))) ) ) ).

% sless_eq_word_numeral_minus_numeral
tff(fact_6274_zero__notin__Suc__image,axiom,
    ! [A2: set(nat)] : ~ member(nat,zero_zero(nat),aa(set(nat),set(nat),image(nat,nat,suc),A2)) ).

% zero_notin_Suc_image
tff(fact_6275_word__sle__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          ( word_sle(A,A3,B3)
        <=> aa(int,$o,ord_less_eq(int,aa(word(A),int,ring_1_signed(A,int),A3)),aa(word(A),int,ring_1_signed(A,int),B3)) ) ) ).

% word_sle_eq
tff(fact_6276_signed_Olift__Suc__antimono__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F2: fun(nat,word(A)),Na: nat,N6: nat] :
          ( ! [N: nat] : word_sle(A,aa(nat,word(A),F2,aa(nat,nat,suc,N)),aa(nat,word(A),F2,N))
         => ( aa(nat,$o,ord_less_eq(nat,Na),N6)
           => word_sle(A,aa(nat,word(A),F2,N6),aa(nat,word(A),F2,Na)) ) ) ) ).

% signed.lift_Suc_antimono_le
tff(fact_6277_signed_Olift__Suc__mono__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F2: fun(nat,word(A)),Na: nat,N6: nat] :
          ( ! [N: nat] : word_sle(A,aa(nat,word(A),F2,N),aa(nat,word(A),F2,aa(nat,nat,suc,N)))
         => ( aa(nat,$o,ord_less_eq(nat,Na),N6)
           => word_sle(A,aa(nat,word(A),F2,Na),aa(nat,word(A),F2,N6)) ) ) ) ).

% signed.lift_Suc_mono_le
tff(fact_6278_word__roti__add,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: int,Na: int,W: word(A)] : word_roti(A,aa(int,int,plus_plus(int,M),Na),W) = word_roti(A,M,word_roti(A,Na,W)) ) ).

% word_roti_add
tff(fact_6279_signed_Olinear,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( word_sle(A,X,Y)
          | word_sle(A,Y,X) ) ) ).

% signed.linear
tff(fact_6280_signed_Onle__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          ( ~ word_sle(A,A3,B3)
        <=> ( word_sle(A,B3,A3)
            & ( B3 != A3 ) ) ) ) ).

% signed.nle_le
tff(fact_6281_signed_Oeq__refl,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ( X = Y )
         => word_sle(A,X,Y) ) ) ).

% signed.eq_refl
tff(fact_6282_signed_Ole__cases,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ~ word_sle(A,X,Y)
         => word_sle(A,Y,X) ) ) ).

% signed.le_cases
tff(fact_6283_signed_Ole__cases3,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A),Z: word(A)] :
          ( ( word_sle(A,X,Y)
           => ~ word_sle(A,Y,Z) )
         => ( ( word_sle(A,Y,X)
             => ~ word_sle(A,X,Z) )
           => ( ( word_sle(A,X,Z)
               => ~ word_sle(A,Z,Y) )
             => ( ( word_sle(A,Z,Y)
                 => ~ word_sle(A,Y,X) )
               => ( ( word_sle(A,Y,Z)
                   => ~ word_sle(A,Z,X) )
                 => ~ ( word_sle(A,Z,X)
                     => ~ word_sle(A,X,Y) ) ) ) ) ) ) ) ).

% signed.le_cases3
tff(fact_6284_signed_Oorder_Otrans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A),C2: word(A)] :
          ( word_sle(A,A3,B3)
         => ( word_sle(A,B3,C2)
           => word_sle(A,A3,C2) ) ) ) ).

% signed.order.trans
tff(fact_6285_signed_Oorder__trans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A),Z: word(A)] :
          ( word_sle(A,X,Y)
         => ( word_sle(A,Y,Z)
           => word_sle(A,X,Z) ) ) ) ).

% signed.order_trans
tff(fact_6286_signed_Oantisym__conv,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A)] :
          ( word_sle(A,Y,X)
         => ( word_sle(A,X,Y)
          <=> ( X = Y ) ) ) ) ).

% signed.antisym_conv
tff(fact_6287_signed_Oorder_Oeq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          ( ( A3 = B3 )
        <=> ( word_sle(A,A3,B3)
            & word_sle(A,B3,A3) ) ) ) ).

% signed.order.eq_iff
tff(fact_6288_signed_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ( X = Y )
        <=> ( word_sle(A,X,Y)
            & word_sle(A,Y,X) ) ) ) ).

% signed.order_eq_iff
tff(fact_6289_signed_Olinorder__wlog,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(word(A),fun(word(A),$o)),A3: word(A),B3: word(A)] :
          ( ! [A4: word(A),B5: word(A)] :
              ( word_sle(A,A4,B5)
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),P,A4),B5) )
         => ( ! [A4: word(A),B5: word(A)] :
                ( aa(word(A),$o,aa(word(A),fun(word(A),$o),P,B5),A4)
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),P,A4),B5) )
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),P,A3),B3) ) ) ) ).

% signed.linorder_wlog
tff(fact_6290_signed_Oorder_Oantisym,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          ( word_sle(A,A3,B3)
         => ( word_sle(A,B3,A3)
           => ( A3 = B3 ) ) ) ) ).

% signed.order.antisym
tff(fact_6291_signed_Oorder__antisym,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( word_sle(A,X,Y)
         => ( word_sle(A,Y,X)
           => ( X = Y ) ) ) ) ).

% signed.order_antisym
tff(fact_6292_signed_Oord__eq__le__trans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A),C2: word(A)] :
          ( ( A3 = B3 )
         => ( word_sle(A,B3,C2)
           => word_sle(A,A3,C2) ) ) ) ).

% signed.ord_eq_le_trans
tff(fact_6293_signed_Oord__le__eq__trans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A),C2: word(A)] :
          ( word_sle(A,A3,B3)
         => ( ( B3 = C2 )
           => word_sle(A,A3,C2) ) ) ) ).

% signed.ord_le_eq_trans
tff(fact_6294_signed_Odual__order_Otrans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: word(A),A3: word(A),C2: word(A)] :
          ( word_sle(A,B3,A3)
         => ( word_sle(A,C2,B3)
           => word_sle(A,C2,A3) ) ) ) ).

% signed.dual_order.trans
tff(fact_6295_signed_Odual__order_Oeq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          ( ( A3 = B3 )
        <=> ( word_sle(A,B3,A3)
            & word_sle(A,A3,B3) ) ) ) ).

% signed.dual_order.eq_iff
tff(fact_6296_signed_Odual__order_Oantisym,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: word(A),A3: word(A)] :
          ( word_sle(A,B3,A3)
         => ( word_sle(A,A3,B3)
           => ( A3 = B3 ) ) ) ) ).

% signed.dual_order.antisym
tff(fact_6297_signed_Ofinite__has__maximal,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: set(word(A))] :
          ( finite_finite2(word(A),A2)
         => ( ( A2 != bot_bot(set(word(A))) )
           => ? [X4: word(A)] :
                ( member(word(A),X4,A2)
                & ! [Xa: word(A)] :
                    ( member(word(A),Xa,A2)
                   => ( word_sle(A,X4,Xa)
                     => ( X4 = Xa ) ) ) ) ) ) ) ).

% signed.finite_has_maximal
tff(fact_6298_signed_Ofinite__has__minimal,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: set(word(A))] :
          ( finite_finite2(word(A),A2)
         => ( ( A2 != bot_bot(set(word(A))) )
           => ? [X4: word(A)] :
                ( member(word(A),X4,A2)
                & ! [Xa: word(A)] :
                    ( member(word(A),Xa,A2)
                   => ( word_sle(A,Xa,X4)
                     => ( X4 = Xa ) ) ) ) ) ) ) ).

% signed.finite_has_minimal
tff(fact_6299_signed_Ofinite__has__maximal2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: set(word(A)),A3: word(A)] :
          ( finite_finite2(word(A),A2)
         => ( member(word(A),A3,A2)
           => ? [X4: word(A)] :
                ( member(word(A),X4,A2)
                & word_sle(A,A3,X4)
                & ! [Xa: word(A)] :
                    ( member(word(A),Xa,A2)
                   => ( word_sle(A,X4,Xa)
                     => ( X4 = Xa ) ) ) ) ) ) ) ).

% signed.finite_has_maximal2
tff(fact_6300_signed_Ofinite__has__minimal2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: set(word(A)),A3: word(A)] :
          ( finite_finite2(word(A),A2)
         => ( member(word(A),A3,A2)
           => ? [X4: word(A)] :
                ( member(word(A),X4,A2)
                & word_sle(A,X4,A3)
                & ! [Xa: word(A)] :
                    ( member(word(A),Xa,A2)
                   => ( word_sle(A,Xa,X4)
                     => ( X4 = Xa ) ) ) ) ) ) ) ).

% signed.finite_has_minimal2
tff(fact_6301_in__image__insert__iff,axiom,
    ! [A: $tType,B2: set(set(A)),X: A,A2: set(A)] :
      ( ! [C7: set(A)] :
          ( member(set(A),C7,B2)
         => ~ member(A,X,C7) )
     => ( member(set(A),A2,aa(set(set(A)),set(set(A)),image(set(A),set(A),insert(A,X)),B2))
      <=> ( member(A,X,A2)
          & member(set(A),aa(set(A),set(A),minus_minus(set(A),A2),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))),B2) ) ) ) ).

% in_image_insert_iff
tff(fact_6302_signed_Ofinite__ranking__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [S: set(A),P: fun(set(A),$o),F2: fun(A,word(B))] :
          ( finite_finite2(A,S)
         => ( aa(set(A),$o,P,bot_bot(set(A)))
           => ( ! [X4: A,S4: set(A)] :
                  ( finite_finite2(A,S4)
                 => ( ! [Y4: A] :
                        ( member(A,Y4,S4)
                       => word_sle(B,aa(A,word(B),F2,Y4),aa(A,word(B),F2,X4)) )
                   => ( aa(set(A),$o,P,S4)
                     => aa(set(A),$o,P,aa(set(A),set(A),insert(A,X4),S4)) ) ) )
             => aa(set(A),$o,P,S) ) ) ) ) ).

% signed.finite_ranking_induct
tff(fact_6303_image__int__atLeastAtMost,axiom,
    ! [A3: nat,B3: nat] : aa(set(nat),set(int),image(nat,int,semiring_1_of_nat(int)),set_or1337092689740270186AtMost(nat,A3,B3)) = set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),A3),aa(nat,int,semiring_1_of_nat(int),B3)) ).

% image_int_atLeastAtMost
tff(fact_6304_image__int__atLeastLessThan,axiom,
    ! [A3: nat,B3: nat] : aa(set(nat),set(int),image(nat,int,semiring_1_of_nat(int)),set_or7035219750837199246ssThan(nat,A3,B3)) = set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),A3),aa(nat,int,semiring_1_of_nat(int),B3)) ).

% image_int_atLeastLessThan
tff(fact_6305_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_6306_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_6307_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),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_6308_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),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_6309_lessThan__Suc__eq__insert__0,axiom,
    ! [Na: nat] : set_ord_lessThan(nat,aa(nat,nat,suc,Na)) = aa(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_6310_atMost__Suc__eq__insert__0,axiom,
    ! [Na: nat] : set_ord_atMost(nat,aa(nat,nat,suc,Na)) = aa(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_6311_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : aa(set(int),set(int),image(int,int,aTP_Lamp_pe(int,fun(int,int),L)),set_or7035219750837199246ssThan(int,zero_zero(int),aa(int,int,minus_minus(int,U),L))) = set_or7035219750837199246ssThan(int,L,U) ).

% image_add_int_atLeastLessThan
tff(fact_6312_word__roti__conv__mod_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: int,W: word(A)] : word_roti(A,Na,W) = word_roti(A,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Na),aa(nat,int,semiring_1_of_nat(int),size_size(word(A),W))),W) ) ).

% word_roti_conv_mod'
tff(fact_6313_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_pf(code_integer,fun(code_integer,code_integer),L)),set_or7035219750837199246ssThan(code_integer,zero_zero(code_integer),aa(code_integer,code_integer,minus_minus(code_integer,U),L))) = set_or7035219750837199246ssThan(code_integer,L,U) ).

% image_add_integer_atLeastLessThan
tff(fact_6314_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( aa(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,nat2(U))) ) ) ).

% image_atLeastZeroLessThan_int
tff(fact_6315_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C2: nat,X: nat,Y: nat] :
      aa(set(nat),set(nat),image(nat,nat,aTP_Lamp_pg(nat,fun(nat,nat),C2)),set_or7035219750837199246ssThan(nat,X,Y)) = $ite(
        aa(nat,$o,ord_less(nat,C2),Y),
        set_or7035219750837199246ssThan(nat,aa(nat,nat,minus_minus(nat,X),C2),aa(nat,nat,minus_minus(nat,Y),C2)),
        $ite(aa(nat,$o,ord_less(nat,X),Y),aa(set(nat),set(nat),insert(nat,zero_zero(nat)),bot_bot(set(nat))),bot_bot(set(nat))) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_6316_word__roti__conv__mod,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: int,W: word(A)] : word_roti(A,Na,W) = word_roti(A,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Na),aa(nat,int,semiring_1_of_nat(int),type_len0_len_of(A,type2(A)))),W) ) ).

% word_roti_conv_mod
tff(fact_6317_word__0__sle__from__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),X),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))))
         => word_sle(A,zero_zero(word(A)),X) ) ) ).

% word_0_sle_from_less
tff(fact_6318_sless__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( word_sless(A,aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(num,word(A),numeral_numeral(word(A)),B3))
        <=> aa(int,$o,ord_less(int,bit_ri4674362597316999326ke_bit(int,aa(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),A3)))),bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),aa(num,int,numeral_numeral(int),B3))) ) ) ).

% sless_word_minus_numeral_numeral
tff(fact_6319_sless__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( word_sless(A,aa(num,word(A),numeral_numeral(word(A)),A3),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3)))
        <=> aa(int,$o,ord_less(int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),aa(num,int,numeral_numeral(int),A3))),bit_ri4674362597316999326ke_bit(int,aa(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),B3)))) ) ) ).

% sless_word_numeral_minus_numeral
tff(fact_6320_word__sless__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( word_sless(A,aa(num,word(A),numeral_numeral(word(A)),A3),aa(num,word(A),numeral_numeral(word(A)),B3))
        <=> ( word_sle(A,aa(num,word(A),numeral_numeral(word(A)),A3),aa(num,word(A),numeral_numeral(word(A)),B3))
            & ( aa(num,word(A),numeral_numeral(word(A)),A3) != aa(num,word(A),numeral_numeral(word(A)),B3) ) ) ) ) ).

% word_sless_no
tff(fact_6321_extra__sle__sless__unfolds_I10_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num] :
          ( word_sless(A,one_one(word(A)),aa(num,word(A),numeral_numeral(word(A)),Na))
        <=> aa(int,$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_6322_extra__sle__sless__unfolds_I12_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num] :
          ( word_sless(A,aa(num,word(A),numeral_numeral(word(A)),Na),one_one(word(A)))
        <=> aa(int,$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_6323_extra__sle__sless__unfolds_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num] :
          ( word_sless(A,zero_zero(word(A)),aa(num,word(A),numeral_numeral(word(A)),Na))
        <=> aa(int,$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_6324_extra__sle__sless__unfolds_I11_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num] :
          ( word_sless(A,aa(num,word(A),numeral_numeral(word(A)),Na),zero_zero(word(A)))
        <=> aa(int,$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_6325_extra__sle__sless__unfolds_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( word_sless(A,zero_zero(word(A)),one_one(word(A)))
      <=> aa(int,$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_6326_extra__sle__sless__unfolds_I9_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( word_sless(A,one_one(word(A)),zero_zero(word(A)))
      <=> aa(int,$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_6327_sless__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( word_sless(A,aa(num,word(A),numeral_numeral(word(A)),A3),aa(num,word(A),numeral_numeral(word(A)),B3))
        <=> aa(int,$o,ord_less(int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),aa(num,int,numeral_numeral(int),A3))),bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),aa(num,int,numeral_numeral(int),B3))) ) ) ).

% sless_word_numeral_numeral
tff(fact_6328_sless__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( word_sless(A,aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3)))
        <=> aa(int,$o,ord_less(int,bit_ri4674362597316999326ke_bit(int,aa(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),A3)))),bit_ri4674362597316999326ke_bit(int,aa(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),B3)))) ) ) ).

% sless_word_minus_numeral_minus_numeral
tff(fact_6329_signed_OleD,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A)] :
          ( word_sle(A,Y,X)
         => ~ word_sless(A,X,Y) ) ) ).

% signed.leD
tff(fact_6330_signed_OleI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ~ word_sless(A,X,Y)
         => word_sle(A,Y,X) ) ) ).

% signed.leI
tff(fact_6331_signed_Onot__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ~ word_sle(A,X,Y)
        <=> word_sless(A,Y,X) ) ) ).

% signed.not_le
tff(fact_6332_signed_Ole__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( word_sle(A,X,Y)
        <=> ( word_sless(A,X,Y)
            | ( X = Y ) ) ) ) ).

% signed.le_less
tff(fact_6333_signed_Onless__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          ( ~ word_sless(A,A3,B3)
        <=> ( ~ word_sle(A,A3,B3)
            | ( A3 = B3 ) ) ) ) ).

% signed.nless_le
tff(fact_6334_signed_Onot__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ~ word_sless(A,X,Y)
        <=> word_sle(A,Y,X) ) ) ).

% signed.not_less
tff(fact_6335_signed_Oless__imp__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( word_sless(A,X,Y)
         => word_sle(A,X,Y) ) ) ).

% signed.less_imp_le
tff(fact_6336_signed_Ole__neq__trans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          ( word_sle(A,A3,B3)
         => ( ( A3 != B3 )
           => word_sless(A,A3,B3) ) ) ) ).

% signed.le_neq_trans
tff(fact_6337_signed_Oantisym__conv1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ~ word_sless(A,X,Y)
         => ( word_sle(A,X,Y)
          <=> ( X = Y ) ) ) ) ).

% signed.antisym_conv1
tff(fact_6338_signed_Oantisym__conv2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( word_sle(A,X,Y)
         => ( ~ word_sless(A,X,Y)
          <=> ( X = Y ) ) ) ) ).

% signed.antisym_conv2
tff(fact_6339_signed_Ole__less__trans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A),Z: word(A)] :
          ( word_sle(A,X,Y)
         => ( word_sless(A,Y,Z)
           => word_sless(A,X,Z) ) ) ) ).

% signed.le_less_trans
tff(fact_6340_signed_Oless__le__trans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A),Z: word(A)] :
          ( word_sless(A,X,Y)
         => ( word_sle(A,Y,Z)
           => word_sless(A,X,Z) ) ) ) ).

% signed.less_le_trans
tff(fact_6341_signed_Ole__less__linear,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( word_sle(A,X,Y)
          | word_sless(A,Y,X) ) ) ).

% signed.le_less_linear
tff(fact_6342_signed_Oless__le__not__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( word_sless(A,X,Y)
        <=> ( word_sle(A,X,Y)
            & ~ word_sle(A,Y,X) ) ) ) ).

% signed.less_le_not_le
tff(fact_6343_signed_Onot__le__imp__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A)] :
          ( ~ word_sle(A,Y,X)
         => word_sless(A,X,Y) ) ) ).

% signed.not_le_imp_less
tff(fact_6344_signed_Ole__imp__less__or__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( word_sle(A,X,Y)
         => ( word_sless(A,X,Y)
            | ( X = Y ) ) ) ) ).

% signed.le_imp_less_or_eq
tff(fact_6345_signed_Oorder_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A),C2: word(A)] :
          ( word_sle(A,A3,B3)
         => ( word_sless(A,B3,C2)
           => word_sless(A,A3,C2) ) ) ) ).

% signed.order.strict_trans1
tff(fact_6346_signed_Oorder_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A),C2: word(A)] :
          ( word_sless(A,A3,B3)
         => ( word_sle(A,B3,C2)
           => word_sless(A,A3,C2) ) ) ) ).

% signed.order.strict_trans2
tff(fact_6347_signed_Oorder_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          ( word_sless(A,A3,B3)
        <=> ( word_sle(A,A3,B3)
            & ~ word_sle(A,B3,A3) ) ) ) ).

% signed.order.strict_iff_not
tff(fact_6348_signed_Oorder_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          ( word_sle(A,A3,B3)
        <=> ( word_sless(A,A3,B3)
            | ( A3 = B3 ) ) ) ) ).

% signed.order.order_iff_strict
tff(fact_6349_signed_Oorder_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          ( word_sless(A,A3,B3)
        <=> ( word_sle(A,A3,B3)
            & ( A3 != B3 ) ) ) ) ).

% signed.order.strict_iff_order
tff(fact_6350_signed_Odual__order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: word(A),A3: word(A),C2: word(A)] :
          ( word_sle(A,B3,A3)
         => ( word_sless(A,C2,B3)
           => word_sless(A,C2,A3) ) ) ) ).

% signed.dual_order.strict_trans1
tff(fact_6351_signed_Odual__order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: word(A),A3: word(A),C2: word(A)] :
          ( word_sless(A,B3,A3)
         => ( word_sle(A,C2,B3)
           => word_sless(A,C2,A3) ) ) ) ).

% signed.dual_order.strict_trans2
tff(fact_6352_signed_Odual__order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: word(A),A3: word(A)] :
          ( word_sless(A,B3,A3)
        <=> ( word_sle(A,B3,A3)
            & ~ word_sle(A,A3,B3) ) ) ) ).

% signed.dual_order.strict_iff_not
tff(fact_6353_signed_Oorder_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          ( word_sless(A,A3,B3)
         => word_sle(A,A3,B3) ) ) ).

% signed.order.strict_implies_order
tff(fact_6354_signed_Odual__order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: word(A),A3: word(A)] :
          ( word_sle(A,B3,A3)
        <=> ( word_sless(A,B3,A3)
            | ( A3 = B3 ) ) ) ) ).

% signed.dual_order.order_iff_strict
tff(fact_6355_signed_Odual__order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: word(A),A3: word(A)] :
          ( word_sless(A,B3,A3)
        <=> ( word_sle(A,B3,A3)
            & ( A3 != B3 ) ) ) ) ).

% signed.dual_order.strict_iff_order
tff(fact_6356_signed_Odual__order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: word(A),A3: word(A)] :
          ( word_sless(A,B3,A3)
         => word_sle(A,B3,A3) ) ) ).

% signed.dual_order.strict_implies_order
tff(fact_6357_signed_Oorder_Onot__eq__order__implies__strict,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          ( ( A3 != B3 )
         => ( word_sle(A,A3,B3)
           => word_sless(A,A3,B3) ) ) ) ).

% signed.order.not_eq_order_implies_strict
tff(fact_6358_signed_Odual__order_Onot__eq__order__implies__strict,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          ( ( A3 != B3 )
         => ( word_sle(A,B3,A3)
           => word_sless(A,B3,A3) ) ) ) ).

% signed.dual_order.not_eq_order_implies_strict
tff(fact_6359_word__sless__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( word_sless(A,X,Y)
        <=> ( word_sle(A,X,Y)
            & ( X != Y ) ) ) ) ).

% word_sless_eq
tff(fact_6360_finite__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: set(word(A))] : finite_finite2(word(A),A2) ) ).

% finite_word
tff(fact_6361_signed_Oinfinite__growing,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X5: set(word(A))] :
          ( ( X5 != bot_bot(set(word(A))) )
         => ( ! [X4: word(A)] :
                ( member(word(A),X4,X5)
               => ? [Xa: word(A)] :
                    ( member(word(A),Xa,X5)
                    & word_sless(A,X4,Xa) ) )
           => ~ finite_finite2(word(A),X5) ) ) ) ).

% signed.infinite_growing
tff(fact_6362_signed_Olift__Suc__mono__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F2: fun(nat,word(A)),Na: nat,N6: nat] :
          ( ! [N: nat] : word_sless(A,aa(nat,word(A),F2,N),aa(nat,word(A),F2,aa(nat,nat,suc,N)))
         => ( aa(nat,$o,ord_less(nat,Na),N6)
           => word_sless(A,aa(nat,word(A),F2,Na),aa(nat,word(A),F2,N6)) ) ) ) ).

% signed.lift_Suc_mono_less
tff(fact_6363_signed_Olift__Suc__mono__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F2: fun(nat,word(A)),Na: nat,M: nat] :
          ( ! [N: nat] : word_sless(A,aa(nat,word(A),F2,N),aa(nat,word(A),F2,aa(nat,nat,suc,N)))
         => ( word_sless(A,aa(nat,word(A),F2,Na),aa(nat,word(A),F2,M))
          <=> aa(nat,$o,ord_less(nat,Na),M) ) ) ) ).

% signed.lift_Suc_mono_less_iff
tff(fact_6364_signed_Odual__order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: word(A),A3: word(A)] :
          ( word_sless(A,B3,A3)
         => ( A3 != B3 ) ) ) ).

% signed.dual_order.strict_implies_not_eq
tff(fact_6365_signed_Oorder_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          ( word_sless(A,A3,B3)
         => ( A3 != B3 ) ) ) ).

% signed.order.strict_implies_not_eq
tff(fact_6366_signed_Odual__order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: word(A),A3: word(A),C2: word(A)] :
          ( word_sless(A,B3,A3)
         => ( word_sless(A,C2,B3)
           => word_sless(A,C2,A3) ) ) ) ).

% signed.dual_order.strict_trans
tff(fact_6367_signed_Onot__less__iff__gr__or__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ~ word_sless(A,X,Y)
        <=> ( word_sless(A,Y,X)
            | ( X = Y ) ) ) ) ).

% signed.not_less_iff_gr_or_eq
tff(fact_6368_signed_Oorder_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A),C2: word(A)] :
          ( word_sless(A,A3,B3)
         => ( word_sless(A,B3,C2)
           => word_sless(A,A3,C2) ) ) ) ).

% signed.order.strict_trans
tff(fact_6369_signed_Olinorder__less__wlog,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(word(A),fun(word(A),$o)),A3: word(A),B3: word(A)] :
          ( ! [A4: word(A),B5: word(A)] :
              ( word_sless(A,A4,B5)
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),P,A4),B5) )
         => ( ! [A4: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),P,A4),A4)
           => ( ! [A4: word(A),B5: word(A)] :
                  ( aa(word(A),$o,aa(word(A),fun(word(A),$o),P,B5),A4)
                 => aa(word(A),$o,aa(word(A),fun(word(A),$o),P,A4),B5) )
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),P,A3),B3) ) ) ) ) ).

% signed.linorder_less_wlog
tff(fact_6370_signed_Oord__less__eq__trans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A),C2: word(A)] :
          ( word_sless(A,A3,B3)
         => ( ( B3 = C2 )
           => word_sless(A,A3,C2) ) ) ) ).

% signed.ord_less_eq_trans
tff(fact_6371_signed_Oord__eq__less__trans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A),C2: word(A)] :
          ( ( A3 = B3 )
         => ( word_sless(A,B3,C2)
           => word_sless(A,A3,C2) ) ) ) ).

% signed.ord_eq_less_trans
tff(fact_6372_signed_Oless__imp__not__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( word_sless(A,X,Y)
         => ~ word_sless(A,Y,X) ) ) ).

% signed.less_imp_not_less
tff(fact_6373_signed_Odual__order_Oirrefl,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : ~ word_sless(A,A3,A3) ) ).

% signed.dual_order.irrefl
tff(fact_6374_signed_Oless__imp__not__eq2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( word_sless(A,X,Y)
         => ( Y != X ) ) ) ).

% signed.less_imp_not_eq2
tff(fact_6375_signed_Oless__imp__not__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( word_sless(A,X,Y)
         => ( X != Y ) ) ) ).

% signed.less_imp_not_eq
tff(fact_6376_signed_Odual__order_Oasym,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: word(A),A3: word(A)] :
          ( word_sless(A,B3,A3)
         => ~ word_sless(A,A3,B3) ) ) ).

% signed.dual_order.asym
tff(fact_6377_signed_Olinorder__cases,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ~ word_sless(A,X,Y)
         => ( ( X != Y )
           => word_sless(A,Y,X) ) ) ) ).

% signed.linorder_cases
tff(fact_6378_signed_Oless__imp__triv,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A),P: $o] :
          ( word_sless(A,X,Y)
         => ( word_sless(A,Y,X)
           => (P) ) ) ) ).

% signed.less_imp_triv
tff(fact_6379_signed_Oantisym__conv3,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A)] :
          ( ~ word_sless(A,Y,X)
         => ( ~ word_sless(A,X,Y)
          <=> ( X = Y ) ) ) ) ).

% signed.antisym_conv3
tff(fact_6380_signed_Oless__not__sym,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( word_sless(A,X,Y)
         => ~ word_sless(A,Y,X) ) ) ).

% signed.less_not_sym
tff(fact_6381_signed_Oless__imp__neq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( word_sless(A,X,Y)
         => ( X != Y ) ) ) ).

% signed.less_imp_neq
tff(fact_6382_signed_Oless__linear,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( word_sless(A,X,Y)
          | ( X = Y )
          | word_sless(A,Y,X) ) ) ).

% signed.less_linear
tff(fact_6383_signed_Oless__irrefl,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : ~ word_sless(A,X,X) ) ).

% signed.less_irrefl
tff(fact_6384_signed_Oorder_Oasym,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          ( word_sless(A,A3,B3)
         => ~ word_sless(A,B3,A3) ) ) ).

% signed.order.asym
tff(fact_6385_signed_Oless__trans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A),Z: word(A)] :
          ( word_sless(A,X,Y)
         => ( word_sless(A,Y,Z)
           => word_sless(A,X,Z) ) ) ) ).

% signed.less_trans
tff(fact_6386_signed_Oless__asym_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          ( word_sless(A,A3,B3)
         => ~ word_sless(A,B3,A3) ) ) ).

% signed.less_asym'
tff(fact_6387_signed_Oless__asym,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( word_sless(A,X,Y)
         => ~ word_sless(A,Y,X) ) ) ).

% signed.less_asym
tff(fact_6388_signed_Oneq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ( X != Y )
        <=> ( word_sless(A,X,Y)
            | word_sless(A,Y,X) ) ) ) ).

% signed.neq_iff
tff(fact_6389_signed_OneqE,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ( X != Y )
         => ( ~ word_sless(A,X,Y)
           => word_sless(A,Y,X) ) ) ) ).

% signed.neqE
tff(fact_6390_word__sless__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          ( word_sless(A,A3,B3)
        <=> aa(int,$o,ord_less(int,aa(word(A),int,ring_1_signed(A,int),A3)),aa(word(A),int,ring_1_signed(A,int),B3)) ) ) ).

% word_sless_alt
tff(fact_6391_signed_Ofinite__linorder__min__induct,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: set(word(A)),P: fun(set(word(A)),$o)] :
          ( finite_finite2(word(A),A2)
         => ( aa(set(word(A)),$o,P,bot_bot(set(word(A))))
           => ( ! [B5: word(A),A8: set(word(A))] :
                  ( finite_finite2(word(A),A8)
                 => ( ! [X2: word(A)] :
                        ( member(word(A),X2,A8)
                       => word_sless(A,B5,X2) )
                   => ( aa(set(word(A)),$o,P,A8)
                     => aa(set(word(A)),$o,P,aa(set(word(A)),set(word(A)),insert(word(A),B5),A8)) ) ) )
             => aa(set(word(A)),$o,P,A2) ) ) ) ) ).

% signed.finite_linorder_min_induct
tff(fact_6392_signed_Ofinite__linorder__max__induct,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: set(word(A)),P: fun(set(word(A)),$o)] :
          ( finite_finite2(word(A),A2)
         => ( aa(set(word(A)),$o,P,bot_bot(set(word(A))))
           => ( ! [B5: word(A),A8: set(word(A))] :
                  ( finite_finite2(word(A),A8)
                 => ( ! [X2: word(A)] :
                        ( member(word(A),X2,A8)
                       => word_sless(A,X2,B5) )
                   => ( aa(set(word(A)),$o,P,A8)
                     => aa(set(word(A)),$o,P,aa(set(word(A)),set(word(A)),insert(word(A),B5),A8)) ) ) )
             => aa(set(word(A)),$o,P,A2) ) ) ) ) ).

% signed.finite_linorder_max_induct
tff(fact_6393_word__sless__sint__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( word_sless(A,X,Y)
         => aa(int,$o,ord_less_eq(int,aa(word(A),int,ring_1_signed(A,int),X)),aa(int,int,minus_minus(int,aa(word(A),int,ring_1_signed(A,int),Y)),one_one(int))) ) ) ).

% word_sless_sint_le
tff(fact_6394_take__bit__word__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num,K: num] : bit_se2584673776208193580ke_bit(word(A),aa(num,nat,numeral_numeral(nat),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)),bit_se2584673776208193580ke_bit(int,ord_min(nat,type_len0_len_of(A,type2(A)),aa(num,nat,numeral_numeral(nat),Na)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))) ) ).

% take_bit_word_minus_numeral
tff(fact_6395_take__bit__word__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,K: num] : 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)),bit_se2584673776208193580ke_bit(int,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_6396_min_Oidem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A] : ord_min(A,A3,A3) = A3 ) ).

% min.idem
tff(fact_6397_min_Oleft__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] : ord_min(A,A3,ord_min(A,A3,B3)) = ord_min(A,A3,B3) ) ).

% min.left_idem
tff(fact_6398_min_Oright__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] : ord_min(A,ord_min(A,A3,B3),B3) = ord_min(A,A3,B3) ) ).

% min.right_idem
tff(fact_6399_min_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),ord_min(A,B3,C2))
        <=> ( aa(A,$o,ord_less_eq(A,A3),B3)
            & aa(A,$o,ord_less_eq(A,A3),C2) ) ) ) ).

% min.bounded_iff
tff(fact_6400_min_Oabsorb2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
         => ( ord_min(A,A3,B3) = B3 ) ) ) ).

% min.absorb2
tff(fact_6401_min_Oabsorb1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( ord_min(A,A3,B3) = A3 ) ) ) ).

% min.absorb1
tff(fact_6402_min__simps_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,B3),A3)
         => ( ord_min(A,A3,B3) = B3 ) ) ) ).

% min_simps(2)
tff(fact_6403_min__simps_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( ord_min(A,A3,B3) = A3 ) ) ) ).

% min_simps(1)
tff(fact_6404_min__less__self__conv_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,ord_min(A,A3,B3)),B3)
        <=> aa(A,$o,ord_less(A,A3),B3) ) ) ).

% min_less_self_conv(2)
tff(fact_6405_min__less__self__conv_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,ord_min(A,A3,B3)),A3)
        <=> aa(A,$o,ord_less(A,B3),A3) ) ) ).

% min_less_self_conv(1)
tff(fact_6406_min__arg__not__ge_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,Na: A] :
          ( ~ aa(A,$o,ord_less(A,ord_min(A,M,Na)),Na)
        <=> ( ord_min(A,M,Na) = Na ) ) ) ).

% min_arg_not_ge(2)
tff(fact_6407_min__arg__not__ge_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,Na: A] :
          ( ~ aa(A,$o,ord_less(A,ord_min(A,M,Na)),M)
        <=> ( ord_min(A,M,Na) = M ) ) ) ).

% min_arg_not_ge(1)
tff(fact_6408_min_Oabsorb3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( ord_min(A,A3,B3) = A3 ) ) ) ).

% min.absorb3
tff(fact_6409_min_Oabsorb4,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,B3),A3)
         => ( ord_min(A,A3,B3) = B3 ) ) ) ).

% min.absorb4
tff(fact_6410_min__less__iff__conj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,X: A,Y: A] :
          ( aa(A,$o,ord_less(A,Z),ord_min(A,X,Y))
        <=> ( aa(A,$o,ord_less(A,Z),X)
            & aa(A,$o,ord_less(A,Z),Y) ) ) ) ).

% min_less_iff_conj
tff(fact_6411_min__bot2,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [X: A] : ord_min(A,X,bot_bot(A)) = bot_bot(A) ) ).

% min_bot2
tff(fact_6412_min__bot,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [X: A] : ord_min(A,bot_bot(A),X) = bot_bot(A) ) ).

% min_bot
tff(fact_6413_min__Suc__Suc,axiom,
    ! [M: nat,Na: nat] : ord_min(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Na)) = aa(nat,nat,suc,ord_min(nat,M,Na)) ).

% min_Suc_Suc
tff(fact_6414_min__0L,axiom,
    ! [Na: nat] : ord_min(nat,zero_zero(nat),Na) = zero_zero(nat) ).

% min_0L
tff(fact_6415_min__0R,axiom,
    ! [Na: nat] : ord_min(nat,Na,zero_zero(nat)) = zero_zero(nat) ).

% min_0R
tff(fact_6416_max__min__same_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),X),ord_min(A,X,Y)) = X ) ).

% max_min_same(1)
tff(fact_6417_max__min__same_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),ord_min(A,X,Y)),X) = X ) ).

% max_min_same(2)
tff(fact_6418_max__min__same_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),ord_min(A,X,Y)),Y) = Y ) ).

% max_min_same(3)
tff(fact_6419_max__min__same_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),Y),ord_min(A,X,Y)) = Y ) ).

% max_min_same(4)
tff(fact_6420_min__minus_H,axiom,
    ! [M: nat,K: nat] : ord_min(nat,aa(nat,nat,minus_minus(nat,M),K),M) = aa(nat,nat,minus_minus(nat,M),K) ).

% min_minus'
tff(fact_6421_min__minus,axiom,
    ! [M: nat,K: nat] : ord_min(nat,M,aa(nat,nat,minus_minus(nat,M),K)) = aa(nat,nat,minus_minus(nat,M),K) ).

% min_minus
tff(fact_6422_min__0__1_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : ord_min(A,aa(num,A,numeral_numeral(A),X),zero_zero(A)) = zero_zero(A) ) ).

% min_0_1(4)
tff(fact_6423_min__0__1_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : ord_min(A,zero_zero(A),aa(num,A,numeral_numeral(A),X)) = zero_zero(A) ) ).

% min_0_1(3)
tff(fact_6424_min__number__of_I1_J,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ord_min(A,aa(num,A,numeral_numeral(A),U),aa(num,A,numeral_numeral(A),V2)) = $ite(aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)),aa(num,A,numeral_numeral(A),U),aa(num,A,numeral_numeral(A),V2)) ) ).

% min_number_of(1)
tff(fact_6425_min__0__1_I1_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ( ord_min(A,zero_zero(A),one_one(A)) = zero_zero(A) ) ) ).

% min_0_1(1)
tff(fact_6426_min__0__1_I2_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ( ord_min(A,one_one(A),zero_zero(A)) = zero_zero(A) ) ) ).

% min_0_1(2)
tff(fact_6427_min__0__1_I6_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : ord_min(A,aa(num,A,numeral_numeral(A),X),one_one(A)) = one_one(A) ) ).

% min_0_1(6)
tff(fact_6428_min__0__1_I5_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : ord_min(A,one_one(A),aa(num,A,numeral_numeral(A),X)) = one_one(A) ) ).

% min_0_1(5)
tff(fact_6429_min__Suc__gt_I1_J,axiom,
    ! [A3: nat,B3: nat] :
      ( aa(nat,$o,ord_less(nat,A3),B3)
     => ( ord_min(nat,aa(nat,nat,suc,A3),B3) = aa(nat,nat,suc,A3) ) ) ).

% min_Suc_gt(1)
tff(fact_6430_min__Suc__gt_I2_J,axiom,
    ! [A3: nat,B3: nat] :
      ( aa(nat,$o,ord_less(nat,A3),B3)
     => ( ord_min(nat,B3,aa(nat,nat,suc,A3)) = aa(nat,nat,suc,A3) ) ) ).

% min_Suc_gt(2)
tff(fact_6431_rev__min__pm1,axiom,
    ! [A3: nat,B3: nat] : aa(nat,nat,plus_plus(nat,aa(nat,nat,minus_minus(nat,A3),B3)),ord_min(nat,B3,A3)) = A3 ).

% rev_min_pm1
tff(fact_6432_rev__min__pm,axiom,
    ! [B3: nat,A3: nat] : aa(nat,nat,plus_plus(nat,ord_min(nat,B3,A3)),aa(nat,nat,minus_minus(nat,A3),B3)) = A3 ).

% rev_min_pm
tff(fact_6433_min__pm1,axiom,
    ! [A3: nat,B3: nat] : aa(nat,nat,plus_plus(nat,aa(nat,nat,minus_minus(nat,A3),B3)),ord_min(nat,A3,B3)) = A3 ).

% min_pm1
tff(fact_6434_min__pm,axiom,
    ! [A3: nat,B3: nat] : aa(nat,nat,plus_plus(nat,ord_min(nat,A3,B3)),aa(nat,nat,minus_minus(nat,A3),B3)) = A3 ).

% min_pm
tff(fact_6435_min__number__of_I2_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ord_min(A,aa(num,A,numeral_numeral(A),U),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = $ite(aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),aa(num,A,numeral_numeral(A),U),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) ) ).

% min_number_of(2)
tff(fact_6436_min__number__of_I3_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ord_min(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)) = $ite(aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)) ) ).

% min_number_of(3)
tff(fact_6437_min__number__of_I4_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          ord_min(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = $ite(aa(A,$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),V2))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) ) ).

% min_number_of(4)
tff(fact_6438_min__Suc__numeral,axiom,
    ! [Na: nat,K: num] : ord_min(nat,aa(nat,nat,suc,Na),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,suc,ord_min(nat,Na,pred_numeral(K))) ).

% min_Suc_numeral
tff(fact_6439_min__numeral__Suc,axiom,
    ! [K: num,Na: nat] : ord_min(nat,aa(num,nat,numeral_numeral(nat),K),aa(nat,nat,suc,Na)) = aa(nat,nat,suc,ord_min(nat,pred_numeral(K),Na)) ).

% min_numeral_Suc
tff(fact_6440_take__bit__word__Suc__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,K: num] : 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)),bit_se2584673776208193580ke_bit(int,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_6441_take__bit__word__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num,K: num] : bit_se2584673776208193580ke_bit(word(A),aa(num,nat,numeral_numeral(nat),Na),aa(num,word(A),numeral_numeral(word(A)),K)) = aa(int,word(A),ring_1_of_int(word(A)),bit_se2584673776208193580ke_bit(int,ord_min(nat,type_len0_len_of(A,type2(A)),aa(num,nat,numeral_numeral(nat),Na)),aa(num,int,numeral_numeral(int),K))) ) ).

% take_bit_word_numeral
tff(fact_6442_min__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X2: A,Xa: A] :
          ord_min(A,X2,Xa) = $ite(aa(A,$o,ord_less_eq(A,X2),Xa),X2,Xa) ) ).

% min_def_raw
tff(fact_6443_min__le__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Z: A] :
          ( aa(A,$o,ord_less_eq(A,ord_min(A,X,Y)),Z)
        <=> ( aa(A,$o,ord_less_eq(A,X),Z)
            | aa(A,$o,ord_less_eq(A,Y),Z) ) ) ) ).

% min_le_iff_disj
tff(fact_6444_min_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,C2: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,B3),C2)
         => aa(A,$o,ord_less_eq(A,ord_min(A,A3,B3)),C2) ) ) ).

% min.coboundedI2
tff(fact_6445_min_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),C2)
         => aa(A,$o,ord_less_eq(A,ord_min(A,A3,B3)),C2) ) ) ).

% min.coboundedI1
tff(fact_6446_min_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
        <=> ( ord_min(A,A3,B3) = B3 ) ) ) ).

% min.absorb_iff2
tff(fact_6447_min_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
        <=> ( ord_min(A,A3,B3) = A3 ) ) ) ).

% min.absorb_iff1
tff(fact_6448_min_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] : aa(A,$o,ord_less_eq(A,ord_min(A,A3,B3)),B3) ) ).

% min.cobounded2
tff(fact_6449_min_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] : aa(A,$o,ord_less_eq(A,ord_min(A,A3,B3)),A3) ) ).

% min.cobounded1
tff(fact_6450_min_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
        <=> ( A3 = ord_min(A,A3,B3) ) ) ) ).

% min.order_iff
tff(fact_6451_min_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,A3),C2)
           => aa(A,$o,ord_less_eq(A,A3),ord_min(A,B3,C2)) ) ) ) ).

% min.boundedI
tff(fact_6452_min_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),ord_min(A,B3,C2))
         => ~ ( aa(A,$o,ord_less_eq(A,A3),B3)
             => ~ aa(A,$o,ord_less_eq(A,A3),C2) ) ) ) ).

% min.boundedE
tff(fact_6453_min_OorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = ord_min(A,A3,B3) )
         => aa(A,$o,ord_less_eq(A,A3),B3) ) ) ).

% min.orderI
tff(fact_6454_min_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( A3 = ord_min(A,A3,B3) ) ) ) ).

% min.orderE
tff(fact_6455_min_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,C2: A,B3: A,D2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),C2)
         => ( aa(A,$o,ord_less_eq(A,B3),D2)
           => aa(A,$o,ord_less_eq(A,ord_min(A,A3,B3)),ord_min(A,C2,D2)) ) ) ) ).

% min.mono
tff(fact_6456_min__absorb2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,ord_less_eq(A,Y),X)
         => ( ord_min(A,X,Y) = Y ) ) ) ).

% min_absorb2
tff(fact_6457_min__absorb1,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,X),Y)
         => ( ord_min(A,X,Y) = X ) ) ) ).

% min_absorb1
tff(fact_6458_min__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B3: A] :
          ord_min(A,A3,B3) = $ite(aa(A,$o,ord_less_eq(A,A3),B3),A3,B3) ) ).

% min_def
tff(fact_6459_min__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [X: A,Y: A,Z: A] : aa(A,A,plus_plus(A,ord_min(A,X,Y)),Z) = ord_min(A,aa(A,A,plus_plus(A,X),Z),aa(A,A,plus_plus(A,Y),Z)) ) ).

% min_add_distrib_left
tff(fact_6460_min__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [X: A,Y: A,Z: A] : aa(A,A,plus_plus(A,X),ord_min(A,Y,Z)) = ord_min(A,aa(A,A,plus_plus(A,X),Y),aa(A,A,plus_plus(A,X),Z)) ) ).

% min_add_distrib_right
tff(fact_6461_min__diff__distrib__left,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A,Y: A,Z: A] : aa(A,A,minus_minus(A,ord_min(A,X,Y)),Z) = ord_min(A,aa(A,A,minus_minus(A,X),Z),aa(A,A,minus_minus(A,Y),Z)) ) ).

% min_diff_distrib_left
tff(fact_6462_min__diff,axiom,
    ! [M: nat,I: nat,Na: nat] : ord_min(nat,aa(nat,nat,minus_minus(nat,M),I),aa(nat,nat,minus_minus(nat,Na),I)) = aa(nat,nat,minus_minus(nat,ord_min(nat,M,Na)),I) ).

% min_diff
tff(fact_6463_nat__mult__min__left,axiom,
    ! [M: nat,Na: nat,Q3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),ord_min(nat,M,Na)),Q3) = 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_6464_nat__mult__min__right,axiom,
    ! [M: nat,Na: nat,Q3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),ord_min(nat,Na,Q3)) = 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_6465_min__max__distrib2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C2: A] : ord_min(A,A3,aa(A,A,aa(A,fun(A,A),ord_max(A),B3),C2)) = aa(A,A,aa(A,fun(A,A),ord_max(A),ord_min(A,A3,B3)),ord_min(A,A3,C2)) ) ).

% min_max_distrib2
tff(fact_6466_min__max__distrib1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,C2: A,A3: A] : ord_min(A,aa(A,A,aa(A,fun(A,A),ord_max(A),B3),C2),A3) = aa(A,A,aa(A,fun(A,A),ord_max(A),ord_min(A,B3,A3)),ord_min(A,C2,A3)) ) ).

% min_max_distrib1
tff(fact_6467_max__min__distrib2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),A3),ord_min(A,B3,C2)) = ord_min(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C2)) ) ).

% max_min_distrib2
tff(fact_6468_max__min__distrib1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,C2: A,A3: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),ord_min(A,B3,C2)),A3) = ord_min(A,aa(A,A,aa(A,fun(A,A),ord_max(A),B3),A3),aa(A,A,aa(A,fun(A,A),ord_max(A),C2),A3)) ) ).

% max_min_distrib1
tff(fact_6469_min_Oassoc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C2: A] : ord_min(A,ord_min(A,A3,B3),C2) = ord_min(A,A3,ord_min(A,B3,C2)) ) ).

% min.assoc
tff(fact_6470_min_Ocommute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] : ord_min(A,A3,B3) = ord_min(A,B3,A3) ) ).

% min.commute
tff(fact_6471_min_Oleft__commute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A,C2: A] : ord_min(A,B3,ord_min(A,A3,C2)) = ord_min(A,A3,ord_min(A,B3,C2)) ) ).

% min.left_commute
tff(fact_6472_of__int__min,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: int,Y: int] : aa(int,A,ring_1_of_int(A),ord_min(int,X,Y)) = ord_min(A,aa(int,A,ring_1_of_int(A),X),aa(int,A,ring_1_of_int(A),Y)) ) ).

% of_int_min
tff(fact_6473_of__nat__min,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: nat,Y: nat] : aa(nat,A,semiring_1_of_nat(A),ord_min(nat,X,Y)) = ord_min(A,aa(nat,A,semiring_1_of_nat(A),X),aa(nat,A,semiring_1_of_nat(A),Y)) ) ).

% of_nat_min
tff(fact_6474_minus__min__eq__max,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [X: A,Y: A] : aa(A,A,uminus_uminus(A),ord_min(A,X,Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y)) ) ).

% minus_min_eq_max
tff(fact_6475_minus__max__eq__min,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [X: A,Y: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)) = ord_min(A,aa(A,A,uminus_uminus(A),X),aa(A,A,uminus_uminus(A),Y)) ) ).

% minus_max_eq_min
tff(fact_6476_min__less__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Z: A] :
          ( aa(A,$o,ord_less(A,ord_min(A,X,Y)),Z)
        <=> ( aa(A,$o,ord_less(A,X),Z)
            | aa(A,$o,ord_less(A,Y),Z) ) ) ) ).

% min_less_iff_disj
tff(fact_6477_min_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less(A,A3),ord_min(A,B3,C2))
         => ~ ( aa(A,$o,ord_less(A,A3),B3)
             => ~ aa(A,$o,ord_less(A,A3),C2) ) ) ) ).

% min.strict_boundedE
tff(fact_6478_min_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
        <=> ( ( A3 = ord_min(A,A3,B3) )
            & ( A3 != B3 ) ) ) ) ).

% min.strict_order_iff
tff(fact_6479_min_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),C2)
         => aa(A,$o,ord_less(A,ord_min(A,A3,B3)),C2) ) ) ).

% min.strict_coboundedI1
tff(fact_6480_min_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,C2: A,A3: A] :
          ( aa(A,$o,ord_less(A,B3),C2)
         => aa(A,$o,ord_less(A,ord_min(A,A3,B3)),C2) ) ) ).

% min.strict_coboundedI2
tff(fact_6481_mask__and__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: nat,B3: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),bit_se2239418461657761734s_mask(word(A),A3)),bit_se2239418461657761734s_mask(word(A),B3)) = bit_se2239418461657761734s_mask(word(A),ord_min(nat,A3,B3)) ) ).

% mask_and_mask
tff(fact_6482_mask__twice,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Na: nat,M: nat] : 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)),X),bit_se2239418461657761734s_mask(word(A),Na))),bit_se2239418461657761734s_mask(word(A),M)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),bit_se2239418461657761734s_mask(word(A),ord_min(nat,M,Na))) ) ).

% mask_twice
tff(fact_6483_max__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P2: A,X: A,Y: A] :
          aa(A,A,aa(A,fun(A,A),times_times(A),P2),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)) = $ite(aa(A,$o,ord_less_eq(A,zero_zero(A)),P2),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P2),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y)),ord_min(A,aa(A,A,aa(A,fun(A,A),times_times(A),P2),X),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y))) ) ).

% max_mult_distrib_left
tff(fact_6484_min__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P2: A,X: A,Y: A] :
          aa(A,A,aa(A,fun(A,A),times_times(A),P2),ord_min(A,X,Y)) = $ite(aa(A,$o,ord_less_eq(A,zero_zero(A)),P2),ord_min(A,aa(A,A,aa(A,fun(A,A),times_times(A),P2),X),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y)),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P2),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y))) ) ).

% min_mult_distrib_left
tff(fact_6485_max__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A,P2: A] :
          aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),P2) = $ite(aa(A,$o,ord_less_eq(A,zero_zero(A)),P2),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P2)),ord_min(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),P2),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P2))) ) ).

% max_mult_distrib_right
tff(fact_6486_min__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A,P2: A] :
          aa(A,A,aa(A,fun(A,A),times_times(A),ord_min(A,X,Y)),P2) = $ite(aa(A,$o,ord_less_eq(A,zero_zero(A)),P2),ord_min(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),P2),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P2)),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P2))) ) ).

% min_mult_distrib_right
tff(fact_6487_min__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A,P2: A] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),ord_min(A,X,Y)),P2) = $ite(aa(A,$o,ord_less_eq(A,zero_zero(A)),P2),ord_min(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),P2),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),P2)),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),P2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),P2))) ) ).

% min_divide_distrib_right
tff(fact_6488_max__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A,P2: A] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),P2) = $ite(aa(A,$o,ord_less_eq(A,zero_zero(A)),P2),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),P2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),P2)),ord_min(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),P2),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),P2))) ) ).

% max_divide_distrib_right
tff(fact_6489_ucast__mask__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Na: nat] : aa(word(B),word(A),semiring_1_unsigned(B,word(A)),bit_se2239418461657761734s_mask(word(B),Na)) = bit_se2239418461657761734s_mask(word(A),ord_min(nat,type_len0_len_of(B,type2(B)),Na)) ) ).

% ucast_mask_eq
tff(fact_6490_mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,M: nat,Na: nat] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M))),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),ord_min(nat,M,Na))) ) ).

% mod_exp_eq
tff(fact_6491_mod__mod__power,axiom,
    ! [K: nat,M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),K),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),K),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),ord_min(nat,M,Na))) ).

% mod_mod_power
tff(fact_6492_Word_Obit__mask__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),bit_se2239418461657761734s_mask(word(A),M)),Na)
        <=> aa(nat,$o,ord_less(nat,Na),ord_min(nat,type_len0_len_of(A,type2(A)),M)) ) ) ).

% Word.bit_mask_iff
tff(fact_6493_uint__mask__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] : aa(word(A),int,semiring_1_unsigned(A,int),bit_se2239418461657761734s_mask(word(A),Na)) = bit_se2239418461657761734s_mask(int,ord_min(nat,type_len0_len_of(A,type2(A)),Na)) ) ).

% uint_mask_eq
tff(fact_6494_unat__mask__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] : aa(word(A),nat,semiring_1_unsigned(A,nat),bit_se2239418461657761734s_mask(word(A),Na)) = bit_se2239418461657761734s_mask(nat,ord_min(nat,type_len0_len_of(A,type2(A)),Na)) ) ).

% unat_mask_eq
tff(fact_6495_and__mask__wi_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: int,Na: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(int,word(A),ring_1_of_int(word(A)),I)),bit_se2239418461657761734s_mask(word(A),Na)) = aa(int,word(A),ring_1_of_int(word(A)),bit_se2584673776208193580ke_bit(int,ord_min(nat,type_len0_len_of(A,type2(A)),Na),I)) ) ).

% and_mask_wi'
tff(fact_6496_mask__mod__exp,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Na: nat,M: nat] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)),one_one(A))),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M)) = aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),ord_min(nat,M,Na))),one_one(A)) ) ).

% mask_mod_exp
tff(fact_6497_bit__slice__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [M: nat,W: word(B),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(B),word(A),slice2(B,A,M),W)),Na)
        <=> ( aa(nat,$o,ord_less(nat,Na),ord_min(nat,type_len0_len_of(A,type2(A)),aa(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)),W),aa(nat,nat,minus_minus(nat,aa(nat,nat,plus_plus(nat,Na),type_len0_len_of(B,type2(B)))),aa(nat,nat,minus_minus(nat,type_len0_len_of(B,type2(B))),M))) ) ) ) ).

% bit_slice_iff
tff(fact_6498_bit__slice1__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [M: nat,W: word(B),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(B),word(A),slice1(B,A,M),W)),Na)
        <=> ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,M),type_len0_len_of(B,type2(B)))),Na)
            & aa(nat,$o,ord_less(nat,Na),ord_min(nat,type_len0_len_of(A,type2(A)),M))
            & aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),W),aa(nat,nat,minus_minus(nat,aa(nat,nat,plus_plus(nat,Na),aa(nat,nat,minus_minus(nat,type_len0_len_of(B,type2(B))),M))),aa(nat,nat,minus_minus(nat,M),type_len0_len_of(B,type2(B))))) ) ) ) ).

% bit_slice1_iff
tff(fact_6499_unat__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] : aa(word(A),nat,semiring_1_unsigned(A,nat),bit_se2239418461657761734s_mask(word(A),Na)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),ord_min(nat,Na,type_len0_len_of(A,type2(A))))),one_one(nat)) ) ).

% unat_mask
tff(fact_6500_drop__bit__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: nat] :
          aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),M),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              aa($o,A,zero_neq_one_of_bool(A),
                ( aa(nat,$o,ord_less_eq(nat,M),Na)
                & bit_se6407376104438227557le_bit(A,type2(A),Na) ))),
            aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,minus_minus(nat,Na),M))) ) ).

% drop_bit_exp_eq
tff(fact_6501_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,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_6502_top__apply,axiom,
    ! [B: $tType,A: $tType] :
      ( top(A)
     => ! [X: B] : aa(B,A,top_top(fun(B,A)),X) = top_top(A) ) ).

% top_apply
tff(fact_6503_UNIV__I,axiom,
    ! [A: $tType,X: A] : member(A,X,top_top(set(A))) ).

% UNIV_I
tff(fact_6504_min__enat__simps_I3_J,axiom,
    ! [Q3: extended_enat] : ord_min(extended_enat,zero_zero(extended_enat),Q3) = zero_zero(extended_enat) ).

% min_enat_simps(3)
tff(fact_6505_min__enat__simps_I2_J,axiom,
    ! [Q3: extended_enat] : ord_min(extended_enat,Q3,zero_zero(extended_enat)) = zero_zero(extended_enat) ).

% min_enat_simps(2)
tff(fact_6506_sup__top__right,axiom,
    ! [A: $tType] :
      ( bounded_lattice_top(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),top_top(A)) = top_top(A) ) ).

% sup_top_right
tff(fact_6507_sup__top__left,axiom,
    ! [A: $tType] :
      ( bounded_lattice_top(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),top_top(A)),X) = top_top(A) ) ).

% sup_top_left
tff(fact_6508_boolean__algebra_Odisj__one__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),top_top(A)) = top_top(A) ) ).

% boolean_algebra.disj_one_right
tff(fact_6509_boolean__algebra_Odisj__one__left,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),top_top(A)),X) = top_top(A) ) ).

% boolean_algebra.disj_one_left
tff(fact_6510_min__top,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [X: A] : ord_min(A,top_top(A),X) = X ) ).

% min_top
tff(fact_6511_min__top2,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [X: A] : ord_min(A,X,top_top(A)) = X ) ).

% min_top2
tff(fact_6512_max__top,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),top_top(A)),X) = top_top(A) ) ).

% max_top
tff(fact_6513_max__top2,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),X),top_top(A)) = top_top(A) ) ).

% max_top2
tff(fact_6514_Collect__const,axiom,
    ! [A: $tType,P: $o] :
      collect(A,aTP_Lamp_ph($o,fun(A,$o),(P))) = $ite((P),top_top(set(A)),bot_bot(set(A))) ).

% Collect_const
tff(fact_6515_finite__Collect__not,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( finite_finite2(A,collect(A,P))
     => ( finite_finite2(A,collect(A,aTP_Lamp_al(fun(A,$o),fun(A,$o),P)))
      <=> finite_finite2(A,top_top(set(A))) ) ) ).

% finite_Collect_not
tff(fact_6516_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_6517_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_6518_possible__bit__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat] :
          ( bit_se6407376104438227557le_bit(word(A),type2(word(A)),M)
        <=> aa(nat,$o,ord_less(nat,M),type_len0_len_of(A,type2(A))) ) ) ).

% possible_bit_word
tff(fact_6519_boolean__algebra_Odisj__cancel__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,uminus_uminus(A),X)) = top_top(A) ) ).

% boolean_algebra.disj_cancel_right
tff(fact_6520_boolean__algebra_Odisj__cancel__left,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),X) = top_top(A) ) ).

% boolean_algebra.disj_cancel_left
tff(fact_6521_sup__compl__top__left2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),Y)) = top_top(A) ) ).

% sup_compl_top_left2
tff(fact_6522_sup__compl__top__left1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) = top_top(A) ) ).

% sup_compl_top_left1
tff(fact_6523_Diff__UNIV,axiom,
    ! [A: $tType,A2: set(A)] : aa(set(A),set(A),minus_minus(set(A),A2),top_top(set(A))) = bot_bot(set(A)) ).

% Diff_UNIV
tff(fact_6524_surj__diff__right,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A] : aa(set(A),set(A),image(A,A,aTP_Lamp_ou(A,fun(A,A),A3)),top_top(set(A))) = top_top(set(A)) ) ).

% surj_diff_right
tff(fact_6525_range__constant,axiom,
    ! [B: $tType,A: $tType,X: A] : aa(set(B),set(A),image(B,A,aTP_Lamp_or(A,fun(B,A),X)),top_top(set(B))) = aa(set(A),set(A),insert(A,X),bot_bot(set(A))) ).

% range_constant
tff(fact_6526_bit__minus__1__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),one_one(A))),Na)
        <=> bit_se6407376104438227557le_bit(A,type2(A),Na) ) ) ).

% bit_minus_1_iff
tff(fact_6527_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,ord_less(nat,zero_zero(nat)),Na) ) ) ) ).

% bit_minus_2_iff
tff(fact_6528_Compl__eq__Diff__UNIV,axiom,
    ! [A: $tType,A2: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A2) = aa(set(A),set(A),minus_minus(set(A),top_top(set(A))),A2) ).

% Compl_eq_Diff_UNIV
tff(fact_6529_Compl__partition2,axiom,
    ! [A: $tType,A2: 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)),A2)),A2) = top_top(set(A)) ).

% Compl_partition2
tff(fact_6530_Compl__partition,axiom,
    ! [A: $tType,A2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),aa(set(A),set(A),uminus_uminus(set(A)),A2)) = top_top(set(A)) ).

% Compl_partition
tff(fact_6531_atLeastAtMost__eq__UNIV__iff,axiom,
    ! [A: $tType] :
      ( bounded_lattice(A)
     => ! [X: A,Y: A] :
          ( ( set_or1337092689740270186AtMost(A,X,Y) = top_top(set(A)) )
        <=> ( ( X = bot_bot(A) )
            & ( Y = top_top(A) ) ) ) ) ).

% atLeastAtMost_eq_UNIV_iff
tff(fact_6532_empty__not__UNIV,axiom,
    ! [A: $tType] : bot_bot(set(A)) != top_top(set(A)) ).

% empty_not_UNIV
tff(fact_6533_insert__UNIV,axiom,
    ! [A: $tType,X: A] : aa(set(A),set(A),insert(A,X),top_top(set(A))) = top_top(set(A)) ).

% insert_UNIV
tff(fact_6534_top_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] :
          ( ( A3 != top_top(A) )
        <=> aa(A,$o,ord_less(A,A3),top_top(A)) ) ) ).

% top.not_eq_extremum
tff(fact_6535_top_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] : ~ aa(A,$o,ord_less(A,top_top(A)),A3) ) ).

% top.extremum_strict
tff(fact_6536_sup__cancel__left1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),A3)),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),B3)) = top_top(A) ) ).

% sup_cancel_left1
tff(fact_6537_sup__cancel__left2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),A3)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),B3)) = top_top(A) ) ).

% sup_cancel_left2
tff(fact_6538_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_6539_UNIV__def,axiom,
    ! [A: $tType] : top_top(set(A)) = collect(A,aTP_Lamp_pi(A,$o)) ).

% UNIV_def
tff(fact_6540_UNIV__witness,axiom,
    ! [A: $tType] :
    ? [X4: A] : member(A,X4,top_top(set(A))) ).

% UNIV_witness
tff(fact_6541_UNIV__eq__I,axiom,
    ! [A: $tType,A2: set(A)] :
      ( ! [X4: A] : member(A,X4,A2)
     => ( top_top(set(A)) = A2 ) ) ).

% UNIV_eq_I
tff(fact_6542_Un__UNIV__right,axiom,
    ! [A: $tType,A2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),top_top(set(A))) = top_top(set(A)) ).

% Un_UNIV_right
tff(fact_6543_Un__UNIV__left,axiom,
    ! [A: $tType,B2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),top_top(set(A))),B2) = top_top(set(A)) ).

% Un_UNIV_left
tff(fact_6544_top__greatest,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] : aa(A,$o,ord_less_eq(A,A3),top_top(A)) ) ).

% top_greatest
tff(fact_6545_top_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less_eq(A,top_top(A)),A3)
        <=> ( A3 = top_top(A) ) ) ) ).

% top.extremum_unique
tff(fact_6546_top_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less_eq(A,top_top(A)),A3)
         => ( A3 = top_top(A) ) ) ) ).

% top.extremum_uniqueI
tff(fact_6547_subset__UNIV,axiom,
    ! [A: $tType,A2: set(A)] : aa(set(A),$o,ord_less_eq(set(A),A2),top_top(set(A))) ).

% subset_UNIV
tff(fact_6548_perfect__space__class_OUNIV__not__singleton,axiom,
    ! [A: $tType] :
      ( topolo8386298272705272623_space(A)
     => ! [X: A] : top_top(set(A)) != aa(set(A),set(A),insert(A,X),bot_bot(set(A))) ) ).

% perfect_space_class.UNIV_not_singleton
tff(fact_6549_possible__bit,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat] : bit_se6407376104438227557le_bit(A,type2(A),Na) ) ).

% possible_bit
tff(fact_6550_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_6551_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_6552_range__subsetD,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),B2: set(A),I: B] :
      ( aa(set(A),$o,ord_less_eq(set(A),aa(set(B),set(A),image(B,A,F2),top_top(set(B)))),B2)
     => member(A,aa(B,A,F2,I),B2) ) ).

% range_subsetD
tff(fact_6553_rangeE,axiom,
    ! [A: $tType,B: $tType,B3: A,F2: fun(B,A)] :
      ( member(A,B3,aa(set(B),set(A),image(B,A,F2),top_top(set(B))))
     => ~ ! [X4: B] : B3 != aa(B,A,F2,X4) ) ).

% rangeE
tff(fact_6554_rangeI,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),X: B] : member(A,aa(B,A,F2,X),aa(set(B),set(A),image(B,A,F2),top_top(set(B)))) ).

% rangeI
tff(fact_6555_range__eqI,axiom,
    ! [A: $tType,B: $tType,B3: A,F2: fun(B,A),X: B] :
      ( ( B3 = aa(B,A,F2,X) )
     => member(A,B3,aa(set(B),set(A),image(B,A,F2),top_top(set(B)))) ) ).

% range_eqI
tff(fact_6556_range__composition,axiom,
    ! [A: $tType,C: $tType,B: $tType,F2: fun(C,A),G: fun(B,C)] : aa(set(B),set(A),image(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_pj(fun(C,A),fun(fun(B,C),fun(B,A)),F2),G)),top_top(set(B))) = aa(set(C),set(A),image(C,A,F2),aa(set(B),set(C),image(B,C,G),top_top(set(B)))) ).

% range_composition
tff(fact_6557_finite__range__imageI,axiom,
    ! [A: $tType,C: $tType,B: $tType,G: fun(B,A),F2: fun(A,C)] :
      ( finite_finite2(A,aa(set(B),set(A),image(B,A,G),top_top(set(B))))
     => finite_finite2(C,aa(set(B),set(C),image(B,C,aa(fun(A,C),fun(B,C),aTP_Lamp_pk(fun(B,A),fun(fun(A,C),fun(B,C)),G),F2)),top_top(set(B)))) ) ).

% finite_range_imageI
tff(fact_6558_sup__shunt,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = top_top(A) )
        <=> aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),X)),Y) ) ) ).

% sup_shunt
tff(fact_6559_range__eq__singletonD,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: A,X: B] :
      ( ( aa(set(B),set(A),image(B,A,F2),top_top(set(B))) = aa(set(A),set(A),insert(A,A3),bot_bot(set(A))) )
     => ( aa(B,A,F2,X) = A3 ) ) ).

% range_eq_singletonD
tff(fact_6560_bit__mask__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),bit_se2239418461657761734s_mask(A,M)),Na)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Na)
            & aa(nat,$o,ord_less(nat,Na),M) ) ) ) ).

% bit_mask_iff
tff(fact_6561_bit__unsigned__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_semiring_bits(A) )
     => ! [W: word(B),Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(word(B),A,semiring_1_unsigned(B,A),W)),Na)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Na)
            & aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),W),Na) ) ) ) ).

% bit_unsigned_iff
tff(fact_6562_bit__of__nat__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(nat,A,semiring_1_of_nat(A),M)),Na)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Na)
            & aa(nat,$o,aa(nat,fun(nat,$o),bit_se5641148757651400278ts_bit(nat),M),Na) ) ) ) ).

% bit_of_nat_iff
tff(fact_6563_possible__bit__def,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Tyrep: itself(A),Na: nat] :
          ( bit_se6407376104438227557le_bit(A,Tyrep,Na)
        <=> ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na) != zero_zero(A) ) ) ) ).

% possible_bit_def
tff(fact_6564_bit__minus__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,uminus_uminus(A),A3)),Na)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Na)
            & ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,minus_minus(A,A3),one_one(A))),Na) ) ) ) ).

% bit_minus_iff
tff(fact_6565_bit__twiddle__min,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: 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)),X),Y)),
              $ite(aa(word(A),$o,ord_less(word(A),X),Y),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))),zero_zero(word(A))))) = ord_min(word(A),X,Y) ) ).

% bit_twiddle_min
tff(fact_6566_fold__possible__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Na: nat] :
          ( ( aa(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_6567_bit__exp__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(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M)),Na)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Na)
            & ( M = Na ) ) ) ) ).

% bit_exp_iff
tff(fact_6568_bit__2__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),one_one(nat))
            & ( Na = one_one(nat) ) ) ) ) ).

% bit_2_iff
tff(fact_6569_bit__not__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,bit_ri4277139882892585799ns_not(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M))),Na)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Na)
            & ( Na != M ) ) ) ) ).

% bit_not_exp_iff
tff(fact_6570_bit__minus__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,uminus_uminus(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M))),Na)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Na)
            & aa(nat,$o,ord_less_eq(nat,M),Na) ) ) ) ).

% bit_minus_exp_iff
tff(fact_6571_range__uint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(set(word(A)),set(int),image(word(A),int,semiring_1_unsigned(A,int)),top_top(set(word(A)))) = set_or7035219750837199246ssThan(int,zero_zero(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% range_uint
tff(fact_6572_UNIV__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( top_top(set(word(A))) = aa(set(int),set(word(A)),image(int,word(A),ring_1_of_int(word(A))),set_or7035219750837199246ssThan(int,zero_zero(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ) ).

% UNIV_eq
tff(fact_6573_ucast__range__less,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ( aa(nat,$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)))) = collect(word(B),aTP_Lamp_pl(word(B),$o)) ) ) ) ).

% ucast_range_less
tff(fact_6574_bit__mask__sub__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M)),one_one(A))),Na)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Na)
            & aa(nat,$o,ord_less(nat,Na),M) ) ) ) ).

% bit_mask_sub_iff
tff(fact_6575_bit__double__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)),Na)
        <=> ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),aa(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_6576_bit__signed__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_ri3973907225187159222ations(A) )
     => ! [W: word(B),Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(word(B),A,ring_1_signed(B,A),W)),Na)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Na)
            & aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),W),ord_min(nat,aa(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_6577_uint__word__rotr__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),word_rotr(A,Na),W)) = bit_concat_bit(aa(nat,nat,minus_minus(nat,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)))),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Na),type_len0_len_of(A,type2(A)))),aa(word(A),int,semiring_1_unsigned(A,int),W)),aa(word(A),int,semiring_1_unsigned(A,int),bit_se2584673776208193580ke_bit(word(A),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Na),type_len0_len_of(A,type2(A))),W))) ) ).

% uint_word_rotr_eq
tff(fact_6578_word__roti_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa2: int,X: int] : word_roti(A,Xa2,aa(int,word(A),word2(A),X)) = aa(int,word(A),word2(A),bit_concat_bit(aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),nat2(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Xa2),aa(nat,int,semiring_1_of_nat(int),type_len0_len_of(A,type2(A)))))),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),nat2(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Xa2),aa(nat,int,semiring_1_of_nat(int),type_len0_len_of(A,type2(A)))))),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),X)),bit_se2584673776208193580ke_bit(int,nat2(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Xa2),aa(nat,int,semiring_1_of_nat(int),type_len0_len_of(A,type2(A))))),X))) ) ).

% word_roti.abs_eq
tff(fact_6579_assn__basic__inequalities_I1_J,axiom,
    top_top(assn) != one_one(assn) ).

% assn_basic_inequalities(1)
tff(fact_6580_assn__basic__inequalities_I5_J,axiom,
    top_top(assn) != bot_bot(assn) ).

% assn_basic_inequalities(5)
tff(fact_6581_range__mult,axiom,
    ! [A3: real] :
      aa(set(real),set(real),image(real,real,aa(real,fun(real,real),times_times(real),A3)),top_top(set(real))) = $ite(A3 = zero_zero(real),aa(set(real),set(real),insert(real,zero_zero(real)),bot_bot(set(real))),top_top(set(real))) ).

% range_mult
tff(fact_6582_Word__eq__word__of__int,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( word2(A) = ring_1_of_int(word(A)) ) ) ).

% Word_eq_word_of_int
tff(fact_6583_Collect__const__case__prod,axiom,
    ! [B: $tType,A: $tType,P: $o] :
      collect(product_prod(A,B),product_case_prod(A,B,$o,aTP_Lamp_pm($o,fun(A,fun(B,$o)),(P)))) = $ite((P),top_top(set(product_prod(A,B))),bot_bot(set(product_prod(A,B)))) ).

% Collect_const_case_prod
tff(fact_6584_top__empty__eq,axiom,
    ! [A: $tType,X2: A] :
      ( aa(A,$o,top_top(fun(A,$o)),X2)
    <=> member(A,X2,top_top(set(A))) ) ).

% top_empty_eq
tff(fact_6585_top__set__def,axiom,
    ! [A: $tType] : top_top(set(A)) = collect(A,top_top(fun(A,$o))) ).

% top_set_def
tff(fact_6586_top__empty__eq2,axiom,
    ! [B: $tType,A: $tType,X2: A,Xa: B] :
      ( aa(B,$o,aa(A,fun(B,$o),top_top(fun(A,fun(B,$o))),X2),Xa)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Xa),top_top(set(product_prod(A,B)))) ) ).

% top_empty_eq2
tff(fact_6587_infinite__UNIV__listI,axiom,
    ! [A: $tType] : ~ finite_finite2(list(A),top_top(set(list(A)))) ).

% infinite_UNIV_listI
tff(fact_6588_infinite__UNIV__int,axiom,
    ~ finite_finite2(int,top_top(set(int))) ).

% infinite_UNIV_int
tff(fact_6589_xor__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa2: int,X: int] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),aa(int,word(A),word2(A),Xa2)),aa(int,word(A),word2(A),X)) = aa(int,word(A),word2(A),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),Xa2),X)) ) ).

% xor_word.abs_eq
tff(fact_6590_mask__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: nat] : bit_se2239418461657761734s_mask(word(A),X) = aa(int,word(A),word2(A),bit_se2239418461657761734s_mask(int,X)) ) ).

% mask_word.abs_eq
tff(fact_6591_push__bit__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa2: nat,X: int] : aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4730199178511100633sh_bit(word(A)),Xa2),aa(int,word(A),word2(A),X)) = aa(int,word(A),word2(A),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),Xa2),X)) ) ).

% push_bit_word.abs_eq
tff(fact_6592_not__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: int] : aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(int,word(A),word2(A),X)) = aa(int,word(A),word2(A),aa(int,int,bit_ri4277139882892585799ns_not(int),X)) ) ).

% not_word.abs_eq
tff(fact_6593_or__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa2: int,X: int] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(int,word(A),word2(A),Xa2)),aa(int,word(A),word2(A),X)) = aa(int,word(A),word2(A),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),Xa2),X)) ) ).

% or_word.abs_eq
tff(fact_6594_word__rotr__word__rotr__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,Na: nat,W: word(A)] : aa(word(A),word(A),word_rotr(A,M),aa(word(A),word(A),word_rotr(A,Na),W)) = aa(word(A),word(A),word_rotr(A,aa(nat,nat,plus_plus(nat,M),Na)),W) ) ).

% word_rotr_word_rotr_eq
tff(fact_6595_plus__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa2: int,X: int] : aa(word(A),word(A),plus_plus(word(A),aa(int,word(A),word2(A),Xa2)),aa(int,word(A),word2(A),X)) = aa(int,word(A),word2(A),aa(int,int,plus_plus(int,Xa2),X)) ) ).

% plus_word.abs_eq
tff(fact_6596_word__rotate_Oword__rot__logs_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,X: word(A),Y: word(A)] : aa(word(A),word(A),word_rotr(A,Na),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),Y)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),word_rotr(A,Na),X)),aa(word(A),word(A),word_rotr(A,Na),Y)) ) ).

% word_rotate.word_rot_logs(4)
tff(fact_6597_word__rotate_Oword__rot__logs_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,X: word(A),Y: word(A)] : aa(word(A),word(A),word_rotr(A,Na),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),X),Y)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(word(A),word(A),word_rotr(A,Na),X)),aa(word(A),word(A),word_rotr(A,Na),Y)) ) ).

% word_rotate.word_rot_logs(6)
tff(fact_6598_word__rotate_Oword__rot__logs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,V2: word(A)] : aa(word(A),word(A),word_rotr(A,Na),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),V2)) = aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(word(A),word(A),word_rotr(A,Na),V2)) ) ).

% word_rotate.word_rot_logs(2)
tff(fact_6599_unset__bit__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa2: nat,X: int] : bit_se2638667681897837118et_bit(word(A),Xa2,aa(int,word(A),word2(A),X)) = aa(int,word(A),word2(A),bit_se2638667681897837118et_bit(int,Xa2,X)) ) ).

% unset_bit_word.abs_eq
tff(fact_6600_word_Oabs__induct,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(word(A),$o),X: word(A)] :
          ( ! [Y3: int] : aa(word(A),$o,P,aa(int,word(A),word2(A),Y3))
         => aa(word(A),$o,P,X) ) ) ).

% word.abs_induct
tff(fact_6601_set__bit__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa2: nat,X: int] : bit_se5668285175392031749et_bit(word(A),Xa2,aa(int,word(A),word2(A),X)) = aa(int,word(A),word2(A),bit_se5668285175392031749et_bit(int,Xa2,X)) ) ).

% set_bit_word.abs_eq
tff(fact_6602_flip__bit__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa2: nat,X: int] : bit_se8732182000553998342ip_bit(word(A),Xa2,aa(int,word(A),word2(A),X)) = aa(int,word(A),word2(A),bit_se8732182000553998342ip_bit(int,Xa2,X)) ) ).

% flip_bit_word.abs_eq
tff(fact_6603_one__word__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( one_one(word(A)) = aa(int,word(A),word2(A),one_one(int)) ) ) ).

% one_word_def
tff(fact_6604_word__rotate_Oword__rot__logs_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,X: word(A),Y: word(A)] : aa(word(A),word(A),word_rotr(A,Na),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),X),Y)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),aa(word(A),word(A),word_rotr(A,Na),X)),aa(word(A),word(A),word_rotr(A,Na),Y)) ) ).

% word_rotate.word_rot_logs(8)
tff(fact_6605_zero__word__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( zero_zero(word(A)) = aa(int,word(A),word2(A),zero_zero(int)) ) ) ).

% zero_word_def
tff(fact_6606_and__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa2: int,X: int] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(int,word(A),word2(A),Xa2)),aa(int,word(A),word2(A),X)) = aa(int,word(A),word2(A),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xa2),X)) ) ).

% and_word.abs_eq
tff(fact_6607_times__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa2: int,X: int] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(int,word(A),word2(A),Xa2)),aa(int,word(A),word2(A),X)) = aa(int,word(A),word2(A),aa(int,int,aa(int,fun(int,int),times_times(int),Xa2),X)) ) ).

% times_word.abs_eq
tff(fact_6608_minus__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa2: int,X: int] : aa(word(A),word(A),minus_minus(word(A),aa(int,word(A),word2(A),Xa2)),aa(int,word(A),word2(A),X)) = aa(int,word(A),word2(A),aa(int,int,minus_minus(int,Xa2),X)) ) ).

% minus_word.abs_eq
tff(fact_6609_uminus__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: int] : aa(word(A),word(A),uminus_uminus(word(A)),aa(int,word(A),word2(A),X)) = aa(int,word(A),word2(A),aa(int,int,uminus_uminus(int),X)) ) ).

% uminus_word.abs_eq
tff(fact_6610_word_Oabs__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: int,Y: int] :
          ( ( aa(int,word(A),word2(A),X) = aa(int,word(A),word2(A),Y) )
        <=> ( bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),X) = bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),Y) ) ) ) ).

% word.abs_eq_iff
tff(fact_6611_size__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: int] : size_size(word(A),aa(int,word(A),word2(A),X)) = type_len0_len_of(A,type2(A)) ) ).

% size_word.abs_eq
tff(fact_6612_word__succ_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: int] : word_succ(A,aa(int,word(A),word2(A),X)) = aa(int,word(A),word2(A),aa(int,int,plus_plus(int,X),one_one(int))) ) ).

% word_succ.abs_eq
tff(fact_6613_word__pred_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: int] : word_pred(A,aa(int,word(A),word2(A),X)) = aa(int,word(A),word2(A),aa(int,int,minus_minus(int,X),one_one(int))) ) ).

% word_pred.abs_eq
tff(fact_6614_Ints__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ( ring_1_Ints(A) = aa(set(int),set(A),image(int,A,ring_1_of_int(A)),top_top(set(int))) ) ) ).

% Ints_def
tff(fact_6615_word__rotr_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa2: nat,X: int] : aa(word(A),word(A),word_rotr(A,Xa2),aa(int,word(A),word2(A),X)) = aa(int,word(A),word2(A),bit_concat_bit(aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xa2),type_len0_len_of(A,type2(A)))),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xa2),type_len0_len_of(A,type2(A)))),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),X)),bit_se2584673776208193580ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xa2),type_len0_len_of(A,type2(A))),X))) ) ).

% word_rotr.abs_eq
tff(fact_6616_int__in__range__abs,axiom,
    ! [Na: nat] : member(int,aa(nat,int,semiring_1_of_nat(int),Na),aa(set(int),set(int),image(int,int,abs_abs(int)),top_top(set(int)))) ).

% int_in_range_abs
tff(fact_6617_word__int__case_Oabs__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(B)
     => ! [Xa2: fun(int,A),X: int] : word_int_case(A,B,Xa2,aa(int,word(B),word2(B),X)) = aa(int,A,Xa2,bit_se2584673776208193580ke_bit(int,type_len0_len_of(B,type2(B)),X)) ) ).

% word_int_case.abs_eq
tff(fact_6618_less__eq__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa2: int,X: int] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(int,word(A),word2(A),Xa2)),aa(int,word(A),word2(A),X))
        <=> aa(int,$o,ord_less_eq(int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),Xa2)),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),X)) ) ) ).

% less_eq_word.abs_eq
tff(fact_6619_less__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa2: int,X: int] :
          ( aa(word(A),$o,ord_less(word(A),aa(int,word(A),word2(A),Xa2)),aa(int,word(A),word2(A),X))
        <=> aa(int,$o,ord_less(int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),Xa2)),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),X)) ) ) ).

% less_word.abs_eq
tff(fact_6620_bit__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: int,X2: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(int,word(A),word2(A),X)),X2)
        <=> ( aa(nat,$o,ord_less(nat,X2),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),X),X2) ) ) ) ).

% bit_word.abs_eq
tff(fact_6621_divide__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa2: int,X: int] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(int,word(A),word2(A),Xa2)),aa(int,word(A),word2(A),X)) = aa(int,word(A),word2(A),aa(int,int,aa(int,fun(int,int),divide_divide(int),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),Xa2)),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),X))) ) ).

% divide_word.abs_eq
tff(fact_6622_modulo__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa2: int,X: int] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),aa(int,word(A),word2(A),Xa2)),aa(int,word(A),word2(A),X)) = aa(int,word(A),word2(A),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),Xa2)),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),X))) ) ).

% modulo_word.abs_eq
tff(fact_6623_take__bit__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa2: nat,X: int] : bit_se2584673776208193580ke_bit(word(A),Xa2,aa(int,word(A),word2(A),X)) = aa(int,word(A),word2(A),bit_se2584673776208193580ke_bit(int,ord_min(nat,type_len0_len_of(A,type2(A)),Xa2),X)) ) ).

% take_bit_word.abs_eq
tff(fact_6624_drop__bit__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa2: nat,X: int] : aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),Xa2),aa(int,word(A),word2(A),X)) = aa(int,word(A),word2(A),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),Xa2),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),X))) ) ).

% drop_bit_word.abs_eq
tff(fact_6625_word__cat_Oabs__eq,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(C)
        & type_len(A) )
     => ! [Xa2: int,X: int] : word_cat(B,C,A,aa(int,word(B),word2(B),Xa2),aa(int,word(C),word2(C),X)) = aa(int,word(A),word2(A),bit_concat_bit(type_len0_len_of(C,type2(C)),X,bit_se2584673776208193580ke_bit(int,type_len0_len_of(B,type2(B)),Xa2))) ) ).

% word_cat.abs_eq
tff(fact_6626_unsigned_Oabs__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & semiring_1(A) )
     => ! [X: int] : aa(word(B),A,semiring_1_unsigned(B,A),aa(int,word(B),word2(B),X)) = aa(nat,A,semiring_1_of_nat(A),nat2(bit_se2584673776208193580ke_bit(int,type_len0_len_of(B,type2(B)),X))) ) ).

% unsigned.abs_eq
tff(fact_6627_range__mod,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(set(nat),set(nat),image(nat,nat,aTP_Lamp_pn(nat,fun(nat,nat),Na)),top_top(set(nat))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),Na) ) ) ).

% range_mod
tff(fact_6628_word__sle_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa2: int,X: int] :
          ( word_sle(A,aa(int,word(A),word2(A),Xa2),aa(int,word(A),word2(A),X))
        <=> aa(int,$o,ord_less_eq(int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))),Xa2)),bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))),X)) ) ) ).

% word_sle.abs_eq
tff(fact_6629_word__sless_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa2: int,X: int] :
          ( word_sless(A,aa(int,word(A),word2(A),Xa2),aa(int,word(A),word2(A),X))
        <=> aa(int,$o,ord_less(int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))),Xa2)),bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))),X)) ) ) ).

% word_sless.abs_eq
tff(fact_6630_signed_Oabs__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_1(A) )
     => ! [X: int] : aa(word(B),A,ring_1_signed(B,A),aa(int,word(B),word2(B),X)) = aa(int,A,ring_1_of_int(A),bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat))),X)) ) ).

% signed.abs_eq
tff(fact_6631_signed__drop__bit_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa2: nat,X: int] : signed_drop_bit(A,Xa2,aa(int,word(A),word2(A),X)) = aa(int,word(A),word2(A),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),Xa2),bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))),X))) ) ).

% signed_drop_bit.abs_eq
tff(fact_6632_bit__word__rotr__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,W: word(A),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),word_rotr(A,M),W)),Na)
        <=> ( aa(nat,$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)),W),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,plus_plus(nat,Na),M)),type_len0_len_of(A,type2(A)))) ) ) ) ).

% bit_word_rotr_iff
tff(fact_6633_signed__modulo__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa2: int,X: int] : signed6721504322012087516modulo(word(A),aa(int,word(A),word2(A),Xa2),aa(int,word(A),word2(A),X)) = aa(int,word(A),word2(A),signed6721504322012087516modulo(int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))),Xa2),bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))),X))) ) ).

% signed_modulo_word.abs_eq
tff(fact_6634_signed__divide__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa2: int,X: int] : signed7115095781618012415divide(word(A),aa(int,word(A),word2(A),Xa2),aa(int,word(A),word2(A),X)) = aa(int,word(A),word2(A),signed7115095781618012415divide(int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))),Xa2),bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))),X))) ) ).

% signed_divide_word.abs_eq
tff(fact_6635_signed__cast_Oabs__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [X: int] : signed_cast(B,A,aa(int,word(B),word2(B),X)) = aa(int,word(A),word2(A),bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat))),X)) ) ).

% signed_cast.abs_eq
tff(fact_6636_the__signed__int_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: int] : the_signed_int(A,aa(int,word(A),word2(A),X)) = bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))),X) ) ).

% the_signed_int.abs_eq
tff(fact_6637_top1I,axiom,
    ! [A: $tType,X: A] : aa(A,$o,top_top(fun(A,$o)),X) ).

% top1I
tff(fact_6638_top2I,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B] : aa(B,$o,aa(A,fun(B,$o),top_top(fun(A,fun(B,$o))),X),Y) ).

% top2I
tff(fact_6639_UNIV__nat__eq,axiom,
    top_top(set(nat)) = aa(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_6640_of__nat_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: nat] : of_nat(A,X) = aa(int,word(A),word2(A),aa(nat,int,semiring_1_of_nat(int),bit_se2584673776208193580ke_bit(nat,type_len0_len_of(A,type2(A)),X))) ) ).

% of_nat.abs_eq
tff(fact_6641_the__nat_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: int] : the_nat(A,aa(int,word(A),word2(A),X)) = nat2(bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),X)) ) ).

% the_nat.abs_eq
tff(fact_6642_cast_Oabs__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [X: int] : cast(B,A,aa(int,word(B),word2(B),X)) = aa(int,word(A),word2(A),bit_se2584673776208193580ke_bit(int,type_len0_len_of(B,type2(B)),X)) ) ).

% cast.abs_eq
tff(fact_6643_Word_Oof__int_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: int] : of_int2(A,X) = aa(int,word(A),word2(A),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),X)) ) ).

% Word.of_int.abs_eq
tff(fact_6644_root__def,axiom,
    ! [Na: nat,X: real] :
      aa(real,real,root(Na),X) = $ite(Na = zero_zero(nat),zero_zero(real),the_inv_into(real,real,top_top(set(real)),aTP_Lamp_po(nat,fun(real,real),Na),X)) ).

% root_def
tff(fact_6645_the__inv__into__def,axiom,
    ! [B: $tType,A: $tType,A2: set(B),F2: fun(B,A),X2: A] : the_inv_into(B,A,A2,F2,X2) = the(B,aa(A,fun(B,$o),aa(fun(B,A),fun(A,fun(B,$o)),aTP_Lamp_pp(set(B),fun(fun(B,A),fun(A,fun(B,$o))),A2),F2),X2)) ).

% the_inv_into_def
tff(fact_6646_word__roti__eq__word__rotr__word__rotl,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: int,W: word(A)] :
          word_roti(A,I,W) = $ite(aa(int,$o,ord_less_eq(int,zero_zero(int)),I),aa(word(A),word(A),word_rotr(A,nat2(I)),W),aa(word(A),word(A),word_rotl(A,nat2(aa(int,int,uminus_uminus(int),I))),W)) ) ).

% word_roti_eq_word_rotr_word_rotl
tff(fact_6647_word__rotl_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa2: nat,X: int] : aa(word(A),word(A),word_rotl(A,Xa2),aa(int,word(A),word2(A),X)) = aa(int,word(A),word2(A),bit_concat_bit(aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xa2),type_len0_len_of(A,type2(A))),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xa2),type_len0_len_of(A,type2(A))))),bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),X)),bit_se2584673776208193580ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xa2),type_len0_len_of(A,type2(A)))),X))) ) ).

% word_rotl.abs_eq
tff(fact_6648_word__rot__lr,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: nat,V2: word(A)] : aa(word(A),word(A),word_rotr(A,K),aa(word(A),word(A),word_rotl(A,K),V2)) = V2 ) ).

% word_rot_lr
tff(fact_6649_word__rot__rl,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: nat,V2: word(A)] : aa(word(A),word(A),word_rotl(A,K),aa(word(A),word(A),word_rotr(A,K),V2)) = V2 ) ).

% word_rot_rl
tff(fact_6650_word__rot__gal,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,V2: word(A),W: word(A)] :
          ( ( aa(word(A),word(A),word_rotr(A,Na),V2) = W )
        <=> ( aa(word(A),word(A),word_rotl(A,Na),W) = V2 ) ) ) ).

% word_rot_gal
tff(fact_6651_word__rot__gal_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Na: nat,V2: word(A)] :
          ( ( W = aa(word(A),word(A),word_rotr(A,Na),V2) )
        <=> ( V2 = aa(word(A),word(A),word_rotl(A,Na),W) ) ) ) ).

% word_rot_gal'
tff(fact_6652_word__rotate_Oword__rot__logs_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,X: word(A),Y: word(A)] : aa(word(A),word(A),word_rotl(A,Na),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),X),Y)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),aa(word(A),word(A),word_rotl(A,Na),X)),aa(word(A),word(A),word_rotl(A,Na),Y)) ) ).

% word_rotate.word_rot_logs(7)
tff(fact_6653_word__rotate_Oword__rot__logs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,V2: word(A)] : aa(word(A),word(A),word_rotl(A,Na),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),V2)) = aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(word(A),word(A),word_rotl(A,Na),V2)) ) ).

% word_rotate.word_rot_logs(1)
tff(fact_6654_word__rotate_Oword__rot__logs_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,X: word(A),Y: word(A)] : aa(word(A),word(A),word_rotl(A,Na),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),X),Y)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(word(A),word(A),word_rotl(A,Na),X)),aa(word(A),word(A),word_rotl(A,Na),Y)) ) ).

% word_rotate.word_rot_logs(5)
tff(fact_6655_word__rotate_Oword__rot__logs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,X: word(A),Y: word(A)] : aa(word(A),word(A),word_rotl(A,Na),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X),Y)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),word_rotl(A,Na),X)),aa(word(A),word(A),word_rotl(A,Na),Y)) ) ).

% word_rotate.word_rot_logs(3)
tff(fact_6656_word__rotx__0,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [I: nat] :
          ( ( aa(word(A),word(A),word_rotr(A,I),zero_zero(word(A))) = zero_zero(word(A)) )
          & ( aa(word(B),word(B),word_rotl(B,I),zero_zero(word(B))) = zero_zero(word(B)) ) ) ) ).

% word_rotx_0
tff(fact_6657_word__rotl__eq__word__rotr,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] : word_rotl(A,Na) = word_rotr(A,aa(nat,nat,minus_minus(nat,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))))) ) ).

% word_rotl_eq_word_rotr
tff(fact_6658_bit__word__rotl__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,W: word(A),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),word_rotl(A,M),W)),Na)
        <=> ( aa(nat,$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)),W),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,plus_plus(nat,Na),aa(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_6659_the__int_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: int] : the_int(A,aa(int,word(A),word2(A),X)) = bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A)),X) ) ).

% the_int.abs_eq
tff(fact_6660_CHAR__pos__iff,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),semiri4206861660011772517g_char(A,type2(A)))
      <=> ? [N3: nat] :
            ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N3)
            & ( aa(nat,A,semiring_1_of_nat(A),N3) = zero_zero(A) ) ) ) ) ).

% CHAR_pos_iff
tff(fact_6661_CHAR__eq__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiri4206861660011772517g_char(A,type2(A)) = zero_zero(nat) ) ) ).

% CHAR_eq_0
tff(fact_6662_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_6663_CHAR__eqI,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [C2: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),C2) = zero_zero(A) )
         => ( ! [X4: nat] :
                ( ( aa(nat,A,semiring_1_of_nat(A),X4) = zero_zero(A) )
               => aa(nat,$o,dvd_dvd(nat,C2),X4) )
           => ( semiri4206861660011772517g_char(A,type2(A)) = C2 ) ) ) ) ).

% CHAR_eqI
tff(fact_6664_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,dvd_dvd(nat,semiri4206861660011772517g_char(A,type2(A))),Na) ) ) ).

% of_nat_eq_0_iff_char_dvd
tff(fact_6665_CHAR__eq0__iff,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( ( semiri4206861660011772517g_char(A,type2(A)) = zero_zero(nat) )
      <=> ! [N3: nat] :
            ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N3)
           => ( aa(nat,A,semiring_1_of_nat(A),N3) != zero_zero(A) ) ) ) ) ).

% CHAR_eq0_iff
tff(fact_6666_CHAR__eq__posI,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [C2: nat] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),C2)
         => ( ( aa(nat,A,semiring_1_of_nat(A),C2) = zero_zero(A) )
           => ( ! [X4: nat] :
                  ( aa(nat,$o,ord_less(nat,zero_zero(nat)),X4)
                 => ( aa(nat,$o,ord_less(nat,X4),C2)
                   => ( aa(nat,A,semiring_1_of_nat(A),X4) != zero_zero(A) ) ) )
             => ( semiri4206861660011772517g_char(A,type2(A)) = C2 ) ) ) ) ) ).

% CHAR_eq_posI
tff(fact_6667_bit__revcast__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [W: word(B),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(B),word(A),revcast(B,A),W)),Na)
        <=> ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))),Na)
            & aa(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)),W),aa(nat,nat,minus_minus(nat,aa(nat,nat,plus_plus(nat,Na),aa(nat,nat,minus_minus(nat,type_len0_len_of(B,type2(B))),type_len0_len_of(A,type2(A))))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B))))) ) ) ) ).

% bit_revcast_iff
tff(fact_6668_word__msb__neg__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: num] :
          ( most_s684356279273892711sb_msb(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),W)))
        <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),W))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))) ) ) ).

% word_msb_neg_numeral
tff(fact_6669_word__msb__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( most_s684356279273892711sb_msb(word(A),one_one(word(A)))
      <=> ( type_len0_len_of(A,type2(A)) = one_one(nat) ) ) ) ).

% word_msb_1
tff(fact_6670_word__msb__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: num] :
          ( most_s684356279273892711sb_msb(word(A),aa(num,word(A),numeral_numeral(word(A)),W))
        <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(num,int,numeral_numeral(int),W)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))) ) ) ).

% word_msb_numeral
tff(fact_6671_word__msb__n1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => most_s684356279273892711sb_msb(word(A),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) ) ).

% word_msb_n1
tff(fact_6672_word__msb__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ~ most_s684356279273892711sb_msb(word(A),zero_zero(word(A))) ) ).

% word_msb_0
tff(fact_6673_word__msb__sint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( most_s684356279273892711sb_msb(word(A),W)
        <=> aa(int,$o,ord_less(int,aa(word(A),int,ring_1_signed(A,int),W)),zero_zero(int)) ) ) ).

% word_msb_sint
tff(fact_6674_msb__word__iff__sless__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( most_s684356279273892711sb_msb(word(A),W)
        <=> word_sless(A,W,zero_zero(word(A))) ) ) ).

% msb_word_iff_sless_0
tff(fact_6675_word__sless__msb__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( word_sless(A,X,Y)
        <=> ( ( most_s684356279273892711sb_msb(word(A),Y)
             => most_s684356279273892711sb_msb(word(A),X) )
            & ( ( most_s684356279273892711sb_msb(word(A),X)
                & ~ most_s684356279273892711sb_msb(word(A),Y) )
              | aa(word(A),$o,ord_less(word(A),X),Y) ) ) ) ) ).

% word_sless_msb_less
tff(fact_6676_revcast__slice1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [W: word(B)] : aa(word(B),word(A),slice1(B,A,size_size(word(A),aa(word(B),word(A),revcast(B,A),W))),W) = aa(word(B),word(A),revcast(B,A),W) ) ).

% revcast_slice1
tff(fact_6677_msb__shift,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( most_s684356279273892711sb_msb(word(A),W)
        <=> ( bit_Sh4282982442137083166shiftr(word(A),W,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))) != zero_zero(word(A)) ) ) ) ).

% msb_shift
tff(fact_6678_msb__word__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( most_s684356279273892711sb_msb(word(A),W)
        <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))) ) ) ).

% msb_word_eq
tff(fact_6679_revcast__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ( revcast(A,B) = slice1(A,B,type_len0_len_of(B,type2(B))) ) ) ).

% revcast_def
tff(fact_6680_msb__word__iff__bit,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( most_s684356279273892711sb_msb(word(A),W)
        <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),aa(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_6681_word__msb__nth,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( most_s684356279273892711sb_msb(word(A),W)
        <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(word(A),int,semiring_1_unsigned(A,int),W)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))) ) ) ).

% word_msb_nth
tff(fact_6682_msb__word__of__int,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: int] :
          ( most_s684356279273892711sb_msb(word(A),aa(int,word(A),ring_1_of_int(word(A)),X))
        <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),X),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))) ) ) ).

% msb_word_of_int
tff(fact_6683_not__msb__from__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V2: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),V2),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))))
         => ~ most_s684356279273892711sb_msb(word(A),V2) ) ) ).

% not_msb_from_less
tff(fact_6684_word__sint__msb__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          aa(word(A),int,ring_1_signed(A,int),X) = aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),X)),
            $ite(most_s684356279273892711sb_msb(word(A),X),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),size_size(word(A),X)),zero_zero(int))) ) ).

% word_sint_msb_eq
tff(fact_6685_msb__big,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] :
          ( most_s684356279273892711sb_msb(word(A),A3)
        <=> aa(word(A),$o,ord_less_eq(word(A),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))),A3) ) ) ).

% msb_big
tff(fact_6686_inj__on__word__of__int,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => inj_on(int,word(A),ring_1_of_int(word(A)),set_or7035219750837199246ssThan(int,zero_zero(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ).

% inj_on_word_of_int
tff(fact_6687_bit__horner__sum__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Bs: list($o),Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),groups4207007520872428315er_sum($o,A,zero_neq_one_of_bool(A),aa(num,A,numeral_numeral(A),bit0(one2)),Bs)),Na)
        <=> ( aa(nat,$o,ord_less(nat,Na),size_size(list($o),Bs))
            & aa(nat,$o,nth($o,Bs),Na) ) ) ) ).

% bit_horner_sum_bit_iff
tff(fact_6688_inj__on__empty,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] : inj_on(A,B,F2,bot_bot(set(A))) ).

% inj_on_empty
tff(fact_6689_inj__mult__left,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A3: A] :
          ( inj_on(A,A,aa(A,fun(A,A),times_times(A),A3),top_top(set(A)))
        <=> ( A3 != zero_zero(A) ) ) ) ).

% inj_mult_left
tff(fact_6690_inj__divide__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A] :
          ( inj_on(A,A,aTP_Lamp_pq(A,fun(A,A),A3),top_top(set(A)))
        <=> ( A3 != zero_zero(A) ) ) ) ).

% inj_divide_right
tff(fact_6691_inj__on__insert,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: A,A2: set(A)] :
      ( inj_on(A,B,F2,aa(set(A),set(A),insert(A,A3),A2))
    <=> ( inj_on(A,B,F2,A2)
        & ~ member(B,aa(A,B,F2,A3),aa(set(A),set(B),image(A,B,F2),aa(set(A),set(A),minus_minus(set(A),A2),aa(set(A),set(A),insert(A,A3),bot_bot(set(A)))))) ) ) ).

% inj_on_insert
tff(fact_6692_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),size_size(list(A),L))) = aa(list(A),set(A),set2(A),L) ).

% nth_image_indices
tff(fact_6693_linorder__injI,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [F2: fun(A,B)] :
          ( ! [X4: A,Y3: A] :
              ( aa(A,$o,ord_less(A,X4),Y3)
             => ( aa(A,B,F2,X4) != aa(A,B,F2,Y3) ) )
         => inj_on(A,B,F2,top_top(set(A))) ) ) ).

% linorder_injI
tff(fact_6694_sorted__list__of__set_Oinj__on,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => inj_on(A,A,aTP_Lamp_pr(A,A),top_top(set(A))) ) ).

% sorted_list_of_set.inj_on
tff(fact_6695_inj__fun,axiom,
    ! [B: $tType,C: $tType,A: $tType,F2: fun(A,B)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => inj_on(A,fun(C,B),aTP_Lamp_ps(fun(A,B),fun(A,fun(C,B)),F2),top_top(set(A))) ) ).

% inj_fun
tff(fact_6696_inj__diff__right,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A] : inj_on(A,A,aTP_Lamp_ou(A,fun(A,A),A3),top_top(set(A))) ) ).

% inj_diff_right
tff(fact_6697_finite__inverse__image,axiom,
    ! [A: $tType,B: $tType,A2: set(A),F2: fun(B,A)] :
      ( finite_finite2(A,A2)
     => ( inj_on(B,A,F2,top_top(set(B)))
       => finite_finite2(B,collect(B,aa(fun(B,A),fun(B,$o),aTP_Lamp_pt(set(A),fun(fun(B,A),fun(B,$o)),A2),F2))) ) ) ).

% finite_inverse_image
tff(fact_6698_finite__Collect,axiom,
    ! [A: $tType,B: $tType,S: set(A),F2: fun(B,A)] :
      ( finite_finite2(A,S)
     => ( inj_on(B,A,F2,top_top(set(B)))
       => finite_finite2(B,collect(B,aa(fun(B,A),fun(B,$o),aTP_Lamp_pt(set(A),fun(fun(B,A),fun(B,$o)),S),F2))) ) ) ).

% finite_Collect
tff(fact_6699_inj__img__insertE,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A2: set(A),X: B,B2: set(B)] :
      ( inj_on(A,B,F2,A2)
     => ( ~ member(B,X,B2)
       => ( ( aa(set(B),set(B),insert(B,X),B2) = aa(set(A),set(B),image(A,B,F2),A2) )
         => ~ ! [X7: A,A6: set(A)] :
                ( ~ member(A,X7,A6)
               => ( ( A2 = aa(set(A),set(A),insert(A,X7),A6) )
                 => ( ( X = aa(A,B,F2,X7) )
                   => ( B2 != aa(set(A),set(B),image(A,B,F2),A6) ) ) ) ) ) ) ) ).

% inj_img_insertE
tff(fact_6700_inj__on__strict__subset,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),B2: set(A),A2: set(A)] :
      ( inj_on(A,B,F2,B2)
     => ( aa(set(A),$o,ord_less(set(A),A2),B2)
       => aa(set(B),$o,ord_less(set(B),aa(set(A),set(B),image(A,B,F2),A2)),aa(set(A),set(B),image(A,B,F2),B2)) ) ) ).

% inj_on_strict_subset
tff(fact_6701_sum_Oimage__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(A,B),A2: set(A)] :
          ( inj_on(A,B,G,A2)
         => ( aa(set(B),B,groups7311177749621191930dd_sum(B,B,aTP_Lamp_pu(B,B)),aa(set(A),set(B),image(A,B,G),A2)) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A2) ) ) ) ).

% sum.image_eq
tff(fact_6702_prod_Oimage__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(A,B),A2: set(A)] :
          ( inj_on(A,B,G,A2)
         => ( groups7121269368397514597t_prod(B,B,aTP_Lamp_pv(B,B),aa(set(A),set(B),image(A,B,G),A2)) = groups7121269368397514597t_prod(A,B,G,A2) ) ) ) ).

% prod.image_eq
tff(fact_6703_inj__on__Un__image__eq__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A2: set(A),B2: set(A)] :
      ( inj_on(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),B2))
     => ( ( aa(set(A),set(B),image(A,B,F2),A2) = aa(set(A),set(B),image(A,B,F2),B2) )
      <=> ( A2 = B2 ) ) ) ).

% inj_on_Un_image_eq_iff
tff(fact_6704_finite__inverse__image__gen,axiom,
    ! [A: $tType,B: $tType,A2: set(A),F2: fun(B,A),D3: set(B)] :
      ( finite_finite2(A,A2)
     => ( inj_on(B,A,F2,D3)
       => finite_finite2(B,collect(B,aa(set(B),fun(B,$o),aa(fun(B,A),fun(set(B),fun(B,$o)),aTP_Lamp_pw(set(A),fun(fun(B,A),fun(set(B),fun(B,$o))),A2),F2),D3))) ) ) ).

% finite_inverse_image_gen
tff(fact_6705_linorder__inj__onI,axiom,
    ! [B: $tType,A: $tType] :
      ( order(A)
     => ! [A2: set(A),F2: fun(A,B)] :
          ( ! [X4: A,Y3: A] :
              ( aa(A,$o,ord_less(A,X4),Y3)
             => ( member(A,X4,A2)
               => ( member(A,Y3,A2)
                 => ( aa(A,B,F2,X4) != aa(A,B,F2,Y3) ) ) ) )
         => ( ! [X4: A,Y3: A] :
                ( member(A,X4,A2)
               => ( member(A,Y3,A2)
                 => ( aa(A,$o,ord_less_eq(A,X4),Y3)
                    | aa(A,$o,ord_less_eq(A,Y3),X4) ) ) )
           => inj_on(A,B,F2,A2) ) ) ) ).

% linorder_inj_onI
tff(fact_6706_inj__on__add_H,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A3: A,A2: set(A)] : inj_on(A,A,aTP_Lamp_px(A,fun(A,A),A3),A2) ) ).

% inj_on_add'
tff(fact_6707_inj__on__id2,axiom,
    ! [A: $tType,A2: set(A)] : inj_on(A,A,aTP_Lamp_oi(A,A),A2) ).

% inj_on_id2
tff(fact_6708_inj__on__mult,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A,A2: set(A)] :
          ( ( A3 != zero_zero(A) )
         => inj_on(A,A,aa(A,fun(A,A),times_times(A),A3),A2) ) ) ).

% inj_on_mult
tff(fact_6709_obtain__list__from__elements,axiom,
    ! [A: $tType,Na: nat,P: fun(A,fun(nat,$o))] :
      ( ! [I3: nat] :
          ( aa(nat,$o,ord_less(nat,I3),Na)
         => ? [Li: A] : aa(nat,$o,aa(A,fun(nat,$o),P,Li),I3) )
     => ~ ! [L2: list(A)] :
            ( ( size_size(list(A),L2) = Na )
           => ~ ! [I2: nat] :
                  ( aa(nat,$o,ord_less(nat,I2),Na)
                 => aa(nat,$o,aa(A,fun(nat,$o),P,aa(nat,A,nth(A,L2),I2)),I2) ) ) ) ).

% obtain_list_from_elements
tff(fact_6710_nth__equalityI,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] :
      ( ( size_size(list(A),Xsa) = size_size(list(A),Ys) )
     => ( ! [I3: nat] :
            ( aa(nat,$o,ord_less(nat,I3),size_size(list(A),Xsa))
           => ( aa(nat,A,nth(A,Xsa),I3) = aa(nat,A,nth(A,Ys),I3) ) )
       => ( Xsa = Ys ) ) ) ).

% nth_equalityI
tff(fact_6711_Skolem__list__nth,axiom,
    ! [A: $tType,K: nat,P: fun(nat,fun(A,$o))] :
      ( ! [I4: nat] :
          ( aa(nat,$o,ord_less(nat,I4),K)
         => ? [X_1: A] : aa(A,$o,aa(nat,fun(A,$o),P,I4),X_1) )
    <=> ? [Xs2: list(A)] :
          ( ( size_size(list(A),Xs2) = K )
          & ! [I4: nat] :
              ( aa(nat,$o,ord_less(nat,I4),K)
             => aa(A,$o,aa(nat,fun(A,$o),P,I4),aa(nat,A,nth(A,Xs2),I4)) ) ) ) ).

% Skolem_list_nth
tff(fact_6712_list__eq__iff__nth__eq,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] :
      ( ( Xsa = Ys )
    <=> ( ( size_size(list(A),Xsa) = size_size(list(A),Ys) )
        & ! [I4: nat] :
            ( aa(nat,$o,ord_less(nat,I4),size_size(list(A),Xsa))
           => ( aa(nat,A,nth(A,Xsa),I4) = aa(nat,A,nth(A,Ys),I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_6713_inj__on__iff__surj,axiom,
    ! [B: $tType,A: $tType,A2: set(A),A7: set(B)] :
      ( ( A2 != bot_bot(set(A)) )
     => ( ? [F6: fun(A,B)] :
            ( inj_on(A,B,F6,A2)
            & aa(set(B),$o,ord_less_eq(set(B),aa(set(A),set(B),image(A,B,F6),A2)),A7) )
      <=> ? [G3: fun(B,A)] : aa(set(B),set(A),image(B,A,G3),A7) = A2 ) ) ).

% inj_on_iff_surj
tff(fact_6714_all__set__conv__nth,axiom,
    ! [A: $tType,L: list(A),P: fun(A,$o)] :
      ( ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),L))
         => aa(A,$o,P,X3) )
    <=> ! [I4: nat] :
          ( aa(nat,$o,ord_less(nat,I4),size_size(list(A),L))
         => aa(A,$o,P,aa(nat,A,nth(A,L),I4)) ) ) ).

% all_set_conv_nth
tff(fact_6715_nth__mem,axiom,
    ! [A: $tType,Na: nat,Xsa: list(A)] :
      ( aa(nat,$o,ord_less(nat,Na),size_size(list(A),Xsa))
     => member(A,aa(nat,A,nth(A,Xsa),Na),aa(list(A),set(A),set2(A),Xsa)) ) ).

% nth_mem
tff(fact_6716_list__ball__nth,axiom,
    ! [A: $tType,Na: nat,Xsa: list(A),P: fun(A,$o)] :
      ( aa(nat,$o,ord_less(nat,Na),size_size(list(A),Xsa))
     => ( ! [X4: A] :
            ( member(A,X4,aa(list(A),set(A),set2(A),Xsa))
           => aa(A,$o,P,X4) )
       => aa(A,$o,P,aa(nat,A,nth(A,Xsa),Na)) ) ) ).

% list_ball_nth
tff(fact_6717_in__set__conv__nth,axiom,
    ! [A: $tType,X: A,Xsa: list(A)] :
      ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
    <=> ? [I4: nat] :
          ( aa(nat,$o,ord_less(nat,I4),size_size(list(A),Xsa))
          & ( aa(nat,A,nth(A,Xsa),I4) = X ) ) ) ).

% in_set_conv_nth
tff(fact_6718_all__nth__imp__all__set,axiom,
    ! [A: $tType,Xsa: list(A),P: fun(A,$o),X: A] :
      ( ! [I3: nat] :
          ( aa(nat,$o,ord_less(nat,I3),size_size(list(A),Xsa))
         => aa(A,$o,P,aa(nat,A,nth(A,Xsa),I3)) )
     => ( member(A,X,aa(list(A),set(A),set2(A),Xsa))
       => aa(A,$o,P,X) ) ) ).

% all_nth_imp_all_set
tff(fact_6719_all__set__conv__all__nth,axiom,
    ! [A: $tType,Xsa: list(A),P: fun(A,$o)] :
      ( ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xsa))
         => aa(A,$o,P,X3) )
    <=> ! [I4: nat] :
          ( aa(nat,$o,ord_less(nat,I4),size_size(list(A),Xsa))
         => aa(A,$o,P,aa(nat,A,nth(A,Xsa),I4)) ) ) ).

% all_set_conv_all_nth
tff(fact_6720_VEBT__internal_Ointhall,axiom,
    ! [A: $tType,Xsa: list(A),P: fun(A,$o),Na: nat] :
      ( ! [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xsa))
         => aa(A,$o,P,X4) )
     => ( aa(nat,$o,ord_less(nat,Na),size_size(list(A),Xsa))
       => aa(A,$o,P,aa(nat,A,nth(A,Xsa),Na)) ) ) ).

% VEBT_internal.inthall
tff(fact_6721_injective__scaleR,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [C2: real] :
          ( ( C2 != zero_zero(real) )
         => inj_on(A,A,real_V8093663219630862766scaleR(A,C2),top_top(set(A))) ) ) ).

% injective_scaleR
tff(fact_6722_in__set__image__conv__nth,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),X: B,L: list(B)] :
      ( member(A,aa(B,A,F2,X),aa(set(B),set(A),image(B,A,F2),aa(list(B),set(B),set2(B),L)))
    <=> ? [I4: nat] :
          ( aa(nat,$o,ord_less(nat,I4),size_size(list(B),L))
          & ( aa(B,A,F2,aa(nat,B,nth(B,L),I4)) = aa(B,A,F2,X) ) ) ) ).

% in_set_image_conv_nth
tff(fact_6723_set__image__eq__pointwiseI,axiom,
    ! [B: $tType,A: $tType,L: list(A),L4: list(A),F2: fun(A,B)] :
      ( ( size_size(list(A),L) = size_size(list(A),L4) )
     => ( ! [I3: nat] :
            ( aa(nat,$o,ord_less(nat,I3),size_size(list(A),L))
           => ( aa(A,B,F2,aa(nat,A,nth(A,L),I3)) = aa(A,B,F2,aa(nat,A,nth(A,L4),I3)) ) )
       => ( aa(set(A),set(B),image(A,B,F2),aa(list(A),set(A),set2(A),L)) = aa(set(A),set(B),image(A,B,F2),aa(list(A),set(A),set2(A),L4)) ) ) ) ).

% set_image_eq_pointwiseI
tff(fact_6724_bit__horner__sum__bit__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bs: list($o),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),groups4207007520872428315er_sum($o,word(A),zero_neq_one_of_bool(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2)),Bs)),Na)
        <=> ( aa(nat,$o,ord_less(nat,Na),ord_min(nat,type_len0_len_of(A,type2(A)),size_size(list($o),Bs)))
            & aa(nat,$o,nth($o,Bs),Na) ) ) ) ).

% bit_horner_sum_bit_word_iff
tff(fact_6725_horner__sum__eq__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F2: fun(B,A),A3: A,Xsa: list(B)] : groups4207007520872428315er_sum(B,A,F2,A3,Xsa) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_py(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F2),A3),Xsa)),set_or7035219750837199246ssThan(nat,zero_zero(nat),size_size(list(B),Xsa))) ) ).

% horner_sum_eq_sum
tff(fact_6726_product__nth,axiom,
    ! [A: $tType,B: $tType,Na: nat,Xsa: list(A),Ys: list(B)] :
      ( aa(nat,$o,ord_less(nat,Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),size_size(list(A),Xsa)),size_size(list(B),Ys)))
     => ( aa(nat,product_prod(A,B),nth(product_prod(A,B),product(A,B,Xsa,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,Xsa),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),size_size(list(B),Ys)))),aa(nat,B,nth(B,Ys),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Na),size_size(list(B),Ys)))) ) ) ).

% product_nth
tff(fact_6727_length__product,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Ys: list(B)] : size_size(list(product_prod(A,B)),product(A,B,Xsa,Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),size_size(list(A),Xsa)),size_size(list(B),Ys)) ).

% length_product
tff(fact_6728_inj__graph,axiom,
    ! [B: $tType,A: $tType] : inj_on(fun(A,B),set(product_prod(A,B)),aTP_Lamp_qa(fun(A,B),set(product_prod(A,B))),top_top(set(fun(A,B)))) ).

% inj_graph
tff(fact_6729_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_6730_inj__signed,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & ring_char_0(B) )
     => inj_on(word(A),B,ring_1_signed(A,B),top_top(set(word(A)))) ) ).

% inj_signed
tff(fact_6731_unat__inj,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => inj_on(word(A),nat,semiring_1_unsigned(A,nat),top_top(set(word(A)))) ) ).

% unat_inj
tff(fact_6732_inj__unsigned,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & semiring_char_0(B) )
     => inj_on(word(A),B,semiring_1_unsigned(A,B),top_top(set(word(A)))) ) ).

% inj_unsigned
tff(fact_6733_signed_Osorted__list__of__set_Oinj__on,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => inj_on(word(A),word(A),aTP_Lamp_qb(word(A),word(A)),top_top(set(word(A)))) ) ).

% signed.sorted_list_of_set.inj_on
tff(fact_6734_inj__singleton,axiom,
    ! [A: $tType,A2: set(A)] : inj_on(A,set(A),aTP_Lamp_qc(A,set(A)),A2) ).

% inj_singleton
tff(fact_6735_swap__inj__on,axiom,
    ! [B: $tType,A: $tType,A2: set(product_prod(A,B))] : inj_on(product_prod(A,B),product_prod(B,A),product_case_prod(A,B,product_prod(B,A),aTP_Lamp_qd(A,fun(B,product_prod(B,A)))),A2) ).

% swap_inj_on
tff(fact_6736_inj__on__diff__nat,axiom,
    ! [N2: set(nat),K: nat] :
      ( ! [N: nat] :
          ( member(nat,N,N2)
         => aa(nat,$o,ord_less_eq(nat,K),N) )
     => inj_on(nat,nat,aTP_Lamp_pg(nat,fun(nat,nat),K),N2) ) ).

% inj_on_diff_nat
tff(fact_6737_inj__on__convol__ident,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),X5: set(A)] : inj_on(A,product_prod(A,B),aTP_Lamp_qe(fun(A,B),fun(A,product_prod(A,B)),F2),X5) ).

% inj_on_convol_ident
tff(fact_6738_inj__Pair_I1_J,axiom,
    ! [B: $tType,A: $tType,C2: fun(A,B),S: set(A)] : inj_on(A,product_prod(A,B),aTP_Lamp_qe(fun(A,B),fun(A,product_prod(A,B)),C2),S) ).

% inj_Pair(1)
tff(fact_6739_inj__Pair_I2_J,axiom,
    ! [B: $tType,A: $tType,C2: fun(A,B),S: set(A)] : inj_on(A,product_prod(B,A),aTP_Lamp_qf(fun(A,B),fun(A,product_prod(B,A)),C2),S) ).

% inj_Pair(2)
tff(fact_6740_inj__Suc,axiom,
    ! [N2: set(nat)] : inj_on(nat,nat,suc,N2) ).

% inj_Suc
tff(fact_6741_inj__on__of__nat,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N2: set(nat)] : inj_on(nat,A,semiring_1_of_nat(A),N2) ) ).

% inj_on_of_nat
tff(fact_6742_finite__imp__nat__seg__image__inj__on,axiom,
    ! [A: $tType,A2: set(A)] :
      ( finite_finite2(A,A2)
     => ? [N: nat,F4: fun(nat,A)] :
          ( ( A2 = aa(set(nat),set(A),image(nat,A,F4),collect(nat,aTP_Lamp_gc(nat,fun(nat,$o),N))) )
          & inj_on(nat,A,F4,collect(nat,aTP_Lamp_gc(nat,fun(nat,$o),N))) ) ) ).

% finite_imp_nat_seg_image_inj_on
tff(fact_6743_finite__imp__inj__to__nat__seg_H,axiom,
    ! [A: $tType,A2: set(A)] :
      ( finite_finite2(A,A2)
     => ~ ! [F4: fun(A,nat)] :
            ( ? [N: nat] : aa(set(A),set(nat),image(A,nat,F4),A2) = collect(nat,aTP_Lamp_gc(nat,fun(nat,$o),N))
           => ~ inj_on(A,nat,F4,A2) ) ) ).

% finite_imp_inj_to_nat_seg'
tff(fact_6744_finite__imp__inj__to__nat__seg,axiom,
    ! [A: $tType,A2: set(A)] :
      ( finite_finite2(A,A2)
     => ? [F4: fun(A,nat),N: nat] :
          ( ( aa(set(A),set(nat),image(A,nat,F4),A2) = collect(nat,aTP_Lamp_gc(nat,fun(nat,$o),N)) )
          & inj_on(A,nat,F4,A2) ) ) ).

% finite_imp_inj_to_nat_seg
tff(fact_6745_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,$o),bit_se5641148757651400278ts_bit(int),groups4207007520872428315er_sum(word(A),int,semiring_1_unsigned(A,int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))),Ws)),Na)
        <=> ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),type_len0_len_of(A,type2(A)))),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_6746_inj__sgn__power,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => inj_on(real,real,aTP_Lamp_po(nat,fun(real,real),Na),top_top(set(real))) ) ).

% inj_sgn_power
tff(fact_6747_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,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_6748_slice__nth,axiom,
    ! [A: $tType,From: nat,To: nat,Xsa: list(A),I: nat] :
      ( aa(nat,$o,ord_less(nat,From),To)
     => ( aa(nat,$o,ord_less_eq(nat,To),size_size(list(A),Xsa))
       => ( aa(nat,$o,ord_less(nat,I),aa(nat,nat,minus_minus(nat,To),From))
         => ( aa(nat,A,nth(A,slice(A,From,To,Xsa)),I) = aa(nat,A,nth(A,Xsa),aa(nat,nat,plus_plus(nat,From),I)) ) ) ) ) ).

% slice_nth
tff(fact_6749_If__the__inv__into__in__Func,axiom,
    ! [B: $tType,A: $tType,G: fun(A,B),C3: set(A),B2: set(A),X: A] :
      ( inj_on(A,B,G,C3)
     => ( aa(set(A),$o,ord_less_eq(set(A),C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B2),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))))
       => member(fun(B,A),aa(A,fun(B,A),aa(set(A),fun(A,fun(B,A)),aTP_Lamp_qg(fun(A,B),fun(set(A),fun(A,fun(B,A))),G),C3),X),bNF_Wellorder_Func(B,A,top_top(set(B)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B2),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))))) ) ) ).

% If_the_inv_into_in_Func
tff(fact_6750_slice__complete,axiom,
    ! [A: $tType,Xsa: list(A)] : slice(A,zero_zero(nat),size_size(list(A),Xsa),Xsa) = Xsa ).

% slice_complete
tff(fact_6751_Func__empty,axiom,
    ! [B: $tType,A: $tType,B2: set(B)] : bNF_Wellorder_Func(A,B,bot_bot(set(A)),B2) = aa(set(fun(A,B)),set(fun(A,B)),insert(fun(A,B),aTP_Lamp_qh(A,B)),bot_bot(set(fun(A,B)))) ).

% Func_empty
tff(fact_6752_horner__sum__uint__exp__Cons__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Ws: list(word(A))] : groups4207007520872428315er_sum(word(A),int,semiring_1_unsigned(A,int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))),aa(list(word(A)),list(word(A)),cons(word(A),W),Ws)) = bit_concat_bit(type_len0_len_of(A,type2(A)),aa(word(A),int,semiring_1_unsigned(A,int),W),groups4207007520872428315er_sum(word(A),int,semiring_1_unsigned(A,int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))),Ws)) ) ).

% horner_sum_uint_exp_Cons_eq
tff(fact_6753_list_Oinject,axiom,
    ! [A: $tType,X21: A,X222: list(A),Y21: A,Y222: list(A)] :
      ( ( aa(list(A),list(A),cons(A,X21),X222) = aa(list(A),list(A),cons(A,Y21),Y222) )
    <=> ( ( X21 = Y21 )
        & ( X222 = Y222 ) ) ) ).

% list.inject
tff(fact_6754_nth__Cons__Suc,axiom,
    ! [A: $tType,X: A,Xsa: list(A),Na: nat] : aa(nat,A,nth(A,aa(list(A),list(A),cons(A,X),Xsa)),aa(nat,nat,suc,Na)) = aa(nat,A,nth(A,Xsa),Na) ).

% nth_Cons_Suc
tff(fact_6755_length__nth__simps_I4_J,axiom,
    ! [A: $tType,X: A,Xsa: list(A),Na: nat] : aa(nat,A,nth(A,aa(list(A),list(A),cons(A,X),Xsa)),aa(nat,nat,suc,Na)) = aa(nat,A,nth(A,Xsa),Na) ).

% length_nth_simps(4)
tff(fact_6756_length__nth__simps_I3_J,axiom,
    ! [A: $tType,X: A,Xsa: list(A)] : aa(nat,A,nth(A,aa(list(A),list(A),cons(A,X),Xsa)),zero_zero(nat)) = X ).

% length_nth_simps(3)
tff(fact_6757_nth__Cons__0,axiom,
    ! [A: $tType,X: A,Xsa: list(A)] : aa(nat,A,nth(A,aa(list(A),list(A),cons(A,X),Xsa)),zero_zero(nat)) = X ).

% nth_Cons_0
tff(fact_6758_list_Osimps_I15_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),cons(A,X21),X222)) = aa(set(A),set(A),insert(A,X21),aa(list(A),set(A),set2(A),X222)) ).

% list.simps(15)
tff(fact_6759_nth__Cons__numeral,axiom,
    ! [A: $tType,X: A,Xsa: list(A),V2: num] : aa(nat,A,nth(A,aa(list(A),list(A),cons(A,X),Xsa)),aa(num,nat,numeral_numeral(nat),V2)) = aa(nat,A,nth(A,Xsa),aa(nat,nat,minus_minus(nat,aa(num,nat,numeral_numeral(nat),V2)),one_one(nat))) ).

% nth_Cons_numeral
tff(fact_6760_nth__Cons__pos,axiom,
    ! [A: $tType,Na: nat,X: A,Xsa: list(A)] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(nat,A,nth(A,aa(list(A),list(A),cons(A,X),Xsa)),Na) = aa(nat,A,nth(A,Xsa),aa(nat,nat,minus_minus(nat,Na),one_one(nat))) ) ) ).

% nth_Cons_pos
tff(fact_6761_inj__on__Cons1,axiom,
    ! [A: $tType,X: A,A2: set(list(A))] : inj_on(list(A),list(A),cons(A,X),A2) ).

% inj_on_Cons1
tff(fact_6762_inj__split__Cons,axiom,
    ! [A: $tType,X5: set(product_prod(list(A),A))] : inj_on(product_prod(list(A),A),list(A),product_case_prod(list(A),A,list(A),aTP_Lamp_qi(list(A),fun(A,list(A)))),X5) ).

% inj_split_Cons
tff(fact_6763_length__Suc__conv,axiom,
    ! [A: $tType,Xsa: list(A),Na: nat] :
      ( ( size_size(list(A),Xsa) = aa(nat,nat,suc,Na) )
    <=> ? [Y6: A,Ys3: list(A)] :
          ( ( Xsa = aa(list(A),list(A),cons(A,Y6),Ys3) )
          & ( size_size(list(A),Ys3) = Na ) ) ) ).

% length_Suc_conv
tff(fact_6764_Suc__length__conv,axiom,
    ! [A: $tType,Na: nat,Xsa: list(A)] :
      ( ( aa(nat,nat,suc,Na) = size_size(list(A),Xsa) )
    <=> ? [Y6: A,Ys3: list(A)] :
          ( ( Xsa = aa(list(A),list(A),cons(A,Y6),Ys3) )
          & ( size_size(list(A),Ys3) = Na ) ) ) ).

% Suc_length_conv
tff(fact_6765_length__nth__simps_I2_J,axiom,
    ! [A: $tType,X: A,Xsa: list(A)] : size_size(list(A),aa(list(A),list(A),cons(A,X),Xsa)) = aa(nat,nat,suc,size_size(list(A),Xsa)) ).

% length_nth_simps(2)
tff(fact_6766_impossible__Cons,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A),X: A] :
      ( aa(nat,$o,ord_less_eq(nat,size_size(list(A),Xsa)),size_size(list(A),Ys))
     => ( Xsa != aa(list(A),list(A),cons(A,X),Ys) ) ) ).

% impossible_Cons
tff(fact_6767_list_Oset__intros_I2_J,axiom,
    ! [A: $tType,Y: A,X222: list(A),X21: A] :
      ( member(A,Y,aa(list(A),set(A),set2(A),X222))
     => member(A,Y,aa(list(A),set(A),set2(A),aa(list(A),list(A),cons(A,X21),X222))) ) ).

% list.set_intros(2)
tff(fact_6768_list_Oset__intros_I1_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : member(A,X21,aa(list(A),set(A),set2(A),aa(list(A),list(A),cons(A,X21),X222))) ).

% list.set_intros(1)
tff(fact_6769_list_Oset__cases,axiom,
    ! [A: $tType,E2: A,A3: list(A)] :
      ( member(A,E2,aa(list(A),set(A),set2(A),A3))
     => ( ! [Z23: list(A)] : A3 != aa(list(A),list(A),cons(A,E2),Z23)
       => ~ ! [Z12: A,Z23: list(A)] :
              ( ( A3 = aa(list(A),list(A),cons(A,Z12),Z23) )
             => ~ member(A,E2,aa(list(A),set(A),set2(A),Z23)) ) ) ) ).

% list.set_cases
tff(fact_6770_set__ConsD,axiom,
    ! [A: $tType,Y: A,X: A,Xsa: list(A)] :
      ( member(A,Y,aa(list(A),set(A),set2(A),aa(list(A),list(A),cons(A,X),Xsa)))
     => ( ( Y = X )
        | member(A,Y,aa(list(A),set(A),set2(A),Xsa)) ) ) ).

% set_ConsD
tff(fact_6771_not__Cons__self2,axiom,
    ! [A: $tType,X: A,Xsa: list(A)] : aa(list(A),list(A),cons(A,X),Xsa) != Xsa ).

% not_Cons_self2
tff(fact_6772_set__subset__Cons,axiom,
    ! [A: $tType,Xsa: list(A),X: A] : aa(set(A),$o,ord_less_eq(set(A),aa(list(A),set(A),set2(A),Xsa)),aa(list(A),set(A),set2(A),aa(list(A),list(A),cons(A,X),Xsa))) ).

% set_subset_Cons
tff(fact_6773_Suc__le__length__iff,axiom,
    ! [A: $tType,Na: nat,Xsa: list(A)] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,Na)),size_size(list(A),Xsa))
    <=> ? [X3: A,Ys3: list(A)] :
          ( ( Xsa = aa(list(A),list(A),cons(A,X3),Ys3) )
          & aa(nat,$o,ord_less_eq(nat,Na),size_size(list(A),Ys3)) ) ) ).

% Suc_le_length_iff
tff(fact_6774_count__list_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Xsa: list(A),Y: A] :
      aa(A,nat,count_list(A,aa(list(A),list(A),cons(A,X),Xsa)),Y) = $ite(X = Y,aa(nat,nat,plus_plus(nat,aa(A,nat,count_list(A,Xsa),Y)),one_one(nat)),aa(A,nat,count_list(A,Xsa),Y)) ).

% count_list.simps(2)
tff(fact_6775_list_Osize_I4_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : size_size(list(A),aa(list(A),list(A),cons(A,X21),X222)) = aa(nat,nat,plus_plus(nat,size_size(list(A),X222)),aa(nat,nat,suc,zero_zero(nat))) ).

% list.size(4)
tff(fact_6776_nth__Cons_H,axiom,
    ! [A: $tType,X: A,Xsa: list(A),Na: nat] :
      aa(nat,A,nth(A,aa(list(A),list(A),cons(A,X),Xsa)),Na) = $ite(Na = zero_zero(nat),X,aa(nat,A,nth(A,Xsa),aa(nat,nat,minus_minus(nat,Na),one_one(nat)))) ).

% nth_Cons'
tff(fact_6777_nth__non__equal__first__eq,axiom,
    ! [A: $tType,X: A,Y: A,Xsa: list(A),Na: nat] :
      ( ( X != Y )
     => ( ( aa(nat,A,nth(A,aa(list(A),list(A),cons(A,X),Xsa)),Na) = Y )
      <=> ( ( aa(nat,A,nth(A,Xsa),aa(nat,nat,minus_minus(nat,Na),one_one(nat))) = Y )
          & aa(nat,$o,ord_less(nat,zero_zero(nat)),Na) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_6778_nth__equal__first__eq,axiom,
    ! [A: $tType,X: A,Xsa: list(A),Na: nat] :
      ( ~ member(A,X,aa(list(A),set(A),set2(A),Xsa))
     => ( aa(nat,$o,ord_less_eq(nat,Na),size_size(list(A),Xsa))
       => ( ( aa(nat,A,nth(A,aa(list(A),list(A),cons(A,X),Xsa)),Na) = X )
        <=> ( Na = zero_zero(nat) ) ) ) ) ).

% nth_equal_first_eq
tff(fact_6779_slice__Cons,axiom,
    ! [A: $tType,Begin: nat,End: nat,X: A,Xsa: list(A)] :
      slice(A,Begin,End,aa(list(A),list(A),cons(A,X),Xsa)) = $ite(
        ( ( Begin = zero_zero(nat) )
        & aa(nat,$o,ord_less(nat,zero_zero(nat)),End) ),
        aa(list(A),list(A),cons(A,X),slice(A,Begin,aa(nat,nat,minus_minus(nat,End),one_one(nat)),Xsa)),
        slice(A,aa(nat,nat,minus_minus(nat,Begin),one_one(nat)),aa(nat,nat,minus_minus(nat,End),one_one(nat)),Xsa) ) ).

% slice_Cons
tff(fact_6780_Func__non__emp,axiom,
    ! [A: $tType,B: $tType,B2: set(A),A2: set(B)] :
      ( ( B2 != bot_bot(set(A)) )
     => ( bNF_Wellorder_Func(B,A,A2,B2) != bot_bot(set(fun(B,A))) ) ) ).

% Func_non_emp
tff(fact_6781_Func__is__emp,axiom,
    ! [A: $tType,B: $tType,A2: set(A),B2: set(B)] :
      ( ( bNF_Wellorder_Func(A,B,A2,B2) = bot_bot(set(fun(A,B))) )
    <=> ( ( A2 != bot_bot(set(A)) )
        & ( B2 = bot_bot(set(B)) ) ) ) ).

% Func_is_emp
tff(fact_6782_hash__code__list__simps_I2_J,axiom,
    ! [A: $tType,H_a: fun(A,uint32),X: A,Xa2: list(A)] : hash_hash_code_list(A,H_a,aa(list(A),list(A),cons(A,X),Xa2)) = aa(uint32,uint32,plus_plus(uint32,aa(uint32,uint32,aa(uint32,fun(uint32,uint32),times_times(uint32),aa(A,uint32,H_a,X)),aa(num,uint32,numeral_numeral(uint32),bit0(bit0(bit1(bit1(bit0(bit1(bit0(bit0(bit0(bit0(bit0(bit1(bit0(bit1(bit0(bit1(bit0(bit0(bit1(bit0(bit0(bit1(bit0(bit1(bit0(bit1(bit1(bit1(bit0(bit0(one2))))))))))))))))))))))))))))))))),aa(uint32,uint32,plus_plus(uint32,aa(uint32,uint32,aa(uint32,fun(uint32,uint32),times_times(uint32),hash_hash_code_list(A,H_a,Xa2)),aa(num,uint32,numeral_numeral(uint32),bit1(bit0(bit1(bit1(bit1(bit0(bit0(bit1(bit0(bit1(bit1(bit0(bit1(bit1(bit0(bit0(bit1(bit1(bit1(bit0(bit0(bit0(bit0(bit1(bit0(bit0(bit0(bit1(bit0(bit0(one2))))))))))))))))))))))))))))))))),aa(num,uint32,numeral_numeral(uint32),bit0(bit1(bit1(bit1(bit0(bit0(bit0(bit0(bit1(bit0(bit1(bit0(bit0(bit0(bit1(bit1(bit1(bit0(bit0(bit1(bit0(bit1(bit1(bit0(bit0(bit1(bit0(bit0(bit0(bit0(one2))))))))))))))))))))))))))))))))) ).

% hash_code_list_simps(2)
tff(fact_6783_Func__map__surj,axiom,
    ! [C: $tType,A: $tType,D: $tType,B: $tType,F1: fun(B,A),A12: set(B),B1: set(A),F22: fun(C,D),B22: set(C),A23: set(D)] :
      ( ( aa(set(B),set(A),image(B,A,F1),A12) = B1 )
     => ( inj_on(C,D,F22,B22)
       => ( aa(set(D),$o,ord_less_eq(set(D),aa(set(C),set(D),image(C,D,F22),B22)),A23)
         => ( ( ( B22 = bot_bot(set(C)) )
             => ( A23 = bot_bot(set(D)) ) )
           => ( bNF_Wellorder_Func(C,A,B22,B1) = aa(set(fun(D,B)),set(fun(C,A)),image(fun(D,B),fun(C,A),bNF_We4925052301507509544nc_map(C,B,A,D,B22,F1,F22)),bNF_Wellorder_Func(D,B,A23,A12)) ) ) ) ) ) ).

% Func_map_surj
tff(fact_6784_upto__aux__rec,axiom,
    ! [I: int,J: int,Js: list(int)] :
      upto_aux(I,J,Js) = $ite(aa(int,$o,ord_less(int,J),I),Js,upto_aux(I,aa(int,int,minus_minus(int,J),one_one(int)),aa(list(int),list(int),cons(int,J),Js))) ).

% upto_aux_rec
tff(fact_6785_image__Collect__subsetI,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F2: fun(A,B),B2: set(B)] :
      ( ! [X4: A] :
          ( aa(A,$o,P,X4)
         => member(B,aa(A,B,F2,X4),B2) )
     => aa(set(B),$o,ord_less_eq(set(B),aa(set(A),set(B),image(A,B,F2),collect(A,P))),B2) ) ).

% image_Collect_subsetI
tff(fact_6786_Collect__restrict,axiom,
    ! [A: $tType,X5: set(A),P: fun(A,$o)] : aa(set(A),$o,ord_less_eq(set(A),collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ao(set(A),fun(fun(A,$o),fun(A,$o)),X5),P))),X5) ).

% Collect_restrict
tff(fact_6787_prop__restrict,axiom,
    ! [A: $tType,X: A,Z5: set(A),X5: set(A),P: fun(A,$o)] :
      ( member(A,X,Z5)
     => ( aa(set(A),$o,ord_less_eq(set(A),Z5),collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ao(set(A),fun(fun(A,$o),fun(A,$o)),X5),P)))
       => aa(A,$o,P,X) ) ) ).

% prop_restrict
tff(fact_6788_subset__emptyI,axiom,
    ! [A: $tType,A2: set(A)] :
      ( ! [X4: A] : ~ member(A,X4,A2)
     => aa(set(A),$o,ord_less_eq(set(A),A2),bot_bot(set(A))) ) ).

% subset_emptyI
tff(fact_6789_insert__subsetI,axiom,
    ! [A: $tType,X: A,A2: set(A),X5: set(A)] :
      ( member(A,X,A2)
     => ( aa(set(A),$o,ord_less_eq(set(A),X5),A2)
       => aa(set(A),$o,ord_less_eq(set(A),aa(set(A),set(A),insert(A,X),X5)),A2) ) ) ).

% insert_subsetI
tff(fact_6790_DERIV__even__real__root,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
       => ( aa(real,$o,ord_less(real,X),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,power_power(real,aa(real,real,root(Na),X)),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ).

% DERIV_even_real_root
tff(fact_6791_DERIV__real__root__generic,axiom,
    ! [Na: nat,X: real,D3: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( ( X != zero_zero(real) )
       => ( ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
           => ( aa(real,$o,ord_less(real,zero_zero(real)),X)
             => ( D3 = aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(nat,real,power_power(real,aa(real,real,root(Na),X)),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat)))))) ) ) )
         => ( ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
             => ( aa(real,$o,ord_less(real,X),zero_zero(real))
               => ( D3 = aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(nat,real,power_power(real,aa(real,real,root(Na),X)),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat))))))) ) ) )
           => ( ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
               => ( D3 = 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,power_power(real,aa(real,real,root(Na),X)),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat)))))) ) )
             => has_field_derivative(real,root(Na),D3,topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_6792_at__within__empty,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A3: A] : topolo174197925503356063within(A,A3,bot_bot(set(A))) = bot_bot(filter(A)) ) ).

% at_within_empty
tff(fact_6793_at__neq__bot,axiom,
    ! [A: $tType] :
      ( topolo8386298272705272623_space(A)
     => ! [A3: A] : topolo174197925503356063within(A,A3,top_top(set(A))) != bot_bot(filter(A)) ) ).

% at_neq_bot
tff(fact_6794_trivial__limit__at__left__real,axiom,
    ! [A: $tType] :
      ( ( dense_order(A)
        & no_bot(A)
        & topolo1944317154257567458pology(A) )
     => ! [X: A] : topolo174197925503356063within(A,X,set_ord_lessThan(A,X)) != bot_bot(filter(A)) ) ).

% trivial_limit_at_left_real
tff(fact_6795_at__discrete,axiom,
    ! [A: $tType] :
      ( topolo8865339358273720382pology(A)
     => ! [X: A,S: set(A)] : topolo174197925503356063within(A,X,S) = bot_bot(filter(A)) ) ).

% at_discrete
tff(fact_6796_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_6797_at__within__union,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [X: A,S: set(A),T3: set(A)] : topolo174197925503356063within(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T3)) = aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),sup_sup(filter(A)),topolo174197925503356063within(A,X,S)),topolo174197925503356063within(A,X,T3)) ) ).

% at_within_union
tff(fact_6798_has__real__derivative__neg__dec__right,axiom,
    ! [F2: fun(real,real),L: real,X: real,S: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,S))
     => ( aa(real,$o,ord_less(real,L),zero_zero(real))
       => ? [D6: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),D6)
            & ! [H3: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),H3)
               => ( member(real,aa(real,real,plus_plus(real,X),H3),S)
                 => ( aa(real,$o,ord_less(real,H3),D6)
                   => aa(real,$o,ord_less(real,aa(real,real,F2,aa(real,real,plus_plus(real,X),H3))),aa(real,real,F2,X)) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
tff(fact_6799_has__real__derivative__pos__inc__right,axiom,
    ! [F2: fun(real,real),L: real,X: real,S: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,S))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),L)
       => ? [D6: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),D6)
            & ! [H3: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),H3)
               => ( member(real,aa(real,real,plus_plus(real,X),H3),S)
                 => ( aa(real,$o,ord_less(real,H3),D6)
                   => aa(real,$o,ord_less(real,aa(real,real,F2,X)),aa(real,real,F2,aa(real,real,plus_plus(real,X),H3))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
tff(fact_6800_has__real__derivative__pos__inc__left,axiom,
    ! [F2: fun(real,real),L: real,X: real,S: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,S))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),L)
       => ? [D6: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),D6)
            & ! [H3: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),H3)
               => ( member(real,aa(real,real,minus_minus(real,X),H3),S)
                 => ( aa(real,$o,ord_less(real,H3),D6)
                   => aa(real,$o,ord_less(real,aa(real,real,F2,aa(real,real,minus_minus(real,X),H3))),aa(real,real,F2,X)) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
tff(fact_6801_has__real__derivative__neg__dec__left,axiom,
    ! [F2: fun(real,real),L: real,X: real,S: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,S))
     => ( aa(real,$o,ord_less(real,L),zero_zero(real))
       => ? [D6: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),D6)
            & ! [H3: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),H3)
               => ( member(real,aa(real,real,minus_minus(real,X),H3),S)
                 => ( aa(real,$o,ord_less(real,H3),D6)
                   => aa(real,$o,ord_less(real,aa(real,real,F2,X)),aa(real,real,F2,aa(real,real,minus_minus(real,X),H3))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
tff(fact_6802_field__differentiable__minus,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F7: A,F3: filter(A)] :
          ( has_field_derivative(A,F2,F7,F3)
         => has_field_derivative(A,aTP_Lamp_qj(fun(A,A),fun(A,A),F2),aa(A,A,uminus_uminus(A),F7),F3) ) ) ).

% field_differentiable_minus
tff(fact_6803_DERIV__minus,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,X: A,Sa: set(A)] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,X,Sa))
         => has_field_derivative(A,aTP_Lamp_qj(fun(A,A),fun(A,A),F2),aa(A,A,uminus_uminus(A),D3),topolo174197925503356063within(A,X,Sa)) ) ) ).

% DERIV_minus
tff(fact_6804_DERIV__sum,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [S: set(A),F2: fun(B,fun(A,B)),F7: fun(C,fun(A,B)),X: C,F3: filter(B)] :
          ( ! [N: A] :
              ( member(A,N,S)
             => has_field_derivative(B,aa(A,fun(B,B),aTP_Lamp_qk(fun(B,fun(A,B)),fun(A,fun(B,B)),F2),N),aa(A,B,aa(C,fun(A,B),F7,X),N),F3) )
         => has_field_derivative(B,aa(fun(B,fun(A,B)),fun(B,B),aTP_Lamp_ql(set(A),fun(fun(B,fun(A,B)),fun(B,B)),S),F2),aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(C,fun(A,B),F7,X)),S),F3) ) ) ).

% DERIV_sum
tff(fact_6805_DERIV__inverse_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,X: A,Sa: set(A)] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,X,Sa))
         => ( ( aa(A,A,F2,X) != zero_zero(A) )
           => has_field_derivative(A,aTP_Lamp_qm(fun(A,A),fun(A,A),F2),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,F2,X))),D3)),aa(A,A,inverse_inverse(A),aa(A,A,F2,X)))),topolo174197925503356063within(A,X,Sa)) ) ) ) ).

% DERIV_inverse'
tff(fact_6806_DERIV__const,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [K: A,F3: filter(A)] : has_field_derivative(A,aTP_Lamp_qn(A,fun(A,A),K),zero_zero(A),F3) ) ).

% DERIV_const
tff(fact_6807_has__field__derivative__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,F3: filter(A),C2: real] :
          ( has_field_derivative(A,F2,D3,F3)
         => has_field_derivative(A,aa(real,fun(A,A),aTP_Lamp_qo(fun(A,A),fun(real,fun(A,A)),F2),C2),aa(A,A,real_V8093663219630862766scaleR(A,C2),D3),F3) ) ) ).

% has_field_derivative_scaleR_right
tff(fact_6808_DERIV__ident,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: filter(A)] : has_field_derivative(A,aTP_Lamp_qp(A,A),one_one(A),F3) ) ).

% DERIV_ident
tff(fact_6809_DERIV__cmult,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,X: A,Sa: set(A),C2: A] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,X,Sa))
         => has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_qq(fun(A,A),fun(A,fun(A,A)),F2),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D3),topolo174197925503356063within(A,X,Sa)) ) ) ).

% DERIV_cmult
tff(fact_6810_DERIV__cmult__right,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,X: A,Sa: set(A),C2: A] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,X,Sa))
         => has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_qr(fun(A,A),fun(A,fun(A,A)),F2),C2),aa(A,A,aa(A,fun(A,A),times_times(A),D3),C2),topolo174197925503356063within(A,X,Sa)) ) ) ).

% DERIV_cmult_right
tff(fact_6811_has__field__derivative__cosh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Db: A,X: A,Sa: set(A)] :
          ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,X,Sa))
         => has_field_derivative(A,aTP_Lamp_qs(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,aa(A,A,G,X))),Db),topolo174197925503356063within(A,X,Sa)) ) ) ).

% has_field_derivative_cosh
tff(fact_6812_has__field__derivative__sinh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Db: A,X: A,Sa: set(A)] :
          ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,X,Sa))
         => has_field_derivative(A,aTP_Lamp_qt(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,aa(A,A,G,X))),Db),topolo174197925503356063within(A,X,Sa)) ) ) ).

% has_field_derivative_sinh
tff(fact_6813_DERIV__cdivide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,X: A,Sa: set(A),C2: A] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,X,Sa))
         => has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_qu(fun(A,A),fun(A,fun(A,A)),F2),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),D3),C2),topolo174197925503356063within(A,X,Sa)) ) ) ).

% DERIV_cdivide
tff(fact_6814_field__differentiable__diff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F7: A,F3: filter(A),G: fun(A,A),G4: A] :
          ( has_field_derivative(A,F2,F7,F3)
         => ( has_field_derivative(A,G,G4,F3)
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_qv(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,minus_minus(A,F7),G4),F3) ) ) ) ).

% field_differentiable_diff
tff(fact_6815_DERIV__diff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,X: A,Sa: set(A),G: fun(A,A),E4: A] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,X,Sa))
         => ( has_field_derivative(A,G,E4,topolo174197925503356063within(A,X,Sa))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_qv(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,minus_minus(A,D3),E4),topolo174197925503356063within(A,X,Sa)) ) ) ) ).

% DERIV_diff
tff(fact_6816_DERIV__mult,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Da: A,X: A,Sa: set(A),G: fun(A,A),Db: A] :
          ( has_field_derivative(A,F2,Da,topolo174197925503356063within(A,X,Sa))
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,X,Sa))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_qw(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),Da),aa(A,A,G,X))),aa(A,A,aa(A,fun(A,A),times_times(A),Db),aa(A,A,F2,X))),topolo174197925503356063within(A,X,Sa)) ) ) ) ).

% DERIV_mult
tff(fact_6817_DERIV__mult_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,X: A,Sa: set(A),G: fun(A,A),E4: A] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,X,Sa))
         => ( has_field_derivative(A,G,E4,topolo174197925503356063within(A,X,Sa))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_qw(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,F2,X)),E4)),aa(A,A,aa(A,fun(A,A),times_times(A),D3),aa(A,A,G,X))),topolo174197925503356063within(A,X,Sa)) ) ) ) ).

% DERIV_mult'
tff(fact_6818_field__differentiable__add,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F7: A,F3: filter(A),G: fun(A,A),G4: A] :
          ( has_field_derivative(A,F2,F7,F3)
         => ( has_field_derivative(A,G,G4,F3)
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_qx(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,plus_plus(A,F7),G4),F3) ) ) ) ).

% field_differentiable_add
tff(fact_6819_DERIV__add,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,X: A,Sa: set(A),G: fun(A,A),E4: A] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,X,Sa))
         => ( has_field_derivative(A,G,E4,topolo174197925503356063within(A,X,Sa))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_qx(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,plus_plus(A,D3),E4),topolo174197925503356063within(A,X,Sa)) ) ) ) ).

% DERIV_add
tff(fact_6820_DERIV__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,X: A,Sa: set(A),G: fun(A,A),E4: A] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,X,Sa))
         => ( has_field_derivative(A,G,E4,topolo174197925503356063within(A,X,Sa))
           => ( ( aa(A,A,G,X) != zero_zero(A) )
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_qy(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),D3),aa(A,A,G,X))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,F2,X)),E4))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,G,X)),aa(A,A,G,X))),topolo174197925503356063within(A,X,Sa)) ) ) ) ) ).

% DERIV_divide
tff(fact_6821_MVT2,axiom,
    ! [A3: real,B3: real,F2: fun(real,real),F7: fun(real,real)] :
      ( aa(real,$o,ord_less(real,A3),B3)
     => ( ! [X4: real] :
            ( aa(real,$o,ord_less_eq(real,A3),X4)
           => ( aa(real,$o,ord_less_eq(real,X4),B3)
             => has_field_derivative(real,F2,aa(real,real,F7,X4),topolo174197925503356063within(real,X4,top_top(set(real)))) ) )
       => ? [Z2: real] :
            ( aa(real,$o,ord_less(real,A3),Z2)
            & aa(real,$o,ord_less(real,Z2),B3)
            & ( aa(real,real,minus_minus(real,aa(real,real,F2,B3)),aa(real,real,F2,A3)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,B3),A3)),aa(real,real,F7,Z2)) ) ) ) ) ).

% MVT2
tff(fact_6822_DERIV__neg__imp__decreasing,axiom,
    ! [A3: real,B3: real,F2: fun(real,real)] :
      ( aa(real,$o,ord_less(real,A3),B3)
     => ( ! [X4: real] :
            ( aa(real,$o,ord_less_eq(real,A3),X4)
           => ( aa(real,$o,ord_less_eq(real,X4),B3)
             => ? [Y4: real] :
                  ( has_field_derivative(real,F2,Y4,topolo174197925503356063within(real,X4,top_top(set(real))))
                  & aa(real,$o,ord_less(real,Y4),zero_zero(real)) ) ) )
       => aa(real,$o,ord_less(real,aa(real,real,F2,B3)),aa(real,real,F2,A3)) ) ) ).

% DERIV_neg_imp_decreasing
tff(fact_6823_DERIV__pos__imp__increasing,axiom,
    ! [A3: real,B3: real,F2: fun(real,real)] :
      ( aa(real,$o,ord_less(real,A3),B3)
     => ( ! [X4: real] :
            ( aa(real,$o,ord_less_eq(real,A3),X4)
           => ( aa(real,$o,ord_less_eq(real,X4),B3)
             => ? [Y4: real] :
                  ( has_field_derivative(real,F2,Y4,topolo174197925503356063within(real,X4,top_top(set(real))))
                  & aa(real,$o,ord_less(real,zero_zero(real)),Y4) ) ) )
       => aa(real,$o,ord_less(real,aa(real,real,F2,A3)),aa(real,real,F2,B3)) ) ) ).

% DERIV_pos_imp_increasing
tff(fact_6824_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A3: real,B3: real,F2: fun(real,real)] :
      ( aa(real,$o,ord_less_eq(real,A3),B3)
     => ( ! [X4: real] :
            ( aa(real,$o,ord_less_eq(real,A3),X4)
           => ( aa(real,$o,ord_less_eq(real,X4),B3)
             => ? [Y4: real] :
                  ( has_field_derivative(real,F2,Y4,topolo174197925503356063within(real,X4,top_top(set(real))))
                  & aa(real,$o,ord_less_eq(real,zero_zero(real)),Y4) ) ) )
       => aa(real,$o,ord_less_eq(real,aa(real,real,F2,A3)),aa(real,real,F2,B3)) ) ) ).

% DERIV_nonneg_imp_nondecreasing
tff(fact_6825_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A3: real,B3: real,F2: fun(real,real)] :
      ( aa(real,$o,ord_less_eq(real,A3),B3)
     => ( ! [X4: real] :
            ( aa(real,$o,ord_less_eq(real,A3),X4)
           => ( aa(real,$o,ord_less_eq(real,X4),B3)
             => ? [Y4: real] :
                  ( has_field_derivative(real,F2,Y4,topolo174197925503356063within(real,X4,top_top(set(real))))
                  & aa(real,$o,ord_less_eq(real,Y4),zero_zero(real)) ) ) )
       => aa(real,$o,ord_less_eq(real,aa(real,real,F2,B3)),aa(real,real,F2,A3)) ) ) ).

% DERIV_nonpos_imp_nonincreasing
tff(fact_6826_DERIV__shift,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Y: A,X: A,Z: A] :
          ( has_field_derivative(A,F2,Y,topolo174197925503356063within(A,aa(A,A,plus_plus(A,X),Z),top_top(set(A))))
        <=> has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_qz(fun(A,A),fun(A,fun(A,A)),F2),Z),Y,topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% DERIV_shift
tff(fact_6827_DERIV__fun__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),M: A,X: A] :
          ( has_field_derivative(A,G,M,topolo174197925503356063within(A,X,top_top(set(A))))
         => has_field_derivative(A,aTP_Lamp_ra(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,exp(A),aa(A,A,G,X))),M),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% DERIV_fun_exp
tff(fact_6828_DERIV__fun__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),M: A,X: A] :
          ( has_field_derivative(A,G,M,topolo174197925503356063within(A,X,top_top(set(A))))
         => has_field_derivative(A,aTP_Lamp_rb(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,aa(A,A,G,X))),M),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% DERIV_fun_sin
tff(fact_6829_DERIV__chain__s,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Sa: set(A),G: fun(A,A),G4: fun(A,A),F2: fun(A,A),F7: A,X: A] :
          ( ! [X4: A] :
              ( member(A,X4,Sa)
             => has_field_derivative(A,G,aa(A,A,G4,X4),topolo174197925503356063within(A,X4,top_top(set(A)))) )
         => ( has_field_derivative(A,F2,F7,topolo174197925503356063within(A,X,top_top(set(A))))
           => ( member(A,aa(A,A,F2,X),Sa)
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_rc(fun(A,A),fun(fun(A,A),fun(A,A)),G),F2),aa(A,A,aa(A,fun(A,A),times_times(A),F7),aa(A,A,G4,aa(A,A,F2,X))),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ) ) ).

% DERIV_chain_s
tff(fact_6830_DERIV__chain3,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [G: fun(A,A),G4: fun(A,A),F2: fun(A,A),F7: A,X: A] :
          ( ! [X4: A] : has_field_derivative(A,G,aa(A,A,G4,X4),topolo174197925503356063within(A,X4,top_top(set(A))))
         => ( has_field_derivative(A,F2,F7,topolo174197925503356063within(A,X,top_top(set(A))))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_rc(fun(A,A),fun(fun(A,A),fun(A,A)),G),F2),aa(A,A,aa(A,fun(A,A),times_times(A),F7),aa(A,A,G4,aa(A,A,F2,X))),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ) ).

% DERIV_chain3
tff(fact_6831_DERIV__chain2,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Da: A,G: fun(A,A),X: A,Db: A,Sa: set(A)] :
          ( has_field_derivative(A,F2,Da,topolo174197925503356063within(A,aa(A,A,G,X),top_top(set(A))))
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,X,Sa))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_rc(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),times_times(A),Da),Db),topolo174197925503356063within(A,X,Sa)) ) ) ) ).

% DERIV_chain2
tff(fact_6832_DERIV__chain_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,X: A,Sa: set(A),G: fun(A,A),E4: A] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,X,Sa))
         => ( has_field_derivative(A,G,E4,topolo174197925503356063within(A,aa(A,A,F2,X),top_top(set(A))))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_rd(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),times_times(A),E4),D3),topolo174197925503356063within(A,X,Sa)) ) ) ) ).

% DERIV_chain'
tff(fact_6833_DERIV__isconst__all,axiom,
    ! [F2: fun(real,real),X: real,Y: real] :
      ( ! [X4: real] : has_field_derivative(real,F2,zero_zero(real),topolo174197925503356063within(real,X4,top_top(set(real))))
     => ( aa(real,real,F2,X) = aa(real,real,F2,Y) ) ) ).

% DERIV_isconst_all
tff(fact_6834_DERIV__ln,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => has_field_derivative(real,ln_ln(real),aa(real,real,inverse_inverse(real),X),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_ln
tff(fact_6835_DERIV__mirror,axiom,
    ! [F2: fun(real,real),Y: real,X: real] :
      ( has_field_derivative(real,F2,Y,topolo174197925503356063within(real,aa(real,real,uminus_uminus(real),X),top_top(set(real))))
    <=> has_field_derivative(real,aTP_Lamp_re(fun(real,real),fun(real,real),F2),aa(real,real,uminus_uminus(real),Y),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_mirror
tff(fact_6836_DERIV__fun__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),M: A,X: A] :
          ( has_field_derivative(A,G,M,topolo174197925503356063within(A,X,top_top(set(A))))
         => has_field_derivative(A,aTP_Lamp_rf(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),sin(A,aa(A,A,G,X)))),M),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% DERIV_fun_cos
tff(fact_6837_DERIV__cos__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K: A,Xa2: A] : has_field_derivative(A,aTP_Lamp_rg(A,fun(A,A),K),aa(A,A,uminus_uminus(A),sin(A,aa(A,A,plus_plus(A,Xa2),K))),topolo174197925503356063within(A,Xa2,top_top(set(A)))) ) ).

% DERIV_cos_add
tff(fact_6838_DERIV__pos__inc__right,axiom,
    ! [F2: fun(real,real),L: real,X: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),L)
       => ? [D6: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),D6)
            & ! [H3: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),H3)
               => ( aa(real,$o,ord_less(real,H3),D6)
                 => aa(real,$o,ord_less(real,aa(real,real,F2,X)),aa(real,real,F2,aa(real,real,plus_plus(real,X),H3))) ) ) ) ) ) ).

% DERIV_pos_inc_right
tff(fact_6839_DERIV__neg__dec__right,axiom,
    ! [F2: fun(real,real),L: real,X: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( aa(real,$o,ord_less(real,L),zero_zero(real))
       => ? [D6: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),D6)
            & ! [H3: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),H3)
               => ( aa(real,$o,ord_less(real,H3),D6)
                 => aa(real,$o,ord_less(real,aa(real,real,F2,aa(real,real,plus_plus(real,X),H3))),aa(real,real,F2,X)) ) ) ) ) ) ).

% DERIV_neg_dec_right
tff(fact_6840_DERIV__local__const,axiom,
    ! [F2: fun(real,real),L: real,X: real,D2: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),D2)
       => ( ! [Y3: real] :
              ( aa(real,$o,ord_less(real,aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,X),Y3))),D2)
             => ( aa(real,real,F2,X) = aa(real,real,F2,Y3) ) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_const
tff(fact_6841_DERIV__pos__inc__left,axiom,
    ! [F2: fun(real,real),L: real,X: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),L)
       => ? [D6: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),D6)
            & ! [H3: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),H3)
               => ( aa(real,$o,ord_less(real,H3),D6)
                 => aa(real,$o,ord_less(real,aa(real,real,F2,aa(real,real,minus_minus(real,X),H3))),aa(real,real,F2,X)) ) ) ) ) ) ).

% DERIV_pos_inc_left
tff(fact_6842_DERIV__neg__dec__left,axiom,
    ! [F2: fun(real,real),L: real,X: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( aa(real,$o,ord_less(real,L),zero_zero(real))
       => ? [D6: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),D6)
            & ! [H3: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),H3)
               => ( aa(real,$o,ord_less(real,H3),D6)
                 => aa(real,$o,ord_less(real,aa(real,real,F2,X)),aa(real,real,F2,aa(real,real,minus_minus(real,X),H3))) ) ) ) ) ) ).

% DERIV_neg_dec_left
tff(fact_6843_deriv__nonneg__imp__mono,axiom,
    ! [A3: real,B3: real,G: fun(real,real),G4: fun(real,real)] :
      ( ! [X4: real] :
          ( member(real,X4,set_or1337092689740270186AtMost(real,A3,B3))
         => has_field_derivative(real,G,aa(real,real,G4,X4),topolo174197925503356063within(real,X4,top_top(set(real)))) )
     => ( ! [X4: real] :
            ( member(real,X4,set_or1337092689740270186AtMost(real,A3,B3))
           => aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,G4,X4)) )
       => ( aa(real,$o,ord_less_eq(real,A3),B3)
         => aa(real,$o,ord_less_eq(real,aa(real,real,G,A3)),aa(real,real,G,B3)) ) ) ) ).

% deriv_nonneg_imp_mono
tff(fact_6844_DERIV__at__within__shift,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Y: A,Z: A,X: A,S: set(A)] :
          ( has_field_derivative(A,F2,Y,topolo174197925503356063within(A,aa(A,A,plus_plus(A,Z),X),aa(set(A),set(A),image(A,A,plus_plus(A,Z)),S)))
        <=> has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_rh(fun(A,A),fun(A,fun(A,A)),F2),Z),Y,topolo174197925503356063within(A,X,S)) ) ) ).

% DERIV_at_within_shift
tff(fact_6845_DERIV__power__Suc,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,X: A,Sa: set(A),Na: nat] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,X,Sa))
         => has_field_derivative(A,aa(nat,fun(A,A),aTP_Lamp_ri(fun(A,A),fun(nat,fun(A,A)),F2),Na),aa(A,A,aa(A,fun(A,A),times_times(A),aa(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),D3),aa(nat,A,power_power(A,aa(A,A,F2,X)),Na))),topolo174197925503356063within(A,X,Sa)) ) ) ).

% DERIV_power_Suc
tff(fact_6846_DERIV__const__average,axiom,
    ! [A3: real,B3: real,V2: fun(real,real),K: real] :
      ( ( A3 != B3 )
     => ( ! [X4: real] : has_field_derivative(real,V2,K,topolo174197925503356063within(real,X4,top_top(set(real))))
       => ( aa(real,real,V2,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,plus_plus(real,A3),B3)),aa(num,real,numeral_numeral(real),bit0(one2)))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,plus_plus(real,aa(real,real,V2,A3)),aa(real,real,V2,B3))),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ) ).

% DERIV_const_average
tff(fact_6847_DERIV__power,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,X: A,Sa: set(A),Na: nat] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,X,Sa))
         => has_field_derivative(A,aa(nat,fun(A,A),aTP_Lamp_rj(fun(A,A),fun(nat,fun(A,A)),F2),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),D3),aa(nat,A,power_power(A,aa(A,A,F2,X)),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(A,X,Sa)) ) ) ).

% DERIV_power
tff(fact_6848_DERIV__inverse,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [X: A,Sa: set(A)] :
          ( ( X != zero_zero(A) )
         => has_field_derivative(A,inverse_inverse(A),aa(A,A,uminus_uminus(A),aa(nat,A,power_power(A,aa(A,A,inverse_inverse(A),X)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(A,X,Sa)) ) ) ).

% DERIV_inverse
tff(fact_6849_DERIV__local__min,axiom,
    ! [F2: fun(real,real),L: real,X: real,D2: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),D2)
       => ( ! [Y3: real] :
              ( aa(real,$o,ord_less(real,aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,X),Y3))),D2)
             => aa(real,$o,ord_less_eq(real,aa(real,real,F2,X)),aa(real,real,F2,Y3)) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_min
tff(fact_6850_DERIV__local__max,axiom,
    ! [F2: fun(real,real),L: real,X: real,D2: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),D2)
       => ( ! [Y3: real] :
              ( aa(real,$o,ord_less(real,aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,X),Y3))),D2)
             => aa(real,$o,ord_less_eq(real,aa(real,real,F2,Y3)),aa(real,real,F2,X)) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_max
tff(fact_6851_DERIV__ln__divide,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => has_field_derivative(real,ln_ln(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),X),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_ln_divide
tff(fact_6852_DERIV__pow,axiom,
    ! [Na: nat,X: real,Sa: set(real)] : has_field_derivative(real,aTP_Lamp_rk(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,power_power(real,X),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(real,X,Sa)) ).

% DERIV_pow
tff(fact_6853_termdiffs__strong__converges__everywhere,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),X: A] :
          ( ! [Y3: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_cv(fun(nat,A),fun(A,fun(nat,A)),C2),Y3))
         => has_field_derivative(A,aTP_Lamp_rl(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_cw(fun(nat,A),fun(A,fun(nat,A)),C2),X)),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% termdiffs_strong_converges_everywhere
tff(fact_6854_at__within__Icc__at,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A3: A,X: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),X)
         => ( aa(A,$o,ord_less(A,X),B3)
           => ( topolo174197925503356063within(A,X,set_or1337092689740270186AtMost(A,A3,B3)) = topolo174197925503356063within(A,X,top_top(set(A))) ) ) ) ) ).

% at_within_Icc_at
tff(fact_6855_DERIV__fun__pow,axiom,
    ! [G: fun(real,real),M: real,X: real,Na: nat] :
      ( has_field_derivative(real,G,M,topolo174197925503356063within(real,X,top_top(set(real))))
     => has_field_derivative(real,aa(nat,fun(real,real),aTP_Lamp_rm(fun(real,real),fun(nat,fun(real,real)),G),Na),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,power_power(real,aa(real,real,G,X)),aa(nat,nat,minus_minus(nat,Na),one_one(nat))))),M),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_fun_pow
tff(fact_6856_at__within__Icc__at__left,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( topolo174197925503356063within(A,B3,set_or1337092689740270186AtMost(A,A3,B3)) = topolo174197925503356063within(A,B3,set_ord_lessThan(A,B3)) ) ) ) ).

% at_within_Icc_at_left
tff(fact_6857_DERIV__quotient,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D2: A,X: A,Sa: set(A),G: fun(A,A),E2: A] :
          ( has_field_derivative(A,F2,D2,topolo174197925503356063within(A,X,Sa))
         => ( has_field_derivative(A,G,E2,topolo174197925503356063within(A,X,Sa))
           => ( ( aa(A,A,G,X) != zero_zero(A) )
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_qy(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),D2),aa(A,A,G,X))),aa(A,A,aa(A,fun(A,A),times_times(A),E2),aa(A,A,F2,X)))),aa(nat,A,power_power(A,aa(A,A,G,X)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(A,X,Sa)) ) ) ) ) ).

% DERIV_quotient
tff(fact_6858_DERIV__inverse__fun,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D2: A,X: A,Sa: set(A)] :
          ( has_field_derivative(A,F2,D2,topolo174197925503356063within(A,X,Sa))
         => ( ( aa(A,A,F2,X) != zero_zero(A) )
           => has_field_derivative(A,aTP_Lamp_qm(fun(A,A),fun(A,A),F2),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,power_power(A,aa(A,A,F2,X)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))))),topolo174197925503356063within(A,X,Sa)) ) ) ) ).

% DERIV_inverse_fun
tff(fact_6859_termdiffs__sums__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K4: real,C2: fun(nat,A),F2: fun(A,A),F7: A,Z: A] :
          ( ! [Z2: A] :
              ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Z2)),K4)
             => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_cv(fun(nat,A),fun(A,fun(nat,A)),C2),Z2)),aa(A,A,F2,Z2)) )
         => ( has_field_derivative(A,F2,F7,topolo174197925503356063within(A,Z,top_top(set(A))))
           => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Z)),K4)
             => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_cw(fun(nat,A),fun(A,fun(nat,A)),C2),Z)),F7) ) ) ) ) ).

% termdiffs_sums_strong
tff(fact_6860_has__real__derivative__powr,axiom,
    ! [Z: real,R3: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Z)
     => has_field_derivative(real,aTP_Lamp_rn(real,fun(real,real),R3),aa(real,real,aa(real,fun(real,real),times_times(real),R3),powr(real,Z,aa(real,real,minus_minus(real,R3),one_one(real)))),topolo174197925503356063within(real,Z,top_top(set(real)))) ) ).

% has_real_derivative_powr
tff(fact_6861_termdiffs,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K4: A,X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_cv(fun(nat,A),fun(A,fun(nat,A)),C2),K4))
         => ( summable(A,aa(A,fun(nat,A),aTP_Lamp_cw(fun(nat,A),fun(A,fun(nat,A)),C2),K4))
           => ( summable(A,aa(A,fun(nat,A),aTP_Lamp_ro(fun(nat,A),fun(A,fun(nat,A)),C2),K4))
             => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,K4))
               => has_field_derivative(A,aTP_Lamp_rl(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_cw(fun(nat,A),fun(A,fun(nat,A)),C2),X)),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ) ) ) ).

% termdiffs
tff(fact_6862_termdiffs__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K4: A,X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_cv(fun(nat,A),fun(A,fun(nat,A)),C2),K4))
         => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,K4))
           => has_field_derivative(A,aTP_Lamp_rl(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_cw(fun(nat,A),fun(A,fun(nat,A)),C2),X)),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ) ).

% termdiffs_strong
tff(fact_6863_termdiffs__strong_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K4: real,C2: fun(nat,A),Z: A] :
          ( ! [Z2: A] :
              ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Z2)),K4)
             => summable(A,aa(A,fun(nat,A),aTP_Lamp_cv(fun(nat,A),fun(A,fun(nat,A)),C2),Z2)) )
         => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Z)),K4)
           => has_field_derivative(A,aTP_Lamp_rl(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_cw(fun(nat,A),fun(A,fun(nat,A)),C2),Z)),topolo174197925503356063within(A,Z,top_top(set(A)))) ) ) ) ).

% termdiffs_strong'
tff(fact_6864_DERIV__fun__powr,axiom,
    ! [G: fun(real,real),M: real,X: real,R3: real] :
      ( has_field_derivative(real,G,M,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,G,X))
       => has_field_derivative(real,aa(real,fun(real,real),aTP_Lamp_rp(fun(real,real),fun(real,fun(real,real)),G),R3),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),R3),powr(real,aa(real,real,G,X),aa(real,real,minus_minus(real,R3),aa(nat,real,semiring_1_of_nat(real),one_one(nat)))))),M),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ).

% DERIV_fun_powr
tff(fact_6865_DERIV__log,axiom,
    ! [X: real,B3: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => has_field_derivative(real,log(B3),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),B3)),X)),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_log
tff(fact_6866_DERIV__powr,axiom,
    ! [G: fun(real,real),M: real,X: real,F2: fun(real,real),R3: real] :
      ( has_field_derivative(real,G,M,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,G,X))
       => ( has_field_derivative(real,F2,R3,topolo174197925503356063within(real,X,top_top(set(real))))
         => has_field_derivative(real,aa(fun(real,real),fun(real,real),aTP_Lamp_rq(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,aa(real,real,G,X),aa(real,real,F2,X))),aa(real,real,plus_plus(real,aa(real,real,aa(real,fun(real,real),times_times(real),R3),aa(real,real,ln_ln(real),aa(real,real,G,X)))),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,F2,X))),aa(real,real,G,X)))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ).

% DERIV_powr
tff(fact_6867_DERIV__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => has_field_derivative(A,tan(A),aa(A,A,inverse_inverse(A),aa(nat,A,power_power(A,cos(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% DERIV_tan
tff(fact_6868_DERIV__real__sqrt,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),X)
     => 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,X))),aa(num,real,numeral_numeral(real),bit0(one2))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_real_sqrt
tff(fact_6869_DERIV__arctan,axiom,
    ! [X: real] : has_field_derivative(real,arctan,aa(real,real,inverse_inverse(real),aa(real,real,plus_plus(real,one_one(real)),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2))))),topolo174197925503356063within(real,X,top_top(set(real)))) ).

% DERIV_arctan
tff(fact_6870_arsinh__real__has__field__derivative,axiom,
    ! [X: real,A2: set(real)] : has_field_derivative(real,arsinh(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,sqrt,aa(real,real,plus_plus(real,aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real)))),topolo174197925503356063within(real,X,A2)) ).

% arsinh_real_has_field_derivative
tff(fact_6871_DERIV__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( sin(A,X) != zero_zero(A) )
         => has_field_derivative(A,cot(A),aa(A,A,uminus_uminus(A),aa(A,A,inverse_inverse(A),aa(nat,A,power_power(A,sin(A,X)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% DERIV_cot
tff(fact_6872_has__field__derivative__tanh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),X: A,Db: A,Sa: set(A)] :
          ( ( cosh(A,aa(A,A,G,X)) != zero_zero(A) )
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,X,Sa))
           => has_field_derivative(A,aTP_Lamp_rr(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,power_power(A,aa(A,A,tanh(A),aa(A,A,G,X))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),Db),topolo174197925503356063within(A,X,Sa)) ) ) ) ).

% has_field_derivative_tanh
tff(fact_6873_DERIV__real__sqrt__generic,axiom,
    ! [X: real,D3: real] :
      ( ( X != zero_zero(real) )
     => ( ( aa(real,$o,ord_less(real,zero_zero(real)),X)
         => ( D3 = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,X))),aa(num,real,numeral_numeral(real),bit0(one2))) ) )
       => ( ( aa(real,$o,ord_less(real,X),zero_zero(real))
           => ( D3 = 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,X)))),aa(num,real,numeral_numeral(real),bit0(one2))) ) )
         => has_field_derivative(real,sqrt,D3,topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_6874_arcosh__real__has__field__derivative,axiom,
    ! [X: real,A2: set(real)] :
      ( aa(real,$o,ord_less(real,one_one(real)),X)
     => 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,minus_minus(real,aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real)))),topolo174197925503356063within(real,X,A2)) ) ).

% arcosh_real_has_field_derivative
tff(fact_6875_artanh__real__has__field__derivative,axiom,
    ! [X: real,A2: set(real)] :
      ( aa(real,$o,ord_less(real,aa(real,real,abs_abs(real),X)),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,minus_minus(real,one_one(real)),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2))))),topolo174197925503356063within(real,X,A2)) ) ).

% artanh_real_has_field_derivative
tff(fact_6876_DERIV__real__root,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),X)
       => 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,power_power(real,aa(real,real,root(Na),X)),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ).

% DERIV_real_root
tff(fact_6877_DERIV__arccos,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,ord_less(real,X),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,minus_minus(real,one_one(real)),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ).

% DERIV_arccos
tff(fact_6878_DERIV__arcsin,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,ord_less(real,X),one_one(real))
       => has_field_derivative(real,arcsin,aa(real,real,inverse_inverse(real),aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ).

% DERIV_arcsin
tff(fact_6879_Maclaurin__all__le,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),X: real,Na: nat] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
     => ( ! [M4: nat,X4: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),X4),topolo174197925503356063within(real,X4,top_top(set(real))))
       => ? [T2: real] :
            ( aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),T2)),aa(real,real,abs_abs(real),X))
            & ( aa(real,real,F2,X) = aa(real,real,plus_plus(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aTP_Lamp_rs(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),X)),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),T2)),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,X),Na))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_6880_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),X: real,Na: nat] :
      ( ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
        & ! [M4: nat,X4: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),X4),topolo174197925503356063within(real,X4,top_top(set(real)))) )
     => ? [T2: real] :
          ( aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),T2)),aa(real,real,abs_abs(real),X))
          & ( aa(real,real,F2,X) = aa(real,real,plus_plus(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aTP_Lamp_rs(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),X)),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),T2)),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,X),Na))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_6881_DERIV__odd__real__root,axiom,
    ! [Na: nat,X: real] :
      ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => ( ( X != 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,power_power(real,aa(real,real,root(Na),X)),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ).

% DERIV_odd_real_root
tff(fact_6882_Maclaurin,axiom,
    ! [H: real,Na: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real)] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),H)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
       => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
         => ( ! [M4: nat,T2: real] :
                ( ( aa(nat,$o,ord_less(nat,M4),Na)
                  & aa(real,$o,ord_less_eq(real,zero_zero(real)),T2)
                  & aa(real,$o,ord_less_eq(real,T2),H) )
               => has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),T2),topolo174197925503356063within(real,T2,top_top(set(real)))) )
           => ? [T2: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),T2)
                & aa(real,$o,ord_less(real,T2),H)
                & ( aa(real,real,F2,H) = aa(real,real,plus_plus(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_rt(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),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),T2)),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,H),Na))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_6883_Maclaurin2,axiom,
    ! [H: real,Diff: fun(nat,fun(real,real)),F2: fun(real,real),Na: nat] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),H)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M4: nat,T2: real] :
              ( ( aa(nat,$o,ord_less(nat,M4),Na)
                & aa(real,$o,ord_less_eq(real,zero_zero(real)),T2)
                & aa(real,$o,ord_less_eq(real,T2),H) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),T2),topolo174197925503356063within(real,T2,top_top(set(real)))) )
         => ? [T2: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),T2)
              & aa(real,$o,ord_less_eq(real,T2),H)
              & ( aa(real,real,F2,H) = aa(real,real,plus_plus(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_rt(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),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),T2)),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,H),Na))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_6884_Maclaurin__minus,axiom,
    ! [H: real,Na: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real)] :
      ( aa(real,$o,ord_less(real,H),zero_zero(real))
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
       => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
         => ( ! [M4: nat,T2: real] :
                ( ( aa(nat,$o,ord_less(nat,M4),Na)
                  & aa(real,$o,ord_less_eq(real,H),T2)
                  & aa(real,$o,ord_less_eq(real,T2),zero_zero(real)) )
               => has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),T2),topolo174197925503356063within(real,T2,top_top(set(real)))) )
           => ? [T2: real] :
                ( aa(real,$o,ord_less(real,H),T2)
                & aa(real,$o,ord_less(real,T2),zero_zero(real))
                & ( aa(real,real,F2,H) = aa(real,real,plus_plus(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_rt(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),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),T2)),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,H),Na))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_6885_Maclaurin__all__lt,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),Na: nat,X: real] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
       => ( ( X != zero_zero(real) )
         => ( ! [M4: nat,X4: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),X4),topolo174197925503356063within(real,X4,top_top(set(real))))
           => ? [T2: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,abs_abs(real),T2))
                & aa(real,$o,ord_less(real,aa(real,real,abs_abs(real),T2)),aa(real,real,abs_abs(real),X))
                & ( aa(real,real,F2,X) = aa(real,real,plus_plus(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aTP_Lamp_rs(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),X)),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),T2)),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,X),Na))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_6886_Maclaurin__bi__le,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),Na: nat,X: real] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
     => ( ! [M4: nat,T2: real] :
            ( ( aa(nat,$o,ord_less(nat,M4),Na)
              & aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),T2)),aa(real,real,abs_abs(real),X)) )
           => has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),T2),topolo174197925503356063within(real,T2,top_top(set(real)))) )
       => ? [T2: real] :
            ( aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),T2)),aa(real,real,abs_abs(real),X))
            & ( aa(real,real,F2,X) = aa(real,real,plus_plus(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aTP_Lamp_rs(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),X)),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),T2)),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,X),Na))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_6887_Taylor__down,axiom,
    ! [Na: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real),A3: real,B3: real,C2: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M4: nat,T2: real] :
              ( ( aa(nat,$o,ord_less(nat,M4),Na)
                & aa(real,$o,ord_less_eq(real,A3),T2)
                & aa(real,$o,ord_less_eq(real,T2),B3) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),T2),topolo174197925503356063within(real,T2,top_top(set(real)))) )
         => ( aa(real,$o,ord_less(real,A3),C2)
           => ( aa(real,$o,ord_less_eq(real,C2),B3)
             => ? [T2: real] :
                  ( aa(real,$o,ord_less(real,A3),T2)
                  & aa(real,$o,ord_less(real,T2),C2)
                  & ( aa(real,real,F2,A3) = aa(real,real,plus_plus(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_ru(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),A3),C2)),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),T2)),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,aa(real,real,minus_minus(real,A3),C2)),Na))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_6888_Taylor__up,axiom,
    ! [Na: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real),A3: real,B3: real,C2: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M4: nat,T2: real] :
              ( ( aa(nat,$o,ord_less(nat,M4),Na)
                & aa(real,$o,ord_less_eq(real,A3),T2)
                & aa(real,$o,ord_less_eq(real,T2),B3) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),T2),topolo174197925503356063within(real,T2,top_top(set(real)))) )
         => ( aa(real,$o,ord_less_eq(real,A3),C2)
           => ( aa(real,$o,ord_less(real,C2),B3)
             => ? [T2: real] :
                  ( aa(real,$o,ord_less(real,C2),T2)
                  & aa(real,$o,ord_less(real,T2),B3)
                  & ( aa(real,real,F2,B3) = aa(real,real,plus_plus(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_ru(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),B3),C2)),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),T2)),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,aa(real,real,minus_minus(real,B3),C2)),Na))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_6889_Taylor,axiom,
    ! [Na: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real),A3: real,B3: real,C2: real,X: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M4: nat,T2: real] :
              ( ( aa(nat,$o,ord_less(nat,M4),Na)
                & aa(real,$o,ord_less_eq(real,A3),T2)
                & aa(real,$o,ord_less_eq(real,T2),B3) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),T2),topolo174197925503356063within(real,T2,top_top(set(real)))) )
         => ( aa(real,$o,ord_less_eq(real,A3),C2)
           => ( aa(real,$o,ord_less_eq(real,C2),B3)
             => ( aa(real,$o,ord_less_eq(real,A3),X)
               => ( aa(real,$o,ord_less_eq(real,X),B3)
                 => ( ( X != C2 )
                   => ? [T2: real] :
                        ( $ite(
                            aa(real,$o,ord_less(real,X),C2),
                            ( aa(real,$o,ord_less(real,X),T2)
                            & aa(real,$o,ord_less(real,T2),C2) ),
                            ( aa(real,$o,ord_less(real,C2),T2)
                            & aa(real,$o,ord_less(real,T2),X) ) )
                        & ( aa(real,real,F2,X) = aa(real,real,plus_plus(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_rv(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),C2),X)),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),T2)),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,aa(real,real,minus_minus(real,X),C2)),Na))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_6890_Maclaurin__lemma2,axiom,
    ! [Na: nat,H: real,Diff: fun(nat,fun(real,real)),K: nat,B2: real] :
      ( ! [M4: nat,T2: real] :
          ( ( aa(nat,$o,ord_less(nat,M4),Na)
            & aa(real,$o,ord_less_eq(real,zero_zero(real)),T2)
            & aa(real,$o,ord_less_eq(real,T2),H) )
         => has_field_derivative(real,aa(nat,fun(real,real),Diff,M4),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M4)),T2),topolo174197925503356063within(real,T2,top_top(set(real)))) )
     => ( ( Na = aa(nat,nat,suc,K) )
       => ! [M2: nat,T6: real] :
            ( ( aa(nat,$o,ord_less(nat,M2),Na)
              & aa(real,$o,ord_less_eq(real,zero_zero(real)),T6)
              & aa(real,$o,ord_less_eq(real,T6),H) )
           => 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_rx(nat,fun(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real)))),Na),Diff),B2),M2),aa(real,real,minus_minus(real,aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),T6)),aa(real,real,plus_plus(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_ry(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Diff),M2),T6)),set_ord_lessThan(nat,aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,M2))))),aa(real,real,aa(real,fun(real,real),times_times(real),B2),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,power_power(real,T6),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,M2)))),semiring_char_0_fact(real,aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,M2))))))),topolo174197925503356063within(real,T6,top_top(set(real)))) ) ) ) ).

% Maclaurin_lemma2
tff(fact_6891_DERIV__arctan__series,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,abs_abs(real),X)),one_one(real))
     => has_field_derivative(real,aTP_Lamp_rz(real,real),suminf(real,aTP_Lamp_sa(real,fun(nat,real),X)),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_arctan_series
tff(fact_6892_DERIV__power__series_H,axiom,
    ! [R: real,F2: fun(nat,real),X0: real] :
      ( ! [X4: real] :
          ( member(real,X4,set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),R),R))
         => summable(real,aa(real,fun(nat,real),aTP_Lamp_sb(fun(nat,real),fun(real,fun(nat,real)),F2),X4)) )
     => ( member(real,X0,set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),R),R))
       => ( aa(real,$o,ord_less(real,zero_zero(real)),R)
         => has_field_derivative(real,aTP_Lamp_sd(fun(nat,real),fun(real,real),F2),suminf(real,aa(real,fun(nat,real),aTP_Lamp_sb(fun(nat,real),fun(real,fun(nat,real)),F2),X0)),topolo174197925503356063within(real,X0,top_top(set(real)))) ) ) ) ).

% DERIV_power_series'
tff(fact_6893_has__derivative__arcsin,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),X: A,G4: fun(A,real),Sa: set(A)] :
          ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,G,X))
         => ( aa(real,$o,ord_less(real,aa(A,real,G,X)),one_one(real))
           => ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,X,Sa))
             => has_derivative(A,real,aTP_Lamp_se(fun(A,real),fun(A,real),G),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_sf(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),X),G4),topolo174197925503356063within(A,X,Sa)) ) ) ) ) ).

% has_derivative_arcsin
tff(fact_6894_greaterThanLessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( member(A,I,set_or5935395276787703475ssThan(A,L,U))
        <=> ( aa(A,$o,ord_less(A,L),I)
            & aa(A,$o,ord_less(A,I),U) ) ) ) ).

% greaterThanLessThan_iff
tff(fact_6895_greaterThanLessThan__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,K: A] :
          ( aa(A,$o,ord_less_eq(A,L),K)
         => ( set_or5935395276787703475ssThan(A,K,L) = bot_bot(set(A)) ) ) ) ).

% greaterThanLessThan_empty
tff(fact_6896_greaterThanLessThan__empty__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A] :
          ( ( set_or5935395276787703475ssThan(A,A3,B3) = bot_bot(set(A)) )
        <=> aa(A,$o,ord_less_eq(A,B3),A3) ) ) ).

% greaterThanLessThan_empty_iff
tff(fact_6897_greaterThanLessThan__empty__iff2,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A] :
          ( ( bot_bot(set(A)) = set_or5935395276787703475ssThan(A,A3,B3) )
        <=> aa(A,$o,ord_less_eq(A,B3),A3) ) ) ).

% greaterThanLessThan_empty_iff2
tff(fact_6898_infinite__Ioo__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A] :
          ( ~ finite_finite2(A,set_or5935395276787703475ssThan(A,A3,B3))
        <=> aa(A,$o,ord_less(A,A3),B3) ) ) ).

% infinite_Ioo_iff
tff(fact_6899_has__derivative__scaleR,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,real),F7: fun(A,real),X: A,Sa: set(A),G: fun(A,B),G4: fun(A,B)] :
          ( has_derivative(A,real,F2,F7,topolo174197925503356063within(A,X,Sa))
         => ( has_derivative(A,B,G,G4,topolo174197925503356063within(A,X,Sa))
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_sg(fun(A,real),fun(fun(A,B),fun(A,B)),F2),G),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,real),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_sh(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F2),F7),X),G),G4),topolo174197925503356063within(A,X,Sa)) ) ) ) ).

% has_derivative_scaleR
tff(fact_6900_has__derivative__compose,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),X: A,Sa: set(A),G: fun(B,C),G4: fun(B,C)] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,Sa))
         => ( has_derivative(B,C,G,G4,topolo174197925503356063within(B,aa(A,B,F2,X),top_top(set(B))))
           => has_derivative(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_si(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),aa(fun(B,C),fun(A,C),aTP_Lamp_si(fun(A,B),fun(fun(B,C),fun(A,C)),F7),G4),topolo174197925503356063within(A,X,Sa)) ) ) ) ).

% has_derivative_compose
tff(fact_6901_has__derivative__in__compose,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),X: A,Sa: set(A),G: fun(B,C),G4: fun(B,C)] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,Sa))
         => ( has_derivative(B,C,G,G4,topolo174197925503356063within(B,aa(A,B,F2,X),aa(set(A),set(B),image(A,B,F2),Sa)))
           => has_derivative(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_si(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),aa(fun(B,C),fun(A,C),aTP_Lamp_si(fun(A,B),fun(fun(B,C),fun(A,C)),F7),G4),topolo174197925503356063within(A,X,Sa)) ) ) ) ).

% has_derivative_in_compose
tff(fact_6902_has__derivative__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),F3: filter(A),G: fun(A,B),G4: fun(A,B)] :
          ( has_derivative(A,B,F2,F7,F3)
         => ( has_derivative(A,B,G,G4,F3)
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_sj(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),aa(fun(A,B),fun(A,B),aTP_Lamp_sj(fun(A,B),fun(fun(A,B),fun(A,B)),F7),G4),F3) ) ) ) ).

% has_derivative_add
tff(fact_6903_has__derivative__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),F3: filter(A),G: fun(A,B),G4: fun(A,B)] :
          ( has_derivative(A,B,F2,F7,F3)
         => ( has_derivative(A,B,G,G4,F3)
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_sk(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),aa(fun(A,B),fun(A,B),aTP_Lamp_sk(fun(A,B),fun(fun(A,B),fun(A,B)),F7),G4),F3) ) ) ) ).

% has_derivative_diff
tff(fact_6904_has__derivative__mult__right,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [G: fun(A,B),G4: fun(A,B),F3: filter(A),X: B] :
          ( has_derivative(A,B,G,G4,F3)
         => has_derivative(A,B,aa(B,fun(A,B),aTP_Lamp_sl(fun(A,B),fun(B,fun(A,B)),G),X),aa(B,fun(A,B),aTP_Lamp_sl(fun(A,B),fun(B,fun(A,B)),G4),X),F3) ) ) ).

% has_derivative_mult_right
tff(fact_6905_has__derivative__mult__left,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [G: fun(A,B),G4: fun(A,B),F3: filter(A),Y: B] :
          ( has_derivative(A,B,G,G4,F3)
         => has_derivative(A,B,aa(B,fun(A,B),aTP_Lamp_sm(fun(A,B),fun(B,fun(A,B)),G),Y),aa(B,fun(A,B),aTP_Lamp_sm(fun(A,B),fun(B,fun(A,B)),G4),Y),F3) ) ) ).

% has_derivative_mult_left
tff(fact_6906_has__derivative__of__real,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [G: fun(A,real),G4: fun(A,real),F3: filter(A)] :
          ( has_derivative(A,real,G,G4,F3)
         => has_derivative(A,B,aTP_Lamp_sn(fun(A,real),fun(A,B),G),aTP_Lamp_sn(fun(A,real),fun(A,B),G4),F3) ) ) ).

% has_derivative_of_real
tff(fact_6907_has__derivative__ident,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: filter(A)] : has_derivative(A,A,aTP_Lamp_so(A,A),aTP_Lamp_so(A,A),F3) ) ).

% has_derivative_ident
tff(fact_6908_has__derivative__scaleR__right,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [G: fun(A,B),G4: fun(A,B),F3: filter(A),R3: real] :
          ( has_derivative(A,B,G,G4,F3)
         => has_derivative(A,B,aa(real,fun(A,B),aTP_Lamp_sp(fun(A,B),fun(real,fun(A,B)),G),R3),aa(real,fun(A,B),aTP_Lamp_sp(fun(A,B),fun(real,fun(A,B)),G4),R3),F3) ) ) ).

% has_derivative_scaleR_right
tff(fact_6909_has__derivative__scaleR__left,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [G: fun(A,real),G4: fun(A,real),F3: filter(A),X: B] :
          ( has_derivative(A,real,G,G4,F3)
         => has_derivative(A,B,aa(B,fun(A,B),aTP_Lamp_sq(fun(A,real),fun(B,fun(A,B)),G),X),aa(B,fun(A,B),aTP_Lamp_sq(fun(A,real),fun(B,fun(A,B)),G4),X),F3) ) ) ).

% has_derivative_scaleR_left
tff(fact_6910_has__derivative__const,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [C2: B,F3: filter(A)] : has_derivative(A,B,aTP_Lamp_sr(B,fun(A,B),C2),aTP_Lamp_ss(A,B),F3) ) ).

% has_derivative_const
tff(fact_6911_has__derivative__sum,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [I5: set(A),F2: fun(A,fun(B,C)),F7: fun(A,fun(B,C)),F3: filter(B)] :
          ( ! [I3: A] :
              ( member(A,I3,I5)
             => has_derivative(B,C,aa(A,fun(B,C),F2,I3),aa(A,fun(B,C),F7,I3),F3) )
         => has_derivative(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_su(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2),aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_su(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F7),F3) ) ) ).

% has_derivative_sum
tff(fact_6912_has__derivative__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),F3: filter(A)] :
          ( has_derivative(A,B,F2,F7,F3)
         => has_derivative(A,B,aTP_Lamp_sv(fun(A,B),fun(A,B),F2),aTP_Lamp_sv(fun(A,B),fun(A,B),F7),F3) ) ) ).

% has_derivative_minus
tff(fact_6913_infinite__Ioo,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ~ finite_finite2(A,set_or5935395276787703475ssThan(A,A3,B3)) ) ) ).

% infinite_Ioo
tff(fact_6914_has__derivative__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),X: A,Sa: set(A),G: fun(A,B),G4: fun(A,B)] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,Sa))
         => ( has_derivative(A,B,G,G4,topolo174197925503356063within(A,X,Sa))
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_sw(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),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_sx(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F2),F7),X),G),G4),topolo174197925503356063within(A,X,Sa)) ) ) ) ).

% has_derivative_mult
tff(fact_6915_has__derivative__zero__unique,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [F3: fun(A,B),X: A] :
          ( has_derivative(A,B,aTP_Lamp_ss(A,B),F3,topolo174197925503356063within(A,X,top_top(set(A))))
         => ! [X2: A] : aa(A,B,F3,X2) = zero_zero(B) ) ) ).

% has_derivative_zero_unique
tff(fact_6916_has__derivative__in__compose2,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Ta: set(A),G: fun(A,B),G4: fun(A,fun(A,B)),F2: fun(C,A),Sa: set(C),X: C,F7: fun(C,A)] :
          ( ! [X4: A] :
              ( member(A,X4,Ta)
             => has_derivative(A,B,G,aa(A,fun(A,B),G4,X4),topolo174197925503356063within(A,X4,Ta)) )
         => ( aa(set(A),$o,ord_less_eq(set(A),aa(set(C),set(A),image(C,A,F2),Sa)),Ta)
           => ( member(C,X,Sa)
             => ( has_derivative(C,A,F2,F7,topolo174197925503356063within(C,X,Sa))
               => has_derivative(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_sy(fun(A,B),fun(fun(C,A),fun(C,B)),G),F2),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_sz(fun(A,fun(A,B)),fun(fun(C,A),fun(C,fun(fun(C,A),fun(C,B)))),G4),F2),X),F7),topolo174197925503356063within(C,X,Sa)) ) ) ) ) ) ).

% has_derivative_in_compose2
tff(fact_6917_has__derivative__exp,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G4: fun(A,real),X: A,Sa: set(A)] :
          ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,X,Sa))
         => has_derivative(A,real,aTP_Lamp_ta(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_tb(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G4),X),topolo174197925503356063within(A,X,Sa)) ) ) ).

% has_derivative_exp
tff(fact_6918_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A,C2: A,D2: A] :
          ( aa(set(A),$o,ord_less_eq(set(A),set_or5935395276787703475ssThan(A,A3,B3)),set_or5935395276787703475ssThan(A,C2,D2))
        <=> ( aa(A,$o,ord_less(A,A3),B3)
           => ( aa(A,$o,ord_less_eq(A,C2),A3)
              & aa(A,$o,ord_less_eq(A,B3),D2) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanLessThan
tff(fact_6919_tanh__real__bounds,axiom,
    ! [X: real] : member(real,aa(real,real,tanh(real),X),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real))) ).

% tanh_real_bounds
tff(fact_6920_has__derivative__sinh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Db: A,X: A,Sa: set(A)] :
          ( has_derivative(A,A,G,aa(A,fun(A,A),times_times(A),Db),topolo174197925503356063within(A,X,Sa))
         => has_derivative(A,A,aTP_Lamp_qt(fun(A,A),fun(A,A),G),aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,aa(A,A,G,X))),Db)),topolo174197925503356063within(A,X,Sa)) ) ) ).

% has_derivative_sinh
tff(fact_6921_has__derivative__cosh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Db: A,X: A,Sa: set(A)] :
          ( has_derivative(A,A,G,aa(A,fun(A,A),times_times(A),Db),topolo174197925503356063within(A,X,Sa))
         => has_derivative(A,A,aTP_Lamp_qs(fun(A,A),fun(A,A),G),aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,aa(A,A,G,X))),Db)),topolo174197925503356063within(A,X,Sa)) ) ) ).

% has_derivative_cosh
tff(fact_6922_has__derivative__sin,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G4: fun(A,real),X: A,Sa: set(A)] :
          ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,X,Sa))
         => has_derivative(A,real,aTP_Lamp_tc(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_td(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G4),X),topolo174197925503356063within(A,X,Sa)) ) ) ).

% has_derivative_sin
tff(fact_6923_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A,C2: A,D2: A] :
          ( aa(set(A),$o,ord_less_eq(set(A),set_or5935395276787703475ssThan(A,A3,B3)),set_or1337092689740270186AtMost(A,C2,D2))
        <=> ( aa(A,$o,ord_less(A,A3),B3)
           => ( aa(A,$o,ord_less_eq(A,C2),A3)
              & aa(A,$o,ord_less_eq(A,B3),D2) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastAtMost_iff
tff(fact_6924_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A,C2: A,D2: A] :
          ( aa(set(A),$o,ord_less_eq(set(A),set_or5935395276787703475ssThan(A,A3,B3)),set_or7035219750837199246ssThan(A,C2,D2))
        <=> ( aa(A,$o,ord_less(A,A3),B3)
           => ( aa(A,$o,ord_less_eq(A,C2),A3)
              & aa(A,$o,ord_less_eq(A,B3),D2) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastLessThan_iff
tff(fact_6925_ivl__disj__un__two_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,ord_less(A,L),M)
         => ( aa(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_6926_atLeastAtMost__diff__ends,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] : aa(set(A),set(A),minus_minus(set(A),set_or1337092689740270186AtMost(A,A3,B3)),aa(set(A),set(A),insert(A,A3),aa(set(A),set(A),insert(A,B3),bot_bot(set(A))))) = set_or5935395276787703475ssThan(A,A3,B3) ) ).

% atLeastAtMost_diff_ends
tff(fact_6927_ivl__disj__un__one_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$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_6928_has__derivative__divide_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),X: A,S: set(A),G: fun(A,B),G4: fun(A,B)] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,S))
         => ( has_derivative(A,B,G,G4,topolo174197925503356063within(A,X,S))
           => ( ( aa(A,B,G,X) != zero_zero(B) )
             => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_te(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),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_tf(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F2),F7),X),G),G4),topolo174197925503356063within(A,X,S)) ) ) ) ) ).

% has_derivative_divide'
tff(fact_6929_has__derivative__inverse,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(B,A),X: B,F7: fun(B,A),S: set(B)] :
          ( ( aa(B,A,F2,X) != zero_zero(A) )
         => ( has_derivative(B,A,F2,F7,topolo174197925503356063within(B,X,S))
           => has_derivative(B,A,aTP_Lamp_tg(fun(B,A),fun(B,A),F2),aa(fun(B,A),fun(B,A),aa(B,fun(fun(B,A),fun(B,A)),aTP_Lamp_th(fun(B,A),fun(B,fun(fun(B,A),fun(B,A))),F2),X),F7),topolo174197925503356063within(B,X,S)) ) ) ) ).

% has_derivative_inverse
tff(fact_6930_has__derivative__inverse_H,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: A,S: set(A)] :
          ( ( X != zero_zero(A) )
         => has_derivative(A,A,inverse_inverse(A),aTP_Lamp_ti(A,fun(A,A),X),topolo174197925503356063within(A,X,S)) ) ) ).

% has_derivative_inverse'
tff(fact_6931_DERIV__compose__FDERIV,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(real,real),F7: real,G: fun(A,real),X: A,G4: fun(A,real),Sa: set(A)] :
          ( has_field_derivative(real,F2,F7,topolo174197925503356063within(real,aa(A,real,G,X),top_top(set(real))))
         => ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,X,Sa))
           => has_derivative(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_tj(fun(real,real),fun(fun(A,real),fun(A,real)),F2),G),aa(fun(A,real),fun(A,real),aTP_Lamp_tk(real,fun(fun(A,real),fun(A,real)),F7),G4),topolo174197925503356063within(A,X,Sa)) ) ) ) ).

% DERIV_compose_FDERIV
tff(fact_6932_has__derivative__cos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G4: fun(A,real),X: A,Sa: set(A)] :
          ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,X,Sa))
         => has_derivative(A,real,aTP_Lamp_tl(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_tm(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G4),X),topolo174197925503356063within(A,X,Sa)) ) ) ).

% has_derivative_cos
tff(fact_6933_DERIV__isconst3,axiom,
    ! [A3: real,B3: real,X: real,Y: real,F2: fun(real,real)] :
      ( aa(real,$o,ord_less(real,A3),B3)
     => ( member(real,X,set_or5935395276787703475ssThan(real,A3,B3))
       => ( member(real,Y,set_or5935395276787703475ssThan(real,A3,B3))
         => ( ! [X4: real] :
                ( member(real,X4,set_or5935395276787703475ssThan(real,A3,B3))
               => has_field_derivative(real,F2,zero_zero(real),topolo174197925503356063within(real,X4,top_top(set(real)))) )
           => ( aa(real,real,F2,X) = aa(real,real,F2,Y) ) ) ) ) ) ).

% DERIV_isconst3
tff(fact_6934_ivl__disj__un__singleton_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$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),insert(A,L),bot_bot(set(A)))),set_or5935395276787703475ssThan(A,L,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(3)
tff(fact_6935_ivl__disj__un__two_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,ord_less_eq(A,L),M)
         => ( aa(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_6936_has__derivative__power,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),X: A,S: set(A),Na: nat] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,S))
         => has_derivative(A,B,aa(nat,fun(A,B),aTP_Lamp_tn(fun(A,B),fun(nat,fun(A,B)),F2),Na),aa(nat,fun(A,B),aa(A,fun(nat,fun(A,B)),aa(fun(A,B),fun(A,fun(nat,fun(A,B))),aTP_Lamp_to(fun(A,B),fun(fun(A,B),fun(A,fun(nat,fun(A,B)))),F2),F7),X),Na),topolo174197925503356063within(A,X,S)) ) ) ).

% has_derivative_power
tff(fact_6937_has__derivative__ln,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),X: A,G4: fun(A,real),Sa: set(A)] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),aa(A,real,G,X))
         => ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,X,Sa))
           => has_derivative(A,real,aTP_Lamp_tp(fun(A,real),fun(A,real),G),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_tq(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),X),G4),topolo174197925503356063within(A,X,Sa)) ) ) ) ).

% has_derivative_ln
tff(fact_6938_has__derivative__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),X: A,S: set(A),G: fun(A,B),G4: fun(A,B)] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,S))
         => ( has_derivative(A,B,G,G4,topolo174197925503356063within(A,X,S))
           => ( ( aa(A,B,G,X) != zero_zero(B) )
             => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tr(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),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_ts(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F2),F7),X),G),G4),topolo174197925503356063within(A,X,S)) ) ) ) ) ).

% has_derivative_divide
tff(fact_6939_has__derivative__prod,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(C) )
     => ! [I5: set(A),F2: fun(A,fun(B,C)),F7: fun(A,fun(B,C)),X: B,S: set(B)] :
          ( ! [I3: A] :
              ( member(A,I3,I5)
             => has_derivative(B,C,aa(A,fun(B,C),F2,I3),aa(A,fun(B,C),F7,I3),topolo174197925503356063within(B,X,S)) )
         => has_derivative(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_tu(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2),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_tw(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,C)))),I5),F2),F7),X),topolo174197925503356063within(B,X,S)) ) ) ).

% has_derivative_prod
tff(fact_6940_has__derivative__powr,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G4: fun(A,real),X: A,X5: set(A),F2: fun(A,real),F7: fun(A,real)] :
          ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,X,X5))
         => ( has_derivative(A,real,F2,F7,topolo174197925503356063within(A,X,X5))
           => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(A,real,G,X))
             => ( member(A,X,X5)
               => has_derivative(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_tx(fun(A,real),fun(fun(A,real),fun(A,real)),G),F2),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_ty(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))))),G),G4),X),F2),F7),topolo174197925503356063within(A,X,X5)) ) ) ) ) ) ).

% has_derivative_powr
tff(fact_6941_has__derivative__real__sqrt,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),X: A,G4: fun(A,real),Sa: set(A)] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),aa(A,real,G,X))
         => ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,X,Sa))
           => has_derivative(A,real,aTP_Lamp_tz(fun(A,real),fun(A,real),G),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_ua(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),X),G4),topolo174197925503356063within(A,X,Sa)) ) ) ) ).

% has_derivative_real_sqrt
tff(fact_6942_has__derivative__arctan,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G4: fun(A,real),X: A,Sa: set(A)] :
          ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,X,Sa))
         => has_derivative(A,real,aTP_Lamp_ub(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_uc(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G4),X),topolo174197925503356063within(A,X,Sa)) ) ) ).

% has_derivative_arctan
tff(fact_6943_has__derivative__tan,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),X: A,G4: fun(A,real),Sa: set(A)] :
          ( ( cos(real,aa(A,real,G,X)) != zero_zero(real) )
         => ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,X,Sa))
           => has_derivative(A,real,aTP_Lamp_ud(fun(A,real),fun(A,real),G),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_ue(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),X),G4),topolo174197925503356063within(A,X,Sa)) ) ) ) ).

% has_derivative_tan
tff(fact_6944_DERIV__series_H,axiom,
    ! [F2: fun(real,fun(nat,real)),F7: fun(real,fun(nat,real)),X0: real,A3: real,B3: real,L5: fun(nat,real)] :
      ( ! [N: nat] : has_field_derivative(real,aa(nat,fun(real,real),aTP_Lamp_uf(fun(real,fun(nat,real)),fun(nat,fun(real,real)),F2),N),aa(nat,real,aa(real,fun(nat,real),F7,X0),N),topolo174197925503356063within(real,X0,top_top(set(real))))
     => ( ! [X4: real] :
            ( member(real,X4,set_or5935395276787703475ssThan(real,A3,B3))
           => summable(real,aa(real,fun(nat,real),F2,X4)) )
       => ( member(real,X0,set_or5935395276787703475ssThan(real,A3,B3))
         => ( summable(real,aa(real,fun(nat,real),F7,X0))
           => ( summable(real,L5)
             => ( ! [N: nat,X4: real,Y3: real] :
                    ( member(real,X4,set_or5935395276787703475ssThan(real,A3,B3))
                   => ( member(real,Y3,set_or5935395276787703475ssThan(real,A3,B3))
                     => aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,aa(nat,real,aa(real,fun(nat,real),F2,X4),N)),aa(nat,real,aa(real,fun(nat,real),F2,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,minus_minus(real,X4),Y3)))) ) )
               => has_field_derivative(real,aTP_Lamp_ug(fun(real,fun(nat,real)),fun(real,real),F2),suminf(real,aa(real,fun(nat,real),F7,X0)),topolo174197925503356063within(real,X0,top_top(set(real)))) ) ) ) ) ) ) ).

% DERIV_series'
tff(fact_6945_has__derivative__arccos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),X: A,G4: fun(A,real),Sa: set(A)] :
          ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,G,X))
         => ( aa(real,$o,ord_less(real,aa(A,real,G,X)),one_one(real))
           => ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,X,Sa))
             => has_derivative(A,real,aTP_Lamp_uh(fun(A,real),fun(A,real),G),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_ui(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),X),G4),topolo174197925503356063within(A,X,Sa)) ) ) ) ) ).

% has_derivative_arccos
tff(fact_6946_has__derivative__floor,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A) )
     => ! [G: fun(B,real),X: B,F2: fun(real,A),G4: fun(B,real),Sa: set(B)] :
          ( topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,aa(B,real,G,X),top_top(set(real))),F2)
         => ( ~ member(A,aa(real,A,F2,aa(B,real,G,X)),ring_1_Ints(A))
           => ( has_derivative(B,real,G,G4,topolo174197925503356063within(B,X,Sa))
             => has_derivative(B,real,aa(fun(real,A),fun(B,real),aTP_Lamp_uj(fun(B,real),fun(fun(real,A),fun(B,real)),G),F2),aTP_Lamp_uk(fun(B,real),fun(B,real),G4),topolo174197925503356063within(B,X,Sa)) ) ) ) ) ).

% has_derivative_floor
tff(fact_6947_termdiffs__aux,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K4: A,X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_ro(fun(nat,A),fun(A,fun(nat,A)),C2),K4))
         => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,K4))
           => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_um(fun(nat,A),fun(A,fun(A,A)),C2),X),topolo7230453075368039082e_nhds(A,zero_zero(A)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% termdiffs_aux
tff(fact_6948_tendsto__const,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [K: B,F3: filter(A)] : filterlim(A,B,aTP_Lamp_un(B,fun(A,B),K),topolo7230453075368039082e_nhds(B,K),F3) ) ).

% tendsto_const
tff(fact_6949_tendsto__ident__at,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A3: A,Sa: set(A)] : filterlim(A,A,aTP_Lamp_uo(A,A),topolo7230453075368039082e_nhds(A,A3),topolo174197925503356063within(A,A3,Sa)) ) ).

% tendsto_ident_at
tff(fact_6950_tendsto__mult__right__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,F2: fun(B,A),L: A,F3: filter(B)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_up(A,fun(fun(B,A),fun(B,A)),C2),F2),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),L),C2)),F3)
          <=> filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F3) ) ) ) ).

% tendsto_mult_right_iff
tff(fact_6951_tendsto__mult__left__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,F2: fun(B,A),L: A,F3: filter(B)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_uq(A,fun(fun(B,A),fun(B,A)),C2),F2),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),L)),F3)
          <=> filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F3) ) ) ) ).

% tendsto_mult_left_iff
tff(fact_6952_power__tendsto__0__iff,axiom,
    ! [A: $tType,Na: nat,F2: fun(A,real),F3: filter(A)] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ur(nat,fun(fun(A,real),fun(A,real)),Na),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
      <=> filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% power_tendsto_0_iff
tff(fact_6953_continuous__ident,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [X: A,S: set(A)] : topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,S),aTP_Lamp_us(A,A)) ) ).

% continuous_ident
tff(fact_6954_has__field__derivative__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,X: A,S: set(A)] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,X,S))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_ut(fun(A,A),fun(A,fun(A,A)),F2),X),topolo7230453075368039082e_nhds(A,D3),topolo174197925503356063within(A,X,S)) ) ) ).

% has_field_derivative_iff
tff(fact_6955_has__field__derivativeD,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,X: A,S: set(A)] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,X,S))
         => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_ut(fun(A,A),fun(A,fun(A,A)),F2),X),topolo7230453075368039082e_nhds(A,D3),topolo174197925503356063within(A,X,S)) ) ) ).

% has_field_derivativeD
tff(fact_6956_real__LIM__sandwich__zero,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(A,real),A3: A,G: fun(A,real)] :
          ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,A3,top_top(set(A))))
         => ( ! [X4: A] :
                ( ( X4 != A3 )
               => aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(A,real,G,X4)) )
           => ( ! [X4: A] :
                  ( ( X4 != A3 )
                 => aa(real,$o,ord_less_eq(real,aa(A,real,G,X4)),aa(A,real,F2,X4)) )
             => filterlim(A,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ) ) ).

% real_LIM_sandwich_zero
tff(fact_6957_isCont__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [X: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,X,top_top(set(A))),F2)
        <=> filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_uu(A,fun(fun(A,B),fun(A,B)),X),F2),topolo7230453075368039082e_nhds(B,aa(A,B,F2,X)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% isCont_iff
tff(fact_6958_LIM__isCont__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),A3: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,A3)),topolo174197925503356063within(A,A3,top_top(set(A))))
        <=> filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_uv(fun(A,B),fun(A,fun(A,B)),F2),A3),topolo7230453075368039082e_nhds(B,aa(A,B,F2,A3)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% LIM_isCont_iff
tff(fact_6959_LIM__offset__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),L5: B,A3: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A3,top_top(set(A))))
         => filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_uv(fun(A,B),fun(A,fun(A,B)),F2),A3),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% LIM_offset_zero
tff(fact_6960_LIM__offset__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),A3: A,L5: B] :
          ( filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_uv(fun(A,B),fun(A,fun(A,B)),F2),A3),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ).

% LIM_offset_zero_cancel
tff(fact_6961_LIM__offset,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),L5: B,A3: A,K: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A3,top_top(set(A))))
         => filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_uw(fun(A,B),fun(A,fun(A,B)),F2),K),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,aa(A,A,minus_minus(A,A3),K),top_top(set(A)))) ) ) ).

% LIM_offset
tff(fact_6962_isCont__Pair,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [A3: A,F2: fun(A,B),G: fun(A,C)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,C,topolo174197925503356063within(A,A3,top_top(set(A))),G)
           => topolo3448309680560233919inuous(A,product_prod(B,C),topolo174197925503356063within(A,A3,top_top(set(A))),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_ux(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F2),G)) ) ) ) ).

% isCont_Pair
tff(fact_6963_filterlim__at__If,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(A,B),G2: filter(B),X: A,P: fun(A,$o),G: fun(A,B)] :
          ( filterlim(A,B,F2,G2,topolo174197925503356063within(A,X,collect(A,P)))
         => ( filterlim(A,B,G,G2,topolo174197925503356063within(A,X,collect(A,aTP_Lamp_uy(fun(A,$o),fun(A,$o),P))))
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,$o),fun(fun(A,B),fun(A,B)),aTP_Lamp_uz(fun(A,B),fun(fun(A,$o),fun(fun(A,B),fun(A,B))),F2),P),G),G2,topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ) ).

% filterlim_at_If
tff(fact_6964_isCont__o2,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & topolo4958980785337419405_space(C)
        & topological_t2_space(A) )
     => ! [A3: A,F2: fun(A,B),G: fun(B,C)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(B,C,topolo174197925503356063within(B,aa(A,B,F2,A3),top_top(set(B))),G)
           => topolo3448309680560233919inuous(A,C,topolo174197925503356063within(A,A3,top_top(set(A))),aa(fun(B,C),fun(A,C),aTP_Lamp_va(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G)) ) ) ) ).

% isCont_o2
tff(fact_6965_LIM__const__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & topolo8386298272705272623_space(A) )
     => ! [K: B,L5: B,A3: A] :
          ( filterlim(A,B,aTP_Lamp_vb(B,fun(A,B),K),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A3,top_top(set(A))))
         => ( K = L5 ) ) ) ).

% LIM_const_eq
tff(fact_6966_tendsto__compose,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(A) )
     => ! [G: fun(A,B),L: A,F2: fun(C,A),F3: filter(C)] :
          ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,aa(A,B,G,L)),topolo174197925503356063within(A,L,top_top(set(A))))
         => ( filterlim(C,A,F2,topolo7230453075368039082e_nhds(A,L),F3)
           => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_vc(fun(A,B),fun(fun(C,A),fun(C,B)),G),F2),topolo7230453075368039082e_nhds(B,aa(A,B,G,L)),F3) ) ) ) ).

% tendsto_compose
tff(fact_6967_LIM__const__not__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo8386298272705272623_space(B)
        & topological_t2_space(A) )
     => ! [K: A,L5: A,A3: B] :
          ( ( K != L5 )
         => ~ filterlim(B,A,aTP_Lamp_vd(A,fun(B,A),K),topolo7230453075368039082e_nhds(A,L5),topolo174197925503356063within(B,A3,top_top(set(B)))) ) ) ).

% LIM_const_not_eq
tff(fact_6968_isCont__tendsto__compose,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topological_t2_space(A) )
     => ! [L: A,G: fun(A,B),F2: fun(C,A),F3: filter(C)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,L,top_top(set(A))),G)
         => ( filterlim(C,A,F2,topolo7230453075368039082e_nhds(A,L),F3)
           => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ve(fun(A,B),fun(fun(C,A),fun(C,B)),G),F2),topolo7230453075368039082e_nhds(B,aa(A,B,G,L)),F3) ) ) ) ).

% isCont_tendsto_compose
tff(fact_6969_continuous__within__compose3,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B)
        & topological_t2_space(C) )
     => ! [F2: fun(C,A),X: C,G: fun(A,B),Sa: set(C)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,aa(C,A,F2,X),top_top(set(A))),G)
         => ( topolo3448309680560233919inuous(C,A,topolo174197925503356063within(C,X,Sa),F2)
           => topolo3448309680560233919inuous(C,B,topolo174197925503356063within(C,X,Sa),aa(fun(A,B),fun(C,B),aTP_Lamp_vf(fun(C,A),fun(fun(A,B),fun(C,B)),F2),G)) ) ) ) ).

% continuous_within_compose3
tff(fact_6970_isCont__norm,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A3: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,top_top(set(A))),aTP_Lamp_vg(fun(A,B),fun(A,real),F2)) ) ) ).

% isCont_norm
tff(fact_6971_isCont__of__real,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [A3: A,G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,top_top(set(A))),G)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aTP_Lamp_vh(fun(A,real),fun(A,B),G)) ) ) ).

% isCont_of_real
tff(fact_6972_isCont__scaleR,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A3: A,F2: fun(A,real),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),G)
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_vi(fun(A,real),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% isCont_scaleR
tff(fact_6973_LIM__not__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo8386298272705272623_space(B)
        & zero(A)
        & topological_t2_space(A) )
     => ! [K: A,A3: B] :
          ( ( K != zero_zero(A) )
         => ~ filterlim(B,A,aTP_Lamp_vj(A,fun(B,A),K),topolo7230453075368039082e_nhds(A,zero_zero(A)),topolo174197925503356063within(B,A3,top_top(set(B)))) ) ) ).

% LIM_not_zero
tff(fact_6974_isCont__LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [A3: A,F2: fun(A,B),G: fun(B,C),L: C] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(B,aa(A,B,F2,A3),top_top(set(B))))
           => ( ? [D5: real] :
                  ( aa(real,$o,ord_less(real,zero_zero(real)),D5)
                  & ! [X4: A] :
                      ( ( ( X4 != A3 )
                        & aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X4),A3))),D5) )
                     => ( aa(A,B,F2,X4) != aa(A,B,F2,A3) ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_vk(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ) ) ).

% isCont_LIM_compose2
tff(fact_6975_continuous__within__compose2,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & topolo4958980785337419405_space(C)
        & topological_t2_space(A) )
     => ! [X: A,Sa: set(A),F2: fun(A,B),G: fun(B,C)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,X,Sa),F2)
         => ( topolo3448309680560233919inuous(B,C,topolo174197925503356063within(B,aa(A,B,F2,X),aa(set(A),set(B),image(A,B,F2),Sa)),G)
           => topolo3448309680560233919inuous(A,C,topolo174197925503356063within(A,X,Sa),aa(fun(B,C),fun(A,C),aTP_Lamp_va(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G)) ) ) ) ).

% continuous_within_compose2
tff(fact_6976_filterlim__sup,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),F3: filter(B),F12: filter(A),F23: filter(A)] :
      ( filterlim(A,B,F2,F3,F12)
     => ( filterlim(A,B,F2,F3,F23)
       => filterlim(A,B,F2,F3,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),sup_sup(filter(A)),F12),F23)) ) ) ).

% filterlim_sup
tff(fact_6977_continuous__trivial__limit,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Net: filter(A),F2: fun(A,B)] :
          ( ( Net = bot_bot(filter(A)) )
         => topolo3448309680560233919inuous(A,B,Net,F2) ) ) ).

% continuous_trivial_limit
tff(fact_6978_continuous__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B)] : topolo3448309680560233919inuous(A,B,bot_bot(filter(A)),F2) ) ).

% continuous_bot
tff(fact_6979_nhds__neq__bot,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A3: A] : topolo7230453075368039082e_nhds(A,A3) != bot_bot(filter(A)) ) ).

% nhds_neq_bot
tff(fact_6980_tendsto__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),A3: B] : filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),bot_bot(filter(A))) ) ).

% tendsto_bot
tff(fact_6981_tendsto__unique,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(B)
     => ! [F3: filter(A),F2: fun(A,B),A3: B,B3: B] :
          ( ( F3 != bot_bot(filter(A)) )
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),F3)
           => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,B3),F3)
             => ( A3 = B3 ) ) ) ) ) ).

% tendsto_unique
tff(fact_6982_tendsto__const__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(B)
     => ! [F3: filter(A),A3: B,B3: B] :
          ( ( F3 != bot_bot(filter(A)) )
         => ( filterlim(A,B,aTP_Lamp_vl(B,fun(A,B),A3),topolo7230453075368039082e_nhds(B,B3),F3)
          <=> ( A3 = B3 ) ) ) ) ).

% tendsto_const_iff
tff(fact_6983_LIM__D,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),L5: B,A3: A,R3: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A3,top_top(set(A))))
         => ( aa(real,$o,ord_less(real,zero_zero(real)),R3)
           => ? [S2: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),S2)
                & ! [X2: A] :
                    ( ( ( X2 != A3 )
                      & aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X2),A3))),S2) )
                   => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(A,B,F2,X2)),L5))),R3) ) ) ) ) ) ).

% LIM_D
tff(fact_6984_LIM__I,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [A3: A,F2: fun(A,B),L5: B] :
          ( ! [R2: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),R2)
             => ? [S6: real] :
                  ( aa(real,$o,ord_less(real,zero_zero(real)),S6)
                  & ! [X4: A] :
                      ( ( ( X4 != A3 )
                        & aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X4),A3))),S6) )
                     => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(A,B,F2,X4)),L5))),R2) ) ) )
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ).

% LIM_I
tff(fact_6985_LIM__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),L5: B,A3: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A3,top_top(set(A))))
        <=> ! [R5: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),R5)
             => ? [S7: real] :
                  ( aa(real,$o,ord_less(real,zero_zero(real)),S7)
                  & ! [X3: A] :
                      ( ( ( X3 != A3 )
                        & aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X3),A3))),S7) )
                     => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(A,B,F2,X3)),L5))),R5) ) ) ) ) ) ).

% LIM_eq
tff(fact_6986_LIM__equal2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [R: real,A3: A,F2: fun(A,B),G: fun(A,B),L: B] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),R)
         => ( ! [X4: A] :
                ( ( X4 != A3 )
               => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X4),A3))),R)
                 => ( aa(A,B,F2,X4) = aa(A,B,G,X4) ) ) )
           => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A3,top_top(set(A))))
             => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ) ) ).

% LIM_equal2
tff(fact_6987_filterlim__mono,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),F23: filter(B),F12: filter(A),F24: filter(B),F13: filter(A)] :
      ( filterlim(A,B,F2,F23,F12)
     => ( aa(filter(B),$o,ord_less_eq(filter(B),F23),F24)
       => ( aa(filter(A),$o,ord_less_eq(filter(A),F13),F12)
         => filterlim(A,B,F2,F24,F13) ) ) ) ).

% filterlim_mono
tff(fact_6988_tendsto__min,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [X5: fun(A,B),X: B,Net: filter(A),Y7: fun(A,B),Y: B] :
          ( filterlim(A,B,X5,topolo7230453075368039082e_nhds(B,X),Net)
         => ( filterlim(A,B,Y7,topolo7230453075368039082e_nhds(B,Y),Net)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vm(fun(A,B),fun(fun(A,B),fun(A,B)),X5),Y7),topolo7230453075368039082e_nhds(B,ord_min(B,X,Y)),Net) ) ) ) ).

% tendsto_min
tff(fact_6989_continuous__min,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [F3: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( topolo3448309680560233919inuous(A,B,F3,G)
           => topolo3448309680560233919inuous(A,B,F3,aa(fun(A,B),fun(A,B),aTP_Lamp_vn(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_min
tff(fact_6990_tendsto__arcosh,axiom,
    ! [A: $tType,F2: fun(A,real),A3: real,F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A3),F3)
     => ( aa(real,$o,ord_less(real,one_one(real)),A3)
       => filterlim(A,real,aTP_Lamp_vo(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,arcosh(real),A3)),F3) ) ) ).

% tendsto_arcosh
tff(fact_6991_continuous__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_vp(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_minus
tff(fact_6992_tendsto__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [F2: fun(A,B),A3: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),F3)
         => filterlim(A,B,aTP_Lamp_vq(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,uminus_uminus(B),A3)),F3) ) ) ).

% tendsto_minus
tff(fact_6993_tendsto__minus__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [F2: fun(A,B),A3: B,F3: filter(A)] :
          ( filterlim(A,B,aTP_Lamp_vq(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,uminus_uminus(B),A3)),F3)
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),F3) ) ) ).

% tendsto_minus_cancel
tff(fact_6994_tendsto__minus__cancel__left,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [F2: fun(A,B),Y: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(B,B,uminus_uminus(B),Y)),F3)
        <=> filterlim(A,B,aTP_Lamp_vq(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,Y),F3) ) ) ).

% tendsto_minus_cancel_left
tff(fact_6995_continuous__sum,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(B)
        & topolo5987344860129210374id_add(C) )
     => ! [I5: set(A),F3: filter(B),F2: fun(A,fun(B,C))] :
          ( ! [I3: A] :
              ( member(A,I3,I5)
             => topolo3448309680560233919inuous(B,C,F3,aa(A,fun(B,C),F2,I3)) )
         => topolo3448309680560233919inuous(B,C,F3,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_vs(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2)) ) ) ).

% continuous_sum
tff(fact_6996_tendsto__sum,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [I5: set(A),F2: fun(A,fun(B,C)),A3: fun(A,C),F3: filter(B)] :
          ( ! [I3: A] :
              ( member(A,I3,I5)
             => filterlim(B,C,aa(A,fun(B,C),F2,I3),topolo7230453075368039082e_nhds(C,aa(A,C,A3,I3)),F3) )
         => filterlim(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_vu(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2),topolo7230453075368039082e_nhds(C,aa(set(A),C,groups7311177749621191930dd_sum(A,C,A3),I5)),F3) ) ) ).

% tendsto_sum
tff(fact_6997_tendsto__null__sum,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [I5: set(A),F2: fun(B,fun(A,C)),F3: filter(B)] :
          ( ! [I3: A] :
              ( member(A,I3,I5)
             => filterlim(B,C,aa(A,fun(B,C),aTP_Lamp_vv(fun(B,fun(A,C)),fun(A,fun(B,C)),F2),I3),topolo7230453075368039082e_nhds(C,zero_zero(C)),F3) )
         => filterlim(B,C,aa(fun(B,fun(A,C)),fun(B,C),aTP_Lamp_vw(set(A),fun(fun(B,fun(A,C)),fun(B,C)),I5),F2),topolo7230453075368039082e_nhds(C,zero_zero(C)),F3) ) ) ).

% tendsto_null_sum
tff(fact_6998_tendsto__null__power,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [F2: fun(A,B),F3: filter(A),Na: nat] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
           => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_vx(fun(A,B),fun(nat,fun(A,B)),F2),Na),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ) ).

% tendsto_null_power
tff(fact_6999_tendsto__log,axiom,
    ! [A: $tType,F2: fun(A,real),A3: real,F3: filter(A),G: fun(A,real),B3: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A3),F3)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B3),F3)
       => ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
         => ( ( A3 != one_one(real) )
           => ( aa(real,$o,ord_less(real,zero_zero(real)),B3)
             => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_vy(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,aa(real,real,log(A3),B3)),F3) ) ) ) ) ) ).

% tendsto_log
tff(fact_7000_tendsto__divide__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [F2: fun(A,B),F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_vz(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ).

% tendsto_divide_zero
tff(fact_7001_tendsto__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [F2: fun(A,B),A3: B,F3: filter(A),G: fun(A,B),B3: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),F3)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,B3),F3)
           => ( ( B3 != zero_zero(B) )
             => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_wa(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),divide_divide(B),A3),B3)),F3) ) ) ) ) ).

% tendsto_divide
tff(fact_7002_tendsto__mult__right__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F2: fun(A,B),F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_wb(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ).

% tendsto_mult_right_zero
tff(fact_7003_tendsto__mult__left__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F2: fun(A,B),F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_wc(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ).

% tendsto_mult_left_zero
tff(fact_7004_tendsto__mult__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F2: fun(A,B),F3: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_wd(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ) ).

% tendsto_mult_zero
tff(fact_7005_tendsto__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),A3: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),F3)
         => ( ( A3 != zero_zero(B) )
           => filterlim(A,B,aTP_Lamp_we(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,inverse_inverse(B),A3)),F3) ) ) ) ).

% tendsto_inverse
tff(fact_7006_tendsto__rabs__zero__cancel,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,aTP_Lamp_wf(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
     => filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ).

% tendsto_rabs_zero_cancel
tff(fact_7007_tendsto__rabs__zero__iff,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,aTP_Lamp_wf(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
    <=> filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ).

% tendsto_rabs_zero_iff
tff(fact_7008_tendsto__rabs__zero,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
     => filterlim(A,real,aTP_Lamp_wf(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ).

% tendsto_rabs_zero
tff(fact_7009_tendsto__tanh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),A3: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),F3)
         => ( ( cosh(B,A3) != zero_zero(B) )
           => filterlim(A,B,aTP_Lamp_wg(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,tanh(B),A3)),F3) ) ) ) ).

% tendsto_tanh
tff(fact_7010_tendsto__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(A,A),A3: A,F3: filter(A)] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,A3),F3)
         => ( ( sin(A,A3) != zero_zero(A) )
           => filterlim(A,A,aTP_Lamp_wh(fun(A,A),fun(A,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,cot(A),A3)),F3) ) ) ) ).

% tendsto_cot
tff(fact_7011_tendsto__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(A,A),A3: A,F3: filter(A)] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,A3),F3)
         => ( ( cos(A,A3) != zero_zero(A) )
           => filterlim(A,A,aTP_Lamp_wi(fun(A,A),fun(A,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,tan(A),A3)),F3) ) ) ) ).

% tendsto_tan
tff(fact_7012_tendsto__ln,axiom,
    ! [A: $tType,F2: fun(A,real),A3: real,F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A3),F3)
     => ( ( A3 != zero_zero(real) )
       => filterlim(A,real,aTP_Lamp_ng(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,ln_ln(real),A3)),F3) ) ) ).

% tendsto_ln
tff(fact_7013_tendsto__powr,axiom,
    ! [A: $tType,F2: fun(A,real),A3: real,F3: filter(A),G: fun(A,real),B3: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A3),F3)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B3),F3)
       => ( ( A3 != zero_zero(real) )
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_wj(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,powr(real,A3,B3)),F3) ) ) ) ).

% tendsto_powr
tff(fact_7014_tendsto__norm__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,real,aTP_Lamp_wk(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ).

% tendsto_norm_zero_cancel
tff(fact_7015_tendsto__norm__zero__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,real,aTP_Lamp_wk(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
        <=> filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ).

% tendsto_norm_zero_iff
tff(fact_7016_tendsto__norm__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => filterlim(A,real,aTP_Lamp_wk(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% tendsto_norm_zero
tff(fact_7017_tendsto__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( ( L != zero_zero(B) )
           => filterlim(A,B,aTP_Lamp_wl(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,sgn_sgn(B),L)),F3) ) ) ) ).

% tendsto_sgn
tff(fact_7018_tendsto__one__prod_H,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [I5: set(A),F2: fun(B,fun(A,C)),F3: filter(B)] :
          ( ! [I3: A] :
              ( member(A,I3,I5)
             => filterlim(B,C,aa(A,fun(B,C),aTP_Lamp_wm(fun(B,fun(A,C)),fun(A,fun(B,C)),F2),I3),topolo7230453075368039082e_nhds(C,one_one(C)),F3) )
         => filterlim(B,C,aa(fun(B,fun(A,C)),fun(B,C),aTP_Lamp_wn(set(A),fun(fun(B,fun(A,C)),fun(B,C)),I5),F2),topolo7230453075368039082e_nhds(C,one_one(C)),F3) ) ) ).

% tendsto_one_prod'
tff(fact_7019_continuous__prod_H,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(B)
        & topolo4987421752381908075d_mult(C) )
     => ! [I5: set(A),F3: filter(B),F2: fun(A,fun(B,C))] :
          ( ! [I3: A] :
              ( member(A,I3,I5)
             => topolo3448309680560233919inuous(B,C,F3,aa(A,fun(B,C),F2,I3)) )
         => topolo3448309680560233919inuous(B,C,F3,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_wp(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2)) ) ) ).

% continuous_prod'
tff(fact_7020_continuous__prod,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(B)
        & real_V4412858255891104859lgebra(C)
        & comm_ring_1(C) )
     => ! [S: set(A),F3: filter(B),F2: fun(A,fun(B,C))] :
          ( ! [I3: A] :
              ( member(A,I3,S)
             => topolo3448309680560233919inuous(B,C,F3,aa(A,fun(B,C),F2,I3)) )
         => topolo3448309680560233919inuous(B,C,F3,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_wr(set(A),fun(fun(A,fun(B,C)),fun(B,C)),S),F2)) ) ) ).

% continuous_prod
tff(fact_7021_tendsto__prod_H,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [I5: set(A),F2: fun(A,fun(B,C)),A3: fun(A,C),F3: filter(B)] :
          ( ! [I3: A] :
              ( member(A,I3,I5)
             => filterlim(B,C,aa(A,fun(B,C),F2,I3),topolo7230453075368039082e_nhds(C,aa(A,C,A3,I3)),F3) )
         => filterlim(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_wt(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2),topolo7230453075368039082e_nhds(C,groups7121269368397514597t_prod(A,C,A3,I5)),F3) ) ) ).

% tendsto_prod'
tff(fact_7022_tendsto__prod,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(C)
        & comm_ring_1(C) )
     => ! [S: set(A),F2: fun(A,fun(B,C)),L5: fun(A,C),F3: filter(B)] :
          ( ! [I3: A] :
              ( member(A,I3,S)
             => filterlim(B,C,aa(A,fun(B,C),F2,I3),topolo7230453075368039082e_nhds(C,aa(A,C,L5,I3)),F3) )
         => filterlim(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_wv(set(A),fun(fun(A,fun(B,C)),fun(B,C)),S),F2),topolo7230453075368039082e_nhds(C,groups7121269368397514597t_prod(A,C,L5,S)),F3) ) ) ).

% tendsto_prod
tff(fact_7023_continuous__power_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1898628316856586783d_mult(B) )
     => ! [F3: filter(A),F2: fun(A,B),G: fun(A,nat)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( topolo3448309680560233919inuous(A,nat,F3,G)
           => topolo3448309680560233919inuous(A,B,F3,aa(fun(A,nat),fun(A,B),aTP_Lamp_ww(fun(A,B),fun(fun(A,nat),fun(A,B)),F2),G)) ) ) ) ).

% continuous_power'
tff(fact_7024_tendsto__power__strong,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [F2: fun(A,B),A3: B,F3: filter(A),G: fun(A,nat),B3: nat] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),F3)
         => ( filterlim(A,nat,G,topolo7230453075368039082e_nhds(nat,B3),F3)
           => filterlim(A,B,aa(fun(A,nat),fun(A,B),aTP_Lamp_wx(fun(A,B),fun(fun(A,nat),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,aa(nat,B,power_power(B,A3),B3)),F3) ) ) ) ).

% tendsto_power_strong
tff(fact_7025_tendsto__power,axiom,
    ! [B: $tType,A: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B) )
     => ! [F2: fun(A,B),A3: B,F3: filter(A),Na: nat] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),F3)
         => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_wy(fun(A,B),fun(nat,fun(A,B)),F2),Na),topolo7230453075368039082e_nhds(B,aa(nat,B,power_power(B,A3),Na)),F3) ) ) ).

% tendsto_power
tff(fact_7026_continuous__power,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [F3: filter(A),F2: fun(A,B),Na: nat] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => topolo3448309680560233919inuous(A,B,F3,aa(nat,fun(A,B),aTP_Lamp_wz(fun(A,B),fun(nat,fun(A,B)),F2),Na)) ) ) ).

% continuous_power
tff(fact_7027_tendsto__real__root,axiom,
    ! [A: $tType,F2: fun(A,real),X: real,F3: filter(A),Na: nat] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,X),F3)
     => filterlim(A,real,aa(nat,fun(A,real),aTP_Lamp_xa(fun(A,real),fun(nat,fun(A,real)),F2),Na),topolo7230453075368039082e_nhds(real,aa(real,real,root(Na),X)),F3) ) ).

% tendsto_real_root
tff(fact_7028_continuous__sinh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_xb(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_sinh
tff(fact_7029_continuous__cosh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_xc(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_cosh
tff(fact_7030_tendsto__sinh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),A3: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),F3)
         => filterlim(A,B,aTP_Lamp_xd(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,sinh(B,A3)),F3) ) ) ).

% tendsto_sinh
tff(fact_7031_tendsto__cosh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),A3: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),F3)
         => filterlim(A,B,aTP_Lamp_xe(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,cosh(B,A3)),F3) ) ) ).

% tendsto_cosh
tff(fact_7032_tendsto__scaleR,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,real),A3: real,F3: filter(A),G: fun(A,B),B3: B] :
          ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A3),F3)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,B3),F3)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_xf(fun(A,real),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,aa(B,B,real_V8093663219630862766scaleR(B,A3),B3)),F3) ) ) ) ).

% tendsto_scaleR
tff(fact_7033_continuous__scaleR,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: filter(A),F2: fun(A,real),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => ( topolo3448309680560233919inuous(A,B,F3,G)
           => topolo3448309680560233919inuous(A,B,F3,aa(fun(A,B),fun(A,B),aTP_Lamp_vi(fun(A,real),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_scaleR
tff(fact_7034_continuous__exp,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_xg(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_exp
tff(fact_7035_tendsto__exp,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),A3: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),F3)
         => filterlim(A,B,aTP_Lamp_xh(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,exp(B),A3)),F3) ) ) ).

% tendsto_exp
tff(fact_7036_continuous__cos,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_xi(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_cos
tff(fact_7037_tendsto__real__sqrt,axiom,
    ! [A: $tType,F2: fun(A,real),X: real,F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,X),F3)
     => filterlim(A,real,aTP_Lamp_xj(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,sqrt,X)),F3) ) ).

% tendsto_real_sqrt
tff(fact_7038_filterlim__compose,axiom,
    ! [B: $tType,A: $tType,C: $tType,G: fun(A,B),F32: filter(B),F23: filter(A),F2: fun(C,A),F12: filter(C)] :
      ( filterlim(A,B,G,F32,F23)
     => ( filterlim(C,A,F2,F23,F12)
       => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_xk(fun(A,B),fun(fun(C,A),fun(C,B)),G),F2),F32,F12) ) ) ).

% filterlim_compose
tff(fact_7039_filterlim__ident,axiom,
    ! [A: $tType,F3: filter(A)] : filterlim(A,A,aTP_Lamp_oi(A,A),F3,F3) ).

% filterlim_ident
tff(fact_7040_tendsto__arctan,axiom,
    ! [A: $tType,F2: fun(A,real),X: real,F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,X),F3)
     => filterlim(A,real,aTP_Lamp_xl(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,arctan,X)),F3) ) ).

% tendsto_arctan
tff(fact_7041_tendsto__of__real,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [G: fun(A,real),A3: real,F3: filter(A)] :
          ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,A3),F3)
         => filterlim(A,B,aTP_Lamp_xm(fun(A,real),fun(A,B),G),topolo7230453075368039082e_nhds(B,aa(real,B,real_Vector_of_real(B),A3)),F3) ) ) ).

% tendsto_of_real
tff(fact_7042_continuous__of__real,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [F3: filter(A),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,G)
         => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_vh(fun(A,real),fun(A,B),G)) ) ) ).

% continuous_of_real
tff(fact_7043_tendsto__of__real__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,real),C2: real,F3: filter(A)] :
          ( filterlim(A,B,aTP_Lamp_xn(fun(A,real),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(real,B,real_Vector_of_real(B),C2)),F3)
        <=> filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F3) ) ) ).

% tendsto_of_real_iff
tff(fact_7044_tendsto__sin,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),A3: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),F3)
         => filterlim(A,B,aTP_Lamp_xo(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,sin(B,A3)),F3) ) ) ).

% tendsto_sin
tff(fact_7045_tendsto__Complex,axiom,
    ! [A: $tType,F2: fun(A,real),A3: real,F3: filter(A),G: fun(A,real),B3: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A3),F3)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B3),F3)
       => filterlim(A,complex,aa(fun(A,real),fun(A,complex),aTP_Lamp_xp(fun(A,real),fun(fun(A,real),fun(A,complex)),F2),G),topolo7230453075368039082e_nhds(complex,complex2(A3,B3)),F3) ) ) ).

% tendsto_Complex
tff(fact_7046_continuous__norm,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_vg(fun(A,B),fun(A,real),F2)) ) ) ).

% continuous_norm
tff(fact_7047_tendsto__norm,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),A3: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),F3)
         => filterlim(A,real,aTP_Lamp_wk(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,real_V7770717601297561774m_norm(B,A3)),F3) ) ) ).

% tendsto_norm
tff(fact_7048_tendsto__cos,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),A3: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),F3)
         => filterlim(A,B,aTP_Lamp_xi(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,cos(B,A3)),F3) ) ) ).

% tendsto_cos
tff(fact_7049_tendsto__arsinh,axiom,
    ! [A: $tType,F2: fun(A,real),A3: real,F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A3),F3)
     => filterlim(A,real,aTP_Lamp_xq(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,arsinh(real),A3)),F3) ) ).

% tendsto_arsinh
tff(fact_7050_continuous__sin,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_xo(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_sin
tff(fact_7051_continuous__const,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topological_t2_space(A) )
     => ! [F3: filter(A),C2: B] : topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_xr(B,fun(A,B),C2)) ) ).

% continuous_const
tff(fact_7052_tendsto__max,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [X5: fun(A,B),X: B,Net: filter(A),Y7: fun(A,B),Y: B] :
          ( filterlim(A,B,X5,topolo7230453075368039082e_nhds(B,X),Net)
         => ( filterlim(A,B,Y7,topolo7230453075368039082e_nhds(B,Y),Net)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_xs(fun(A,B),fun(fun(A,B),fun(A,B)),X5),Y7),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),ord_max(B),X),Y)),Net) ) ) ) ).

% tendsto_max
tff(fact_7053_continuous__max,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [F3: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( topolo3448309680560233919inuous(A,B,F3,G)
           => topolo3448309680560233919inuous(A,B,F3,aa(fun(A,B),fun(A,B),aTP_Lamp_xt(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_max
tff(fact_7054_tendsto__of__int__ceiling,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( ring_1(C)
        & topolo4958980785337419405_space(C)
        & archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( ~ member(B,L,ring_1_Ints(B))
           => filterlim(A,C,aTP_Lamp_xu(fun(A,B),fun(A,C),F2),topolo7230453075368039082e_nhds(C,aa(int,C,ring_1_of_int(C),archimedean_ceiling(B,L))),F3) ) ) ) ).

% tendsto_of_int_ceiling
tff(fact_7055_tendsto__of__int__floor,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( ring_1(C)
        & topolo4958980785337419405_space(C)
        & archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( ~ member(B,L,ring_1_Ints(B))
           => filterlim(A,C,aTP_Lamp_xv(fun(A,B),fun(A,C),F2),topolo7230453075368039082e_nhds(C,aa(int,C,ring_1_of_int(C),archim6421214686448440834_floor(B,L))),F3) ) ) ) ).

% tendsto_of_int_floor
tff(fact_7056_tendsto__rabs,axiom,
    ! [A: $tType,F2: fun(A,real),L: real,F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,L),F3)
     => filterlim(A,real,aTP_Lamp_wf(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,abs_abs(real),L)),F3) ) ).

% tendsto_rabs
tff(fact_7057_tendsto__Pair,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),A3: B,F3: filter(A),G: fun(A,C),B3: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),F3)
         => ( filterlim(A,C,G,topolo7230453075368039082e_nhds(C,B3),F3)
           => filterlim(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_xw(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F2),G),topolo7230453075368039082e_nhds(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B3)),F3) ) ) ) ).

% tendsto_Pair
tff(fact_7058_continuous__Pair,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [F3: filter(A),F2: fun(A,B),G: fun(A,C)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( topolo3448309680560233919inuous(A,C,F3,G)
           => topolo3448309680560233919inuous(A,product_prod(B,C),F3,aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_ux(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F2),G)) ) ) ) ).

% continuous_Pair
tff(fact_7059_tendsto__mult__one,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [F2: fun(A,B),F3: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,one_one(B)),F3)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,one_one(B)),F3)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_xx(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,one_one(B)),F3) ) ) ) ).

% tendsto_mult_one
tff(fact_7060_tendsto__mult__right,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_xy(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),times_times(B),L),C2)),F3) ) ) ).

% tendsto_mult_right
tff(fact_7061_tendsto__mult__left,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_xz(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),times_times(B),C2),L)),F3) ) ) ).

% tendsto_mult_left
tff(fact_7062_tendsto__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [F2: fun(A,B),A3: B,F3: filter(A),G: fun(A,B),B3: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),F3)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,B3),F3)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ya(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),times_times(B),A3),B3)),F3) ) ) ) ).

% tendsto_mult
tff(fact_7063_continuous__mult__right,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [F3: filter(A),F2: fun(A,B),C2: B] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => topolo3448309680560233919inuous(A,B,F3,aa(B,fun(A,B),aTP_Lamp_yb(fun(A,B),fun(B,fun(A,B)),F2),C2)) ) ) ).

% continuous_mult_right
tff(fact_7064_continuous__mult__left,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [F3: filter(A),F2: fun(A,B),C2: B] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => topolo3448309680560233919inuous(A,B,F3,aa(B,fun(A,B),aTP_Lamp_yc(fun(A,B),fun(B,fun(A,B)),F2),C2)) ) ) ).

% continuous_mult_left
tff(fact_7065_continuous__mult_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4211221413907600880p_mult(B) )
     => ! [F3: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( topolo3448309680560233919inuous(A,B,F3,G)
           => topolo3448309680560233919inuous(A,B,F3,aa(fun(A,B),fun(A,B),aTP_Lamp_yd(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_mult'
tff(fact_7066_continuous__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [F3: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( topolo3448309680560233919inuous(A,B,F3,G)
           => topolo3448309680560233919inuous(A,B,F3,aa(fun(A,B),fun(A,B),aTP_Lamp_ye(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_mult
tff(fact_7067_tendsto__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [F2: fun(A,B),A3: B,F3: filter(A),G: fun(A,B),B3: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),F3)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,B3),F3)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yf(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,aa(B,B,minus_minus(B,A3),B3)),F3) ) ) ) ).

% tendsto_diff
tff(fact_7068_continuous__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [F3: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( topolo3448309680560233919inuous(A,B,F3,G)
           => topolo3448309680560233919inuous(A,B,F3,aa(fun(A,B),fun(A,B),aTP_Lamp_yg(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_diff
tff(fact_7069_continuous__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [F3: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( topolo3448309680560233919inuous(A,B,F3,G)
           => topolo3448309680560233919inuous(A,B,F3,aa(fun(A,B),fun(A,B),aTP_Lamp_yh(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_add
tff(fact_7070_tendsto__add,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [F2: fun(A,B),A3: B,F3: filter(A),G: fun(A,B),B3: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),F3)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,B3),F3)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yi(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,aa(B,B,plus_plus(B,A3),B3)),F3) ) ) ) ).

% tendsto_add
tff(fact_7071_tendsto__add__const__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [C2: B,F2: fun(A,B),D2: B,F3: filter(A)] :
          ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yj(B,fun(fun(A,B),fun(A,B)),C2),F2),topolo7230453075368039082e_nhds(B,aa(B,B,plus_plus(B,C2),D2)),F3)
        <=> filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,D2),F3) ) ) ).

% tendsto_add_const_iff
tff(fact_7072_tendsto__add__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [F2: fun(A,B),F3: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yi(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ) ).

% tendsto_add_zero
tff(fact_7073_Lim__transform__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),G: fun(A,B),F3: filter(A),A3: B] :
          ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yk(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),F3)
          <=> filterlim(A,B,G,topolo7230453075368039082e_nhds(B,A3),F3) ) ) ) ).

% Lim_transform_eq
tff(fact_7074_LIM__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_yl(fun(A,B),fun(B,fun(A,B)),F2),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3) ) ) ).

% LIM_zero_cancel
tff(fact_7075_Lim__transform2,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),A3: B,F3: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),F3)
         => ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yk(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
           => filterlim(A,B,G,topolo7230453075368039082e_nhds(B,A3),F3) ) ) ) ).

% Lim_transform2
tff(fact_7076_Lim__transform,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [G: fun(A,B),A3: B,F3: filter(A),F2: fun(A,B)] :
          ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,A3),F3)
         => ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ym(fun(A,B),fun(fun(A,B),fun(A,B)),G),F2),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),F3) ) ) ) ).

% Lim_transform
tff(fact_7077_LIM__zero__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_yl(fun(A,B),fun(B,fun(A,B)),F2),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
        <=> filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3) ) ) ).

% LIM_zero_iff
tff(fact_7078_LIM__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_yl(fun(A,B),fun(B,fun(A,B)),F2),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ).

% LIM_zero
tff(fact_7079_tendsto__artanh,axiom,
    ! [A: $tType,F2: fun(A,real),A3: real,F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A3),F3)
     => ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),A3)
       => ( aa(real,$o,ord_less(real,A3),one_one(real))
         => filterlim(A,real,aTP_Lamp_yn(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,artanh(real),A3)),F3) ) ) ) ).

% tendsto_artanh
tff(fact_7080_filterlim__top,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),F3: filter(A)] : filterlim(A,B,F2,top_top(filter(B)),F3) ).

% filterlim_top
tff(fact_7081_DERIV__LIM__iff,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [F2: fun(A,A),A3: A,D3: A] :
          ( filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_yo(fun(A,A),fun(A,fun(A,A)),F2),A3),topolo7230453075368039082e_nhds(A,D3),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_yp(fun(A,A),fun(A,fun(A,A)),F2),A3),topolo7230453075368039082e_nhds(A,D3),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ).

% DERIV_LIM_iff
tff(fact_7082_LIM__fun__less__zero,axiom,
    ! [F2: fun(real,real),L: real,C2: real] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C2,top_top(set(real))))
     => ( aa(real,$o,ord_less(real,L),zero_zero(real))
       => ? [R2: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),R2)
            & ! [X2: real] :
                ( ( ( X2 != C2 )
                  & aa(real,$o,ord_less(real,aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,C2),X2))),R2) )
               => aa(real,$o,ord_less(real,aa(real,real,F2,X2)),zero_zero(real)) ) ) ) ) ).

% LIM_fun_less_zero
tff(fact_7083_LIM__fun__not__zero,axiom,
    ! [F2: fun(real,real),L: real,C2: real] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C2,top_top(set(real))))
     => ( ( L != zero_zero(real) )
       => ? [R2: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),R2)
            & ! [X2: real] :
                ( ( ( X2 != C2 )
                  & aa(real,$o,ord_less(real,aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,C2),X2))),R2) )
               => ( aa(real,real,F2,X2) != zero_zero(real) ) ) ) ) ) ).

% LIM_fun_not_zero
tff(fact_7084_LIM__fun__gt__zero,axiom,
    ! [F2: fun(real,real),L: real,C2: real] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C2,top_top(set(real))))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),L)
       => ? [R2: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),R2)
            & ! [X2: real] :
                ( ( ( X2 != C2 )
                  & aa(real,$o,ord_less(real,aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,C2),X2))),R2) )
               => aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,F2,X2)) ) ) ) ) ).

% LIM_fun_gt_zero
tff(fact_7085_LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),B3: B,A3: A,G: fun(B,C),C2: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,B3),topolo174197925503356063within(A,A3,top_top(set(A))))
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(B,B3,top_top(set(B))))
           => ( ? [D5: real] :
                  ( aa(real,$o,ord_less(real,zero_zero(real)),D5)
                  & ! [X4: A] :
                      ( ( ( X4 != A3 )
                        & aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X4),A3))),D5) )
                     => ( aa(A,B,F2,X4) != B3 ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_vk(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ) ) ).

% LIM_compose2
tff(fact_7086_isCont__real__sqrt,axiom,
    ! [X: real] : topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),sqrt) ).

% isCont_real_sqrt
tff(fact_7087_isCont__real__root,axiom,
    ! [X: real,Na: nat] : topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),root(Na)) ).

% isCont_real_root
tff(fact_7088_continuous__at__within__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [A3: A,Sa: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,Sa),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,Sa),G)
           => ( ( aa(A,B,G,A3) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,Sa),aa(fun(A,B),fun(A,B),aTP_Lamp_yq(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% continuous_at_within_divide
tff(fact_7089_isCont__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [A3: A,F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),G)
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_yh(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% isCont_add
tff(fact_7090_isCont__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [A3: A,F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),G)
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_ye(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% isCont_mult
tff(fact_7091_isCont__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A3: A,F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),G)
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_yr(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% isCont_diff
tff(fact_7092_isCont__power,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [A3: A,F2: fun(A,B),Na: nat] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aa(nat,fun(A,B),aTP_Lamp_wz(fun(A,B),fun(nat,fun(A,B)),F2),Na)) ) ) ).

% isCont_power
tff(fact_7093_isCont__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A3: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aTP_Lamp_ys(fun(A,B),fun(A,B),F2)) ) ) ).

% isCont_minus
tff(fact_7094_continuous__at__within__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [A3: A,Sa: set(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,Sa),F2)
         => ( ( aa(A,B,F2,A3) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,Sa),aTP_Lamp_yt(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_at_within_inverse
tff(fact_7095_isCont__sum,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(B)
        & topolo5987344860129210374id_add(C) )
     => ! [A2: set(A),A3: B,F2: fun(A,fun(B,C))] :
          ( ! [X4: A] :
              ( member(A,X4,A2)
             => topolo3448309680560233919inuous(B,C,topolo174197925503356063within(B,A3,top_top(set(B))),aa(A,fun(B,C),F2,X4)) )
         => topolo3448309680560233919inuous(B,C,topolo174197925503356063within(B,A3,top_top(set(B))),aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_vs(set(A),fun(fun(A,fun(B,C)),fun(B,C)),A2),F2)) ) ) ).

% isCont_sum
tff(fact_7096_continuous__at__within__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A3: A,Sa: set(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,Sa),F2)
         => ( ( aa(A,B,F2,A3) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,Sa),aTP_Lamp_yu(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_at_within_sgn
tff(fact_7097_isCont__cos_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [A3: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aTP_Lamp_xi(fun(A,B),fun(A,B),F2)) ) ) ).

% isCont_cos'
tff(fact_7098_isCont__sin_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [A3: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aTP_Lamp_xo(fun(A,B),fun(A,B),F2)) ) ) ).

% isCont_sin'
tff(fact_7099_isCont__exp_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [A3: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aTP_Lamp_xg(fun(A,B),fun(A,B),F2)) ) ) ).

% isCont_exp'
tff(fact_7100_isCont__pochhammer,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Z: A,Na: nat] : topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Z,top_top(set(A))),aTP_Lamp_yv(nat,fun(A,A),Na)) ) ).

% isCont_pochhammer
tff(fact_7101_DERIV__def,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,X: A] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,X,top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_yw(fun(A,A),fun(A,fun(A,A)),F2),X),topolo7230453075368039082e_nhds(A,D3),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% DERIV_def
tff(fact_7102_DERIV__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,X: A] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,X,top_top(set(A))))
         => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_yw(fun(A,A),fun(A,fun(A,A)),F2),X),topolo7230453075368039082e_nhds(A,D3),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% DERIV_D
tff(fact_7103_lim__exp__minus__1,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => filterlim(A,A,aTP_Lamp_yx(A,A),topolo7230453075368039082e_nhds(A,one_one(A)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ).

% lim_exp_minus_1
tff(fact_7104_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : set_or7035219750837199246ssThan(int,aa(int,int,plus_plus(int,L),one_one(int)),U) = set_or5935395276787703475ssThan(int,L,U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_int
tff(fact_7105_lemma__termdiff4,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [K: real,F2: fun(A,B),K4: real] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),K)
         => ( ! [H2: A] :
                ( ( H2 != zero_zero(A) )
               => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,H2)),K)
                 => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(A,B,F2,H2))),aa(real,real,aa(real,fun(real,real),times_times(real),K4),real_V7770717601297561774m_norm(A,H2))) ) )
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% lemma_termdiff4
tff(fact_7106_isCont__inverse__function2,axiom,
    ! [A3: real,X: real,B3: real,G: fun(real,real),F2: fun(real,real)] :
      ( aa(real,$o,ord_less(real,A3),X)
     => ( aa(real,$o,ord_less(real,X),B3)
       => ( ! [Z2: real] :
              ( aa(real,$o,ord_less_eq(real,A3),Z2)
             => ( aa(real,$o,ord_less_eq(real,Z2),B3)
               => ( aa(real,real,G,aa(real,real,F2,Z2)) = Z2 ) ) )
         => ( ! [Z2: real] :
                ( aa(real,$o,ord_less_eq(real,A3),Z2)
               => ( aa(real,$o,ord_less_eq(real,Z2),B3)
                 => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z2,top_top(set(real))),F2) ) )
           => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,aa(real,real,F2,X),top_top(set(real))),G) ) ) ) ) ).

% isCont_inverse_function2
tff(fact_7107_field__has__derivative__at,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,X: A] :
          ( has_derivative(A,A,F2,aa(A,fun(A,A),times_times(A),D3),topolo174197925503356063within(A,X,top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_yw(fun(A,A),fun(A,fun(A,A)),F2),X),topolo7230453075368039082e_nhds(A,D3),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% field_has_derivative_at
tff(fact_7108_isCont__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [A3: A,F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),G)
           => ( ( aa(A,B,G,A3) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_yq(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% isCont_divide
tff(fact_7109_isCont__ln,axiom,
    ! [X: real] :
      ( ( X != zero_zero(real) )
     => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),ln_ln(real)) ) ).

% isCont_ln
tff(fact_7110_isCont__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A3: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => ( ( aa(A,B,F2,A3) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aTP_Lamp_yu(fun(A,B),fun(A,B),F2)) ) ) ) ).

% isCont_sgn
tff(fact_7111_filterlim__at__to__0,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(A,B),F3: filter(B),A3: A] :
          ( filterlim(A,B,F2,F3,topolo174197925503356063within(A,A3,top_top(set(A))))
        <=> filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_yy(fun(A,B),fun(A,fun(A,B)),F2),A3),F3,topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% filterlim_at_to_0
tff(fact_7112_continuous__within__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Sa: set(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,Sa),F2)
         => ( ( cos(A,aa(A,A,F2,X)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,Sa),aTP_Lamp_wi(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_within_tan
tff(fact_7113_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or7035219750837199246ssThan(code_integer,aa(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_7114_continuous__within__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Sa: set(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,Sa),F2)
         => ( ( sin(A,aa(A,A,F2,X)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,Sa),aTP_Lamp_wh(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_within_cot
tff(fact_7115_continuous__at__within__tanh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [X: A,A2: set(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,X,A2),F2)
         => ( ( cosh(B,aa(A,B,F2,X)) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,X,A2),aTP_Lamp_yz(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_at_within_tanh
tff(fact_7116_isCont__has__Ub,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A3: real,B3: real,F2: fun(real,A)] :
          ( aa(real,$o,ord_less_eq(real,A3),B3)
         => ( ! [X4: real] :
                ( ( aa(real,$o,ord_less_eq(real,A3),X4)
                  & aa(real,$o,ord_less_eq(real,X4),B3) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X4,top_top(set(real))),F2) )
           => ? [M8: A] :
                ( ! [X2: real] :
                    ( ( aa(real,$o,ord_less_eq(real,A3),X2)
                      & aa(real,$o,ord_less_eq(real,X2),B3) )
                   => aa(A,$o,ord_less_eq(A,aa(real,A,F2,X2)),M8) )
                & ! [N7: A] :
                    ( aa(A,$o,ord_less(A,N7),M8)
                   => ? [X4: real] :
                        ( aa(real,$o,ord_less_eq(real,A3),X4)
                        & aa(real,$o,ord_less_eq(real,X4),B3)
                        & aa(A,$o,ord_less(A,N7),aa(real,A,F2,X4)) ) ) ) ) ) ) ).

% isCont_has_Ub
tff(fact_7117_isCont__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),tan(A)) ) ) ).

% isCont_tan
tff(fact_7118_isCont__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( sin(A,X) != zero_zero(A) )
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),cot(A)) ) ) ).

% isCont_cot
tff(fact_7119_isCont__tanh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cosh(A,X) != zero_zero(A) )
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),tanh(A)) ) ) ).

% isCont_tanh
tff(fact_7120_powser__limit__0__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Sa: real,A3: fun(nat,A),F2: fun(A,A)] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),Sa)
         => ( ! [X4: A] :
                ( ( X4 != zero_zero(A) )
               => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,X4)),Sa)
                 => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_cv(fun(nat,A),fun(A,fun(nat,A)),A3),X4)),aa(A,A,F2,X4)) ) )
           => filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,aa(nat,A,A3,zero_zero(nat))),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% powser_limit_0_strong
tff(fact_7121_powser__limit__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Sa: real,A3: fun(nat,A),F2: fun(A,A)] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),Sa)
         => ( ! [X4: A] :
                ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,X4)),Sa)
               => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_cv(fun(nat,A),fun(A,fun(nat,A)),A3),X4)),aa(A,A,F2,X4)) )
           => filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,aa(nat,A,A3,zero_zero(nat))),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% powser_limit_0
tff(fact_7122_lemma__termdiff5,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_Vector_banach(B) )
     => ! [K: real,F2: fun(nat,real),G: fun(A,fun(nat,B))] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),K)
         => ( summable(real,F2)
           => ( ! [H2: A,N: nat] :
                  ( ( H2 != zero_zero(A) )
                 => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,H2)),K)
                   => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(nat,B,aa(A,fun(nat,B),G,H2),N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,F2,N)),real_V7770717601297561774m_norm(A,H2))) ) )
             => filterlim(A,B,aTP_Lamp_za(fun(A,fun(nat,B)),fun(A,B),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ) ).

% lemma_termdiff5
tff(fact_7123_isCont__tan_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [A3: A,F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => ( ( cos(A,aa(A,A,F2,A3)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A3,top_top(set(A))),aTP_Lamp_wi(fun(A,A),fun(A,A),F2)) ) ) ) ).

% isCont_tan'
tff(fact_7124_isCont__arcosh,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),X)
     => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),arcosh(real)) ) ).

% isCont_arcosh
tff(fact_7125_LIM__cos__div__sin,axiom,
    filterlim(real,real,aTP_Lamp_zb(real,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))),top_top(set(real)))) ).

% LIM_cos_div_sin
tff(fact_7126_isCont__cot_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [A3: A,F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => ( ( sin(A,aa(A,A,F2,A3)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A3,top_top(set(A))),aTP_Lamp_wh(fun(A,A),fun(A,A),F2)) ) ) ) ).

% isCont_cot'
tff(fact_7127_DERIV__inverse__function,axiom,
    ! [F2: fun(real,real),D3: real,G: fun(real,real),X: real,A3: real,B3: real] :
      ( has_field_derivative(real,F2,D3,topolo174197925503356063within(real,aa(real,real,G,X),top_top(set(real))))
     => ( ( D3 != zero_zero(real) )
       => ( aa(real,$o,ord_less(real,A3),X)
         => ( aa(real,$o,ord_less(real,X),B3)
           => ( ! [Y3: real] :
                  ( aa(real,$o,ord_less(real,A3),Y3)
                 => ( aa(real,$o,ord_less(real,Y3),B3)
                   => ( aa(real,real,F2,aa(real,real,G,Y3)) = Y3 ) ) )
             => ( topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),G)
               => has_field_derivative(real,G,aa(real,real,inverse_inverse(real),D3),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ) ) ) ).

% DERIV_inverse_function
tff(fact_7128_isCont__polynom,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A3: A,C2: fun(nat,A),Na: nat] : topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A3,top_top(set(A))),aa(nat,fun(A,A),aTP_Lamp_zc(fun(nat,A),fun(nat,fun(A,A)),C2),Na)) ) ).

% isCont_polynom
tff(fact_7129_isCont__powser__converges__everywhere,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),X: A] :
          ( ! [Y3: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_cv(fun(nat,A),fun(A,fun(nat,A)),C2),Y3))
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),aTP_Lamp_rl(fun(nat,A),fun(A,A),C2)) ) ) ).

% isCont_powser_converges_everywhere
tff(fact_7130_isCont__arccos,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,ord_less(real,X),one_one(real))
       => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),arccos) ) ) ).

% isCont_arccos
tff(fact_7131_isCont__arcsin,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,ord_less(real,X),one_one(real))
       => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),arcsin) ) ) ).

% isCont_arcsin
tff(fact_7132_LIM__less__bound,axiom,
    ! [B3: real,X: real,F2: fun(real,real)] :
      ( aa(real,$o,ord_less(real,B3),X)
     => ( ! [X4: real] :
            ( member(real,X4,set_or5935395276787703475ssThan(real,B3,X))
           => aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,F2,X4)) )
       => ( topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),F2)
         => aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,F2,X)) ) ) ) ).

% LIM_less_bound
tff(fact_7133_isCont__artanh,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,ord_less(real,X),one_one(real))
       => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),artanh(real)) ) ) ).

% isCont_artanh
tff(fact_7134_isCont__inverse__function,axiom,
    ! [D2: real,X: real,G: fun(real,real),F2: fun(real,real)] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),D2)
     => ( ! [Z2: real] :
            ( aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,Z2),X))),D2)
           => ( aa(real,real,G,aa(real,real,F2,Z2)) = Z2 ) )
       => ( ! [Z2: real] :
              ( aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,Z2),X))),D2)
             => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z2,top_top(set(real))),F2) )
         => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,aa(real,real,F2,X),top_top(set(real))),G) ) ) ) ).

% isCont_inverse_function
tff(fact_7135_GMVT_H,axiom,
    ! [A3: real,B3: real,F2: fun(real,real),G: fun(real,real),G4: fun(real,real),F7: fun(real,real)] :
      ( aa(real,$o,ord_less(real,A3),B3)
     => ( ! [Z2: real] :
            ( aa(real,$o,ord_less_eq(real,A3),Z2)
           => ( aa(real,$o,ord_less_eq(real,Z2),B3)
             => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z2,top_top(set(real))),F2) ) )
       => ( ! [Z2: real] :
              ( aa(real,$o,ord_less_eq(real,A3),Z2)
             => ( aa(real,$o,ord_less_eq(real,Z2),B3)
               => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z2,top_top(set(real))),G) ) )
         => ( ! [Z2: real] :
                ( aa(real,$o,ord_less(real,A3),Z2)
               => ( aa(real,$o,ord_less(real,Z2),B3)
                 => has_field_derivative(real,G,aa(real,real,G4,Z2),topolo174197925503356063within(real,Z2,top_top(set(real)))) ) )
           => ( ! [Z2: real] :
                  ( aa(real,$o,ord_less(real,A3),Z2)
                 => ( aa(real,$o,ord_less(real,Z2),B3)
                   => has_field_derivative(real,F2,aa(real,real,F7,Z2),topolo174197925503356063within(real,Z2,top_top(set(real)))) ) )
             => ? [C5: real] :
                  ( aa(real,$o,ord_less(real,A3),C5)
                  & aa(real,$o,ord_less(real,C5),B3)
                  & ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,aa(real,real,F2,B3)),aa(real,real,F2,A3))),aa(real,real,G4,C5)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,aa(real,real,G,B3)),aa(real,real,G,A3))),aa(real,real,F7,C5)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_7136_floor__has__real__derivative,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A) )
     => ! [X: real,F2: fun(real,A)] :
          ( topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X,top_top(set(real))),F2)
         => ( ~ member(A,aa(real,A,F2,X),ring_1_Ints(A))
           => has_field_derivative(real,aTP_Lamp_zd(fun(real,A),fun(real,real),F2),zero_zero(real),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ).

% floor_has_real_derivative
tff(fact_7137_isCont__powser,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K4: A,X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_cv(fun(nat,A),fun(A,fun(nat,A)),C2),K4))
         => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,K4))
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),aTP_Lamp_rl(fun(nat,A),fun(A,A),C2)) ) ) ) ).

% isCont_powser
tff(fact_7138_isCont__powser_H,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B)
        & topological_t2_space(A) )
     => ! [A3: A,F2: fun(A,B),C2: fun(nat,B),K4: B] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => ( summable(B,aa(B,fun(nat,B),aTP_Lamp_ze(fun(nat,B),fun(B,fun(nat,B)),C2),K4))
           => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(B,aa(A,B,F2,A3))),real_V7770717601297561774m_norm(B,K4))
             => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aa(fun(nat,B),fun(A,B),aTP_Lamp_zg(fun(A,B),fun(fun(nat,B),fun(A,B)),F2),C2)) ) ) ) ) ).

% isCont_powser'
tff(fact_7139_summable__Leibniz_I3_J,axiom,
    ! [A3: fun(nat,real)] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A3)
       => ( aa(real,$o,ord_less(real,aa(nat,real,A3,zero_zero(nat))),zero_zero(real))
         => ! [N10: nat] : member(real,suminf(real,aTP_Lamp_zh(fun(nat,real),fun(nat,real),A3)),set_or1337092689740270186AtMost(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_zh(fun(nat,real),fun(nat,real),A3)),set_ord_lessThan(nat,aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N10)),one_one(nat)))),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_zh(fun(nat,real),fun(nat,real),A3)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N10))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_7140_summable__Leibniz_I2_J,axiom,
    ! [A3: fun(nat,real)] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A3)
       => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(nat,real,A3,zero_zero(nat)))
         => ! [N10: nat] : member(real,suminf(real,aTP_Lamp_zh(fun(nat,real),fun(nat,real),A3)),set_or1337092689740270186AtMost(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_zh(fun(nat,real),fun(nat,real),A3)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N10))),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_zh(fun(nat,real),fun(nat,real),A3)),set_ord_lessThan(nat,aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N10)),one_one(nat)))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_7141_trivial__limit__sequentially,axiom,
    at_top(nat) != bot_bot(filter(nat)) ).

% trivial_limit_sequentially
tff(fact_7142_tendsto__zero__mult__left__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,A3: fun(nat,A)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_zi(A,fun(fun(nat,A),fun(nat,A)),C2),A3),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
          <=> filterlim(nat,A,A3,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ) ).

% tendsto_zero_mult_left_iff
tff(fact_7143_tendsto__zero__mult__right__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,A3: fun(nat,A)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_zj(A,fun(fun(nat,A),fun(nat,A)),C2),A3),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
          <=> filterlim(nat,A,A3,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ) ).

% tendsto_zero_mult_right_iff
tff(fact_7144_tendsto__zero__divide__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,A3: fun(nat,A)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_zk(A,fun(fun(nat,A),fun(nat,A)),C2),A3),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
          <=> filterlim(nat,A,A3,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ) ).

% tendsto_zero_divide_iff
tff(fact_7145_filterlim__Suc,axiom,
    filterlim(nat,nat,suc,at_top(nat),at_top(nat)) ).

% filterlim_Suc
tff(fact_7146_approx__from__below__dense__linorder,axiom,
    ! [A: $tType] :
      ( ( dense_linorder(A)
        & topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [Y: A,X: A] :
          ( aa(A,$o,ord_less(A,Y),X)
         => ? [U2: fun(nat,A)] :
              ( ! [N10: nat] : aa(A,$o,ord_less(A,aa(nat,A,U2,N10)),X)
              & filterlim(nat,A,U2,topolo7230453075368039082e_nhds(A,X),at_top(nat)) ) ) ) ).

% approx_from_below_dense_linorder
tff(fact_7147_approx__from__above__dense__linorder,axiom,
    ! [A: $tType] :
      ( ( dense_linorder(A)
        & topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [X: A,Y: A] :
          ( aa(A,$o,ord_less(A,X),Y)
         => ? [U2: fun(nat,A)] :
              ( ! [N10: nat] : aa(A,$o,ord_less(A,X),aa(nat,A,U2,N10))
              & filterlim(nat,A,U2,topolo7230453075368039082e_nhds(A,X),at_top(nat)) ) ) ) ).

% approx_from_above_dense_linorder
tff(fact_7148_continuous__real__root,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real),Na: nat] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => topolo3448309680560233919inuous(A,real,F3,aa(nat,fun(A,real),aTP_Lamp_zl(fun(A,real),fun(nat,fun(A,real)),F2),Na)) ) ) ).

% continuous_real_root
tff(fact_7149_continuous__arsinh,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_zm(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_arsinh
tff(fact_7150_continuous__real__sqrt,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_zn(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_real_sqrt
tff(fact_7151_filterlim__sequentially__Suc,axiom,
    ! [A: $tType,F2: fun(nat,A),F3: filter(A)] :
      ( filterlim(nat,A,aTP_Lamp_zo(fun(nat,A),fun(nat,A),F2),F3,at_top(nat))
    <=> filterlim(nat,A,F2,F3,at_top(nat)) ) ).

% filterlim_sequentially_Suc
tff(fact_7152_continuous__arctan,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_zp(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_arctan
tff(fact_7153_LIMSEQ__Suc,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),L: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => filterlim(nat,A,aTP_Lamp_zq(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% LIMSEQ_Suc
tff(fact_7154_LIMSEQ__imp__Suc,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),L: A] :
          ( filterlim(nat,A,aTP_Lamp_zq(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% LIMSEQ_imp_Suc
tff(fact_7155_LIMSEQ__const__iff,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [K: A,L: A] :
          ( filterlim(nat,A,aTP_Lamp_zr(A,fun(nat,A),K),topolo7230453075368039082e_nhds(A,L),at_top(nat))
        <=> ( K = L ) ) ) ).

% LIMSEQ_const_iff
tff(fact_7156_continuous__rabs,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_zs(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_rabs
tff(fact_7157_seq__offset__neg,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),L: A,K: nat] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_zt(fun(nat,A),fun(nat,fun(nat,A)),F2),K),topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% seq_offset_neg
tff(fact_7158_LIMSEQ__offset,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),K: nat,A3: A] :
          ( filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_zu(fun(nat,A),fun(nat,fun(nat,A)),F2),K),topolo7230453075368039082e_nhds(A,A3),at_top(nat))
         => filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,A3),at_top(nat)) ) ) ).

% LIMSEQ_offset
tff(fact_7159_LIMSEQ__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),A3: A,K: nat] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,A3),at_top(nat))
         => filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_zu(fun(nat,A),fun(nat,fun(nat,A)),F2),K),topolo7230453075368039082e_nhds(A,A3),at_top(nat)) ) ) ).

% LIMSEQ_ignore_initial_segment
tff(fact_7160_less__filter__def,axiom,
    ! [A: $tType,F3: filter(A),F8: filter(A)] :
      ( aa(filter(A),$o,ord_less(filter(A),F3),F8)
    <=> ( aa(filter(A),$o,ord_less_eq(filter(A),F3),F8)
        & ~ aa(filter(A),$o,ord_less_eq(filter(A),F8),F3) ) ) ).

% less_filter_def
tff(fact_7161_trivial__limit__at__top__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( at_top(A) != bot_bot(filter(A)) ) ) ).

% trivial_limit_at_top_linorder
tff(fact_7162_summable__LIMSEQ__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% summable_LIMSEQ_zero
tff(fact_7163_isCont__rabs,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A3: A,F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,top_top(set(A))),aTP_Lamp_zs(fun(A,real),fun(A,real),F2)) ) ) ).

% isCont_rabs
tff(fact_7164_continuous__at__within__powr,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A3: A,Sa: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,Sa),F2)
         => ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,Sa),G)
           => ( ( aa(A,real,F2,A3) != zero_zero(real) )
             => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,Sa),aa(fun(A,real),fun(A,real),aTP_Lamp_zv(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ).

% continuous_at_within_powr
tff(fact_7165_continuous__within__ln,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [X: A,Sa: set(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,X,Sa),F2)
         => ( ( aa(A,real,F2,X) != zero_zero(real) )
           => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,X,Sa),aTP_Lamp_zw(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_within_ln
tff(fact_7166_mult__nat__right__at__top,axiom,
    ! [C2: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),C2)
     => filterlim(nat,nat,aTP_Lamp_zx(nat,fun(nat,nat),C2),at_top(nat),at_top(nat)) ) ).

% mult_nat_right_at_top
tff(fact_7167_mult__nat__left__at__top,axiom,
    ! [C2: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),C2)
     => filterlim(nat,nat,aa(nat,fun(nat,nat),times_times(nat),C2),at_top(nat),at_top(nat)) ) ).

% mult_nat_left_at_top
tff(fact_7168_LIMSEQ__lessThan__iff__atMost,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(set(nat),A),X: A] :
          ( filterlim(nat,A,aTP_Lamp_zy(fun(set(nat),A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,X),at_top(nat))
        <=> filterlim(nat,A,aTP_Lamp_zz(fun(set(nat),A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,X),at_top(nat)) ) ) ).

% LIMSEQ_lessThan_iff_atMost
tff(fact_7169_LIMSEQ__root,axiom,
    filterlim(nat,real,aTP_Lamp_aaa(nat,real),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat)) ).

% LIMSEQ_root
tff(fact_7170_isCont__powr,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A3: A,F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,top_top(set(A))),G)
           => ( ( aa(A,real,F2,A3) != zero_zero(real) )
             => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,top_top(set(A))),aa(fun(A,real),fun(A,real),aTP_Lamp_zv(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ).

% isCont_powr
tff(fact_7171_isCont__ln_H,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [X: A,F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,X,top_top(set(A))),F2)
         => ( ( aa(A,real,F2,X) != zero_zero(real) )
           => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,X,top_top(set(A))),aTP_Lamp_zw(fun(A,real),fun(A,real),F2)) ) ) ) ).

% isCont_ln'
tff(fact_7172_lim__const__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A3: A] : filterlim(nat,A,aTP_Lamp_aab(A,fun(nat,A),A3),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_const_over_n
tff(fact_7173_LIMSEQ__linear,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [X5: fun(nat,A),X: A,L: nat] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,X),at_top(nat))
         => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),L)
           => filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aac(fun(nat,A),fun(nat,fun(nat,A)),X5),L),topolo7230453075368039082e_nhds(A,X),at_top(nat)) ) ) ) ).

% LIMSEQ_linear
tff(fact_7174_lim__inverse__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_aad(nat,A),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_inverse_n
tff(fact_7175_LIMSEQ__SEQ__conv2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A3: A,F2: fun(A,B),L: B] :
          ( ! [S4: fun(nat,A)] :
              ( ( ! [N10: nat] : aa(nat,A,S4,N10) != A3
                & filterlim(nat,A,S4,topolo7230453075368039082e_nhds(A,A3),at_top(nat)) )
             => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_aae(fun(A,B),fun(fun(nat,A),fun(nat,B)),F2),S4),topolo7230453075368039082e_nhds(B,L),at_top(nat)) )
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ).

% LIMSEQ_SEQ_conv2
tff(fact_7176_LIMSEQ__SEQ__conv1,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),L: B,A3: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A3,top_top(set(A))))
         => ! [S8: fun(nat,A)] :
              ( ( ! [N: nat] : aa(nat,A,S8,N) != A3
                & filterlim(nat,A,S8,topolo7230453075368039082e_nhds(A,A3),at_top(nat)) )
             => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_aaf(fun(A,B),fun(fun(nat,A),fun(nat,B)),F2),S8),topolo7230453075368039082e_nhds(B,L),at_top(nat)) ) ) ) ).

% LIMSEQ_SEQ_conv1
tff(fact_7177_LIMSEQ__SEQ__conv,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A3: A,X5: fun(A,B),L5: B] :
          ( ! [S9: fun(nat,A)] :
              ( ( ! [N3: nat] : aa(nat,A,S9,N3) != A3
                & filterlim(nat,A,S9,topolo7230453075368039082e_nhds(A,A3),at_top(nat)) )
             => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_aae(fun(A,B),fun(fun(nat,A),fun(nat,B)),X5),S9),topolo7230453075368039082e_nhds(B,L5),at_top(nat)) )
        <=> filterlim(A,B,X5,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ).

% LIMSEQ_SEQ_conv
tff(fact_7178_telescope__summable,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,C2),at_top(nat))
         => summable(A,aTP_Lamp_aag(fun(nat,A),fun(nat,A),F2)) ) ) ).

% telescope_summable
tff(fact_7179_telescope__summable_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,C2),at_top(nat))
         => summable(A,aTP_Lamp_aah(fun(nat,A),fun(nat,A),F2)) ) ) ).

% telescope_summable'
tff(fact_7180_nested__sequence__unique,axiom,
    ! [F2: fun(nat,real),G: fun(nat,real)] :
      ( ! [N: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,F2,N)),aa(nat,real,F2,aa(nat,nat,suc,N)))
     => ( ! [N: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,G,aa(nat,nat,suc,N))),aa(nat,real,G,N))
       => ( ! [N: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,F2,N)),aa(nat,real,G,N))
         => ( filterlim(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_aai(fun(nat,real),fun(fun(nat,real),fun(nat,real)),F2),G),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => ? [L2: real] :
                ( ! [N10: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,F2,N10)),L2)
                & filterlim(nat,real,F2,topolo7230453075368039082e_nhds(real,L2),at_top(nat))
                & ! [N10: nat] : aa(real,$o,ord_less_eq(real,L2),aa(nat,real,G,N10))
                & filterlim(nat,real,G,topolo7230453075368039082e_nhds(real,L2),at_top(nat)) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_7181_LIMSEQ__inverse__zero,axiom,
    ! [X5: fun(nat,real)] :
      ( ! [R2: real] :
        ? [N7: nat] :
        ! [N: nat] :
          ( aa(nat,$o,ord_less_eq(nat,N7),N)
         => aa(real,$o,ord_less(real,R2),aa(nat,real,X5,N)) )
     => filterlim(nat,real,aTP_Lamp_aaj(fun(nat,real),fun(nat,real),X5),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_inverse_zero
tff(fact_7182_lim__inverse__n_H,axiom,
    filterlim(nat,real,aTP_Lamp_aak(nat,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ).

% lim_inverse_n'
tff(fact_7183_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim(nat,real,aTP_Lamp_aal(nat,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_7184_LIMSEQ__root__const,axiom,
    ! [C2: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
     => filterlim(nat,real,aTP_Lamp_aam(real,fun(nat,real),C2),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat)) ) ).

% LIMSEQ_root_const
tff(fact_7185_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R3: real] : filterlim(nat,real,aTP_Lamp_aan(real,fun(nat,real),R3),topolo7230453075368039082e_nhds(real,R3),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_7186_sums__def,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A),Sa: A] :
          ( aa(A,$o,sums(A,F2),Sa)
        <=> filterlim(nat,A,aTP_Lamp_aao(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,Sa),at_top(nat)) ) ) ).

% sums_def
tff(fact_7187_sums__def__le,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A),Sa: A] :
          ( aa(A,$o,sums(A,F2),Sa)
        <=> filterlim(nat,A,aTP_Lamp_aap(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,Sa),at_top(nat)) ) ) ).

% sums_def_le
tff(fact_7188_continuous__at__within__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A3: A,Sa: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,Sa),F2)
         => ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,Sa),G)
           => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(A,real,F2,A3))
             => ( ( aa(A,real,F2,A3) != one_one(real) )
               => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(A,real,G,A3))
                 => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,Sa),aa(fun(A,real),fun(A,real),aTP_Lamp_aaq(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% continuous_at_within_log
tff(fact_7189_increasing__LIMSEQ,axiom,
    ! [F2: fun(nat,real),L: real] :
      ( ! [N: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,F2,N)),aa(nat,real,F2,aa(nat,nat,suc,N)))
     => ( ! [N: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,F2,N)),L)
       => ( ! [E: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),E)
             => ? [N10: nat] : aa(real,$o,ord_less_eq(real,L),aa(real,real,plus_plus(real,aa(nat,real,F2,N10)),E)) )
         => filterlim(nat,real,F2,topolo7230453075368039082e_nhds(real,L),at_top(nat)) ) ) ) ).

% increasing_LIMSEQ
tff(fact_7190_lim__1__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_aar(nat,A),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_1_over_n
tff(fact_7191_LIMSEQ__Suc__n__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_aas(nat,A),topolo7230453075368039082e_nhds(A,one_one(A)),at_top(nat)) ) ).

% LIMSEQ_Suc_n_over_n
tff(fact_7192_LIMSEQ__n__over__Suc__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_aat(nat,A),topolo7230453075368039082e_nhds(A,one_one(A)),at_top(nat)) ) ).

% LIMSEQ_n_over_Suc_n
tff(fact_7193_telescope__sums,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,C2),at_top(nat))
         => aa(A,$o,sums(A,aTP_Lamp_aag(fun(nat,A),fun(nat,A),F2)),aa(A,A,minus_minus(A,C2),aa(nat,A,F2,zero_zero(nat)))) ) ) ).

% telescope_sums
tff(fact_7194_telescope__sums_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,C2),at_top(nat))
         => aa(A,$o,sums(A,aTP_Lamp_aah(fun(nat,A),fun(nat,A),F2)),aa(A,A,minus_minus(A,aa(nat,A,F2,zero_zero(nat))),C2)) ) ) ).

% telescope_sums'
tff(fact_7195_LIMSEQ__realpow__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less(real,X),one_one(real))
       => filterlim(nat,real,power_power(real,X),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ) ).

% LIMSEQ_realpow_zero
tff(fact_7196_LIMSEQ__divide__realpow__zero,axiom,
    ! [X: real,A3: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),X)
     => filterlim(nat,real,aa(real,fun(nat,real),aTP_Lamp_aau(real,fun(real,fun(nat,real)),X),A3),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_7197_LIMSEQ__abs__realpow__zero,axiom,
    ! [C2: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,abs_abs(real),C2)),one_one(real))
     => filterlim(nat,real,power_power(real,aa(real,real,abs_abs(real),C2)),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_abs_realpow_zero
tff(fact_7198_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C2: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,abs_abs(real),C2)),one_one(real))
     => filterlim(nat,real,power_power(real,C2),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_abs_realpow_zero2
tff(fact_7199_LIMSEQ__inverse__realpow__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),X)
     => filterlim(nat,real,aTP_Lamp_aav(real,fun(nat,real),X),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_7200_sums__def_H,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A),Sa: A] :
          ( aa(A,$o,sums(A,F2),Sa)
        <=> filterlim(nat,A,aTP_Lamp_aaw(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,Sa),at_top(nat)) ) ) ).

% sums_def'
tff(fact_7201_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R3: real] : filterlim(nat,real,aTP_Lamp_aax(real,fun(nat,real),R3),topolo7230453075368039082e_nhds(real,R3),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_7202_root__test__convergence,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),X: real] :
          ( filterlim(nat,real,aTP_Lamp_aay(fun(nat,A),fun(nat,real),F2),topolo7230453075368039082e_nhds(real,X),at_top(nat))
         => ( aa(real,$o,ord_less(real,X),one_one(real))
           => summable(A,F2) ) ) ) ).

% root_test_convergence
tff(fact_7203_summable__LIMSEQ,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => filterlim(nat,A,aTP_Lamp_aaz(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,suminf(A,F2)),at_top(nat)) ) ) ).

% summable_LIMSEQ
tff(fact_7204_summable__LIMSEQ_H,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => filterlim(nat,A,aTP_Lamp_aba(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,suminf(A,F2)),at_top(nat)) ) ) ).

% summable_LIMSEQ'
tff(fact_7205_isCont__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A3: A,F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,top_top(set(A))),G)
           => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(A,real,F2,A3))
             => ( ( aa(A,real,F2,A3) != one_one(real) )
               => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(A,real,G,A3))
                 => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,top_top(set(A))),aa(fun(A,real),fun(A,real),aTP_Lamp_aaq(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% isCont_log
tff(fact_7206_LIMSEQ__D,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),L5: A,R3: real] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
         => ( aa(real,$o,ord_less(real,zero_zero(real)),R3)
           => ? [No: nat] :
              ! [N10: nat] :
                ( aa(nat,$o,ord_less_eq(nat,No),N10)
               => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X5,N10)),L5))),R3) ) ) ) ) ).

% LIMSEQ_D
tff(fact_7207_LIMSEQ__I,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),L5: A] :
          ( ! [R2: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),R2)
             => ? [No2: nat] :
                ! [N: nat] :
                  ( aa(nat,$o,ord_less_eq(nat,No2),N)
                 => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X5,N)),L5))),R2) ) )
         => filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat)) ) ) ).

% LIMSEQ_I
tff(fact_7208_LIMSEQ__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),L5: A] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
        <=> ! [R5: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),R5)
             => ? [No3: nat] :
                ! [N3: nat] :
                  ( aa(nat,$o,ord_less_eq(nat,No3),N3)
                 => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X5,N3)),L5))),R5) ) ) ) ) ).

% LIMSEQ_iff
tff(fact_7209_LIMSEQ__power__zero,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [X: A] :
          ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,X)),one_one(real))
         => filterlim(nat,A,power_power(A,X),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% LIMSEQ_power_zero
tff(fact_7210_tendsto__exp__limit__sequentially,axiom,
    ! [X: real] : filterlim(nat,real,aTP_Lamp_abb(real,fun(nat,real),X),topolo7230453075368039082e_nhds(real,aa(real,real,exp(real),X)),at_top(nat)) ).

% tendsto_exp_limit_sequentially
tff(fact_7211_tendsto__power__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [F2: fun(A,nat),F3: filter(A),X: B] :
          ( filterlim(A,nat,F2,at_top(nat),F3)
         => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(B,X)),one_one(real))
           => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_abc(fun(A,nat),fun(B,fun(A,B)),F2),X),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ) ).

% tendsto_power_zero
tff(fact_7212_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R3: real] : filterlim(nat,real,aTP_Lamp_abd(real,fun(nat,real),R3),topolo7230453075368039082e_nhds(real,R3),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_7213_LIMSEQ__norm__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( ! [N: nat] : aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(nat,A,F2,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,F2,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% LIMSEQ_norm_0
tff(fact_7214_summable__Leibniz_I1_J,axiom,
    ! [A3: fun(nat,real)] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A3)
       => summable(real,aTP_Lamp_zh(fun(nat,real),fun(nat,real),A3)) ) ) ).

% summable_Leibniz(1)
tff(fact_7215_field__derivative__lim__unique,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Df: A,Z: A,Sa: fun(nat,A),A3: A] :
          ( has_field_derivative(A,F2,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_abe(fun(A,A),fun(A,fun(fun(nat,A),fun(nat,A))),F2),Z),Sa),topolo7230453075368039082e_nhds(A,A3),at_top(nat))
               => ( Df = A3 ) ) ) ) ) ) ).

% field_derivative_lim_unique
tff(fact_7216_powser__times__n__limit__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [X: A] :
          ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,X)),one_one(real))
         => filterlim(nat,A,aTP_Lamp_abf(A,fun(nat,A),X),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% powser_times_n_limit_0
tff(fact_7217_lim__n__over__pown,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( aa(real,$o,ord_less(real,one_one(real)),real_V7770717601297561774m_norm(A,X))
         => filterlim(nat,A,aTP_Lamp_abg(A,fun(nat,A),X),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% lim_n_over_pown
tff(fact_7218_summable,axiom,
    ! [A3: fun(nat,real)] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N: nat] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(nat,real,A3,N))
       => ( ! [N: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,A3,aa(nat,nat,suc,N))),aa(nat,real,A3,N))
         => summable(real,aTP_Lamp_zh(fun(nat,real),fun(nat,real),A3)) ) ) ) ).

% summable
tff(fact_7219_cos__diff__limit__1,axiom,
    ! [Theta: fun(nat,real),Theta2: real] :
      ( filterlim(nat,real,aa(real,fun(nat,real),aTP_Lamp_abh(fun(nat,real),fun(real,fun(nat,real)),Theta),Theta2),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat))
     => ~ ! [K2: fun(nat,int)] : ~ filterlim(nat,real,aa(fun(nat,int),fun(nat,real),aTP_Lamp_abi(fun(nat,real),fun(fun(nat,int),fun(nat,real)),Theta),K2),topolo7230453075368039082e_nhds(real,Theta2),at_top(nat)) ) ).

% cos_diff_limit_1
tff(fact_7220_cos__limit__1,axiom,
    ! [Theta: fun(nat,real)] :
      ( filterlim(nat,real,aTP_Lamp_abj(fun(nat,real),fun(nat,real),Theta),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat))
     => ? [K2: fun(nat,int)] : filterlim(nat,real,aa(fun(nat,int),fun(nat,real),aTP_Lamp_abi(fun(nat,real),fun(fun(nat,int),fun(nat,real)),Theta),K2),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% cos_limit_1
tff(fact_7221_summable__Leibniz_I4_J,axiom,
    ! [A3: fun(nat,real)] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A3)
       => filterlim(nat,real,aTP_Lamp_abk(fun(nat,real),fun(nat,real),A3),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_zh(fun(nat,real),fun(nat,real),A3))),at_top(nat)) ) ) ).

% summable_Leibniz(4)
tff(fact_7222_zeroseq__arctan__series,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,abs_abs(real),X)),one_one(real))
     => filterlim(nat,real,aTP_Lamp_av(real,fun(nat,real),X),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% zeroseq_arctan_series
tff(fact_7223_summable__Leibniz_H_I3_J,axiom,
    ! [A3: fun(nat,real)] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N: nat] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(nat,real,A3,N))
       => ( ! [N: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,A3,aa(nat,nat,suc,N))),aa(nat,real,A3,N))
         => filterlim(nat,real,aTP_Lamp_abk(fun(nat,real),fun(nat,real),A3),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_zh(fun(nat,real),fun(nat,real),A3))),at_top(nat)) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_7224_summable__Leibniz_H_I2_J,axiom,
    ! [A3: fun(nat,real),Na: nat] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N: nat] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(nat,real,A3,N))
       => ( ! [N: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,A3,aa(nat,nat,suc,N))),aa(nat,real,A3,N))
         => aa(real,$o,ord_less_eq(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_zh(fun(nat,real),fun(nat,real),A3)),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_zh(fun(nat,real),fun(nat,real),A3))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_7225_sums__alternating__upper__lower,axiom,
    ! [A3: fun(nat,real)] :
      ( ! [N: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,A3,aa(nat,nat,suc,N))),aa(nat,real,A3,N))
     => ( ! [N: nat] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(nat,real,A3,N))
       => ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
         => ? [L2: real] :
              ( ! [N10: nat] : aa(real,$o,ord_less_eq(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_zh(fun(nat,real),fun(nat,real),A3)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N10)))),L2)
              & filterlim(nat,real,aTP_Lamp_abk(fun(nat,real),fun(nat,real),A3),topolo7230453075368039082e_nhds(real,L2),at_top(nat))
              & ! [N10: nat] : aa(real,$o,ord_less_eq(real,L2),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_zh(fun(nat,real),fun(nat,real),A3)),set_ord_lessThan(nat,aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N10)),one_one(nat)))))
              & filterlim(nat,real,aTP_Lamp_abl(fun(nat,real),fun(nat,real),A3),topolo7230453075368039082e_nhds(real,L2),at_top(nat)) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_7226_summable__Leibniz_I5_J,axiom,
    ! [A3: fun(nat,real)] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A3)
       => filterlim(nat,real,aTP_Lamp_abl(fun(nat,real),fun(nat,real),A3),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_zh(fun(nat,real),fun(nat,real),A3))),at_top(nat)) ) ) ).

% summable_Leibniz(5)
tff(fact_7227_summable__Leibniz_H_I4_J,axiom,
    ! [A3: fun(nat,real),Na: nat] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N: nat] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(nat,real,A3,N))
       => ( ! [N: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,A3,aa(nat,nat,suc,N))),aa(nat,real,A3,N))
         => aa(real,$o,ord_less_eq(real,suminf(real,aTP_Lamp_zh(fun(nat,real),fun(nat,real),A3))),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_zh(fun(nat,real),fun(nat,real),A3)),set_ord_lessThan(nat,aa(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_7228_summable__Leibniz_H_I5_J,axiom,
    ! [A3: fun(nat,real)] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N: nat] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(nat,real,A3,N))
       => ( ! [N: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,A3,aa(nat,nat,suc,N))),aa(nat,real,A3,N))
         => filterlim(nat,real,aTP_Lamp_abl(fun(nat,real),fun(nat,real),A3),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_zh(fun(nat,real),fun(nat,real),A3))),at_top(nat)) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_7229_has__derivative__at2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),X: A] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,F7)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_abm(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F2),F7),X),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ) ).

% has_derivative_at2
tff(fact_7230_has__derivative__at,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),D3: fun(A,B),X: A] :
          ( has_derivative(A,B,F2,D3,topolo174197925503356063within(A,X,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,D3)
            & filterlim(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_abn(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),F2),D3),X),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% has_derivative_at
tff(fact_7231_filterlim__int__sequentially,axiom,
    filterlim(nat,int,semiring_1_of_nat(int),at_top(int),at_top(nat)) ).

% filterlim_int_sequentially
tff(fact_7232_exp__at__top,axiom,
    filterlim(real,real,exp(real),at_top(real),at_top(real)) ).

% exp_at_top
tff(fact_7233_sqrt__at__top,axiom,
    filterlim(real,real,sqrt,at_top(real),at_top(real)) ).

% sqrt_at_top
tff(fact_7234_ln__at__top,axiom,
    filterlim(real,real,ln_ln(real),at_top(real),at_top(real)) ).

% ln_at_top
tff(fact_7235_bounded__linear_Ocontinuous,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F3: filter(C),G: fun(C,A)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( topolo3448309680560233919inuous(C,A,F3,G)
           => topolo3448309680560233919inuous(C,B,F3,aa(fun(C,A),fun(C,B),aTP_Lamp_abo(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G)) ) ) ) ).

% bounded_linear.continuous
tff(fact_7236_bounded__linear_Otendsto,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [F2: fun(A,B),G: fun(C,A),A3: A,F3: filter(C)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( filterlim(C,A,G,topolo7230453075368039082e_nhds(A,A3),F3)
           => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_abp(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G),topolo7230453075368039082e_nhds(B,aa(A,B,F2,A3)),F3) ) ) ) ).

% bounded_linear.tendsto
tff(fact_7237_filterlim__at__top__mult__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,at_top(real),F3)
     => ( filterlim(A,real,G,at_top(real),F3)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_abq(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F3) ) ) ).

% filterlim_at_top_mult_at_top
tff(fact_7238_filterlim__at__top__add__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,at_top(real),F3)
     => ( filterlim(A,real,G,at_top(real),F3)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_abr(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F3) ) ) ).

% filterlim_at_top_add_at_top
tff(fact_7239_bounded__linear_OCauchy,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),X5: fun(nat,A)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( topolo3814608138187158403Cauchy(A,X5)
           => topolo3814608138187158403Cauchy(B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_abs(fun(A,B),fun(fun(nat,A),fun(nat,B)),F2),X5)) ) ) ) ).

% bounded_linear.Cauchy
tff(fact_7240_bounded__linear__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => real_V3181309239436604168linear(A,B,aTP_Lamp_sv(fun(A,B),fun(A,B),F2)) ) ) ).

% bounded_linear_minus
tff(fact_7241_bounded__linear__sum,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [I5: set(A),F2: fun(A,fun(B,C))] :
          ( ! [I3: A] :
              ( member(A,I3,I5)
             => real_V3181309239436604168linear(B,C,aa(A,fun(B,C),F2,I3)) )
         => real_V3181309239436604168linear(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_su(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2)) ) ) ).

% bounded_linear_sum
tff(fact_7242_bounded__linear__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => real_V3181309239436604168linear(A,B,aTP_Lamp_ss(A,B)) ) ).

% bounded_linear_zero
tff(fact_7243_bounded__linear__scaleR__left,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] : real_V3181309239436604168linear(real,A,aTP_Lamp_abt(A,fun(real,A),X)) ) ).

% bounded_linear_scaleR_left
tff(fact_7244_bounded__linear__const__scaleR,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [G: fun(A,B),R3: real] :
          ( real_V3181309239436604168linear(A,B,G)
         => real_V3181309239436604168linear(A,B,aa(real,fun(A,B),aTP_Lamp_sp(fun(A,B),fun(real,fun(A,B)),G),R3)) ) ) ).

% bounded_linear_const_scaleR
tff(fact_7245_bounded__linear__scaleR__const,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [G: fun(A,real),X: B] :
          ( real_V3181309239436604168linear(A,real,G)
         => real_V3181309239436604168linear(A,B,aa(B,fun(A,B),aTP_Lamp_sq(fun(A,real),fun(B,fun(A,B)),G),X)) ) ) ).

% bounded_linear_scaleR_const
tff(fact_7246_bounded__linear__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R3: real] : real_V3181309239436604168linear(A,A,real_V8093663219630862766scaleR(A,R3)) ) ).

% bounded_linear_scaleR_right
tff(fact_7247_bounded__linear_Osums,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),X5: fun(nat,A),A3: A] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( aa(A,$o,sums(A,X5),A3)
           => aa(B,$o,sums(B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_abs(fun(A,B),fun(fun(nat,A),fun(nat,B)),F2),X5)),aa(A,B,F2,A3)) ) ) ) ).

% bounded_linear.sums
tff(fact_7248_bounded__linear__compose,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),G: fun(C,A)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( real_V3181309239436604168linear(C,A,G)
           => real_V3181309239436604168linear(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_sy(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G)) ) ) ) ).

% bounded_linear_compose
tff(fact_7249_bounded__linear__ident,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => real_V3181309239436604168linear(A,A,aTP_Lamp_so(A,A)) ) ).

% bounded_linear_ident
tff(fact_7250_bounded__linear_Osummable,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),X5: fun(nat,A)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( summable(A,X5)
           => summable(B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_abs(fun(A,B),fun(fun(nat,A),fun(nat,B)),F2),X5)) ) ) ) ).

% bounded_linear.summable
tff(fact_7251_bounded__linear__of__real,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => real_V3181309239436604168linear(real,A,real_Vector_of_real(A)) ) ).

% bounded_linear_of_real
tff(fact_7252_bounded__linear__mult__right,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [X: A] : real_V3181309239436604168linear(A,A,aa(A,fun(A,A),times_times(A),X)) ) ).

% bounded_linear_mult_right
tff(fact_7253_bounded__linear__mult__const,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & real_V822414075346904944vector(A) )
     => ! [G: fun(A,B),Y: B] :
          ( real_V3181309239436604168linear(A,B,G)
         => real_V3181309239436604168linear(A,B,aa(B,fun(A,B),aTP_Lamp_sm(fun(A,B),fun(B,fun(A,B)),G),Y)) ) ) ).

% bounded_linear_mult_const
tff(fact_7254_bounded__linear__const__mult,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & real_V822414075346904944vector(A) )
     => ! [G: fun(A,B),X: B] :
          ( real_V3181309239436604168linear(A,B,G)
         => real_V3181309239436604168linear(A,B,aa(B,fun(A,B),aTP_Lamp_sl(fun(A,B),fun(B,fun(A,B)),G),X)) ) ) ).

% bounded_linear_const_mult
tff(fact_7255_bounded__linear__mult__left,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Y: A] : real_V3181309239436604168linear(A,A,aTP_Lamp_abu(A,fun(A,A),Y)) ) ).

% bounded_linear_mult_left
tff(fact_7256_bounded__linear__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Y: A] : real_V3181309239436604168linear(A,A,aTP_Lamp_abv(A,fun(A,A),Y)) ) ).

% bounded_linear_divide
tff(fact_7257_bounded__linear__sub,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( real_V3181309239436604168linear(A,B,G)
           => real_V3181309239436604168linear(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_sk(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% bounded_linear_sub
tff(fact_7258_bounded__linear__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( real_V3181309239436604168linear(A,B,G)
           => real_V3181309239436604168linear(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_sj(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% bounded_linear_add
tff(fact_7259_bounded__linear_Osuminf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),X5: fun(nat,A)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( summable(A,X5)
           => ( aa(A,B,F2,suminf(A,X5)) = suminf(B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_abs(fun(A,B),fun(fun(nat,A),fun(nat,B)),F2),X5)) ) ) ) ) ).

% bounded_linear.suminf
tff(fact_7260_bounded__linear_Ohas__derivative,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [F2: fun(A,B),G: fun(C,A),G4: fun(C,A),F3: filter(C)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( has_derivative(C,A,G,G4,F3)
           => has_derivative(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_sy(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G),aa(fun(C,A),fun(C,B),aTP_Lamp_sy(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G4),F3) ) ) ) ).

% bounded_linear.has_derivative
tff(fact_7261_bounded__linear_Otendsto__zero,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),G: fun(C,A),F3: filter(C)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( filterlim(C,A,G,topolo7230453075368039082e_nhds(A,zero_zero(A)),F3)
           => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_abp(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ) ).

% bounded_linear.tendsto_zero
tff(fact_7262_bounded__linear_OisCont,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),A3: C,G: fun(C,A)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( topolo3448309680560233919inuous(C,A,topolo174197925503356063within(C,A3,top_top(set(C))),G)
           => topolo3448309680560233919inuous(C,B,topolo174197925503356063within(C,A3,top_top(set(C))),aa(fun(C,A),fun(C,B),aTP_Lamp_abo(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G)) ) ) ) ).

% bounded_linear.isCont
tff(fact_7263_filterlim__tendsto__add__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F3)
     => ( filterlim(A,real,G,at_top(real),F3)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_abr(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F3) ) ) ).

% filterlim_tendsto_add_at_top
tff(fact_7264_filterlim__real__sequentially,axiom,
    filterlim(nat,real,semiring_1_of_nat(real),at_top(real),at_top(nat)) ).

% filterlim_real_sequentially
tff(fact_7265_bounded__linear_Ononneg__bounded,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ? [K7: real] :
              ( aa(real,$o,ord_less_eq(real,zero_zero(real)),K7)
              & ! [X2: A] : aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(A,B,F2,X2))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X2)),K7)) ) ) ) ).

% bounded_linear.nonneg_bounded
tff(fact_7266_has__derivative__within__singleton__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),G: fun(A,B),X: A] :
          ( has_derivative(A,B,F2,G,topolo174197925503356063within(A,X,aa(set(A),set(A),insert(A,X),bot_bot(set(A)))))
        <=> real_V3181309239436604168linear(A,B,G) ) ) ).

% has_derivative_within_singleton_iff
tff(fact_7267_filterlim__pow__at__top,axiom,
    ! [A: $tType,Na: nat,F2: fun(A,real),F3: filter(A)] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( filterlim(A,real,F2,at_top(real),F3)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ur(nat,fun(fun(A,real),fun(A,real)),Na),F2),at_top(real),F3) ) ) ).

% filterlim_pow_at_top
tff(fact_7268_tanh__real__at__top,axiom,
    filterlim(real,real,tanh(real),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(real)) ).

% tanh_real_at_top
tff(fact_7269_real__tendsto__divide__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F3)
     => ( filterlim(A,real,G,at_top(real),F3)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_abw(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% real_tendsto_divide_at_top
tff(fact_7270_artanh__real__at__left__1,axiom,
    filterlim(real,real,artanh(real),at_top(real),topolo174197925503356063within(real,one_one(real),set_ord_lessThan(real,one_one(real)))) ).

% artanh_real_at_left_1
tff(fact_7271_tendsto__inverse__0__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,at_top(real),F3)
     => filterlim(A,real,aTP_Lamp_abx(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ).

% tendsto_inverse_0_at_top
tff(fact_7272_filterlim__int__of__nat__at__topD,axiom,
    ! [A: $tType,F2: fun(int,A),F3: filter(A)] :
      ( filterlim(nat,A,aTP_Lamp_aby(fun(int,A),fun(nat,A),F2),F3,at_top(nat))
     => filterlim(int,A,F2,F3,at_top(int)) ) ).

% filterlim_int_of_nat_at_topD
tff(fact_7273_bounded__linear_Opos__bounded,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ? [K7: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),K7)
              & ! [X2: A] : aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(A,B,F2,X2))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X2)),K7)) ) ) ) ).

% bounded_linear.pos_bounded
tff(fact_7274_filterlim__sequentially__iff__filterlim__real,axiom,
    ! [A: $tType,F2: fun(A,nat),F3: filter(A)] :
      ( filterlim(A,nat,F2,at_top(nat),F3)
    <=> filterlim(A,real,aTP_Lamp_abz(fun(A,nat),fun(A,real),F2),at_top(real),F3) ) ).

% filterlim_sequentially_iff_filterlim_real
tff(fact_7275_filterlim__tendsto__pos__mult__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
       => ( filterlim(A,real,G,at_top(real),F3)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_abq(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F3) ) ) ) ).

% filterlim_tendsto_pos_mult_at_top
tff(fact_7276_filterlim__at__top__mult__tendsto__pos,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
       => ( filterlim(A,real,G,at_top(real),F3)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_aca(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F3) ) ) ) ).

% filterlim_at_top_mult_tendsto_pos
tff(fact_7277_tendsto__neg__powr,axiom,
    ! [A: $tType,Sa: real,F2: fun(A,real),F3: filter(A)] :
      ( aa(real,$o,ord_less(real,Sa),zero_zero(real))
     => ( filterlim(A,real,F2,at_top(real),F3)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_acb(real,fun(fun(A,real),fun(A,real)),Sa),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% tendsto_neg_powr
tff(fact_7278_ln__x__over__x__tendsto__0,axiom,
    filterlim(real,real,aTP_Lamp_acc(real,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(real)) ).

% ln_x_over_x_tendsto_0
tff(fact_7279_tendsto__at__topI__sequentially,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [F2: fun(real,A),Y: A] :
          ( ! [X8: fun(nat,real)] :
              ( filterlim(nat,real,X8,at_top(real),at_top(nat))
             => filterlim(nat,A,aa(fun(nat,real),fun(nat,A),aTP_Lamp_acd(fun(real,A),fun(fun(nat,real),fun(nat,A)),F2),X8),topolo7230453075368039082e_nhds(A,Y),at_top(nat)) )
         => filterlim(real,A,F2,topolo7230453075368039082e_nhds(A,Y),at_top(real)) ) ) ).

% tendsto_at_topI_sequentially
tff(fact_7280_tendsto__power__div__exp__0,axiom,
    ! [K: nat] : filterlim(real,real,aTP_Lamp_ace(nat,fun(real,real),K),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(real)) ).

% tendsto_power_div_exp_0
tff(fact_7281_tendsto__exp__limit__at__top,axiom,
    ! [X: real] : filterlim(real,real,aTP_Lamp_acf(real,fun(real,real),X),topolo7230453075368039082e_nhds(real,aa(real,real,exp(real),X)),at_top(real)) ).

% tendsto_exp_limit_at_top
tff(fact_7282_filterlim__tan__at__left,axiom,
    filterlim(real,real,tan(real),at_top(real),topolo174197925503356063within(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))),set_ord_lessThan(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))) ).

% filterlim_tan_at_left
tff(fact_7283_tendsto__arctan__at__top,axiom,
    filterlim(real,real,arctan,topolo7230453075368039082e_nhds(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),at_top(real)) ).

% tendsto_arctan_at_top
tff(fact_7284_has__derivative__iff__norm,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),X: A,Sa: set(A)] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,Sa))
        <=> ( real_V3181309239436604168linear(A,B,F7)
            & filterlim(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_acg(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),F2),F7),X),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,X,Sa)) ) ) ) ).

% has_derivative_iff_norm
tff(fact_7285_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B3: real,F2: fun(real,real),Flim: real] :
      ( ! [X4: real] :
          ( aa(real,$o,ord_less_eq(real,B3),X4)
         => ? [Y4: real] :
              ( has_field_derivative(real,F2,Y4,topolo174197925503356063within(real,X4,top_top(set(real))))
              & aa(real,$o,ord_less(real,Y4),zero_zero(real)) ) )
     => ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,Flim),at_top(real))
       => aa(real,$o,ord_less(real,Flim),aa(real,real,F2,B3)) ) ) ).

% DERIV_neg_imp_decreasing_at_top
tff(fact_7286_has__derivative__at__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),X: A,Sa: set(A)] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,Sa))
        <=> ( real_V3181309239436604168linear(A,B,F7)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_ach(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F2),F7),X),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,X,Sa)) ) ) ) ).

% has_derivative_at_within
tff(fact_7287_has__derivativeI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F7: fun(A,B),X: A,F2: fun(A,B),Sa: set(A)] :
          ( real_V3181309239436604168linear(A,B,F7)
         => ( filterlim(A,B,aa(fun(A,B),fun(A,B),aa(A,fun(fun(A,B),fun(A,B)),aTP_Lamp_aci(fun(A,B),fun(A,fun(fun(A,B),fun(A,B))),F7),X),F2),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,X,Sa))
           => has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,Sa)) ) ) ) ).

% has_derivativeI
tff(fact_7288_has__derivative__iff__Ex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),X: A] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,F7)
            & ? [E3: fun(A,B)] :
                ( ! [H4: A] : aa(A,B,F2,aa(A,A,plus_plus(A,X),H4)) = aa(B,B,plus_plus(B,aa(B,B,plus_plus(B,aa(A,B,F2,X)),aa(A,B,F7,H4))),aa(A,B,E3,H4))
                & filterlim(A,real,aTP_Lamp_acj(fun(A,B),fun(A,real),E3),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ) ).

% has_derivative_iff_Ex
tff(fact_7289_has__derivative__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),X: A,Sa: set(A)] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,Sa))
        <=> ( real_V3181309239436604168linear(A,B,F7)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_abm(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F2),F7),X),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,X,Sa)) ) ) ) ).

% has_derivative_within
tff(fact_7290_has__derivative__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),F3: filter(A)] :
          ( has_derivative(A,B,F2,F7,F3)
        <=> ( real_V3181309239436604168linear(A,B,F7)
            & filterlim(A,B,aa(filter(A),fun(A,B),aa(fun(A,B),fun(filter(A),fun(A,B)),aTP_Lamp_ack(fun(A,B),fun(fun(A,B),fun(filter(A),fun(A,B))),F2),F7),F3),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ) ).

% has_derivative_def
tff(fact_7291_has__derivative__at__within__iff__Ex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [X: A,S: set(A),F2: fun(A,B),F7: fun(A,B)] :
          ( member(A,X,S)
         => ( topolo1002775350975398744n_open(A,S)
           => ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,S))
            <=> ( real_V3181309239436604168linear(A,B,F7)
                & ? [E3: fun(A,B)] :
                    ( ! [H4: A] :
                        ( member(A,aa(A,A,plus_plus(A,X),H4),S)
                       => ( aa(A,B,F2,aa(A,A,plus_plus(A,X),H4)) = aa(B,B,plus_plus(B,aa(B,B,plus_plus(B,aa(A,B,F2,X)),aa(A,B,F7,H4))),aa(A,B,E3,H4)) ) )
                    & filterlim(A,real,aTP_Lamp_acj(fun(A,B),fun(A,real),E3),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ) ) ) ).

% has_derivative_at_within_iff_Ex
tff(fact_7292_lim__const,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A3: A] : topolo3827282254853284352ce_Lim(nat,A,at_top(nat),aTP_Lamp_zr(A,fun(nat,A),A3)) = A3 ) ).

% lim_const
tff(fact_7293_open__empty,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => topolo1002775350975398744n_open(A,bot_bot(set(A))) ) ).

% open_empty
tff(fact_7294_open__Un,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),T3: set(A)] :
          ( topolo1002775350975398744n_open(A,S)
         => ( topolo1002775350975398744n_open(A,T3)
           => topolo1002775350975398744n_open(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T3)) ) ) ) ).

% open_Un
tff(fact_7295_open__right,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S: set(A),X: A,Y: A] :
          ( topolo1002775350975398744n_open(A,S)
         => ( member(A,X,S)
           => ( aa(A,$o,ord_less(A,X),Y)
             => ? [B5: A] :
                  ( aa(A,$o,ord_less(A,X),B5)
                  & aa(set(A),$o,ord_less_eq(set(A),set_or7035219750837199246ssThan(A,X,B5)),S) ) ) ) ) ) ).

% open_right
tff(fact_7296_not__open__singleton,axiom,
    ! [A: $tType] :
      ( topolo8386298272705272623_space(A)
     => ! [X: A] : ~ topolo1002775350975398744n_open(A,aa(set(A),set(A),insert(A,X),bot_bot(set(A)))) ) ).

% not_open_singleton
tff(fact_7297_open__Collect__conj,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [P: fun(A,$o),Q: fun(A,$o)] :
          ( topolo1002775350975398744n_open(A,collect(A,P))
         => ( topolo1002775350975398744n_open(A,collect(A,Q))
           => topolo1002775350975398744n_open(A,collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_acl(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q))) ) ) ) ).

% open_Collect_conj
tff(fact_7298_open__Collect__disj,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [P: fun(A,$o),Q: fun(A,$o)] :
          ( topolo1002775350975398744n_open(A,collect(A,P))
         => ( topolo1002775350975398744n_open(A,collect(A,Q))
           => topolo1002775350975398744n_open(A,collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_acm(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q))) ) ) ) ).

% open_Collect_disj
tff(fact_7299_open__Collect__const,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [P: $o] : topolo1002775350975398744n_open(A,collect(A,aTP_Lamp_acn($o,fun(A,$o),(P)))) ) ).

% open_Collect_const
tff(fact_7300_Lim__ident__at,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [X: A,Sa: set(A)] :
          ( ( topolo174197925503356063within(A,X,Sa) != bot_bot(filter(A)) )
         => ( topolo3827282254853284352ce_Lim(A,A,topolo174197925503356063within(A,X,Sa),aTP_Lamp_us(A,A)) = X ) ) ) ).

% Lim_ident_at
tff(fact_7301_tendsto__Lim,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(B)
     => ! [Net: filter(A),F2: fun(A,B),L: B] :
          ( ( Net != bot_bot(filter(A)) )
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),Net)
           => ( topolo3827282254853284352ce_Lim(A,B,Net,F2) = L ) ) ) ) ).

% tendsto_Lim
tff(fact_7302_continuous__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [F3: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( topolo3448309680560233919inuous(A,B,F3,G)
           => ( ( aa(A,B,G,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_us(A,A))) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,F3,aa(fun(A,B),fun(A,B),aTP_Lamp_yq(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% continuous_divide
tff(fact_7303_continuous__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( ( aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_us(A,A))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_yt(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_inverse
tff(fact_7304_continuous__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
        <=> filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_us(A,A)))),F3) ) ) ).

% continuous_def
tff(fact_7305_continuous__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( ( aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_us(A,A))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_yu(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_sgn
tff(fact_7306_t2__space__class_OLim__def,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(A)
     => ! [A2: filter(B),F2: fun(B,A)] : topolo3827282254853284352ce_Lim(B,A,A2,F2) = the(A,aa(fun(B,A),fun(A,$o),aTP_Lamp_aco(filter(B),fun(fun(B,A),fun(A,$o)),A2),F2)) ) ).

% t2_space_class.Lim_def
tff(fact_7307_continuous__powr,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => ( topolo3448309680560233919inuous(A,real,F3,G)
           => ( ( aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_us(A,A))) != zero_zero(real) )
             => topolo3448309680560233919inuous(A,real,F3,aa(fun(A,real),fun(A,real),aTP_Lamp_zv(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ).

% continuous_powr
tff(fact_7308_continuous__ln,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => ( ( aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_us(A,A))) != zero_zero(real) )
           => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_zw(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_ln
tff(fact_7309_at__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A3: A] :
          ( ( topolo174197925503356063within(A,A3,top_top(set(A))) = bot_bot(filter(A)) )
        <=> topolo1002775350975398744n_open(A,aa(set(A),set(A),insert(A,A3),bot_bot(set(A)))) ) ) ).

% at_eq_bot_iff
tff(fact_7310_suminf__eq__lim,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [F2: fun(nat,A)] : suminf(A,F2) = topolo3827282254853284352ce_Lim(nat,A,at_top(nat),aTP_Lamp_aaz(fun(nat,A),fun(nat,A),F2)) ) ).

% suminf_eq_lim
tff(fact_7311_Topological__Spaces_Olim__def,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [X5: fun(nat,A)] : topolo3827282254853284352ce_Lim(nat,A,at_top(nat),X5) = the(A,aTP_Lamp_acp(fun(nat,A),fun(A,$o),X5)) ) ).

% Topological_Spaces.lim_def
tff(fact_7312_continuous__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F3: filter(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,F3,F2)
         => ( ( cos(A,aa(A,A,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_acq(A,A)))) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,F3,aTP_Lamp_wi(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_tan
tff(fact_7313_continuous__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F3: filter(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,F3,F2)
         => ( ( sin(A,aa(A,A,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_acq(A,A)))) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,F3,aTP_Lamp_wh(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_cot
tff(fact_7314_continuous__tanh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( ( cosh(B,aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_us(A,A)))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_yz(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_tanh
tff(fact_7315_continuous__arcosh,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => ( aa(real,$o,ord_less(real,one_one(real)),aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_us(A,A))))
           => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_acr(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_arcosh
tff(fact_7316_continuous__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => ( topolo3448309680560233919inuous(A,real,F3,G)
           => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_us(A,A))))
             => ( ( aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_us(A,A))) != one_one(real) )
               => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(A,real,G,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_us(A,A))))
                 => topolo3448309680560233919inuous(A,real,F3,aa(fun(A,real),fun(A,real),aTP_Lamp_aaq(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% continuous_log
tff(fact_7317_continuous__artanh,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => ( member(real,aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_us(A,A))),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real)))
           => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_acs(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_artanh
tff(fact_7318_tendsto__offset__zero__iff,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & topolo4958980785337419405_space(C)
        & zero(A) )
     => ! [A3: B,S: set(B),F2: fun(B,C),L5: C] :
          ( nO_MATCH(A,B,zero_zero(A),A3)
         => ( member(B,A3,S)
           => ( topolo1002775350975398744n_open(B,S)
             => ( filterlim(B,C,F2,topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,A3,S))
              <=> filterlim(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_act(B,fun(fun(B,C),fun(B,C)),A3),F2),topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,zero_zero(B),top_top(set(B)))) ) ) ) ) ) ).

% tendsto_offset_zero_iff
tff(fact_7319_has__derivativeI__sandwich,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [E2: real,F7: fun(A,B),Sa: set(A),X: A,F2: fun(A,B),H5: fun(A,real)] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),E2)
         => ( real_V3181309239436604168linear(A,B,F7)
           => ( ! [Y3: A] :
                  ( member(A,Y3,Sa)
                 => ( ( Y3 != X )
                   => ( aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,Y3,X)),E2)
                     => aa(real,$o,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(B,B,minus_minus(B,aa(A,B,F2,Y3)),aa(A,B,F2,X))),aa(A,B,F7,aa(A,A,minus_minus(A,Y3),X))))),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Y3),X)))),aa(A,real,H5,Y3)) ) ) )
             => ( filterlim(A,real,H5,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,X,Sa))
               => has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,Sa)) ) ) ) ) ) ).

% has_derivativeI_sandwich
tff(fact_7320_dist__eq__0__iff,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y: A] :
          ( ( real_V557655796197034286t_dist(A,X,Y) = zero_zero(real) )
        <=> ( X = Y ) ) ) ).

% dist_eq_0_iff
tff(fact_7321_dist__self,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A] : real_V557655796197034286t_dist(A,X,X) = zero_zero(real) ) ).

% dist_self
tff(fact_7322_dist__0__norm,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] : real_V557655796197034286t_dist(A,zero_zero(A),X) = real_V7770717601297561774m_norm(A,X) ) ).

% dist_0_norm
tff(fact_7323_zero__less__dist__iff,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y: A] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),real_V557655796197034286t_dist(A,X,Y))
        <=> ( X != Y ) ) ) ).

% zero_less_dist_iff
tff(fact_7324_dist__le__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y: A] :
          ( aa(real,$o,ord_less_eq(real,real_V557655796197034286t_dist(A,X,Y)),zero_zero(real))
        <=> ( X = Y ) ) ) ).

% dist_le_zero_iff
tff(fact_7325_div__add__self2__no__field,axiom,
    ! [A: $tType,B: $tType] :
      ( ( euclid4440199948858584721cancel(B)
        & field(A) )
     => ! [X: A,B3: B,A3: B] :
          ( nO_MATCH(A,B,X,B3)
         => ( ( B3 != zero_zero(B) )
           => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,plus_plus(B,A3),B3)),B3) = aa(B,B,plus_plus(B,aa(B,B,aa(B,fun(B,B),divide_divide(B),A3),B3)),one_one(B)) ) ) ) ) ).

% div_add_self2_no_field
tff(fact_7326_div__add__self1__no__field,axiom,
    ! [A: $tType,B: $tType] :
      ( ( euclid4440199948858584721cancel(B)
        & field(A) )
     => ! [X: A,B3: B,A3: B] :
          ( nO_MATCH(A,B,X,B3)
         => ( ( B3 != zero_zero(B) )
           => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,plus_plus(B,B3),A3)),B3) = aa(B,B,plus_plus(B,aa(B,B,aa(B,fun(B,B),divide_divide(B),A3),B3)),one_one(B)) ) ) ) ) ).

% div_add_self1_no_field
tff(fact_7327_open__dist,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [S: set(A)] :
          ( topolo1002775350975398744n_open(A,S)
        <=> ! [X3: A] :
              ( member(A,X3,S)
             => ? [E3: real] :
                  ( aa(real,$o,ord_less(real,zero_zero(real)),E3)
                  & ! [Y6: A] :
                      ( aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,Y6,X3)),E3)
                     => member(A,Y6,S) ) ) ) ) ) ).

% open_dist
tff(fact_7328_open__ball,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,D2: real] : topolo1002775350975398744n_open(A,collect(A,aa(real,fun(A,$o),aTP_Lamp_acu(A,fun(real,fun(A,$o)),X),D2))) ) ).

% open_ball
tff(fact_7329_continuous__dist,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [F3: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( topolo3448309680560233919inuous(A,B,F3,G)
           => topolo3448309680560233919inuous(A,real,F3,aa(fun(A,B),fun(A,real),aTP_Lamp_acv(fun(A,B),fun(fun(A,B),fun(A,real)),F2),G)) ) ) ) ).

% continuous_dist
tff(fact_7330_zero__le__dist,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y: A] : aa(real,$o,ord_less_eq(real,zero_zero(real)),real_V557655796197034286t_dist(A,X,Y)) ) ).

% zero_le_dist
tff(fact_7331_norm__conv__dist,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] : real_V7770717601297561774m_norm(A,X) = real_V557655796197034286t_dist(A,X,zero_zero(A)) ) ).

% norm_conv_dist
tff(fact_7332_dist__triangle__less__add,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X12: A,Y: A,E1: real,X22: A,E22: real] :
          ( aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X12,Y)),E1)
         => ( aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X22,Y)),E22)
           => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X12,X22)),aa(real,real,plus_plus(real,E1),E22)) ) ) ) ).

% dist_triangle_less_add
tff(fact_7333_dist__triangle__lt,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Z: A,Y: A,E2: real] :
          ( aa(real,$o,ord_less(real,aa(real,real,plus_plus(real,real_V557655796197034286t_dist(A,X,Z)),real_V557655796197034286t_dist(A,Y,Z))),E2)
         => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X,Y)),E2) ) ) ).

% dist_triangle_lt
tff(fact_7334_dist__commute__lessI,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Y: A,X: A,E2: real] :
          ( aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,Y,X)),E2)
         => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X,Y)),E2) ) ) ).

% dist_commute_lessI
tff(fact_7335_dist__not__less__zero,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y: A] : ~ aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X,Y)),zero_zero(real)) ) ).

% dist_not_less_zero
tff(fact_7336_dist__pos__lt,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y: A] :
          ( ( X != Y )
         => aa(real,$o,ord_less(real,zero_zero(real)),real_V557655796197034286t_dist(A,X,Y)) ) ) ).

% dist_pos_lt
tff(fact_7337_has__field__derivative__transform__within,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F7: A,A3: A,S: set(A),D2: real,G: fun(A,A)] :
          ( has_field_derivative(A,F2,F7,topolo174197925503356063within(A,A3,S))
         => ( aa(real,$o,ord_less(real,zero_zero(real)),D2)
           => ( member(A,A3,S)
             => ( ! [X4: A] :
                    ( member(A,X4,S)
                   => ( aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X4,A3)),D2)
                     => ( aa(A,A,F2,X4) = aa(A,A,G,X4) ) ) )
               => has_field_derivative(A,G,F7,topolo174197925503356063within(A,A3,S)) ) ) ) ) ) ).

% has_field_derivative_transform_within
tff(fact_7338_has__derivative__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),X: A,Sa: set(A),D2: real,G: fun(A,B)] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,Sa))
         => ( aa(real,$o,ord_less(real,zero_zero(real)),D2)
           => ( member(A,X,Sa)
             => ( ! [X7: A] :
                    ( member(A,X7,Sa)
                   => ( aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X7,X)),D2)
                     => ( aa(A,B,F2,X7) = aa(A,B,G,X7) ) ) )
               => has_derivative(A,B,G,F7,topolo174197925503356063within(A,X,Sa)) ) ) ) ) ) ).

% has_derivative_transform_within
tff(fact_7339_Cauchy__def,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X5)
        <=> ! [E3: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),E3)
             => ? [M9: nat] :
                ! [M3: nat] :
                  ( aa(nat,$o,ord_less_eq(nat,M9),M3)
                 => ! [N3: nat] :
                      ( aa(nat,$o,ord_less_eq(nat,M9),N3)
                     => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,aa(nat,A,X5,M3),aa(nat,A,X5,N3))),E3) ) ) ) ) ) ).

% Cauchy_def
tff(fact_7340_Cauchy__altdef2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Sa: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,Sa)
        <=> ! [E3: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),E3)
             => ? [N8: nat] :
                ! [N3: nat] :
                  ( aa(nat,$o,ord_less_eq(nat,N8),N3)
                 => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,aa(nat,A,Sa,N3),aa(nat,A,Sa,N8))),E3) ) ) ) ) ).

% Cauchy_altdef2
tff(fact_7341_metric__CauchyD,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A),E2: real] :
          ( topolo3814608138187158403Cauchy(A,X5)
         => ( aa(real,$o,ord_less(real,zero_zero(real)),E2)
           => ? [M8: nat] :
              ! [M2: nat] :
                ( aa(nat,$o,ord_less_eq(nat,M8),M2)
               => ! [N10: nat] :
                    ( aa(nat,$o,ord_less_eq(nat,M8),N10)
                   => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,aa(nat,A,X5,M2),aa(nat,A,X5,N10))),E2) ) ) ) ) ) ).

% metric_CauchyD
tff(fact_7342_metric__CauchyI,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A)] :
          ( ! [E: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),E)
             => ? [M10: nat] :
                ! [M4: nat] :
                  ( aa(nat,$o,ord_less_eq(nat,M10),M4)
                 => ! [N: nat] :
                      ( aa(nat,$o,ord_less_eq(nat,M10),N)
                     => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,aa(nat,A,X5,M4),aa(nat,A,X5,N))),E) ) ) )
         => topolo3814608138187158403Cauchy(A,X5) ) ) ).

% metric_CauchyI
tff(fact_7343_tendsto__dist,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A),G: fun(A,B),M: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,M),F3)
           => filterlim(A,real,aa(fun(A,B),fun(A,real),aTP_Lamp_acw(fun(A,B),fun(fun(A,B),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,real_V557655796197034286t_dist(B,L,M)),F3) ) ) ) ).

% tendsto_dist
tff(fact_7344_distrib__left__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring(B)
     => ! [X: A,Y: A,A3: B,B3: B,C2: B] :
          ( nO_MATCH(A,B,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y),A3)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),A3),aa(B,B,plus_plus(B,B3),C2)) = aa(B,B,plus_plus(B,aa(B,B,aa(B,fun(B,B),times_times(B),A3),B3)),aa(B,B,aa(B,fun(B,B),times_times(B),A3),C2)) ) ) ) ).

% distrib_left_NO_MATCH
tff(fact_7345_distrib__right__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring(B)
     => ! [X: A,Y: A,C2: B,A3: B,B3: B] :
          ( nO_MATCH(A,B,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y),C2)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,plus_plus(B,A3),B3)),C2) = aa(B,B,plus_plus(B,aa(B,B,aa(B,fun(B,B),times_times(B),A3),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),B3),C2)) ) ) ) ).

% distrib_right_NO_MATCH
tff(fact_7346_left__diff__distrib__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( ring(B)
     => ! [X: A,Y: A,C2: B,A3: B,B3: B] :
          ( nO_MATCH(A,B,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y),C2)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,minus_minus(B,A3),B3)),C2) = aa(B,B,minus_minus(B,aa(B,B,aa(B,fun(B,B),times_times(B),A3),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),B3),C2)) ) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_7347_right__diff__distrib__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( ring(B)
     => ! [X: A,Y: A,A3: B,B3: B,C2: B] :
          ( nO_MATCH(A,B,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y),A3)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),A3),aa(B,B,minus_minus(B,B3),C2)) = aa(B,B,minus_minus(B,aa(B,B,aa(B,fun(B,B),times_times(B),A3),B3)),aa(B,B,aa(B,fun(B,B),times_times(B),A3),C2)) ) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_7348_dist__triangle__half__r,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Y: A,X12: A,E2: real,X22: A] :
          ( aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,Y,X12)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E2),aa(num,real,numeral_numeral(real),bit0(one2))))
         => ( aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,Y,X22)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E2),aa(num,real,numeral_numeral(real),bit0(one2))))
           => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X12,X22)),E2) ) ) ) ).

% dist_triangle_half_r
tff(fact_7349_dist__triangle__half__l,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X12: A,Y: A,E2: real,X22: A] :
          ( aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X12,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E2),aa(num,real,numeral_numeral(real),bit0(one2))))
         => ( aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X22,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E2),aa(num,real,numeral_numeral(real),bit0(one2))))
           => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X12,X22)),E2) ) ) ) ).

% dist_triangle_half_l
tff(fact_7350_dist__triangle__third,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X12: A,X22: A,E2: real,X32: A,X42: A] :
          ( aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X12,X22)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E2),aa(num,real,numeral_numeral(real),bit1(one2))))
         => ( aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X22,X32)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E2),aa(num,real,numeral_numeral(real),bit1(one2))))
           => ( aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X32,X42)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E2),aa(num,real,numeral_numeral(real),bit1(one2))))
             => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X12,X42)),E2) ) ) ) ) ).

% dist_triangle_third
tff(fact_7351_Lim__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),L: B,X: A,S: set(A),D2: real,G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,X,S))
         => ( aa(real,$o,ord_less(real,zero_zero(real)),D2)
           => ( ! [X7: A] :
                  ( member(A,X7,S)
                 => ( aa(real,$o,ord_less(real,zero_zero(real)),real_V557655796197034286t_dist(A,X7,X))
                   => ( aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X7,X)),D2)
                     => ( aa(A,B,F2,X7) = aa(A,B,G,X7) ) ) ) )
             => filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,X,S)) ) ) ) ) ).

% Lim_transform_within
tff(fact_7352_filterlim__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [G: fun(A,B),G2: filter(B),X: A,S: set(A),F3: filter(B),D2: real,F2: fun(A,B)] :
          ( filterlim(A,B,G,G2,topolo174197925503356063within(A,X,S))
         => ( aa(filter(B),$o,ord_less_eq(filter(B),G2),F3)
           => ( aa(real,$o,ord_less(real,zero_zero(real)),D2)
             => ( ! [X7: A] :
                    ( member(A,X7,S)
                   => ( aa(real,$o,ord_less(real,zero_zero(real)),real_V557655796197034286t_dist(A,X7,X))
                     => ( aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X7,X)),D2)
                       => ( aa(A,B,F2,X7) = aa(A,B,G,X7) ) ) ) )
               => filterlim(A,B,F2,F3,topolo174197925503356063within(A,X,S)) ) ) ) ) ) ).

% filterlim_transform_within
tff(fact_7353_Cauchy__altdef,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [F2: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,F2)
        <=> ! [E3: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),E3)
             => ? [M9: nat] :
                ! [M3: nat] :
                  ( aa(nat,$o,ord_less_eq(nat,M9),M3)
                 => ! [N3: nat] :
                      ( aa(nat,$o,ord_less(nat,M3),N3)
                     => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,aa(nat,A,F2,M3),aa(nat,A,F2,N3))),E3) ) ) ) ) ) ).

% Cauchy_altdef
tff(fact_7354_CauchyI_H,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A)] :
          ( ! [E: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),E)
             => ? [M10: nat] :
                ! [M4: nat] :
                  ( aa(nat,$o,ord_less_eq(nat,M10),M4)
                 => ! [N: nat] :
                      ( aa(nat,$o,ord_less(nat,M4),N)
                     => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,aa(nat,A,X5,M4),aa(nat,A,X5,N))),E) ) ) )
         => topolo3814608138187158403Cauchy(A,X5) ) ) ).

% CauchyI'
tff(fact_7355_dist__of__nat,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [M: nat,Na: nat] : real_V557655796197034286t_dist(A,aa(nat,A,semiring_1_of_nat(A),M),aa(nat,A,semiring_1_of_nat(A),Na)) = aa(int,real,ring_1_of_int(real),aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),Na)))) ) ).

% dist_of_nat
tff(fact_7356_tendsto__dist__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
        <=> filterlim(A,real,aa(B,fun(A,real),aTP_Lamp_acx(fun(A,B),fun(B,fun(A,real)),F2),L),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% tendsto_dist_iff
tff(fact_7357_LIM__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [F2: fun(A,B),L5: B,A3: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A3,top_top(set(A))))
        <=> ! [R5: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),R5)
             => ? [S7: real] :
                  ( aa(real,$o,ord_less(real,zero_zero(real)),S7)
                  & ! [X3: A] :
                      ( ( ( X3 != A3 )
                        & aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X3,A3)),S7) )
                     => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(B,aa(A,B,F2,X3),L5)),R5) ) ) ) ) ) ).

% LIM_def
tff(fact_7358_metric__LIM__D,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [F2: fun(A,B),L5: B,A3: A,R3: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A3,top_top(set(A))))
         => ( aa(real,$o,ord_less(real,zero_zero(real)),R3)
           => ? [S2: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),S2)
                & ! [X2: A] :
                    ( ( ( X2 != A3 )
                      & aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X2,A3)),S2) )
                   => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(B,aa(A,B,F2,X2),L5)),R3) ) ) ) ) ) ).

% metric_LIM_D
tff(fact_7359_metric__LIM__I,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [A3: A,F2: fun(A,B),L5: B] :
          ( ! [R2: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),R2)
             => ? [S6: real] :
                  ( aa(real,$o,ord_less(real,zero_zero(real)),S6)
                  & ! [X4: A] :
                      ( ( ( X4 != A3 )
                        & aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X4,A3)),S6) )
                     => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(B,aa(A,B,F2,X4),L5)),R2) ) ) )
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ).

% metric_LIM_I
tff(fact_7360_metric__LIM__equal2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [G: fun(A,B),L: B,A3: A,R: real,F2: fun(A,B)] :
          ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A3,top_top(set(A))))
         => ( aa(real,$o,ord_less(real,zero_zero(real)),R)
           => ( ! [X4: A] :
                  ( ( X4 != A3 )
                 => ( aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X4,A3)),R)
                   => ( aa(A,B,F2,X4) = aa(A,B,G,X4) ) ) )
             => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ) ) ).

% metric_LIM_equal2
tff(fact_7361_metric__LIMSEQ__D,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A),L5: A,R3: real] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
         => ( aa(real,$o,ord_less(real,zero_zero(real)),R3)
           => ? [No: nat] :
              ! [N10: nat] :
                ( aa(nat,$o,ord_less_eq(nat,No),N10)
               => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,aa(nat,A,X5,N10),L5)),R3) ) ) ) ) ).

% metric_LIMSEQ_D
tff(fact_7362_metric__LIMSEQ__I,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A),L5: A] :
          ( ! [R2: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),R2)
             => ? [No2: nat] :
                ! [N: nat] :
                  ( aa(nat,$o,ord_less_eq(nat,No2),N)
                 => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,aa(nat,A,X5,N),L5)),R2) ) )
         => filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat)) ) ) ).

% metric_LIMSEQ_I
tff(fact_7363_lim__sequentially,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A),L5: A] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
        <=> ! [R5: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),R5)
             => ? [No3: nat] :
                ! [N3: nat] :
                  ( aa(nat,$o,ord_less_eq(nat,No3),N3)
                 => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,aa(nat,A,X5,N3),L5)),R5) ) ) ) ) ).

% lim_sequentially
tff(fact_7364_power__minus_H,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: A,Na: nat] :
          ( nO_MATCH(A,A,one_one(A),X)
         => ( aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),X)),Na) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),Na)),aa(nat,A,power_power(A,X),Na)) ) ) ) ).

% power_minus'
tff(fact_7365_metric__Cauchy__iff2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X5)
        <=> ! [J3: nat] :
            ? [M9: nat] :
            ! [M3: nat] :
              ( aa(nat,$o,ord_less_eq(nat,M9),M3)
             => ! [N3: nat] :
                  ( aa(nat,$o,ord_less_eq(nat,M9),N3)
                 => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,aa(nat,A,X5,M3),aa(nat,A,X5,N3))),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,J3)))) ) ) ) ) ).

% metric_Cauchy_iff2
tff(fact_7366_metric__LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),B3: B,A3: A,G: fun(B,C),C2: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,B3),topolo174197925503356063within(A,A3,top_top(set(A))))
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(B,B3,top_top(set(B))))
           => ( ? [D5: real] :
                  ( aa(real,$o,ord_less(real,zero_zero(real)),D5)
                  & ! [X4: A] :
                      ( ( ( X4 != A3 )
                        & aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X4,A3)),D5) )
                     => ( aa(A,B,F2,X4) != B3 ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_acy(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ) ) ).

% metric_LIM_compose2
tff(fact_7367_metric__isCont__LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [A3: A,F2: fun(A,B),G: fun(B,C),L: C] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(B,aa(A,B,F2,A3),top_top(set(B))))
           => ( ? [D5: real] :
                  ( aa(real,$o,ord_less(real,zero_zero(real)),D5)
                  & ! [X4: A] :
                      ( ( ( X4 != A3 )
                        & aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X4,A3)),D5) )
                     => ( aa(A,B,F2,X4) != aa(A,B,F2,A3) ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_acy(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ) ) ).

% metric_isCont_LIM_compose2
tff(fact_7368_LIMSEQ__iff__nz,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A),L5: A] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
        <=> ! [R5: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),R5)
             => ? [No3: nat] :
                  ( aa(nat,$o,ord_less(nat,zero_zero(nat)),No3)
                  & ! [N3: nat] :
                      ( aa(nat,$o,ord_less_eq(nat,No3),N3)
                     => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,aa(nat,A,X5,N3),L5)),R5) ) ) ) ) ) ).

% LIMSEQ_iff_nz
tff(fact_7369_LIM__offset__zero__iff,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & topolo4958980785337419405_space(C)
        & zero(A) )
     => ! [A3: B,F2: fun(B,C),L5: C] :
          ( nO_MATCH(A,B,zero_zero(A),A3)
         => ( filterlim(B,C,F2,topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,A3,top_top(set(B))))
          <=> filterlim(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_act(B,fun(fun(B,C),fun(B,C)),A3),F2),topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,zero_zero(B),top_top(set(B)))) ) ) ) ).

% LIM_offset_zero_iff
tff(fact_7370_filterlim__pow__at__bot__even,axiom,
    ! [Na: nat,F2: fun(real,real),F3: filter(real)] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( filterlim(real,real,F2,at_bot(real),F3)
       => ( aa(nat,$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_acz(nat,fun(fun(real,real),fun(real,real)),Na),F2),at_top(real),F3) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_7371_tendsto__exp__limit__at__right,axiom,
    ! [X: real] : filterlim(real,real,aTP_Lamp_ada(real,fun(real,real),X),topolo7230453075368039082e_nhds(real,aa(real,real,exp(real),X)),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ).

% tendsto_exp_limit_at_right
tff(fact_7372_greaterThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,K: A] :
          ( member(A,I,set_ord_greaterThan(A,K))
        <=> aa(A,$o,ord_less(A,K),I) ) ) ).

% greaterThan_iff
tff(fact_7373_trivial__limit__at__right__real,axiom,
    ! [A: $tType] :
      ( ( dense_order(A)
        & no_top(A)
        & topolo1944317154257567458pology(A) )
     => ! [X: A] : topolo174197925503356063within(A,X,set_ord_greaterThan(A,X)) != bot_bot(filter(A)) ) ).

% trivial_limit_at_right_real
tff(fact_7374_trivial__limit__at__bot__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( at_bot(A) != bot_bot(filter(A)) ) ) ).

% trivial_limit_at_bot_linorder
tff(fact_7375_greaterThan__non__empty,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [X: A] : set_ord_greaterThan(A,X) != bot_bot(set(A)) ) ).

% greaterThan_non_empty
tff(fact_7376_greaterThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A] : set_ord_greaterThan(A,L) = collect(A,ord_less(A,L)) ) ).

% greaterThan_def
tff(fact_7377_ln__at__0,axiom,
    filterlim(real,real,ln_ln(real),at_bot(real),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ).

% ln_at_0
tff(fact_7378_artanh__real__at__right__1,axiom,
    filterlim(real,real,artanh(real),at_bot(real),topolo174197925503356063within(real,aa(real,real,uminus_uminus(real),one_one(real)),set_ord_greaterThan(real,aa(real,real,uminus_uminus(real),one_one(real))))) ).

% artanh_real_at_right_1
tff(fact_7379_at__within__Icc__at__right,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( topolo174197925503356063within(A,A3,set_or1337092689740270186AtMost(A,A3,B3)) = topolo174197925503356063within(A,A3,set_ord_greaterThan(A,A3)) ) ) ) ).

% at_within_Icc_at_right
tff(fact_7380_trivial__limit__at__right__top,axiom,
    ! [A: $tType] :
      ( ( order_top(A)
        & topolo1944317154257567458pology(A) )
     => ( topolo174197925503356063within(A,top_top(A),set_ord_greaterThan(A,top_top(A))) = bot_bot(filter(A)) ) ) ).

% trivial_limit_at_right_top
tff(fact_7381_filterlim__uminus__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,at_bot(real),F3)
    <=> filterlim(A,real,aTP_Lamp_adb(fun(A,real),fun(A,real),F2),at_top(real),F3) ) ).

% filterlim_uminus_at_bot
tff(fact_7382_filterlim__uminus__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,at_top(real),F3)
    <=> filterlim(A,real,aTP_Lamp_adb(fun(A,real),fun(A,real),F2),at_bot(real),F3) ) ).

% filterlim_uminus_at_top
tff(fact_7383_filterlim__at__bot__mirror,axiom,
    ! [A: $tType,F2: fun(real,A),F3: filter(A)] :
      ( filterlim(real,A,F2,F3,at_bot(real))
    <=> filterlim(real,A,aTP_Lamp_adc(fun(real,A),fun(real,A),F2),F3,at_top(real)) ) ).

% filterlim_at_bot_mirror
tff(fact_7384_filterlim__at__top__mirror,axiom,
    ! [A: $tType,F2: fun(real,A),F3: filter(A)] :
      ( filterlim(real,A,F2,F3,at_top(real))
    <=> filterlim(real,A,aTP_Lamp_adc(fun(real,A),fun(real,A),F2),F3,at_bot(real)) ) ).

% filterlim_at_top_mirror
tff(fact_7385_filterlim__uminus__at__bot__at__top,axiom,
    filterlim(real,real,uminus_uminus(real),at_bot(real),at_top(real)) ).

% filterlim_uminus_at_bot_at_top
tff(fact_7386_filterlim__uminus__at__top__at__bot,axiom,
    filterlim(real,real,uminus_uminus(real),at_top(real),at_bot(real)) ).

% filterlim_uminus_at_top_at_bot
tff(fact_7387_filterlim__at__right__to__0,axiom,
    ! [A: $tType,F2: fun(real,A),F3: filter(A),A3: real] :
      ( filterlim(real,A,F2,F3,topolo174197925503356063within(real,A3,set_ord_greaterThan(real,A3)))
    <=> filterlim(real,A,aa(real,fun(real,A),aTP_Lamp_add(fun(real,A),fun(real,fun(real,A)),F2),A3),F3,topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ) ).

% filterlim_at_right_to_0
tff(fact_7388_filterlim__at__left__to__right,axiom,
    ! [A: $tType,F2: fun(real,A),F3: filter(A),A3: real] :
      ( filterlim(real,A,F2,F3,topolo174197925503356063within(real,A3,set_ord_lessThan(real,A3)))
    <=> filterlim(real,A,aTP_Lamp_adc(fun(real,A),fun(real,A),F2),F3,topolo174197925503356063within(real,aa(real,real,uminus_uminus(real),A3),set_ord_greaterThan(real,aa(real,real,uminus_uminus(real),A3)))) ) ).

% filterlim_at_left_to_right
tff(fact_7389_filterlim__tan__at__right,axiom,
    filterlim(real,real,tan(real),at_bot(real),topolo174197925503356063within(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)))),set_ord_greaterThan(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))))))) ).

% filterlim_tan_at_right
tff(fact_7390_exp__at__bot,axiom,
    filterlim(real,real,exp(real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_bot(real)) ).

% exp_at_bot
tff(fact_7391_filterlim__times__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linordered_field(B)
        & topolo1944317154257567458pology(B) )
     => ! [F2: fun(A,B),P2: B,F12: filter(A),C2: B,L: B] :
          ( filterlim(A,B,F2,topolo174197925503356063within(B,P2,set_ord_greaterThan(B,P2)),F12)
         => ( aa(B,$o,ord_less(B,zero_zero(B)),C2)
           => ( ( L = aa(B,B,aa(B,fun(B,B),times_times(B),C2),P2) )
             => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_ade(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo174197925503356063within(B,L,set_ord_greaterThan(B,L)),F12) ) ) ) ) ).

% filterlim_times_pos
tff(fact_7392_filterlim__at__right__to__top,axiom,
    ! [A: $tType,F2: fun(real,A),F3: filter(A)] :
      ( filterlim(real,A,F2,F3,topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
    <=> filterlim(real,A,aTP_Lamp_adf(fun(real,A),fun(real,A),F2),F3,at_top(real)) ) ).

% filterlim_at_right_to_top
tff(fact_7393_filterlim__at__top__to__right,axiom,
    ! [A: $tType,F2: fun(real,A),F3: filter(A)] :
      ( filterlim(real,A,F2,F3,at_top(real))
    <=> filterlim(real,A,aTP_Lamp_adf(fun(real,A),fun(real,A),F2),F3,topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ) ).

% filterlim_at_top_to_right
tff(fact_7394_filterlim__inverse__at__right__top,axiom,
    filterlim(real,real,inverse_inverse(real),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))),at_top(real)) ).

% filterlim_inverse_at_right_top
tff(fact_7395_filterlim__inverse__at__top__right,axiom,
    filterlim(real,real,inverse_inverse(real),at_top(real),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ).

% filterlim_inverse_at_top_right
tff(fact_7396_log__inj,axiom,
    ! [B3: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),B3)
     => inj_on(real,real,log(B3),set_ord_greaterThan(real,zero_zero(real))) ) ).

% log_inj
tff(fact_7397_tanh__real__at__bot,axiom,
    filterlim(real,real,tanh(real),topolo7230453075368039082e_nhds(real,aa(real,real,uminus_uminus(real),one_one(real))),at_bot(real)) ).

% tanh_real_at_bot
tff(fact_7398_filterlim__tendsto__pos__mult__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
       => ( filterlim(A,real,G,at_bot(real),F3)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_abq(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_bot(real),F3) ) ) ) ).

% filterlim_tendsto_pos_mult_at_bot
tff(fact_7399_tendsto__arcosh__at__left__1,axiom,
    filterlim(real,real,arcosh(real),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,one_one(real),set_ord_greaterThan(real,one_one(real)))) ).

% tendsto_arcosh_at_left_1
tff(fact_7400_tendsto__at__botI__sequentially,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [F2: fun(real,A),Y: A] :
          ( ! [X8: fun(nat,real)] :
              ( filterlim(nat,real,X8,at_bot(real),at_top(nat))
             => filterlim(nat,A,aa(fun(nat,real),fun(nat,A),aTP_Lamp_acd(fun(real,A),fun(fun(nat,real),fun(nat,A)),F2),X8),topolo7230453075368039082e_nhds(A,Y),at_top(nat)) )
         => filterlim(real,A,F2,topolo7230453075368039082e_nhds(A,Y),at_bot(real)) ) ) ).

% tendsto_at_botI_sequentially
tff(fact_7401_filterlim__inverse__at__bot__neg,axiom,
    filterlim(real,real,inverse_inverse(real),at_bot(real),topolo174197925503356063within(real,zero_zero(real),set_ord_lessThan(real,zero_zero(real)))) ).

% filterlim_inverse_at_bot_neg
tff(fact_7402_filterlim__tendsto__neg__mult__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F3)
     => ( aa(real,$o,ord_less(real,C2),zero_zero(real))
       => ( filterlim(A,real,G,at_top(real),F3)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_abq(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_bot(real),F3) ) ) ) ).

% filterlim_tendsto_neg_mult_at_bot
tff(fact_7403_isCont__If__ge,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A3: A,G: fun(A,B),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,set_ord_lessThan(A,A3)),G)
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,G,A3)),topolo174197925503356063within(A,A3,set_ord_greaterThan(A,A3)))
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_adg(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A3),G),F2)) ) ) ) ).

% isCont_If_ge
tff(fact_7404_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B3: real,F2: fun(real,real),Flim: real] :
      ( ! [X4: real] :
          ( aa(real,$o,ord_less_eq(real,X4),B3)
         => ? [Y4: real] :
              ( has_field_derivative(real,F2,Y4,topolo174197925503356063within(real,X4,top_top(set(real))))
              & aa(real,$o,ord_less(real,zero_zero(real)),Y4) ) )
     => ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,Flim),at_bot(real))
       => aa(real,$o,ord_less(real,Flim),aa(real,real,F2,B3)) ) ) ).

% DERIV_pos_imp_increasing_at_bot
tff(fact_7405_filterlim__pow__at__bot__odd,axiom,
    ! [Na: nat,F2: fun(real,real),F3: filter(real)] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( filterlim(real,real,F2,at_bot(real),F3)
       => ( ~ aa(nat,$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_acz(nat,fun(fun(real,real),fun(real,real)),Na),F2),at_bot(real),F3) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_7406_tendsto__arctan__at__bot,axiom,
    filterlim(real,real,arctan,topolo7230453075368039082e_nhds(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))))),at_bot(real)) ).

% tendsto_arctan_at_bot
tff(fact_7407_lhopital__left__at__top,axiom,
    ! [G: fun(real,real),X: real,G4: fun(real,real),F2: fun(real,real),F7: fun(real,real),Y: real] :
      ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
     => ( eventually(real,aTP_Lamp_adh(fun(real,real),fun(real,$o),G4),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_adi(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F7),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_adi(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G4),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_adj(fun(real,real),fun(fun(real,real),fun(real,real)),G4),F7),topolo7230453075368039082e_nhds(real,Y),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_adj(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),topolo7230453075368039082e_nhds(real,Y),topolo174197925503356063within(real,X,set_ord_lessThan(real,X))) ) ) ) ) ) ).

% lhopital_left_at_top
tff(fact_7408_lhopital__right__at__top,axiom,
    ! [G: fun(real,real),X: real,G4: fun(real,real),F2: fun(real,real),F7: fun(real,real),Y: real] :
      ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
     => ( eventually(real,aTP_Lamp_adh(fun(real,real),fun(real,$o),G4),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_adi(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F7),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_adi(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G4),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_adj(fun(real,real),fun(fun(real,real),fun(real,real)),G4),F7),topolo7230453075368039082e_nhds(real,Y),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_adj(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),topolo7230453075368039082e_nhds(real,Y),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X))) ) ) ) ) ) ).

% lhopital_right_at_top
tff(fact_7409_eventually__top,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( eventually(A,P,top_top(filter(A)))
    <=> ! [X_1: A] : aa(A,$o,P,X_1) ) ).

% eventually_top
tff(fact_7410_eventually__const,axiom,
    ! [A: $tType,F3: filter(A),P: $o] :
      ( ( F3 != bot_bot(filter(A)) )
     => ( eventually(A,aTP_Lamp_ph($o,fun(A,$o),(P)),F3)
      <=> (P) ) ) ).

% eventually_const
tff(fact_7411_eventually__at__right__field,axiom,
    ! [A: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [P: fun(A,$o),X: A] :
          ( eventually(A,P,topolo174197925503356063within(A,X,set_ord_greaterThan(A,X)))
        <=> ? [B9: A] :
              ( aa(A,$o,ord_less(A,X),B9)
              & ! [Y6: A] :
                  ( aa(A,$o,ord_less(A,X),Y6)
                 => ( aa(A,$o,ord_less(A,Y6),B9)
                   => aa(A,$o,P,Y6) ) ) ) ) ) ).

% eventually_at_right_field
tff(fact_7412_eventually__at__right,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X: A,Y: A,P: fun(A,$o)] :
          ( aa(A,$o,ord_less(A,X),Y)
         => ( eventually(A,P,topolo174197925503356063within(A,X,set_ord_greaterThan(A,X)))
          <=> ? [B9: A] :
                ( aa(A,$o,ord_less(A,X),B9)
                & ! [Y6: A] :
                    ( aa(A,$o,ord_less(A,X),Y6)
                   => ( aa(A,$o,ord_less(A,Y6),B9)
                     => aa(A,$o,P,Y6) ) ) ) ) ) ) ).

% eventually_at_right
tff(fact_7413_eventually__at__bot__dense,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [P: fun(A,$o)] :
          ( eventually(A,P,at_bot(A))
        <=> ? [N8: A] :
            ! [N3: A] :
              ( aa(A,$o,ord_less(A,N3),N8)
             => aa(A,$o,P,N3) ) ) ) ).

% eventually_at_bot_dense
tff(fact_7414_eventually__gt__at__bot,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [C2: A] : eventually(A,aTP_Lamp_adk(A,fun(A,$o),C2),at_bot(A)) ) ).

% eventually_gt_at_bot
tff(fact_7415_eventually__at__bot__not__equal,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [C2: A] : eventually(A,aTP_Lamp_adl(A,fun(A,$o),C2),at_bot(A)) ) ).

% eventually_at_bot_not_equal
tff(fact_7416_eventually__at__bot__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o)] :
          ( eventually(A,P,at_bot(A))
        <=> ? [N8: A] :
            ! [N3: A] :
              ( aa(A,$o,ord_less_eq(A,N3),N8)
             => aa(A,$o,P,N3) ) ) ) ).

% eventually_at_bot_linorder
tff(fact_7417_eventually__le__at__bot,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A] : eventually(A,aTP_Lamp_adm(A,fun(A,$o),C2),at_bot(A)) ) ).

% eventually_le_at_bot
tff(fact_7418_filterlim__at__within__not__equal,axiom,
    ! [B: $tType,A: $tType] :
      ( topological_t2_space(B)
     => ! [F2: fun(A,B),A3: B,Sa: set(B),F3: filter(A),B3: B] :
          ( filterlim(A,B,F2,topolo174197925503356063within(B,A3,Sa),F3)
         => eventually(A,aa(B,fun(A,$o),aa(set(B),fun(B,fun(A,$o)),aTP_Lamp_adn(fun(A,B),fun(set(B),fun(B,fun(A,$o))),F2),Sa),B3),F3) ) ) ).

% filterlim_at_within_not_equal
tff(fact_7419_sequentially__imp__eventually__within,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [Sa: set(A),A3: A,P: fun(A,$o)] :
          ( ! [F4: fun(nat,A)] :
              ( ( ! [N10: nat] :
                    ( member(A,aa(nat,A,F4,N10),Sa)
                    & ( aa(nat,A,F4,N10) != A3 ) )
                & filterlim(nat,A,F4,topolo7230453075368039082e_nhds(A,A3),at_top(nat)) )
             => eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_ado(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),P),F4),at_top(nat)) )
         => eventually(A,P,topolo174197925503356063within(A,A3,Sa)) ) ) ).

% sequentially_imp_eventually_within
tff(fact_7420_sequentially__imp__eventually__at,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [A3: A,P: fun(A,$o)] :
          ( ! [F4: fun(nat,A)] :
              ( ( ! [N10: nat] : aa(nat,A,F4,N10) != A3
                & filterlim(nat,A,F4,topolo7230453075368039082e_nhds(A,A3),at_top(nat)) )
             => eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_ado(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),P),F4),at_top(nat)) )
         => eventually(A,P,topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ).

% sequentially_imp_eventually_at
tff(fact_7421_filterlim__at__top__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [Q: fun(A,$o),F2: fun(A,B),P: fun(B,$o),G: fun(B,A)] :
          ( ! [X4: A,Y3: A] :
              ( aa(A,$o,Q,X4)
             => ( aa(A,$o,Q,Y3)
               => ( aa(A,$o,ord_less_eq(A,X4),Y3)
                 => aa(B,$o,ord_less_eq(B,aa(A,B,F2,X4)),aa(A,B,F2,Y3)) ) ) )
         => ( ! [X4: B] :
                ( aa(B,$o,P,X4)
               => ( aa(A,B,F2,aa(B,A,G,X4)) = X4 ) )
           => ( ! [X4: B] :
                  ( aa(B,$o,P,X4)
                 => aa(A,$o,Q,aa(B,A,G,X4)) )
             => ( eventually(A,Q,at_top(A))
               => ( eventually(B,P,at_top(B))
                 => filterlim(A,B,F2,at_top(B),at_top(A)) ) ) ) ) ) ) ).

% filterlim_at_top_at_top
tff(fact_7422_Lim__transform__eventually,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_adp(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G),F3)
           => filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),F3) ) ) ) ).

% Lim_transform_eventually
tff(fact_7423_eventually__compose__filterlim,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F3: filter(A),F2: fun(B,A),G2: filter(B)] :
      ( eventually(A,P,F3)
     => ( filterlim(B,A,F2,F3,G2)
       => eventually(B,aa(fun(B,A),fun(B,$o),aTP_Lamp_adq(fun(A,$o),fun(fun(B,A),fun(B,$o)),P),F2),G2) ) ) ).

% eventually_compose_filterlim
tff(fact_7424_filterlim__cong,axiom,
    ! [A: $tType,B: $tType,F12: filter(A),F13: filter(A),F23: filter(B),F24: filter(B),F2: fun(B,A),G: fun(B,A)] :
      ( ( F12 = F13 )
     => ( ( F23 = F24 )
       => ( eventually(B,aa(fun(B,A),fun(B,$o),aTP_Lamp_adr(fun(B,A),fun(fun(B,A),fun(B,$o)),F2),G),F23)
         => ( filterlim(B,A,F2,F12,F23)
          <=> filterlim(B,A,G,F13,F24) ) ) ) ) ).

% filterlim_cong
tff(fact_7425_filterlim__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),F23: filter(B),F12: filter(A)] :
      ( filterlim(A,B,F2,F23,F12)
    <=> ! [P5: fun(B,$o)] :
          ( eventually(B,P5,F23)
         => eventually(A,aa(fun(B,$o),fun(A,$o),aTP_Lamp_ads(fun(A,B),fun(fun(B,$o),fun(A,$o)),F2),P5),F12) ) ) ).

% filterlim_iff
tff(fact_7426_eventually__nhds__iff__sequentially,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [P: fun(A,$o),A3: A] :
          ( eventually(A,P,topolo7230453075368039082e_nhds(A,A3))
        <=> ! [F6: fun(nat,A)] :
              ( filterlim(nat,A,F6,topolo7230453075368039082e_nhds(A,A3),at_top(nat))
             => eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_ado(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),P),F6),at_top(nat)) ) ) ) ).

% eventually_nhds_iff_sequentially
tff(fact_7427_tendsto__eventually,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),L: B,Net: filter(A)] :
          ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_adt(fun(A,B),fun(B,fun(A,$o)),F2),L),Net)
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),Net) ) ) ).

% tendsto_eventually
tff(fact_7428_tendsto__imp__eventually__ne,axiom,
    ! [B: $tType,A: $tType] :
      ( topological_t1_space(B)
     => ! [F2: fun(A,B),C2: B,F3: filter(A),C8: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F3)
         => ( ( C2 != C8 )
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_adu(fun(A,B),fun(B,fun(A,$o)),F2),C8),F3) ) ) ) ).

% tendsto_imp_eventually_ne
tff(fact_7429_tendsto__discrete,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo8865339358273720382pology(B)
     => ! [F2: fun(A,B),Y: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Y),F3)
        <=> eventually(A,aa(B,fun(A,$o),aTP_Lamp_adv(fun(A,B),fun(B,fun(A,$o)),F2),Y),F3) ) ) ).

% tendsto_discrete
tff(fact_7430_tendsto__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),G: fun(A,B),F3: filter(A),C2: B] :
          ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_adp(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G),F3)
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F3)
          <=> filterlim(A,B,G,topolo7230453075368039082e_nhds(B,C2),F3) ) ) ) ).

% tendsto_cong
tff(fact_7431_filterlim__mono__eventually,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),F3: filter(B),G2: filter(A),F8: filter(B),G5: filter(A),F7: fun(A,B)] :
      ( filterlim(A,B,F2,F3,G2)
     => ( aa(filter(B),$o,ord_less_eq(filter(B),F3),F8)
       => ( aa(filter(A),$o,ord_less_eq(filter(A),G5),G2)
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_adw(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),F7),G5)
           => filterlim(A,B,F7,F8,G5) ) ) ) ) ).

% filterlim_mono_eventually
tff(fact_7432_eventually__ge__at__top,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A] : eventually(A,ord_less_eq(A,C2),at_top(A)) ) ).

% eventually_ge_at_top
tff(fact_7433_eventually__at__top__not__equal,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [C2: A] : eventually(A,aTP_Lamp_adx(A,fun(A,$o),C2),at_top(A)) ) ).

% eventually_at_top_not_equal
tff(fact_7434_eventually__gt__at__top,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [C2: A] : eventually(A,ord_less(A,C2),at_top(A)) ) ).

% eventually_gt_at_top
tff(fact_7435_eventually__at__top__dense,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [P: fun(A,$o)] :
          ( eventually(A,P,at_top(A))
        <=> ? [N8: A] :
            ! [N3: A] :
              ( aa(A,$o,ord_less(A,N8),N3)
             => aa(A,$o,P,N3) ) ) ) ).

% eventually_at_top_dense
tff(fact_7436_eventually__at__top__linorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,P: fun(A,$o)] :
          ( ! [X4: A] :
              ( aa(A,$o,ord_less_eq(A,C2),X4)
             => aa(A,$o,P,X4) )
         => eventually(A,P,at_top(A)) ) ) ).

% eventually_at_top_linorderI
tff(fact_7437_eventually__at__top__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o)] :
          ( eventually(A,P,at_top(A))
        <=> ? [N8: A] :
            ! [N3: A] :
              ( aa(A,$o,ord_less_eq(A,N8),N3)
             => aa(A,$o,P,N3) ) ) ) ).

% eventually_at_top_linorder
tff(fact_7438_eventually__at__left__field,axiom,
    ! [A: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [P: fun(A,$o),X: A] :
          ( eventually(A,P,topolo174197925503356063within(A,X,set_ord_lessThan(A,X)))
        <=> ? [B9: A] :
              ( aa(A,$o,ord_less(A,B9),X)
              & ! [Y6: A] :
                  ( aa(A,$o,ord_less(A,B9),Y6)
                 => ( aa(A,$o,ord_less(A,Y6),X)
                   => aa(A,$o,P,Y6) ) ) ) ) ) ).

% eventually_at_left_field
tff(fact_7439_eventually__at__left,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Y: A,X: A,P: fun(A,$o)] :
          ( aa(A,$o,ord_less(A,Y),X)
         => ( eventually(A,P,topolo174197925503356063within(A,X,set_ord_lessThan(A,X)))
          <=> ? [B9: A] :
                ( aa(A,$o,ord_less(A,B9),X)
                & ! [Y6: A] :
                    ( aa(A,$o,ord_less(A,B9),Y6)
                   => ( aa(A,$o,ord_less(A,Y6),X)
                     => aa(A,$o,P,Y6) ) ) ) ) ) ) ).

% eventually_at_left
tff(fact_7440_eventually__nhds__in__open,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Sa: set(A),X: A] :
          ( topolo1002775350975398744n_open(A,Sa)
         => ( member(A,X,Sa)
           => eventually(A,aTP_Lamp_ady(set(A),fun(A,$o),Sa),topolo7230453075368039082e_nhds(A,X)) ) ) ) ).

% eventually_nhds_in_open
tff(fact_7441_eventually__at__filter,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [P: fun(A,$o),A3: A,Sa: set(A)] :
          ( eventually(A,P,topolo174197925503356063within(A,A3,Sa))
        <=> eventually(A,aa(set(A),fun(A,$o),aa(A,fun(set(A),fun(A,$o)),aTP_Lamp_adz(fun(A,$o),fun(A,fun(set(A),fun(A,$o))),P),A3),Sa),topolo7230453075368039082e_nhds(A,A3)) ) ) ).

% eventually_at_filter
tff(fact_7442_eventually__eventually,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [P: fun(A,$o),X: A] :
          ( eventually(A,aTP_Lamp_aea(fun(A,$o),fun(A,$o),P),topolo7230453075368039082e_nhds(A,X))
        <=> eventually(A,P,topolo7230453075368039082e_nhds(A,X)) ) ) ).

% eventually_eventually
tff(fact_7443_t1__space__nhds,axiom,
    ! [A: $tType] :
      ( topological_t1_space(A)
     => ! [X: A,Y: A] :
          ( ( X != Y )
         => eventually(A,aTP_Lamp_aeb(A,fun(A,$o),Y),topolo7230453075368039082e_nhds(A,X)) ) ) ).

% t1_space_nhds
tff(fact_7444_filter__leD,axiom,
    ! [A: $tType,F3: filter(A),F8: filter(A),P: fun(A,$o)] :
      ( aa(filter(A),$o,ord_less_eq(filter(A),F3),F8)
     => ( eventually(A,P,F8)
       => eventually(A,P,F3) ) ) ).

% filter_leD
tff(fact_7445_filter__leI,axiom,
    ! [A: $tType,F8: filter(A),F3: filter(A)] :
      ( ! [P6: fun(A,$o)] :
          ( eventually(A,P6,F8)
         => eventually(A,P6,F3) )
     => aa(filter(A),$o,ord_less_eq(filter(A),F3),F8) ) ).

% filter_leI
tff(fact_7446_le__filter__def,axiom,
    ! [A: $tType,F3: filter(A),F8: filter(A)] :
      ( aa(filter(A),$o,ord_less_eq(filter(A),F3),F8)
    <=> ! [P5: fun(A,$o)] :
          ( eventually(A,P5,F8)
         => eventually(A,P5,F3) ) ) ).

% le_filter_def
tff(fact_7447_has__field__derivative__cong__eventually,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),G: fun(A,A),X: A,S: set(A),U: A] :
          ( eventually(A,aa(fun(A,A),fun(A,$o),aTP_Lamp_aec(fun(A,A),fun(fun(A,A),fun(A,$o)),F2),G),topolo174197925503356063within(A,X,S))
         => ( ( aa(A,A,F2,X) = aa(A,A,G,X) )
           => ( has_field_derivative(A,F2,U,topolo174197925503356063within(A,X,S))
            <=> has_field_derivative(A,G,U,topolo174197925503356063within(A,X,S)) ) ) ) ) ).

% has_field_derivative_cong_eventually
tff(fact_7448_eventually__sup,axiom,
    ! [A: $tType,P: fun(A,$o),F3: filter(A),F8: filter(A)] :
      ( eventually(A,P,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),sup_sup(filter(A)),F3),F8))
    <=> ( eventually(A,P,F3)
        & eventually(A,P,F8) ) ) ).

% eventually_sup
tff(fact_7449_eventuallyI,axiom,
    ! [A: $tType,P: fun(A,$o),F3: filter(A)] :
      ( ! [X4: A] : aa(A,$o,P,X4)
     => eventually(A,P,F3) ) ).

% eventuallyI
tff(fact_7450_filter__eq__iff,axiom,
    ! [A: $tType,F3: filter(A),F8: filter(A)] :
      ( ( F3 = F8 )
    <=> ! [P5: fun(A,$o)] :
          ( eventually(A,P5,F3)
        <=> eventually(A,P5,F8) ) ) ).

% filter_eq_iff
tff(fact_7451_eventually__mono,axiom,
    ! [A: $tType,P: fun(A,$o),F3: filter(A),Q: fun(A,$o)] :
      ( eventually(A,P,F3)
     => ( ! [X4: A] :
            ( aa(A,$o,P,X4)
           => aa(A,$o,Q,X4) )
       => eventually(A,Q,F3) ) ) ).

% eventually_mono
tff(fact_7452_not__eventuallyD,axiom,
    ! [A: $tType,P: fun(A,$o),F3: filter(A)] :
      ( ~ eventually(A,P,F3)
     => ? [X4: A] : ~ aa(A,$o,P,X4) ) ).

% not_eventuallyD
tff(fact_7453_always__eventually,axiom,
    ! [A: $tType,P: fun(A,$o),F3: filter(A)] :
      ( ! [X_12: A] : aa(A,$o,P,X_12)
     => eventually(A,P,F3) ) ).

% always_eventually
tff(fact_7454_not__eventually__impI,axiom,
    ! [A: $tType,P: fun(A,$o),F3: filter(A),Q: fun(A,$o)] :
      ( eventually(A,P,F3)
     => ( ~ eventually(A,Q,F3)
       => ~ eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_aj(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F3) ) ) ).

% not_eventually_impI
tff(fact_7455_eventually__conj__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),F3: filter(A)] :
      ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_nj(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F3)
    <=> ( eventually(A,P,F3)
        & eventually(A,Q,F3) ) ) ).

% eventually_conj_iff
tff(fact_7456_eventually__rev__mp,axiom,
    ! [A: $tType,P: fun(A,$o),F3: filter(A),Q: fun(A,$o)] :
      ( eventually(A,P,F3)
     => ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_aj(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F3)
       => eventually(A,Q,F3) ) ) ).

% eventually_rev_mp
tff(fact_7457_eventually__subst,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),F3: filter(A)] :
      ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_aed(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F3)
     => ( eventually(A,P,F3)
      <=> eventually(A,Q,F3) ) ) ).

% eventually_subst
tff(fact_7458_eventually__elim2,axiom,
    ! [A: $tType,P: fun(A,$o),F3: filter(A),Q: fun(A,$o),R: fun(A,$o)] :
      ( eventually(A,P,F3)
     => ( eventually(A,Q,F3)
       => ( ! [I3: A] :
              ( aa(A,$o,P,I3)
             => ( aa(A,$o,Q,I3)
               => aa(A,$o,R,I3) ) )
         => eventually(A,R,F3) ) ) ) ).

% eventually_elim2
tff(fact_7459_eventually__conj,axiom,
    ! [A: $tType,P: fun(A,$o),F3: filter(A),Q: fun(A,$o)] :
      ( eventually(A,P,F3)
     => ( eventually(A,Q,F3)
       => eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_nj(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F3) ) ) ).

% eventually_conj
tff(fact_7460_eventually__True,axiom,
    ! [A: $tType,F3: filter(A)] : eventually(A,aTP_Lamp_pi(A,$o),F3) ).

% eventually_True
tff(fact_7461_eventually__mp,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),F3: filter(A)] :
      ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_aj(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F3)
     => ( eventually(A,P,F3)
       => eventually(A,Q,F3) ) ) ).

% eventually_mp
tff(fact_7462_eventually__frequently__const__simps_I3_J,axiom,
    ! [A: $tType,P: fun(A,$o),C3: $o,F3: filter(A)] :
      ( eventually(A,aa($o,fun(A,$o),aTP_Lamp_aee(fun(A,$o),fun($o,fun(A,$o)),P),(C3)),F3)
    <=> ( eventually(A,P,F3)
        | (C3) ) ) ).

% eventually_frequently_const_simps(3)
tff(fact_7463_eventually__frequently__const__simps_I4_J,axiom,
    ! [A: $tType,C3: $o,P: fun(A,$o),F3: filter(A)] :
      ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_aef($o,fun(fun(A,$o),fun(A,$o)),(C3)),P),F3)
    <=> ( (C3)
        | eventually(A,P,F3) ) ) ).

% eventually_frequently_const_simps(4)
tff(fact_7464_eventually__frequently__const__simps_I6_J,axiom,
    ! [A: $tType,C3: $o,P: fun(A,$o),F3: filter(A)] :
      ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_aeg($o,fun(fun(A,$o),fun(A,$o)),(C3)),P),F3)
    <=> ( (C3)
       => eventually(A,P,F3) ) ) ).

% eventually_frequently_const_simps(6)
tff(fact_7465_has__derivative__transform__eventually,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),X: A,Sa: set(A),G: fun(A,B)] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,Sa))
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_aeh(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G),topolo174197925503356063within(A,X,Sa))
           => ( ( aa(A,B,F2,X) = aa(A,B,G,X) )
             => ( member(A,X,Sa)
               => has_derivative(A,B,G,F7,topolo174197925503356063within(A,X,Sa)) ) ) ) ) ) ).

% has_derivative_transform_eventually
tff(fact_7466_trivial__limit__def,axiom,
    ! [A: $tType,F3: filter(A)] :
      ( ( F3 = bot_bot(filter(A)) )
    <=> eventually(A,aTP_Lamp_ai(A,$o),F3) ) ).

% trivial_limit_def
tff(fact_7467_eventually__const__iff,axiom,
    ! [A: $tType,P: $o,F3: filter(A)] :
      ( eventually(A,aTP_Lamp_ph($o,fun(A,$o),(P)),F3)
    <=> ( (P)
        | ( F3 = bot_bot(filter(A)) ) ) ) ).

% eventually_const_iff
tff(fact_7468_False__imp__not__eventually,axiom,
    ! [A: $tType,P: fun(A,$o),Net: filter(A)] :
      ( ! [X4: A] : ~ aa(A,$o,P,X4)
     => ( ( Net != bot_bot(filter(A)) )
       => ~ eventually(A,P,Net) ) ) ).

% False_imp_not_eventually
tff(fact_7469_eventually__happens_H,axiom,
    ! [A: $tType,F3: filter(A),P: fun(A,$o)] :
      ( ( F3 != bot_bot(filter(A)) )
     => ( eventually(A,P,F3)
       => ? [X_12: A] : aa(A,$o,P,X_12) ) ) ).

% eventually_happens'
tff(fact_7470_eventually__happens,axiom,
    ! [A: $tType,P: fun(A,$o),Net: filter(A)] :
      ( eventually(A,P,Net)
     => ( ( Net = bot_bot(filter(A)) )
        | ? [X_12: A] : aa(A,$o,P,X_12) ) ) ).

% eventually_happens
tff(fact_7471_eventually__bot,axiom,
    ! [A: $tType,P: fun(A,$o)] : eventually(A,P,bot_bot(filter(A))) ).

% eventually_bot
tff(fact_7472_eventually__nhds__top,axiom,
    ! [A: $tType] :
      ( ( order_top(A)
        & topolo1944317154257567458pology(A) )
     => ! [B3: A,P: fun(A,$o)] :
          ( aa(A,$o,ord_less(A,B3),top_top(A))
         => ( eventually(A,P,topolo7230453075368039082e_nhds(A,top_top(A)))
          <=> ? [B9: A] :
                ( aa(A,$o,ord_less(A,B9),top_top(A))
                & ! [Z6: A] :
                    ( aa(A,$o,ord_less(A,B9),Z6)
                   => aa(A,$o,P,Z6) ) ) ) ) ) ).

% eventually_nhds_top
tff(fact_7473_eventually__Lim__ident__at,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [P: fun(A,fun(A,$o)),X: A,X5: set(A)] :
          ( eventually(A,aa(A,fun(A,$o),P,topolo3827282254853284352ce_Lim(A,A,topolo174197925503356063within(A,X,X5),aTP_Lamp_us(A,A))),topolo174197925503356063within(A,X,X5))
        <=> eventually(A,aa(A,fun(A,$o),P,X),topolo174197925503356063within(A,X,X5)) ) ) ).

% eventually_Lim_ident_at
tff(fact_7474_tendsto__sandwich,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),G: fun(A,B),Net: filter(A),H: fun(A,B),C2: B] :
          ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_aei(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G),Net)
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_aei(fun(A,B),fun(fun(A,B),fun(A,$o)),G),H),Net)
           => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),Net)
             => ( filterlim(A,B,H,topolo7230453075368039082e_nhds(B,C2),Net)
               => filterlim(A,B,G,topolo7230453075368039082e_nhds(B,C2),Net) ) ) ) ) ) ).

% tendsto_sandwich
tff(fact_7475_order__tendstoD_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),Y: B,F3: filter(A),A3: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Y),F3)
         => ( aa(B,$o,ord_less(B,Y),A3)
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_aej(fun(A,B),fun(B,fun(A,$o)),F2),A3),F3) ) ) ) ).

% order_tendstoD(2)
tff(fact_7476_order__tendstoD_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),Y: B,F3: filter(A),A3: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Y),F3)
         => ( aa(B,$o,ord_less(B,A3),Y)
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_aek(fun(A,B),fun(B,fun(A,$o)),F2),A3),F3) ) ) ) ).

% order_tendstoD(1)
tff(fact_7477_order__tendstoI,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Y: A,F2: fun(B,A),F3: filter(B)] :
          ( ! [A4: A] :
              ( aa(A,$o,ord_less(A,A4),Y)
             => eventually(B,aa(A,fun(B,$o),aTP_Lamp_ael(fun(B,A),fun(A,fun(B,$o)),F2),A4),F3) )
         => ( ! [A4: A] :
                ( aa(A,$o,ord_less(A,Y),A4)
               => eventually(B,aa(A,fun(B,$o),aTP_Lamp_aem(fun(B,A),fun(A,fun(B,$o)),F2),A4),F3) )
           => filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,Y),F3) ) ) ) ).

% order_tendstoI
tff(fact_7478_order__tendsto__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),X: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,X),F3)
        <=> ( ! [L3: B] :
                ( aa(B,$o,ord_less(B,L3),X)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_aek(fun(A,B),fun(B,fun(A,$o)),F2),L3),F3) )
            & ! [U3: B] :
                ( aa(B,$o,ord_less(B,X),U3)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_aej(fun(A,B),fun(B,fun(A,$o)),F2),U3),F3) ) ) ) ) ).

% order_tendsto_iff
tff(fact_7479_filterlim__at__top__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F3: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,at_top(B),F3)
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_aen(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G),F3)
           => filterlim(A,B,G,at_top(B),F3) ) ) ) ).

% filterlim_at_top_mono
tff(fact_7480_filterlim__at__top__ge,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_top(B),F3)
        <=> ! [Z7: B] :
              ( aa(B,$o,ord_less_eq(B,C2),Z7)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_aeo(fun(A,B),fun(B,fun(A,$o)),F2),Z7),F3) ) ) ) ).

% filterlim_at_top_ge
tff(fact_7481_filterlim__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,B,F2,at_top(B),F3)
        <=> ! [Z7: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_aeo(fun(A,B),fun(B,fun(A,$o)),F2),Z7),F3) ) ) ).

% filterlim_at_top
tff(fact_7482_filterlim__at__top__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,B,F2,at_top(B),F3)
        <=> ! [Z7: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_aep(fun(A,B),fun(B,fun(A,$o)),F2),Z7),F3) ) ) ).

% filterlim_at_top_dense
tff(fact_7483_filterlim__at,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),B3: B,Sa: set(B),F3: filter(A)] :
          ( filterlim(A,B,F2,topolo174197925503356063within(B,B3,Sa),F3)
        <=> ( eventually(A,aa(set(B),fun(A,$o),aa(B,fun(set(B),fun(A,$o)),aTP_Lamp_aeq(fun(A,B),fun(B,fun(set(B),fun(A,$o))),F2),B3),Sa),F3)
            & filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,B3),F3) ) ) ) ).

% filterlim_at
tff(fact_7484_eventually__at__right__less,axiom,
    ! [A: $tType] :
      ( ( no_top(A)
        & topolo1944317154257567458pology(A) )
     => ! [X: A] : eventually(A,ord_less(A,X),topolo174197925503356063within(A,X,set_ord_greaterThan(A,X))) ) ).

% eventually_at_right_less
tff(fact_7485_has__field__derivative__cong__ev,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [X: A,Y: A,S: set(A),F2: fun(A,A),G: fun(A,A),U: A,V2: A,Ta: set(A)] :
          ( ( X = Y )
         => ( eventually(A,aa(fun(A,A),fun(A,$o),aa(fun(A,A),fun(fun(A,A),fun(A,$o)),aTP_Lamp_aer(set(A),fun(fun(A,A),fun(fun(A,A),fun(A,$o))),S),F2),G),topolo7230453075368039082e_nhds(A,X))
           => ( ( U = V2 )
             => ( ( S = Ta )
               => ( member(A,X,S)
                 => ( has_field_derivative(A,F2,U,topolo174197925503356063within(A,X,S))
                  <=> has_field_derivative(A,G,V2,topolo174197925503356063within(A,Y,Ta)) ) ) ) ) ) ) ) ).

% has_field_derivative_cong_ev
tff(fact_7486_tendsto__def,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
        <=> ! [S9: set(B)] :
              ( topolo1002775350975398744n_open(B,S9)
             => ( member(B,L,S9)
               => eventually(A,aa(set(B),fun(A,$o),aTP_Lamp_aes(fun(A,B),fun(set(B),fun(A,$o)),F2),S9),F3) ) ) ) ) ).

% tendsto_def
tff(fact_7487_topological__tendstoD,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A),S: set(B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( topolo1002775350975398744n_open(B,S)
           => ( member(B,L,S)
             => eventually(A,aa(set(B),fun(A,$o),aTP_Lamp_aes(fun(A,B),fun(set(B),fun(A,$o)),F2),S),F3) ) ) ) ) ).

% topological_tendstoD
tff(fact_7488_topological__tendstoI,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [L: A,F2: fun(B,A),F3: filter(B)] :
          ( ! [S4: set(A)] :
              ( topolo1002775350975398744n_open(A,S4)
             => ( member(A,L,S4)
               => eventually(B,aa(set(A),fun(B,$o),aTP_Lamp_aet(fun(B,A),fun(set(A),fun(B,$o)),F2),S4),F3) ) )
         => filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F3) ) ) ).

% topological_tendstoI
tff(fact_7489_filterlim__at__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,B,F2,at_bot(B),F3)
        <=> ! [Z7: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_aeu(fun(A,B),fun(B,fun(A,$o)),F2),Z7),F3) ) ) ).

% filterlim_at_bot
tff(fact_7490_filterlim__at__bot__le,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_bot(B),F3)
        <=> ! [Z7: B] :
              ( aa(B,$o,ord_less_eq(B,Z7),C2)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_aeu(fun(A,B),fun(B,fun(A,$o)),F2),Z7),F3) ) ) ) ).

% filterlim_at_bot_le
tff(fact_7491_filterlim__at__bot__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( ( dense_linorder(B)
        & no_bot(B) )
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,B,F2,at_bot(B),F3)
        <=> ! [Z7: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_aev(fun(A,B),fun(B,fun(A,$o)),F2),Z7),F3) ) ) ).

% filterlim_at_bot_dense
tff(fact_7492_real__tendsto__sandwich,axiom,
    ! [A: $tType,F2: fun(A,real),G: fun(A,real),Net: filter(A),H: fun(A,real),C2: real] :
      ( eventually(A,aa(fun(A,real),fun(A,$o),aTP_Lamp_aew(fun(A,real),fun(fun(A,real),fun(A,$o)),F2),G),Net)
     => ( eventually(A,aa(fun(A,real),fun(A,$o),aTP_Lamp_aew(fun(A,real),fun(fun(A,real),fun(A,$o)),G),H),Net)
       => ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),Net)
         => ( filterlim(A,real,H,topolo7230453075368039082e_nhds(real,C2),Net)
           => filterlim(A,real,G,topolo7230453075368039082e_nhds(real,C2),Net) ) ) ) ) ).

% real_tendsto_sandwich
tff(fact_7493_eventually__at,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [P: fun(A,$o),A3: A,S: set(A)] :
          ( eventually(A,P,topolo174197925503356063within(A,A3,S))
        <=> ? [D4: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),D4)
              & ! [X3: A] :
                  ( member(A,X3,S)
                 => ( ( ( X3 != A3 )
                      & aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X3,A3)),D4) )
                   => aa(A,$o,P,X3) ) ) ) ) ) ).

% eventually_at
tff(fact_7494_eventually__nhds__metric,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [P: fun(A,$o),A3: A] :
          ( eventually(A,P,topolo7230453075368039082e_nhds(A,A3))
        <=> ? [D4: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),D4)
              & ! [X3: A] :
                  ( aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X3,A3)),D4)
                 => aa(A,$o,P,X3) ) ) ) ) ).

% eventually_nhds_metric
tff(fact_7495_eventually__at__leftI,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A3: A,B3: A,P: fun(A,$o)] :
          ( ! [X4: A] :
              ( member(A,X4,set_or5935395276787703475ssThan(A,A3,B3))
             => aa(A,$o,P,X4) )
         => ( aa(A,$o,ord_less(A,A3),B3)
           => eventually(A,P,topolo174197925503356063within(A,B3,set_ord_lessThan(A,B3))) ) ) ) ).

% eventually_at_leftI
tff(fact_7496_eventually__at__rightI,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A3: A,B3: A,P: fun(A,$o)] :
          ( ! [X4: A] :
              ( member(A,X4,set_or5935395276787703475ssThan(A,A3,B3))
             => aa(A,$o,P,X4) )
         => ( aa(A,$o,ord_less(A,A3),B3)
           => eventually(A,P,topolo174197925503356063within(A,A3,set_ord_greaterThan(A,A3))) ) ) ) ).

% eventually_at_rightI
tff(fact_7497_eventually__at__to__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [P: fun(A,$o),A3: A] :
          ( eventually(A,P,topolo174197925503356063within(A,A3,top_top(set(A))))
        <=> eventually(A,aa(A,fun(A,$o),aTP_Lamp_aex(fun(A,$o),fun(A,fun(A,$o)),P),A3),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% eventually_at_to_0
tff(fact_7498_decreasing__tendsto,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [L: B,F2: fun(A,B),F3: filter(A)] :
          ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_aey(B,fun(fun(A,B),fun(A,$o)),L),F2),F3)
         => ( ! [X4: B] :
                ( aa(B,$o,ord_less(B,L),X4)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_aej(fun(A,B),fun(B,fun(A,$o)),F2),X4),F3) )
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3) ) ) ) ).

% decreasing_tendsto
tff(fact_7499_increasing__tendsto,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_aez(fun(A,B),fun(B,fun(A,$o)),F2),L),F3)
         => ( ! [X4: B] :
                ( aa(B,$o,ord_less(B,X4),L)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_aek(fun(A,B),fun(B,fun(A,$o)),F2),X4),F3) )
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3) ) ) ) ).

% increasing_tendsto
tff(fact_7500_filterlim__at__top__gt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F2: fun(A,B),F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_top(B),F3)
        <=> ! [Z7: B] :
              ( aa(B,$o,ord_less(B,C2),Z7)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_afa(fun(A,B),fun(B,fun(A,$o)),F2),Z7),F3) ) ) ) ).

% filterlim_at_top_gt
tff(fact_7501_filterlim__atI,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),C2: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F3)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_afb(fun(A,B),fun(B,fun(A,$o)),F2),C2),F3)
           => filterlim(A,B,F2,topolo174197925503356063within(B,C2,top_top(set(B))),F3) ) ) ) ).

% filterlim_atI
tff(fact_7502_LIM__compose__eventually,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),B3: B,A3: A,G: fun(B,C),C2: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,B3),topolo174197925503356063within(A,A3,top_top(set(A))))
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(B,B3,top_top(set(B))))
           => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_afc(fun(A,B),fun(B,fun(A,$o)),F2),B3),topolo174197925503356063within(A,A3,top_top(set(A))))
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_afd(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ) ) ).

% LIM_compose_eventually
tff(fact_7503_tendsto__compose__eventually,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [G: fun(A,B),M: B,L: A,F2: fun(C,A),F3: filter(C)] :
          ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,M),topolo174197925503356063within(A,L,top_top(set(A))))
         => ( filterlim(C,A,F2,topolo7230453075368039082e_nhds(A,L),F3)
           => ( eventually(C,aa(fun(C,A),fun(C,$o),aTP_Lamp_afe(A,fun(fun(C,A),fun(C,$o)),L),F2),F3)
             => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_vc(fun(A,B),fun(fun(C,A),fun(C,B)),G),F2),topolo7230453075368039082e_nhds(B,M),F3) ) ) ) ) ).

% tendsto_compose_eventually
tff(fact_7504_isCont__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),G: fun(A,B),X: A] :
          ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_aff(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G),topolo7230453075368039082e_nhds(A,X))
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,X,top_top(set(A))),F2)
          <=> topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,X,top_top(set(A))),G) ) ) ) ).

% isCont_cong
tff(fact_7505_DERIV__cong__ev,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [X: A,Y: A,F2: fun(A,A),G: fun(A,A),U: A,V2: A] :
          ( ( X = Y )
         => ( eventually(A,aa(fun(A,A),fun(A,$o),aTP_Lamp_aec(fun(A,A),fun(fun(A,A),fun(A,$o)),F2),G),topolo7230453075368039082e_nhds(A,X))
           => ( ( U = V2 )
             => ( has_field_derivative(A,F2,U,topolo174197925503356063within(A,X,top_top(set(A))))
              <=> has_field_derivative(A,G,V2,topolo174197925503356063within(A,Y,top_top(set(A)))) ) ) ) ) ) ).

% DERIV_cong_ev
tff(fact_7506_filterlim__at__bot__lt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F2: fun(A,B),F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_bot(B),F3)
        <=> ! [Z7: B] :
              ( aa(B,$o,ord_less(B,Z7),C2)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_afg(fun(A,B),fun(B,fun(A,$o)),F2),Z7),F3) ) ) ) ).

% filterlim_at_bot_lt
tff(fact_7507_tendsto__upperbound,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [F2: fun(A,B),X: B,F3: filter(A),A3: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,X),F3)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_afh(fun(A,B),fun(B,fun(A,$o)),F2),A3),F3)
           => ( ( F3 != bot_bot(filter(A)) )
             => aa(B,$o,ord_less_eq(B,X),A3) ) ) ) ) ).

% tendsto_upperbound
tff(fact_7508_tendsto__lowerbound,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [F2: fun(A,B),X: B,F3: filter(A),A3: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,X),F3)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_afi(fun(A,B),fun(B,fun(A,$o)),F2),A3),F3)
           => ( ( F3 != bot_bot(filter(A)) )
             => aa(B,$o,ord_less_eq(B,A3),X) ) ) ) ) ).

% tendsto_lowerbound
tff(fact_7509_tendsto__le,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [F3: filter(A),F2: fun(A,B),X: B,G: fun(A,B),Y: B] :
          ( ( F3 != bot_bot(filter(A)) )
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,X),F3)
           => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,Y),F3)
             => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_afj(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G),F3)
               => aa(B,$o,ord_less_eq(B,Y),X) ) ) ) ) ) ).

% tendsto_le
tff(fact_7510_metric__tendsto__imp__tendsto,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(C)
        & real_V7819770556892013058_space(B) )
     => ! [F2: fun(A,B),A3: B,F3: filter(A),G: fun(A,C),B3: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),F3)
         => ( eventually(A,aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aa(B,fun(fun(A,C),fun(C,fun(A,$o))),aTP_Lamp_afk(fun(A,B),fun(B,fun(fun(A,C),fun(C,fun(A,$o)))),F2),A3),G),B3),F3)
           => filterlim(A,C,G,topolo7230453075368039082e_nhds(C,B3),F3) ) ) ) ).

% metric_tendsto_imp_tendsto
tff(fact_7511_greaterThan__0,axiom,
    set_ord_greaterThan(nat,zero_zero(nat)) = aa(set(nat),set(nat),image(nat,nat,suc),top_top(set(nat))) ).

% greaterThan_0
tff(fact_7512_eventually__floor__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( ~ member(B,L,ring_1_Ints(B))
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_afl(fun(A,B),fun(B,fun(A,$o)),F2),L),F3) ) ) ) ).

% eventually_floor_eq
tff(fact_7513_eventually__ceiling__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( ~ member(B,L,ring_1_Ints(B))
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_afm(fun(A,B),fun(B,fun(A,$o)),F2),L),F3) ) ) ) ).

% eventually_ceiling_eq
tff(fact_7514_eventually__at__right__to__0,axiom,
    ! [P: fun(real,$o),A3: real] :
      ( eventually(real,P,topolo174197925503356063within(real,A3,set_ord_greaterThan(real,A3)))
    <=> eventually(real,aa(real,fun(real,$o),aTP_Lamp_afn(fun(real,$o),fun(real,fun(real,$o)),P),A3),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ) ).

% eventually_at_right_to_0
tff(fact_7515_greaterThan__Suc,axiom,
    ! [K: nat] : set_ord_greaterThan(nat,aa(nat,nat,suc,K)) = aa(set(nat),set(nat),minus_minus(set(nat),set_ord_greaterThan(nat,K)),aa(set(nat),set(nat),insert(nat,aa(nat,nat,suc,K)),bot_bot(set(nat)))) ).

% greaterThan_Suc
tff(fact_7516_eventually__at__left__to__right,axiom,
    ! [P: fun(real,$o),A3: real] :
      ( eventually(real,P,topolo174197925503356063within(real,A3,set_ord_lessThan(real,A3)))
    <=> eventually(real,aTP_Lamp_afo(fun(real,$o),fun(real,$o),P),topolo174197925503356063within(real,aa(real,real,uminus_uminus(real),A3),set_ord_greaterThan(real,aa(real,real,uminus_uminus(real),A3)))) ) ).

% eventually_at_left_to_right
tff(fact_7517_continuous__arcosh__strong,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => ( eventually(A,aTP_Lamp_afp(fun(A,real),fun(A,$o),F2),F3)
           => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_acr(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_arcosh_strong
tff(fact_7518_eventually__at__right__real,axiom,
    ! [A3: real,B3: real] :
      ( aa(real,$o,ord_less(real,A3),B3)
     => eventually(real,aa(real,fun(real,$o),aTP_Lamp_afq(real,fun(real,fun(real,$o)),A3),B3),topolo174197925503356063within(real,A3,set_ord_greaterThan(real,A3))) ) ).

% eventually_at_right_real
tff(fact_7519_eventually__at__left__real,axiom,
    ! [B3: real,A3: real] :
      ( aa(real,$o,ord_less(real,B3),A3)
     => eventually(real,aa(real,fun(real,$o),aTP_Lamp_afq(real,fun(real,fun(real,$o)),B3),A3),topolo174197925503356063within(real,A3,set_ord_lessThan(real,A3))) ) ).

% eventually_at_left_real
tff(fact_7520_eventually__at__le,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [P: fun(A,$o),A3: A,S: set(A)] :
          ( eventually(A,P,topolo174197925503356063within(A,A3,S))
        <=> ? [D4: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),D4)
              & ! [X3: A] :
                  ( member(A,X3,S)
                 => ( ( ( X3 != A3 )
                      & aa(real,$o,ord_less_eq(real,real_V557655796197034286t_dist(A,X3,A3)),D4) )
                   => aa(A,$o,P,X3) ) ) ) ) ) ).

% eventually_at_le
tff(fact_7521_tendsto__imp__filterlim__at__left,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),L5: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),F3)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_aej(fun(A,B),fun(B,fun(A,$o)),F2),L5),F3)
           => filterlim(A,B,F2,topolo174197925503356063within(B,L5,set_ord_lessThan(B,L5)),F3) ) ) ) ).

% tendsto_imp_filterlim_at_left
tff(fact_7522_tendsto__imp__filterlim__at__right,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),L5: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),F3)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_aek(fun(A,B),fun(B,fun(A,$o)),F2),L5),F3)
           => filterlim(A,B,F2,topolo174197925503356063within(B,L5,set_ord_greaterThan(B,L5)),F3) ) ) ) ).

% tendsto_imp_filterlim_at_right
tff(fact_7523_tendsto__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
        <=> ! [E3: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),E3)
             => eventually(A,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_afr(fun(A,B),fun(B,fun(real,fun(A,$o))),F2),L),E3),F3) ) ) ) ).

% tendsto_iff
tff(fact_7524_tendstoI,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( ! [E: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),E)
             => eventually(A,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_afr(fun(A,B),fun(B,fun(real,fun(A,$o))),F2),L),E),F3) )
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3) ) ) ).

% tendstoI
tff(fact_7525_tendstoD,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A),E2: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( aa(real,$o,ord_less(real,zero_zero(real)),E2)
           => eventually(A,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_afr(fun(A,B),fun(B,fun(real,fun(A,$o))),F2),L),E2),F3) ) ) ) ).

% tendstoD
tff(fact_7526_eventually__at__right__to__top,axiom,
    ! [P: fun(real,$o)] :
      ( eventually(real,P,topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
    <=> eventually(real,aTP_Lamp_afs(fun(real,$o),fun(real,$o),P),at_top(real)) ) ).

% eventually_at_right_to_top
tff(fact_7527_eventually__at__top__to__right,axiom,
    ! [P: fun(real,$o)] :
      ( eventually(real,P,at_top(real))
    <=> eventually(real,aTP_Lamp_afs(fun(real,$o),fun(real,$o),P),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ) ).

% eventually_at_top_to_right
tff(fact_7528_tendsto__arcosh__strong,axiom,
    ! [A: $tType,F2: fun(A,real),A3: real,F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A3),F3)
     => ( aa(real,$o,ord_less_eq(real,one_one(real)),A3)
       => ( eventually(A,aTP_Lamp_aft(fun(A,real),fun(A,$o),F2),F3)
         => filterlim(A,real,aTP_Lamp_vo(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,arcosh(real),A3)),F3) ) ) ) ).

% tendsto_arcosh_strong
tff(fact_7529_filterlim__at__top__at__left,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & linorder(B) )
     => ! [Q: fun(A,$o),F2: fun(A,B),P: fun(B,$o),G: fun(B,A),A3: A] :
          ( ! [X4: A,Y3: A] :
              ( aa(A,$o,Q,X4)
             => ( aa(A,$o,Q,Y3)
               => ( aa(A,$o,ord_less_eq(A,X4),Y3)
                 => aa(B,$o,ord_less_eq(B,aa(A,B,F2,X4)),aa(A,B,F2,Y3)) ) ) )
         => ( ! [X4: B] :
                ( aa(B,$o,P,X4)
               => ( aa(A,B,F2,aa(B,A,G,X4)) = X4 ) )
           => ( ! [X4: B] :
                  ( aa(B,$o,P,X4)
                 => aa(A,$o,Q,aa(B,A,G,X4)) )
             => ( eventually(A,Q,topolo174197925503356063within(A,A3,set_ord_lessThan(A,A3)))
               => ( ! [B5: A] :
                      ( aa(A,$o,Q,B5)
                     => aa(A,$o,ord_less(A,B5),A3) )
                 => ( eventually(B,P,at_top(B))
                   => filterlim(A,B,F2,at_top(B),topolo174197925503356063within(A,A3,set_ord_lessThan(A,A3))) ) ) ) ) ) ) ) ).

% filterlim_at_top_at_left
tff(fact_7530_filterlim__at__bot__at__right,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & linorder(B) )
     => ! [Q: fun(A,$o),F2: fun(A,B),P: fun(B,$o),G: fun(B,A),A3: A] :
          ( ! [X4: A,Y3: A] :
              ( aa(A,$o,Q,X4)
             => ( aa(A,$o,Q,Y3)
               => ( aa(A,$o,ord_less_eq(A,X4),Y3)
                 => aa(B,$o,ord_less_eq(B,aa(A,B,F2,X4)),aa(A,B,F2,Y3)) ) ) )
         => ( ! [X4: B] :
                ( aa(B,$o,P,X4)
               => ( aa(A,B,F2,aa(B,A,G,X4)) = X4 ) )
           => ( ! [X4: B] :
                  ( aa(B,$o,P,X4)
                 => aa(A,$o,Q,aa(B,A,G,X4)) )
             => ( eventually(A,Q,topolo174197925503356063within(A,A3,set_ord_greaterThan(A,A3)))
               => ( ! [B5: A] :
                      ( aa(A,$o,Q,B5)
                     => aa(A,$o,ord_less(A,A3),B5) )
                 => ( eventually(B,P,at_bot(B))
                   => filterlim(A,B,F2,at_bot(B),topolo174197925503356063within(A,A3,set_ord_greaterThan(A,A3))) ) ) ) ) ) ) ) ).

% filterlim_at_bot_at_right
tff(fact_7531_filterlim__at__withinI,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),C2: B,F3: filter(A),A2: set(B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F3)
         => ( eventually(A,aa(set(B),fun(A,$o),aa(B,fun(set(B),fun(A,$o)),aTP_Lamp_afu(fun(A,B),fun(B,fun(set(B),fun(A,$o))),F2),C2),A2),F3)
           => filterlim(A,B,F2,topolo174197925503356063within(B,C2,A2),F3) ) ) ) ).

% filterlim_at_withinI
tff(fact_7532_tendsto__0__le,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F3: filter(A),G: fun(A,C),K4: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => ( eventually(A,aa(real,fun(A,$o),aa(fun(A,C),fun(real,fun(A,$o)),aTP_Lamp_afv(fun(A,B),fun(fun(A,C),fun(real,fun(A,$o))),F2),G),K4),F3)
           => filterlim(A,C,G,topolo7230453075368039082e_nhds(C,zero_zero(C)),F3) ) ) ) ).

% tendsto_0_le
tff(fact_7533_eventually__floor__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( ~ member(B,L,ring_1_Ints(B))
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_afw(fun(A,B),fun(B,fun(A,$o)),F2),L),F3) ) ) ) ).

% eventually_floor_less
tff(fact_7534_eventually__less__ceiling,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( ~ member(B,L,ring_1_Ints(B))
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_afx(fun(A,B),fun(B,fun(A,$o)),F2),L),F3) ) ) ) ).

% eventually_less_ceiling
tff(fact_7535_tendsto__powr_H,axiom,
    ! [A: $tType,F2: fun(A,real),A3: real,F3: filter(A),G: fun(A,real),B3: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A3),F3)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B3),F3)
       => ( ( ( A3 != zero_zero(real) )
            | ( aa(real,$o,ord_less(real,zero_zero(real)),B3)
              & eventually(A,aTP_Lamp_afy(fun(A,real),fun(A,$o),F2),F3) ) )
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_wj(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,powr(real,A3,B3)),F3) ) ) ) ).

% tendsto_powr'
tff(fact_7536_tendsto__powr2,axiom,
    ! [A: $tType,F2: fun(A,real),A3: real,F3: filter(A),G: fun(A,real),B3: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A3),F3)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B3),F3)
       => ( eventually(A,aTP_Lamp_afy(fun(A,real),fun(A,$o),F2),F3)
         => ( aa(real,$o,ord_less(real,zero_zero(real)),B3)
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_wj(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,powr(real,A3,B3)),F3) ) ) ) ) ).

% tendsto_powr2
tff(fact_7537_tendsto__zero__powrI,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A),G: fun(A,real),B3: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B3),F3)
       => ( eventually(A,aTP_Lamp_afy(fun(A,real),fun(A,$o),F2),F3)
         => ( aa(real,$o,ord_less(real,zero_zero(real)),B3)
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_wj(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ) ) ).

% tendsto_zero_powrI
tff(fact_7538_LIM__at__top__divide,axiom,
    ! [A: $tType,F2: fun(A,real),A3: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A3),F3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
       => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
         => ( eventually(A,aTP_Lamp_afz(fun(A,real),fun(A,$o),G),F3)
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_abw(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F3) ) ) ) ) ).

% LIM_at_top_divide
tff(fact_7539_filterlim__inverse__at__top__iff,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( eventually(A,aTP_Lamp_afz(fun(A,real),fun(A,$o),F2),F3)
     => ( filterlim(A,real,aTP_Lamp_abx(fun(A,real),fun(A,real),F2),at_top(real),F3)
      <=> filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% filterlim_inverse_at_top_iff
tff(fact_7540_filterlim__inverse__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
     => ( eventually(A,aTP_Lamp_afz(fun(A,real),fun(A,$o),F2),F3)
       => filterlim(A,real,aTP_Lamp_abx(fun(A,real),fun(A,real),F2),at_top(real),F3) ) ) ).

% filterlim_inverse_at_top
tff(fact_7541_filterlim__inverse__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
     => ( eventually(A,aTP_Lamp_aga(fun(A,real),fun(A,$o),F2),F3)
       => filterlim(A,real,aTP_Lamp_abx(fun(A,real),fun(A,real),F2),at_bot(real),F3) ) ) ).

% filterlim_inverse_at_bot
tff(fact_7542_lhopital__at__top__at__top,axiom,
    ! [F2: fun(real,real),A3: real,G: fun(real,real),F7: fun(real,real),G4: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,A3,top_top(set(real))))
     => ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,A3,top_top(set(real))))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_adi(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F7),topolo174197925503356063within(real,A3,top_top(set(real))))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_adi(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G4),topolo174197925503356063within(real,A3,top_top(set(real))))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_agb(fun(real,real),fun(fun(real,real),fun(real,real)),F7),G4),at_top(real),topolo174197925503356063within(real,A3,top_top(set(real))))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_agb(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_top(real),topolo174197925503356063within(real,A3,top_top(set(real)))) ) ) ) ) ) ).

% lhopital_at_top_at_top
tff(fact_7543_lhopital,axiom,
    ! [F2: fun(real,real),X: real,G: fun(real,real),G4: fun(real,real),F7: fun(real,real),F3: filter(real)] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,X,top_top(set(real))))
     => ( filterlim(real,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,X,top_top(set(real))))
       => ( eventually(real,aTP_Lamp_adh(fun(real,real),fun(real,$o),G),topolo174197925503356063within(real,X,top_top(set(real))))
         => ( eventually(real,aTP_Lamp_adh(fun(real,real),fun(real,$o),G4),topolo174197925503356063within(real,X,top_top(set(real))))
           => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_adi(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F7),topolo174197925503356063within(real,X,top_top(set(real))))
             => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_adi(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G4),topolo174197925503356063within(real,X,top_top(set(real))))
               => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_adj(fun(real,real),fun(fun(real,real),fun(real,real)),G4),F7),F3,topolo174197925503356063within(real,X,top_top(set(real))))
                 => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_agb(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),F3,topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ) ) ) ) ).

% lhopital
tff(fact_7544_lhopital__right__at__top__at__top,axiom,
    ! [F2: fun(real,real),A3: real,G: fun(real,real),F7: fun(real,real),G4: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,A3,set_ord_greaterThan(real,A3)))
     => ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,A3,set_ord_greaterThan(real,A3)))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_adi(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F7),topolo174197925503356063within(real,A3,set_ord_greaterThan(real,A3)))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_adi(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G4),topolo174197925503356063within(real,A3,set_ord_greaterThan(real,A3)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_agb(fun(real,real),fun(fun(real,real),fun(real,real)),F7),G4),at_top(real),topolo174197925503356063within(real,A3,set_ord_greaterThan(real,A3)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_agb(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_top(real),topolo174197925503356063within(real,A3,set_ord_greaterThan(real,A3))) ) ) ) ) ) ).

% lhopital_right_at_top_at_top
tff(fact_7545_lhopital__at__top__at__bot,axiom,
    ! [F2: fun(real,real),A3: real,G: fun(real,real),F7: fun(real,real),G4: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,A3,top_top(set(real))))
     => ( filterlim(real,real,G,at_bot(real),topolo174197925503356063within(real,A3,top_top(set(real))))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_adi(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F7),topolo174197925503356063within(real,A3,top_top(set(real))))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_adi(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G4),topolo174197925503356063within(real,A3,top_top(set(real))))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_agb(fun(real,real),fun(fun(real,real),fun(real,real)),F7),G4),at_bot(real),topolo174197925503356063within(real,A3,top_top(set(real))))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_agb(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_bot(real),topolo174197925503356063within(real,A3,top_top(set(real)))) ) ) ) ) ) ).

% lhopital_at_top_at_bot
tff(fact_7546_lhopital__left__at__top__at__top,axiom,
    ! [F2: fun(real,real),A3: real,G: fun(real,real),F7: fun(real,real),G4: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,A3,set_ord_lessThan(real,A3)))
     => ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,A3,set_ord_lessThan(real,A3)))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_adi(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F7),topolo174197925503356063within(real,A3,set_ord_lessThan(real,A3)))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_adi(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G4),topolo174197925503356063within(real,A3,set_ord_lessThan(real,A3)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_agb(fun(real,real),fun(fun(real,real),fun(real,real)),F7),G4),at_top(real),topolo174197925503356063within(real,A3,set_ord_lessThan(real,A3)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_agb(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_top(real),topolo174197925503356063within(real,A3,set_ord_lessThan(real,A3))) ) ) ) ) ) ).

% lhopital_left_at_top_at_top
tff(fact_7547_lhopital__at__top,axiom,
    ! [G: fun(real,real),X: real,G4: fun(real,real),F2: fun(real,real),F7: fun(real,real),Y: real] :
      ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,X,top_top(set(real))))
     => ( eventually(real,aTP_Lamp_adh(fun(real,real),fun(real,$o),G4),topolo174197925503356063within(real,X,top_top(set(real))))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_adi(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F7),topolo174197925503356063within(real,X,top_top(set(real))))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_adi(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G4),topolo174197925503356063within(real,X,top_top(set(real))))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_adj(fun(real,real),fun(fun(real,real),fun(real,real)),G4),F7),topolo7230453075368039082e_nhds(real,Y),topolo174197925503356063within(real,X,top_top(set(real))))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_adj(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),topolo7230453075368039082e_nhds(real,Y),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ) ) ).

% lhopital_at_top
tff(fact_7548_lhospital__at__top__at__top,axiom,
    ! [G: fun(real,real),G4: fun(real,real),F2: fun(real,real),F7: fun(real,real),X: real] :
      ( filterlim(real,real,G,at_top(real),at_top(real))
     => ( eventually(real,aTP_Lamp_adh(fun(real,real),fun(real,$o),G4),at_top(real))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_adi(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F7),at_top(real))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_adi(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G4),at_top(real))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_adj(fun(real,real),fun(fun(real,real),fun(real,real)),G4),F7),topolo7230453075368039082e_nhds(real,X),at_top(real))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_adj(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),topolo7230453075368039082e_nhds(real,X),at_top(real)) ) ) ) ) ) ).

% lhospital_at_top_at_top
tff(fact_7549_lhopital__right,axiom,
    ! [F2: fun(real,real),X: real,G: fun(real,real),G4: fun(real,real),F7: fun(real,real),F3: filter(real)] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
     => ( filterlim(real,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
       => ( eventually(real,aTP_Lamp_adh(fun(real,real),fun(real,$o),G),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
         => ( eventually(real,aTP_Lamp_adh(fun(real,real),fun(real,$o),G4),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
           => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_adi(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F7),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
             => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_adi(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G4),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
               => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_adj(fun(real,real),fun(fun(real,real),fun(real,real)),G4),F7),F3,topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
                 => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_agb(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),F3,topolo174197925503356063within(real,X,set_ord_greaterThan(real,X))) ) ) ) ) ) ) ) ).

% lhopital_right
tff(fact_7550_lhopital__right__0,axiom,
    ! [F0: fun(real,real),G0: fun(real,real),G4: fun(real,real),F7: fun(real,real),F3: filter(real)] :
      ( filterlim(real,real,F0,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
     => ( filterlim(real,real,G0,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
       => ( eventually(real,aTP_Lamp_adh(fun(real,real),fun(real,$o),G0),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
         => ( eventually(real,aTP_Lamp_adh(fun(real,real),fun(real,$o),G4),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
           => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_adi(fun(real,real),fun(fun(real,real),fun(real,$o)),F0),F7),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
             => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_adi(fun(real,real),fun(fun(real,real),fun(real,$o)),G0),G4),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
               => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_adj(fun(real,real),fun(fun(real,real),fun(real,real)),G4),F7),F3,topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
                 => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_agb(fun(real,real),fun(fun(real,real),fun(real,real)),F0),G0),F3,topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ) ) ) ) ) ) ) ).

% lhopital_right_0
tff(fact_7551_lhopital__left,axiom,
    ! [F2: fun(real,real),X: real,G: fun(real,real),G4: fun(real,real),F7: fun(real,real),F3: filter(real)] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
     => ( filterlim(real,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
       => ( eventually(real,aTP_Lamp_adh(fun(real,real),fun(real,$o),G),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
         => ( eventually(real,aTP_Lamp_adh(fun(real,real),fun(real,$o),G4),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
           => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_adi(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F7),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
             => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_adi(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G4),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
               => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_adj(fun(real,real),fun(fun(real,real),fun(real,real)),G4),F7),F3,topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
                 => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_agb(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),F3,topolo174197925503356063within(real,X,set_ord_lessThan(real,X))) ) ) ) ) ) ) ) ).

% lhopital_left
tff(fact_7552_lhopital__right__at__top__at__bot,axiom,
    ! [F2: fun(real,real),A3: real,G: fun(real,real),F7: fun(real,real),G4: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,A3,set_ord_greaterThan(real,A3)))
     => ( filterlim(real,real,G,at_bot(real),topolo174197925503356063within(real,A3,set_ord_greaterThan(real,A3)))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_adi(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F7),topolo174197925503356063within(real,A3,set_ord_greaterThan(real,A3)))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_adi(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G4),topolo174197925503356063within(real,A3,set_ord_greaterThan(real,A3)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_agb(fun(real,real),fun(fun(real,real),fun(real,real)),F7),G4),at_bot(real),topolo174197925503356063within(real,A3,set_ord_greaterThan(real,A3)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_agb(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_bot(real),topolo174197925503356063within(real,A3,set_ord_greaterThan(real,A3))) ) ) ) ) ) ).

% lhopital_right_at_top_at_bot
tff(fact_7553_lhopital__left__at__top__at__bot,axiom,
    ! [F2: fun(real,real),A3: real,G: fun(real,real),F7: fun(real,real),G4: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,A3,set_ord_lessThan(real,A3)))
     => ( filterlim(real,real,G,at_bot(real),topolo174197925503356063within(real,A3,set_ord_lessThan(real,A3)))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_adi(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F7),topolo174197925503356063within(real,A3,set_ord_lessThan(real,A3)))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_adi(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G4),topolo174197925503356063within(real,A3,set_ord_lessThan(real,A3)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_agb(fun(real,real),fun(fun(real,real),fun(real,real)),F7),G4),at_bot(real),topolo174197925503356063within(real,A3,set_ord_lessThan(real,A3)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_agb(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_bot(real),topolo174197925503356063within(real,A3,set_ord_lessThan(real,A3))) ) ) ) ) ) ).

% lhopital_left_at_top_at_bot
tff(fact_7554_lhopital__right__0__at__top,axiom,
    ! [G: fun(real,real),G4: fun(real,real),F2: fun(real,real),F7: fun(real,real),X: real] :
      ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
     => ( eventually(real,aTP_Lamp_adh(fun(real,real),fun(real,$o),G4),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_adi(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F7),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_adi(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G4),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_adj(fun(real,real),fun(fun(real,real),fun(real,real)),G4),F7),topolo7230453075368039082e_nhds(real,X),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_adj(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),topolo7230453075368039082e_nhds(real,X),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ) ) ) ) ) ).

% lhopital_right_0_at_top
tff(fact_7555_polyfun__extremal,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [C2: fun(nat,A),K: nat,Na: nat,B2: real] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( aa(nat,$o,ord_less_eq(nat,one_one(nat)),K)
           => ( aa(nat,$o,ord_less_eq(nat,K),Na)
             => eventually(A,aa(real,fun(A,$o),aa(nat,fun(real,fun(A,$o)),aTP_Lamp_agc(fun(nat,A),fun(nat,fun(real,fun(A,$o))),C2),Na),B2),at_infinity(A)) ) ) ) ) ).

% polyfun_extremal
tff(fact_7556_Bfun__metric__def,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( bfun(A,B,F2,F3)
        <=> ? [Y6: B,K6: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),K6)
              & eventually(A,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_agd(fun(A,B),fun(B,fun(real,fun(A,$o))),F2),Y6),K6),F3) ) ) ) ).

% Bfun_metric_def
tff(fact_7557_eventually__sequentially__Suc,axiom,
    ! [P: fun(nat,$o)] :
      ( eventually(nat,aTP_Lamp_age(fun(nat,$o),fun(nat,$o),P),at_top(nat))
    <=> eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentially_Suc
tff(fact_7558_eventually__sequentially__seg,axiom,
    ! [P: fun(nat,$o),K: nat] :
      ( eventually(nat,aa(nat,fun(nat,$o),aTP_Lamp_agf(fun(nat,$o),fun(nat,fun(nat,$o)),P),K),at_top(nat))
    <=> eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentially_seg
tff(fact_7559_Bseq__eventually__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(nat,A),G: fun(nat,B)] :
          ( eventually(nat,aa(fun(nat,B),fun(nat,$o),aTP_Lamp_agg(fun(nat,A),fun(fun(nat,B),fun(nat,$o)),F2),G),at_top(nat))
         => ( bfun(nat,B,G,at_top(nat))
           => bfun(nat,A,F2,at_top(nat)) ) ) ) ).

% Bseq_eventually_mono
tff(fact_7560_le__sequentially,axiom,
    ! [F3: filter(nat)] :
      ( aa(filter(nat),$o,ord_less_eq(filter(nat),F3),at_top(nat))
    <=> ! [N8: nat] : eventually(nat,ord_less_eq(nat,N8),F3) ) ).

% le_sequentially
tff(fact_7561_sequentially__offset,axiom,
    ! [P: fun(nat,$o),K: nat] :
      ( eventually(nat,P,at_top(nat))
     => eventually(nat,aa(nat,fun(nat,$o),aTP_Lamp_agf(fun(nat,$o),fun(nat,fun(nat,$o)),P),K),at_top(nat)) ) ).

% sequentially_offset
tff(fact_7562_summable__cong,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),G: fun(nat,A)] :
          ( eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_agh(fun(nat,A),fun(fun(nat,A),fun(nat,$o)),F2),G),at_top(nat))
         => ( summable(A,F2)
          <=> summable(A,G) ) ) ) ).

% summable_cong
tff(fact_7563_eventually__False__sequentially,axiom,
    ~ eventually(nat,aTP_Lamp_agi(nat,$o),at_top(nat)) ).

% eventually_False_sequentially
tff(fact_7564_eventually__sequentially,axiom,
    ! [P: fun(nat,$o)] :
      ( eventually(nat,P,at_top(nat))
    <=> ? [N8: nat] :
        ! [N3: nat] :
          ( aa(nat,$o,ord_less_eq(nat,N8),N3)
         => aa(nat,$o,P,N3) ) ) ).

% eventually_sequentially
tff(fact_7565_eventually__sequentiallyI,axiom,
    ! [C2: nat,P: fun(nat,$o)] :
      ( ! [X4: nat] :
          ( aa(nat,$o,ord_less_eq(nat,C2),X4)
         => aa(nat,$o,P,X4) )
     => eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentiallyI
tff(fact_7566_eventually__not__equal__at__infinity,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A3: A] : eventually(A,aTP_Lamp_agj(A,fun(A,$o),A3),at_infinity(A)) ) ).

% eventually_not_equal_at_infinity
tff(fact_7567_Bfun__const,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [C2: B,F3: filter(A)] : bfun(A,B,aTP_Lamp_agk(B,fun(A,B),C2),F3) ) ).

% Bfun_const
tff(fact_7568_Bseq__minus__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( bfun(nat,A,aTP_Lamp_bj(fun(nat,A),fun(nat,A),X5),at_top(nat))
        <=> bfun(nat,A,X5,at_top(nat)) ) ) ).

% Bseq_minus_iff
tff(fact_7569_Bseq__subseq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),G: fun(nat,nat)] :
          ( bfun(nat,A,F2,at_top(nat))
         => bfun(nat,A,aa(fun(nat,nat),fun(nat,A),aTP_Lamp_agl(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),F2),G),at_top(nat)) ) ) ).

% Bseq_subseq
tff(fact_7570_Bseq__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( bfun(nat,A,aTP_Lamp_bw(fun(nat,A),fun(nat,A),F2),at_top(nat))
        <=> bfun(nat,A,F2,at_top(nat)) ) ) ).

% Bseq_Suc_iff
tff(fact_7571_Bseq__mult,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(nat,A),G: fun(nat,A)] :
          ( bfun(nat,A,F2,at_top(nat))
         => ( bfun(nat,A,G,at_top(nat))
           => bfun(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_agm(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G),at_top(nat)) ) ) ) ).

% Bseq_mult
tff(fact_7572_Bseq__add,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( bfun(nat,A,F2,at_top(nat))
         => bfun(nat,A,aa(A,fun(nat,A),aTP_Lamp_agn(fun(nat,A),fun(A,fun(nat,A)),F2),C2),at_top(nat)) ) ) ).

% Bseq_add
tff(fact_7573_Bseq__offset,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A),K: nat] :
          ( bfun(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ago(fun(nat,A),fun(nat,fun(nat,A)),X5),K),at_top(nat))
         => bfun(nat,A,X5,at_top(nat)) ) ) ).

% Bseq_offset
tff(fact_7574_Bseq__add__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( bfun(nat,A,aa(A,fun(nat,A),aTP_Lamp_agn(fun(nat,A),fun(A,fun(nat,A)),F2),C2),at_top(nat))
        <=> bfun(nat,A,F2,at_top(nat)) ) ) ).

% Bseq_add_iff
tff(fact_7575_Bseq__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A),K: nat] :
          ( bfun(nat,A,X5,at_top(nat))
         => bfun(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ago(fun(nat,A),fun(nat,fun(nat,A)),X5),K),at_top(nat)) ) ) ).

% Bseq_ignore_initial_segment
tff(fact_7576_filterlim__at__infinity__imp__eventually__ne,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_infinity(B),F3)
         => eventually(A,aa(B,fun(A,$o),aTP_Lamp_agp(fun(A,B),fun(B,fun(A,$o)),F2),C2),F3) ) ) ).

% filterlim_at_infinity_imp_eventually_ne
tff(fact_7577_countable__basis__at__decseq,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [X: A] :
          ~ ! [A8: fun(nat,set(A))] :
              ( ! [I2: nat] : topolo1002775350975398744n_open(A,aa(nat,set(A),A8,I2))
             => ( ! [I2: nat] : member(A,X,aa(nat,set(A),A8,I2))
               => ~ ! [S8: set(A)] :
                      ( topolo1002775350975398744n_open(A,S8)
                     => ( member(A,X,S8)
                       => eventually(nat,aa(set(A),fun(nat,$o),aTP_Lamp_agq(fun(nat,set(A)),fun(set(A),fun(nat,$o)),A8),S8),at_top(nat)) ) ) ) ) ) ).

% countable_basis_at_decseq
tff(fact_7578_Bseq__cmult__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,F2: fun(nat,A)] :
          ( ( C2 != zero_zero(A) )
         => ( bfun(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ay(A,fun(fun(nat,A),fun(nat,A)),C2),F2),at_top(nat))
          <=> bfun(nat,A,F2,at_top(nat)) ) ) ) ).

% Bseq_cmult_iff
tff(fact_7579_filterlim__real__at__infinity__sequentially,axiom,
    filterlim(nat,real,semiring_1_of_nat(real),at_infinity(real),at_top(nat)) ).

% filterlim_real_at_infinity_sequentially
tff(fact_7580_tendsto__of__nat,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => filterlim(nat,A,semiring_1_of_nat(A),at_infinity(A),at_top(nat)) ) ).

% tendsto_of_nat
tff(fact_7581_not__tendsto__and__filterlim__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: filter(A),F2: fun(A,B),C2: B] :
          ( ( F3 != bot_bot(filter(A)) )
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F3)
           => ~ filterlim(A,B,F2,at_infinity(B),F3) ) ) ) ).

% not_tendsto_and_filterlim_at_infinity
tff(fact_7582_tendsto__add__filterlim__at__infinity_H,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A),G: fun(A,B),C2: B] :
          ( filterlim(A,B,F2,at_infinity(B),F3)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,C2),F3)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_agr(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),at_infinity(B),F3) ) ) ) ).

% tendsto_add_filterlim_at_infinity'
tff(fact_7583_tendsto__add__filterlim__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),C2: B,F3: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F3)
         => ( filterlim(A,B,G,at_infinity(B),F3)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_agr(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),at_infinity(B),F3) ) ) ) ).

% tendsto_add_filterlim_at_infinity
tff(fact_7584_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_7585_summable__comparison__test__ev,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( eventually(nat,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_ags(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),F2),G),at_top(nat))
         => ( summable(real,G)
           => summable(A,F2) ) ) ) ).

% summable_comparison_test_ev
tff(fact_7586_BseqD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( bfun(nat,A,X5,at_top(nat))
         => ? [K7: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),K7)
              & ! [N10: nat] : aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(nat,A,X5,N10))),K7) ) ) ) ).

% BseqD
tff(fact_7587_BseqE,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( bfun(nat,A,X5,at_top(nat))
         => ~ ! [K7: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),K7)
               => ~ ! [N10: nat] : aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(nat,A,X5,N10))),K7) ) ) ) ).

% BseqE
tff(fact_7588_BseqI,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [K4: real,X5: fun(nat,A)] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),K4)
         => ( ! [N: nat] : aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(nat,A,X5,N))),K4)
           => bfun(nat,A,X5,at_top(nat)) ) ) ) ).

% BseqI
tff(fact_7589_Bseq__def,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( bfun(nat,A,X5,at_top(nat))
        <=> ? [K6: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),K6)
              & ! [N3: nat] : aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(nat,A,X5,N3))),K6) ) ) ) ).

% Bseq_def
tff(fact_7590_Bseq__iff1a,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( bfun(nat,A,X5,at_top(nat))
        <=> ? [N8: nat] :
            ! [N3: nat] : aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(nat,A,X5,N3))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N8))) ) ) ).

% Bseq_iff1a
tff(fact_7591_Bseq__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( bfun(nat,A,X5,at_top(nat))
        <=> ? [N8: nat] :
            ! [N3: nat] : aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(nat,A,X5,N3))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N8))) ) ) ).

% Bseq_iff
tff(fact_7592_Bseq__realpow,axiom,
    ! [X: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X)
     => ( aa(real,$o,ord_less_eq(real,X),one_one(real))
       => bfun(nat,real,power_power(real,X),at_top(nat)) ) ) ).

% Bseq_realpow
tff(fact_7593_tendsto__mult__filterlim__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [F2: fun(A,B),C2: B,F3: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F3)
         => ( ( C2 != zero_zero(B) )
           => ( filterlim(A,B,G,at_infinity(B),F3)
             => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_agt(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),at_infinity(B),F3) ) ) ) ) ).

% tendsto_mult_filterlim_at_infinity
tff(fact_7594_filterlim__power__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),F3: filter(A),Na: nat] :
          ( filterlim(A,B,F2,at_infinity(B),F3)
         => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
           => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_agu(fun(A,B),fun(nat,fun(A,B)),F2),Na),at_infinity(B),F3) ) ) ) ).

% filterlim_power_at_infinity
tff(fact_7595_tendsto__divide__0,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),C2: B,F3: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F3)
         => ( filterlim(A,B,G,at_infinity(B),F3)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_agv(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ) ).

% tendsto_divide_0
tff(fact_7596_filterlim__at__infinity__imp__norm__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,B,F2,at_infinity(B),F3)
         => filterlim(A,real,aTP_Lamp_wk(fun(A,B),fun(A,real),F2),at_top(real),F3) ) ) ).

% filterlim_at_infinity_imp_norm_at_top
tff(fact_7597_filterlim__norm__at__top__imp__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,real,aTP_Lamp_wk(fun(A,B),fun(A,real),F2),at_top(real),F3)
         => filterlim(A,B,F2,at_infinity(B),F3) ) ) ).

% filterlim_norm_at_top_imp_at_infinity
tff(fact_7598_filterlim__at__infinity__conv__norm__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),G2: filter(A)] :
          ( filterlim(A,B,F2,at_infinity(B),G2)
        <=> filterlim(A,real,aTP_Lamp_wk(fun(A,B),fun(A,real),F2),at_top(real),G2) ) ) ).

% filterlim_at_infinity_conv_norm_at_top
tff(fact_7599_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_7600_eventually__at__infinity__pos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [P2: fun(A,$o)] :
          ( eventually(A,P2,at_infinity(A))
        <=> ? [B9: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),B9)
              & ! [X3: A] :
                  ( aa(real,$o,ord_less_eq(real,B9),real_V7770717601297561774m_norm(A,X3))
                 => aa(A,$o,P2,X3) ) ) ) ) ).

% eventually_at_infinity_pos
tff(fact_7601_BfunI,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),K4: real,F3: filter(A)] :
          ( eventually(A,aa(real,fun(A,$o),aTP_Lamp_agw(fun(A,B),fun(real,fun(A,$o)),F2),K4),F3)
         => bfun(A,B,F2,F3) ) ) ).

% BfunI
tff(fact_7602_filterlim__at__infinity__imp__filterlim__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,at_infinity(real),F3)
     => ( eventually(A,aTP_Lamp_afz(fun(A,real),fun(A,$o),F2),F3)
       => filterlim(A,real,F2,at_top(real),F3) ) ) ).

% filterlim_at_infinity_imp_filterlim_at_top
tff(fact_7603_filterlim__at__infinity__imp__filterlim__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,at_infinity(real),F3)
     => ( eventually(A,aTP_Lamp_aga(fun(A,real),fun(A,$o),F2),F3)
       => filterlim(A,real,F2,at_bot(real),F3) ) ) ).

% filterlim_at_infinity_imp_filterlim_at_bot
tff(fact_7604_lim__infinity__imp__sequentially,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(real,A),L: A] :
          ( filterlim(real,A,F2,topolo7230453075368039082e_nhds(A,L),at_infinity(real))
         => filterlim(nat,A,aTP_Lamp_agx(fun(real,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% lim_infinity_imp_sequentially
tff(fact_7605_filterlim__inverse__at__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [G: fun(A,B),F3: filter(A)] :
          ( filterlim(A,B,aTP_Lamp_we(fun(A,B),fun(A,B),G),topolo174197925503356063within(B,zero_zero(B),top_top(set(B))),F3)
        <=> filterlim(A,B,G,at_infinity(B),F3) ) ) ).

% filterlim_inverse_at_iff
tff(fact_7606_Bseq__iff3,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( bfun(nat,A,X5,at_top(nat))
        <=> ? [K3: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),K3)
              & ? [N8: nat] :
                ! [N3: nat] : aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,plus_plus(A,aa(nat,A,X5,N3)),aa(A,A,uminus_uminus(A),aa(nat,A,X5,N8))))),K3) ) ) ) ).

% Bseq_iff3
tff(fact_7607_Bseq__iff2,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( bfun(nat,A,X5,at_top(nat))
        <=> ? [K3: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),K3)
              & ? [X3: A] :
                ! [N3: nat] : aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,plus_plus(A,aa(nat,A,X5,N3)),aa(A,A,uminus_uminus(A),X3)))),K3) ) ) ) ).

% Bseq_iff2
tff(fact_7608_filterlim__divide__at__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),C2: A,F3: filter(A),G: fun(A,A)] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,C2),F3)
         => ( filterlim(A,A,G,topolo174197925503356063within(A,zero_zero(A),top_top(set(A))),F3)
           => ( ( C2 != zero_zero(A) )
             => filterlim(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_qy(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),at_infinity(A),F3) ) ) ) ) ).

% filterlim_divide_at_infinity
tff(fact_7609_filterlim__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [C2: real,F2: fun(A,B),F3: filter(A)] :
          ( aa(real,$o,ord_less_eq(real,zero_zero(real)),C2)
         => ( filterlim(A,B,F2,at_infinity(B),F3)
          <=> ! [R5: real] :
                ( aa(real,$o,ord_less(real,C2),R5)
               => eventually(A,aa(real,fun(A,$o),aTP_Lamp_agy(fun(A,B),fun(real,fun(A,$o)),F2),R5),F3) ) ) ) ) ).

% filterlim_at_infinity
tff(fact_7610_Bfun__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),A3: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),F3)
         => ( ( A3 != zero_zero(B) )
           => bfun(A,B,aTP_Lamp_we(fun(A,B),fun(A,B),F2),F3) ) ) ) ).

% Bfun_inverse
tff(fact_7611_filterlim__realpow__sequentially__gt1,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: A] :
          ( aa(real,$o,ord_less(real,one_one(real)),real_V7770717601297561774m_norm(A,X))
         => filterlim(nat,A,power_power(A,X),at_infinity(A),at_top(nat)) ) ) ).

% filterlim_realpow_sequentially_gt1
tff(fact_7612_lim__zero__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),L: A] :
          ( filterlim(A,A,aTP_Lamp_agz(fun(A,A),fun(A,A),F2),topolo7230453075368039082e_nhds(A,L),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
         => filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,L),at_infinity(A)) ) ) ).

% lim_zero_infinity
tff(fact_7613_Bfun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( bfun(A,B,F2,F3)
        <=> ? [K6: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),K6)
              & eventually(A,aa(real,fun(A,$o),aTP_Lamp_agw(fun(A,B),fun(real,fun(A,$o)),F2),K6),F3) ) ) ) ).

% Bfun_def
tff(fact_7614_BfunE,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( bfun(A,B,F2,F3)
         => ~ ! [B4: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),B4)
               => ~ eventually(A,aa(real,fun(A,$o),aTP_Lamp_agw(fun(A,B),fun(real,fun(A,$o)),F2),B4),F3) ) ) ) ).

% BfunE
tff(fact_7615_summable__Cauchy_H,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( eventually(nat,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_aha(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),F2),G),at_top(nat))
         => ( filterlim(nat,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => summable(A,F2) ) ) ) ).

% summable_Cauchy'
tff(fact_7616_sequentially__imp__eventually__at__right,axiom,
    ! [A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [A3: A,B3: A,P: fun(A,$o)] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( ! [F4: fun(nat,A)] :
                ( ! [N10: nat] : aa(A,$o,ord_less(A,A3),aa(nat,A,F4,N10))
               => ( ! [N10: nat] : aa(A,$o,ord_less(A,aa(nat,A,F4,N10)),B3)
                 => ( order_antimono(nat,A,F4)
                   => ( filterlim(nat,A,F4,topolo7230453075368039082e_nhds(A,A3),at_top(nat))
                     => eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_ahb(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),P),F4),at_top(nat)) ) ) ) )
           => eventually(A,P,topolo174197925503356063within(A,A3,set_ord_greaterThan(A,A3))) ) ) ) ).

% sequentially_imp_eventually_at_right
tff(fact_7617_decseq__const,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [K: A] : order_antimono(nat,A,aTP_Lamp_ahc(A,fun(nat,A),K)) ) ).

% decseq_const
tff(fact_7618_eventually__all__finite,axiom,
    ! [A: $tType,B: $tType] :
      ( finite_finite(A)
     => ! [P: fun(B,fun(A,$o)),Net: filter(B)] :
          ( ! [Y3: A] : eventually(B,aa(A,fun(B,$o),aTP_Lamp_ahd(fun(B,fun(A,$o)),fun(A,fun(B,$o)),P),Y3),Net)
         => eventually(B,aTP_Lamp_ahe(fun(B,fun(A,$o)),fun(B,$o),P),Net) ) ) ).

% eventually_all_finite
tff(fact_7619_antimonoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),X: A,Y: A] :
          ( order_antimono(A,B,F2)
         => ( aa(A,$o,ord_less_eq(A,X),Y)
           => aa(B,$o,ord_less_eq(B,aa(A,B,F2,Y)),aa(A,B,F2,X)) ) ) ) ).

% antimonoD
tff(fact_7620_antimonoE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),X: A,Y: A] :
          ( order_antimono(A,B,F2)
         => ( aa(A,$o,ord_less_eq(A,X),Y)
           => aa(B,$o,ord_less_eq(B,aa(A,B,F2,Y)),aa(A,B,F2,X)) ) ) ) ).

% antimonoE
tff(fact_7621_antimonoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( ! [X4: A,Y3: A] :
              ( aa(A,$o,ord_less_eq(A,X4),Y3)
             => aa(B,$o,ord_less_eq(B,aa(A,B,F2,Y3)),aa(A,B,F2,X4)) )
         => order_antimono(A,B,F2) ) ) ).

% antimonoI
tff(fact_7622_antimono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( order_antimono(A,B,F2)
        <=> ! [X3: A,Y6: A] :
              ( aa(A,$o,ord_less_eq(A,X3),Y6)
             => aa(B,$o,ord_less_eq(B,aa(A,B,F2,Y6)),aa(A,B,F2,X3)) ) ) ) ).

% antimono_def
tff(fact_7623_finite__set__of__finite__funs,axiom,
    ! [A: $tType,B: $tType,A2: set(A),B2: set(B),D2: B] :
      ( finite_finite2(A,A2)
     => ( finite_finite2(B,B2)
       => finite_finite2(fun(A,B),collect(fun(A,B),aa(B,fun(fun(A,B),$o),aa(set(B),fun(B,fun(fun(A,B),$o)),aTP_Lamp_ahf(set(A),fun(set(B),fun(B,fun(fun(A,B),$o))),A2),B2),D2))) ) ) ).

% finite_set_of_finite_funs
tff(fact_7624_eventually__all__ge__at__top,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o)] :
          ( eventually(A,P,at_top(A))
         => eventually(A,aTP_Lamp_ahg(fun(A,$o),fun(A,$o),P),at_top(A)) ) ) ).

% eventually_all_ge_at_top
tff(fact_7625_min__of__antimono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F2: fun(A,B),X: A,Y: A] :
          ( order_antimono(A,B,F2)
         => ( ord_min(B,aa(A,B,F2,X),aa(A,B,F2,Y)) = aa(A,B,F2,aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)) ) ) ) ).

% min_of_antimono
tff(fact_7626_max__of__antimono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F2: fun(A,B),X: A,Y: A] :
          ( order_antimono(A,B,F2)
         => ( aa(B,B,aa(B,fun(B,B),ord_max(B),aa(A,B,F2,X)),aa(A,B,F2,Y)) = aa(A,B,F2,ord_min(A,X,Y)) ) ) ) ).

% max_of_antimono
tff(fact_7627_tendsto__at__right__sequentially,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A3: A,B3: A,X5: fun(A,B),L5: B] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( ! [S4: fun(nat,A)] :
                ( ! [N10: nat] : aa(A,$o,ord_less(A,A3),aa(nat,A,S4,N10))
               => ( ! [N10: nat] : aa(A,$o,ord_less(A,aa(nat,A,S4,N10)),B3)
                 => ( order_antimono(nat,A,S4)
                   => ( filterlim(nat,A,S4,topolo7230453075368039082e_nhds(A,A3),at_top(nat))
                     => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_ahh(fun(A,B),fun(fun(nat,A),fun(nat,B)),X5),S4),topolo7230453075368039082e_nhds(B,L5),at_top(nat)) ) ) ) )
           => filterlim(A,B,X5,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A3,set_ord_greaterThan(A,A3))) ) ) ) ).

% tendsto_at_right_sequentially
tff(fact_7628_summable__bounded__partials,axiom,
    ! [A: $tType] :
      ( ( real_V8037385150606011577_space(A)
        & real_V822414075346904944vector(A) )
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( eventually(nat,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_ahi(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),F2),G),at_top(nat))
         => ( filterlim(nat,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => summable(A,F2) ) ) ) ).

% summable_bounded_partials
tff(fact_7629_Greatest__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o)] : order_Greatest(A,P) = the(A,aTP_Lamp_ahj(fun(A,$o),fun(A,$o),P)) ) ).

% Greatest_def
tff(fact_7630_greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( member(A,I,set_or3652927894154168847AtMost(A,L,U))
        <=> ( aa(A,$o,ord_less(A,L),I)
            & aa(A,$o,ord_less_eq(A,I),U) ) ) ) ).

% greaterThanAtMost_iff
tff(fact_7631_greaterThanAtMost__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,K: A] :
          ( aa(A,$o,ord_less_eq(A,L),K)
         => ( set_or3652927894154168847AtMost(A,K,L) = bot_bot(set(A)) ) ) ) ).

% greaterThanAtMost_empty
tff(fact_7632_greaterThanAtMost__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A,L: A] :
          ( ( bot_bot(set(A)) = set_or3652927894154168847AtMost(A,K,L) )
        <=> ~ aa(A,$o,ord_less(A,K),L) ) ) ).

% greaterThanAtMost_empty_iff2
tff(fact_7633_greaterThanAtMost__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A,L: A] :
          ( ( set_or3652927894154168847AtMost(A,K,L) = bot_bot(set(A)) )
        <=> ~ aa(A,$o,ord_less(A,K),L) ) ) ).

% greaterThanAtMost_empty_iff
tff(fact_7634_infinite__Ioc__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A] :
          ( ~ finite_finite2(A,set_or3652927894154168847AtMost(A,A3,B3))
        <=> aa(A,$o,ord_less(A,A3),B3) ) ) ).

% infinite_Ioc_iff
tff(fact_7635_GreatestI__nat,axiom,
    ! [P: fun(nat,$o),K: nat,B3: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [Y3: nat] :
            ( aa(nat,$o,P,Y3)
           => aa(nat,$o,ord_less_eq(nat,Y3),B3) )
       => aa(nat,$o,P,order_Greatest(nat,P)) ) ) ).

% GreatestI_nat
tff(fact_7636_Greatest__le__nat,axiom,
    ! [P: fun(nat,$o),K: nat,B3: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [Y3: nat] :
            ( aa(nat,$o,P,Y3)
           => aa(nat,$o,ord_less_eq(nat,Y3),B3) )
       => aa(nat,$o,ord_less_eq(nat,K),order_Greatest(nat,P)) ) ) ).

% Greatest_le_nat
tff(fact_7637_GreatestI__ex__nat,axiom,
    ! [P: fun(nat,$o),B3: nat] :
      ( ? [X_13: nat] : aa(nat,$o,P,X_13)
     => ( ! [Y3: nat] :
            ( aa(nat,$o,P,Y3)
           => aa(nat,$o,ord_less_eq(nat,Y3),B3) )
       => aa(nat,$o,P,order_Greatest(nat,P)) ) ) ).

% GreatestI_ex_nat
tff(fact_7638_ivl__disj__un__two_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,ord_less_eq(A,L),M)
         => ( aa(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_7639_infinite__Ioc,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ~ finite_finite2(A,set_or3652927894154168847AtMost(A,A3,B3)) ) ) ).

% infinite_Ioc
tff(fact_7640_ivl__disj__un__two_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,ord_less_eq(A,L),M)
         => ( aa(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_7641_open__left,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S: set(A),X: A,Y: A] :
          ( topolo1002775350975398744n_open(A,S)
         => ( member(A,X,S)
           => ( aa(A,$o,ord_less(A,Y),X)
             => ? [B5: A] :
                  ( aa(A,$o,ord_less(A,B5),X)
                  & aa(set(A),$o,ord_less_eq(set(A),set_or3652927894154168847AtMost(A,B5,X)),S) ) ) ) ) ) ).

% open_left
tff(fact_7642_ivl__disj__un__one_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$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_7643_ivl__disj__un__one_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,ord_less_eq(A,L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,U)),set_ord_greaterThan(A,U)) = set_ord_greaterThan(A,L) ) ) ) ).

% ivl_disj_un_one(5)
tff(fact_7644_Greatest__equality,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o),X: A] :
          ( aa(A,$o,P,X)
         => ( ! [Y3: A] :
                ( aa(A,$o,P,Y3)
               => aa(A,$o,ord_less_eq(A,Y3),X) )
           => ( order_Greatest(A,P) = X ) ) ) ) ).

% Greatest_equality
tff(fact_7645_GreatestI2__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o),X: A,Q: fun(A,$o)] :
          ( aa(A,$o,P,X)
         => ( ! [Y3: A] :
                ( aa(A,$o,P,Y3)
               => aa(A,$o,ord_less_eq(A,Y3),X) )
           => ( ! [X4: A] :
                  ( aa(A,$o,P,X4)
                 => ( ! [Y4: A] :
                        ( aa(A,$o,P,Y4)
                       => aa(A,$o,ord_less_eq(A,Y4),X4) )
                   => aa(A,$o,Q,X4) ) )
             => aa(A,$o,Q,order_Greatest(A,P)) ) ) ) ) ).

% GreatestI2_order
tff(fact_7646_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A,C2: A,D2: A] :
          ( aa(set(A),$o,ord_less_eq(set(A),set_or3652927894154168847AtMost(A,A3,B3)),set_or1337092689740270186AtMost(A,C2,D2))
        <=> ( aa(A,$o,ord_less(A,A3),B3)
           => ( aa(A,$o,ord_less_eq(A,C2),A3)
              & aa(A,$o,ord_less_eq(A,B3),D2) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastAtMost_iff
tff(fact_7647_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A,C2: A,D2: A] :
          ( aa(set(A),$o,ord_less_eq(set(A),set_or3652927894154168847AtMost(A,A3,B3)),set_or7035219750837199246ssThan(A,C2,D2))
        <=> ( aa(A,$o,ord_less(A,A3),B3)
           => ( aa(A,$o,ord_less_eq(A,C2),A3)
              & aa(A,$o,ord_less(A,B3),D2) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastLessThan_iff
tff(fact_7648_ivl__disj__un__two__touch_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,ord_less(A,L),M)
         => ( aa(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_7649_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A,C2: A,D2: A] :
          ( aa(set(A),$o,ord_less_eq(set(A),set_or5935395276787703475ssThan(A,A3,B3)),set_or3652927894154168847AtMost(A,C2,D2))
        <=> ( aa(A,$o,ord_less(A,A3),B3)
           => ( aa(A,$o,ord_less_eq(A,C2),A3)
              & aa(A,$o,ord_less_eq(A,B3),D2) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanAtMost_iff
tff(fact_7650_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] : set_or3652927894154168847AtMost(A,A3,B3) = aa(set(A),set(A),minus_minus(set(A),set_or1337092689740270186AtMost(A,A3,B3)),aa(set(A),set(A),insert(A,A3),bot_bot(set(A)))) ) ).

% greaterThanAtMost_eq_atLeastAtMost_diff
tff(fact_7651_ivl__disj__un__two_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,ord_less_eq(A,L),M)
         => ( aa(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_7652_ivl__disj__un__two__touch_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,ord_less(A,L),M)
         => ( aa(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_7653_ivl__disj__un__singleton_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$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),insert(A,L),bot_bot(set(A)))),set_or3652927894154168847AtMost(A,L,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(5)
tff(fact_7654_ivl__disj__un__singleton_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$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),insert(A,U),bot_bot(set(A)))) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(4)
tff(fact_7655_ivl__disj__un__two_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,ord_less(A,L),M)
         => ( aa(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_7656_cauchy__filter__metric,axiom,
    ! [A: $tType] :
      ( ( real_V768167426530841204y_dist(A)
        & topolo7287701948861334536_space(A) )
     => ! [F3: filter(A)] :
          ( topolo6773858410816713723filter(A,F3)
        <=> ! [E3: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),E3)
             => ? [P5: fun(A,$o)] :
                  ( eventually(A,P5,F3)
                  & ! [X3: A,Y6: A] :
                      ( ( aa(A,$o,P5,X3)
                        & aa(A,$o,P5,Y6) )
                     => aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,X3,Y6)),E3) ) ) ) ) ) ).

% cauchy_filter_metric
tff(fact_7657_GMVT,axiom,
    ! [A3: real,B3: real,F2: fun(real,real),G: fun(real,real)] :
      ( aa(real,$o,ord_less(real,A3),B3)
     => ( ! [X4: real] :
            ( ( aa(real,$o,ord_less_eq(real,A3),X4)
              & aa(real,$o,ord_less_eq(real,X4),B3) )
           => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X4,top_top(set(real))),F2) )
       => ( ! [X4: real] :
              ( ( aa(real,$o,ord_less(real,A3),X4)
                & aa(real,$o,ord_less(real,X4),B3) )
             => differentiable(real,real,F2,topolo174197925503356063within(real,X4,top_top(set(real)))) )
         => ( ! [X4: real] :
                ( ( aa(real,$o,ord_less_eq(real,A3),X4)
                  & aa(real,$o,ord_less_eq(real,X4),B3) )
               => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X4,top_top(set(real))),G) )
           => ( ! [X4: real] :
                  ( ( aa(real,$o,ord_less(real,A3),X4)
                    & aa(real,$o,ord_less(real,X4),B3) )
                 => differentiable(real,real,G,topolo174197925503356063within(real,X4,top_top(set(real)))) )
             => ? [G_c: real,F_c: real,C5: real] :
                  ( has_field_derivative(real,G,G_c,topolo174197925503356063within(real,C5,top_top(set(real))))
                  & has_field_derivative(real,F2,F_c,topolo174197925503356063within(real,C5,top_top(set(real))))
                  & aa(real,$o,ord_less(real,A3),C5)
                  & aa(real,$o,ord_less(real,C5),B3)
                  & ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,aa(real,real,F2,B3)),aa(real,real,F2,A3))),G_c) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,aa(real,real,G,B3)),aa(real,real,G,A3))),F_c) ) ) ) ) ) ) ) ).

% GMVT
tff(fact_7658_differentiable__cmult__right__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Q3: fun(A,B),C2: B,Ta: A] :
          ( differentiable(A,B,aa(B,fun(A,B),aTP_Lamp_ahk(fun(A,B),fun(B,fun(A,B)),Q3),C2),topolo174197925503356063within(A,Ta,top_top(set(A))))
        <=> ( ( C2 = zero_zero(B) )
            | differentiable(A,B,Q3,topolo174197925503356063within(A,Ta,top_top(set(A)))) ) ) ) ).

% differentiable_cmult_right_iff
tff(fact_7659_differentiable__cmult__left__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [C2: B,Q3: fun(A,B),Ta: A] :
          ( differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ahl(B,fun(fun(A,B),fun(A,B)),C2),Q3),topolo174197925503356063within(A,Ta,top_top(set(A))))
        <=> ( ( C2 = zero_zero(B) )
            | differentiable(A,B,Q3,topolo174197925503356063within(A,Ta,top_top(set(A)))) ) ) ) ).

% differentiable_cmult_left_iff
tff(fact_7660_differentiable__sum,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Sa: set(A),F2: fun(A,fun(B,C)),Net: filter(B)] :
          ( finite_finite2(A,Sa)
         => ( ! [X4: A] :
                ( member(A,X4,Sa)
               => differentiable(B,C,aa(A,fun(B,C),F2,X4),Net) )
           => differentiable(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_su(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Sa),F2),Net) ) ) ) ).

% differentiable_sum
tff(fact_7661_differentiable__compose,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [F2: fun(A,B),G: fun(C,A),X: C,Sa: set(C)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,aa(C,A,G,X),top_top(set(A))))
         => ( differentiable(C,A,G,topolo174197925503356063within(C,X,Sa))
           => differentiable(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_sy(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G),topolo174197925503356063within(C,X,Sa)) ) ) ) ).

% differentiable_compose
tff(fact_7662_differentiable__power,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),X: A,Sa: set(A),Na: nat] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,X,Sa))
         => differentiable(A,B,aa(nat,fun(A,B),aTP_Lamp_tn(fun(A,B),fun(nat,fun(A,B)),F2),Na),topolo174197925503356063within(A,X,Sa)) ) ) ).

% differentiable_power
tff(fact_7663_differentiable__scaleR,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,real),X: A,Sa: set(A),G: fun(A,B)] :
          ( differentiable(A,real,F2,topolo174197925503356063within(A,X,Sa))
         => ( differentiable(A,B,G,topolo174197925503356063within(A,X,Sa))
           => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_sg(fun(A,real),fun(fun(A,B),fun(A,B)),F2),G),topolo174197925503356063within(A,X,Sa)) ) ) ) ).

% differentiable_scaleR
tff(fact_7664_differentiable__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [F2: fun(A,B),X: A,Sa: set(A),G: fun(A,B)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,X,Sa))
         => ( differentiable(A,B,G,topolo174197925503356063within(A,X,Sa))
           => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_sw(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo174197925503356063within(A,X,Sa)) ) ) ) ).

% differentiable_mult
tff(fact_7665_differentiable__in__compose,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [F2: fun(A,B),G: fun(C,A),X: C,Sa: set(C)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,aa(C,A,G,X),aa(set(C),set(A),image(C,A,G),Sa)))
         => ( differentiable(C,A,G,topolo174197925503356063within(C,X,Sa))
           => differentiable(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_sy(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G),topolo174197925503356063within(C,X,Sa)) ) ) ) ).

% differentiable_in_compose
tff(fact_7666_differentiable__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( differentiable(A,B,F2,F3)
         => differentiable(A,B,aTP_Lamp_sv(fun(A,B),fun(A,B),F2),F3) ) ) ).

% differentiable_minus
tff(fact_7667_differentiable__ident,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: filter(A)] : differentiable(A,A,aTP_Lamp_so(A,A),F3) ) ).

% differentiable_ident
tff(fact_7668_differentiable__const,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [A3: B,F3: filter(A)] : differentiable(A,B,aTP_Lamp_sr(B,fun(A,B),A3),F3) ) ).

% differentiable_const
tff(fact_7669_differentiable__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F3: filter(A),G: fun(A,B)] :
          ( differentiable(A,B,F2,F3)
         => ( differentiable(A,B,G,F3)
           => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_sk(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),F3) ) ) ) ).

% differentiable_diff
tff(fact_7670_differentiable__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F3: filter(A),G: fun(A,B)] :
          ( differentiable(A,B,F2,F3)
         => ( differentiable(A,B,G,F3)
           => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_sj(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),F3) ) ) ) ).

% differentiable_add
tff(fact_7671_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : set_or1337092689740270186AtMost(int,aa(int,int,plus_plus(int,L),one_one(int)),U) = set_or3652927894154168847AtMost(int,L,U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_int
tff(fact_7672_differentiable__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),X: A,Sa: set(A),G: fun(A,B)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,X,Sa))
         => ( differentiable(A,B,G,topolo174197925503356063within(A,X,Sa))
           => ( ( aa(A,B,G,X) != zero_zero(B) )
             => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_te(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo174197925503356063within(A,X,Sa)) ) ) ) ) ).

% differentiable_divide
tff(fact_7673_differentiable__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),X: A,Sa: set(A)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,X,Sa))
         => ( ( aa(A,B,F2,X) != zero_zero(B) )
           => differentiable(A,B,aTP_Lamp_ahm(fun(A,B),fun(A,B),F2),topolo174197925503356063within(A,X,Sa)) ) ) ) ).

% differentiable_inverse
tff(fact_7674_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or1337092689740270186AtMost(code_integer,aa(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_7675_word__atLeastAtMost__Suc__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: word(A),U: word(A)] :
          ( ( M != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
         => ( set_or3652927894154168847AtMost(word(A),M,U) = set_or1337092689740270186AtMost(word(A),aa(word(A),word(A),plus_plus(word(A),M),one_one(word(A))),U) ) ) ) ).

% word_atLeastAtMost_Suc_greaterThanAtMost
tff(fact_7676_word__range__minus__1_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          ( ( A3 != zero_zero(word(A)) )
         => ( set_or3652927894154168847AtMost(word(A),aa(word(A),word(A),minus_minus(word(A),A3),one_one(word(A))),B3) = set_or1337092689740270186AtMost(word(A),A3,B3) ) ) ) ).

% word_range_minus_1'
tff(fact_7677_MVT,axiom,
    ! [A3: real,B3: real,F2: fun(real,real)] :
      ( aa(real,$o,ord_less(real,A3),B3)
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A3,B3),F2)
       => ( ! [X4: real] :
              ( aa(real,$o,ord_less(real,A3),X4)
             => ( aa(real,$o,ord_less(real,X4),B3)
               => differentiable(real,real,F2,topolo174197925503356063within(real,X4,top_top(set(real)))) ) )
         => ? [L2: real,Z2: real] :
              ( aa(real,$o,ord_less(real,A3),Z2)
              & aa(real,$o,ord_less(real,Z2),B3)
              & has_field_derivative(real,F2,L2,topolo174197925503356063within(real,Z2,top_top(set(real))))
              & ( aa(real,real,minus_minus(real,aa(real,real,F2,B3)),aa(real,real,F2,A3)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,B3),A3)),L2) ) ) ) ) ) ).

% MVT
tff(fact_7678_interval__cases,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [S: set(A)] :
          ( ! [A4: A,B5: A,X4: A] :
              ( member(A,A4,S)
             => ( member(A,B5,S)
               => ( aa(A,$o,ord_less_eq(A,A4),X4)
                 => ( aa(A,$o,ord_less_eq(A,X4),B5)
                   => member(A,X4,S) ) ) ) )
         => ? [A4: A,B5: A] :
              ( ( S = bot_bot(set(A)) )
              | ( S = top_top(set(A)) )
              | ( S = set_ord_lessThan(A,B5) )
              | ( S = set_ord_atMost(A,B5) )
              | ( S = set_ord_greaterThan(A,A4) )
              | ( S = set_ord_atLeast(A,A4) )
              | ( S = set_or5935395276787703475ssThan(A,A4,B5) )
              | ( S = set_or3652927894154168847AtMost(A,A4,B5) )
              | ( S = set_or7035219750837199246ssThan(A,A4,B5) )
              | ( S = set_or1337092689740270186AtMost(A,A4,B5) ) ) ) ) ).

% interval_cases
tff(fact_7679_atLeast__empty__triv,axiom,
    ! [A: $tType] : set_ord_atLeast(set(A),bot_bot(set(A))) = top_top(set(set(A))) ).

% atLeast_empty_triv
tff(fact_7680_atLeast__0,axiom,
    set_ord_atLeast(nat,zero_zero(nat)) = top_top(set(nat)) ).

% atLeast_0
tff(fact_7681_continuous__on__compose2,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Ta: set(A),G: fun(A,B),Sa: set(C),F2: fun(C,A)] :
          ( topolo81223032696312382ous_on(A,B,Ta,G)
         => ( topolo81223032696312382ous_on(C,A,Sa,F2)
           => ( aa(set(A),$o,ord_less_eq(set(A),aa(set(C),set(A),image(C,A,F2),Sa)),Ta)
             => topolo81223032696312382ous_on(C,B,Sa,aa(fun(C,A),fun(C,B),aTP_Lamp_ahn(fun(A,B),fun(fun(C,A),fun(C,B)),G),F2)) ) ) ) ) ).

% continuous_on_compose2
tff(fact_7682_atLeast__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A] : set_ord_atLeast(A,L) = collect(A,ord_less_eq(A,L)) ) ).

% atLeast_def
tff(fact_7683_continuous__on__arcosh,axiom,
    ! [A2: set(real)] :
      ( aa(set(real),$o,ord_less_eq(set(real),A2),set_ord_atLeast(real,one_one(real)))
     => topolo81223032696312382ous_on(real,real,A2,arcosh(real)) ) ).

% continuous_on_arcosh
tff(fact_7684_ex__gt__or__lt,axiom,
    ! [A: $tType] :
      ( condit5016429287641298734tinuum(A)
     => ! [A3: A] :
        ? [B5: A] :
          ( aa(A,$o,ord_less(A,A3),B5)
          | aa(A,$o,ord_less(A,B5),A3) ) ) ).

% ex_gt_or_lt
tff(fact_7685_continuous__on__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Sa: set(A),F2: fun(A,A)] :
          ( topolo81223032696312382ous_on(A,A,Sa,F2)
         => ( ! [X4: A] :
                ( member(A,X4,Sa)
               => ( cos(A,aa(A,A,F2,X4)) != zero_zero(A) ) )
           => topolo81223032696312382ous_on(A,A,Sa,aTP_Lamp_wi(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_on_tan
tff(fact_7686_continuous__on__dist,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [Sa: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,Sa,F2)
         => ( topolo81223032696312382ous_on(A,B,Sa,G)
           => topolo81223032696312382ous_on(A,real,Sa,aa(fun(A,B),fun(A,real),aTP_Lamp_aho(fun(A,B),fun(fun(A,B),fun(A,real)),F2),G)) ) ) ) ).

% continuous_on_dist
tff(fact_7687_atLeast__eq__UNIV__iff,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [X: A] :
          ( ( set_ord_atLeast(A,X) = top_top(set(A)) )
        <=> ( X = bot_bot(A) ) ) ) ).

% atLeast_eq_UNIV_iff
tff(fact_7688_continuous__on__sing,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [X: A,F2: fun(A,B)] : topolo81223032696312382ous_on(A,B,aa(set(A),set(A),insert(A,X),bot_bot(set(A))),F2) ) ).

% continuous_on_sing
tff(fact_7689_not__empty__eq__Ici__eq__empty,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [L: A] : bot_bot(set(A)) != set_ord_atLeast(A,L) ) ).

% not_empty_eq_Ici_eq_empty
tff(fact_7690_continuous__on__empty,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B)] : topolo81223032696312382ous_on(A,B,bot_bot(set(A)),F2) ) ).

% continuous_on_empty
tff(fact_7691_continuous__inj__imp__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo8458572112393995274pology(A)
        & topolo1944317154257567458pology(B) )
     => ! [A3: A,X: A,B3: A,F2: fun(A,B)] :
          ( aa(A,$o,ord_less(A,A3),X)
         => ( aa(A,$o,ord_less(A,X),B3)
           => ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A3,B3),F2)
             => ( inj_on(A,B,F2,set_or1337092689740270186AtMost(A,A3,B3))
               => ( ( aa(B,$o,ord_less(B,aa(A,B,F2,A3)),aa(A,B,F2,X))
                    & aa(B,$o,ord_less(B,aa(A,B,F2,X)),aa(A,B,F2,B3)) )
                  | ( aa(B,$o,ord_less(B,aa(A,B,F2,B3)),aa(A,B,F2,X))
                    & aa(B,$o,ord_less(B,aa(A,B,F2,X)),aa(A,B,F2,A3)) ) ) ) ) ) ) ) ).

% continuous_inj_imp_mono
tff(fact_7692_continuous__on__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Sa: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,Sa,F2)
         => ( ! [X4: A] :
                ( member(A,X4,Sa)
               => ( aa(A,B,F2,X4) != zero_zero(B) ) )
           => topolo81223032696312382ous_on(A,B,Sa,aTP_Lamp_ahp(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_on_inverse
tff(fact_7693_continuous__on__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Sa: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,Sa,F2)
         => ( ! [X4: A] :
                ( member(A,X4,Sa)
               => ( aa(A,B,F2,X4) != zero_zero(B) ) )
           => topolo81223032696312382ous_on(A,B,Sa,aTP_Lamp_ahq(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_on_sgn
tff(fact_7694_continuous__on__powr,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Sa: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,Sa,F2)
         => ( topolo81223032696312382ous_on(A,real,Sa,G)
           => ( ! [X4: A] :
                  ( member(A,X4,Sa)
                 => ( aa(A,real,F2,X4) != zero_zero(real) ) )
             => topolo81223032696312382ous_on(A,real,Sa,aa(fun(A,real),fun(A,real),aTP_Lamp_ahr(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ).

% continuous_on_powr
tff(fact_7695_continuous__on__ln,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Sa: set(A),F2: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,Sa,F2)
         => ( ! [X4: A] :
                ( member(A,X4,Sa)
               => ( aa(A,real,F2,X4) != zero_zero(real) ) )
           => topolo81223032696312382ous_on(A,real,Sa,aTP_Lamp_ahs(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_on_ln
tff(fact_7696_continuous__on__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [Sa: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,Sa,F2)
         => ( topolo81223032696312382ous_on(A,B,Sa,G)
           => ( ! [X4: A] :
                  ( member(A,X4,Sa)
                 => ( aa(A,B,G,X4) != zero_zero(B) ) )
             => topolo81223032696312382ous_on(A,B,Sa,aa(fun(A,B),fun(A,B),aTP_Lamp_aht(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% continuous_on_divide
tff(fact_7697_continuous__on__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [Sa: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,Sa,F2)
         => ( topolo81223032696312382ous_on(A,B,Sa,G)
           => topolo81223032696312382ous_on(A,B,Sa,aa(fun(A,B),fun(A,B),aTP_Lamp_ahu(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_on_add
tff(fact_7698_continuous__on__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [Sa: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,Sa,F2)
         => ( topolo81223032696312382ous_on(A,B,Sa,G)
           => topolo81223032696312382ous_on(A,B,Sa,aa(fun(A,B),fun(A,B),aTP_Lamp_ahv(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_on_diff
tff(fact_7699_continuous__on__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Sa: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,Sa,F2)
         => ( topolo81223032696312382ous_on(A,B,Sa,G)
           => topolo81223032696312382ous_on(A,B,Sa,aa(fun(A,B),fun(A,B),aTP_Lamp_ahw(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_on_mult
tff(fact_7700_continuous__on__mult_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4211221413907600880p_mult(B) )
     => ! [A2: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,A2,F2)
         => ( topolo81223032696312382ous_on(A,B,A2,G)
           => topolo81223032696312382ous_on(A,B,A2,aa(fun(A,B),fun(A,B),aTP_Lamp_ahx(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_on_mult'
tff(fact_7701_continuous__on__mult__left,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topolo4958980785337419405_space(A) )
     => ! [Sa: set(A),F2: fun(A,B),C2: B] :
          ( topolo81223032696312382ous_on(A,B,Sa,F2)
         => topolo81223032696312382ous_on(A,B,Sa,aa(B,fun(A,B),aTP_Lamp_ahy(fun(A,B),fun(B,fun(A,B)),F2),C2)) ) ) ).

% continuous_on_mult_left
tff(fact_7702_continuous__on__mult__right,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topolo4958980785337419405_space(A) )
     => ! [Sa: set(A),F2: fun(A,B),C2: B] :
          ( topolo81223032696312382ous_on(A,B,Sa,F2)
         => topolo81223032696312382ous_on(A,B,Sa,aa(B,fun(A,B),aTP_Lamp_ahz(fun(A,B),fun(B,fun(A,B)),F2),C2)) ) ) ).

% continuous_on_mult_right
tff(fact_7703_continuous__on__Pair,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Sa: set(A),F2: fun(A,B),G: fun(A,C)] :
          ( topolo81223032696312382ous_on(A,B,Sa,F2)
         => ( topolo81223032696312382ous_on(A,C,Sa,G)
           => topolo81223032696312382ous_on(A,product_prod(B,C),Sa,aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_aia(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F2),G)) ) ) ) ).

% continuous_on_Pair
tff(fact_7704_continuous__on__rabs,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Sa: set(A),F2: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,Sa,F2)
         => topolo81223032696312382ous_on(A,real,Sa,aTP_Lamp_aib(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_on_rabs
tff(fact_7705_continuous__on__max,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [A2: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,A2,F2)
         => ( topolo81223032696312382ous_on(A,B,A2,G)
           => topolo81223032696312382ous_on(A,B,A2,aa(fun(A,B),fun(A,B),aTP_Lamp_aic(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_on_max
tff(fact_7706_continuous__on__arsinh_H,axiom,
    ! [A2: set(real),F2: fun(real,real)] :
      ( topolo81223032696312382ous_on(real,real,A2,F2)
     => topolo81223032696312382ous_on(real,real,A2,aTP_Lamp_aid(fun(real,real),fun(real,real),F2)) ) ).

% continuous_on_arsinh'
tff(fact_7707_continuous__on__id,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Sa: set(A)] : topolo81223032696312382ous_on(A,A,Sa,aTP_Lamp_uo(A,A)) ) ).

% continuous_on_id
tff(fact_7708_continuous__on__const,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(A) )
     => ! [Sa: set(A),C2: B] : topolo81223032696312382ous_on(A,B,Sa,aTP_Lamp_aie(B,fun(A,B),C2)) ) ).

% continuous_on_const
tff(fact_7709_continuous__on__arctan,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Sa: set(A),F2: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,Sa,F2)
         => topolo81223032696312382ous_on(A,real,Sa,aTP_Lamp_aif(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_on_arctan
tff(fact_7710_continuous__on__real__sqrt,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Sa: set(A),F2: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,Sa,F2)
         => topolo81223032696312382ous_on(A,real,Sa,aTP_Lamp_aig(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_on_real_sqrt
tff(fact_7711_continuous__on__norm,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Sa: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,Sa,F2)
         => topolo81223032696312382ous_on(A,real,Sa,aTP_Lamp_aih(fun(A,B),fun(A,real),F2)) ) ) ).

% continuous_on_norm
tff(fact_7712_continuous__on__of__real,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [Sa: set(A),G: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,Sa,G)
         => topolo81223032696312382ous_on(A,B,Sa,aTP_Lamp_aii(fun(A,real),fun(A,B),G)) ) ) ).

% continuous_on_of_real
tff(fact_7713_continuous__on__sin,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Sa: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,Sa,F2)
         => topolo81223032696312382ous_on(A,B,Sa,aTP_Lamp_xo(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_on_sin
tff(fact_7714_continuous__on__cos,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Sa: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,Sa,F2)
         => topolo81223032696312382ous_on(A,B,Sa,aTP_Lamp_xi(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_on_cos
tff(fact_7715_continuous__on__pochhammer_H,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & topolo4958980785337419405_space(A) )
     => ! [Sa: set(A),F2: fun(A,B),Na: nat] :
          ( topolo81223032696312382ous_on(A,B,Sa,F2)
         => topolo81223032696312382ous_on(A,B,Sa,aa(nat,fun(A,B),aTP_Lamp_aij(fun(A,B),fun(nat,fun(A,B)),F2),Na)) ) ) ).

% continuous_on_pochhammer'
tff(fact_7716_continuous__on__pochhammer,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A2: set(A),Na: nat] : topolo81223032696312382ous_on(A,A,A2,aTP_Lamp_yv(nat,fun(A,A),Na)) ) ).

% continuous_on_pochhammer
tff(fact_7717_continuous__on__exp,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Sa: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,Sa,F2)
         => topolo81223032696312382ous_on(A,B,Sa,aTP_Lamp_aik(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_on_exp
tff(fact_7718_continuous__on__scaleR,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Sa: set(A),F2: fun(A,real),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,real,Sa,F2)
         => ( topolo81223032696312382ous_on(A,B,Sa,G)
           => topolo81223032696312382ous_on(A,B,Sa,aa(fun(A,B),fun(A,B),aTP_Lamp_ail(fun(A,real),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_on_scaleR
tff(fact_7719_continuous__on__cosh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [A2: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,A2,F2)
         => topolo81223032696312382ous_on(A,B,A2,aTP_Lamp_aim(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_on_cosh
tff(fact_7720_continuous__on__sinh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [A2: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,A2,F2)
         => topolo81223032696312382ous_on(A,B,A2,aTP_Lamp_ain(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_on_sinh
tff(fact_7721_continuous__on__real__root,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Sa: set(A),F2: fun(A,real),Na: nat] :
          ( topolo81223032696312382ous_on(A,real,Sa,F2)
         => topolo81223032696312382ous_on(A,real,Sa,aa(nat,fun(A,real),aTP_Lamp_aio(fun(A,real),fun(nat,fun(A,real)),F2),Na)) ) ) ).

% continuous_on_real_root
tff(fact_7722_continuous__on__power,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topolo4958980785337419405_space(A) )
     => ! [Sa: set(A),F2: fun(A,B),Na: nat] :
          ( topolo81223032696312382ous_on(A,B,Sa,F2)
         => topolo81223032696312382ous_on(A,B,Sa,aa(nat,fun(A,B),aTP_Lamp_aip(fun(A,B),fun(nat,fun(A,B)),F2),Na)) ) ) ).

% continuous_on_power
tff(fact_7723_continuous__on__power_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1898628316856586783d_mult(B) )
     => ! [A2: set(A),F2: fun(A,B),G: fun(A,nat)] :
          ( topolo81223032696312382ous_on(A,B,A2,F2)
         => ( topolo81223032696312382ous_on(A,nat,A2,G)
           => topolo81223032696312382ous_on(A,B,A2,aa(fun(A,nat),fun(A,B),aTP_Lamp_aiq(fun(A,B),fun(fun(A,nat),fun(A,B)),F2),G)) ) ) ) ).

% continuous_on_power'
tff(fact_7724_continuous__on__prod,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & real_V4412858255891104859lgebra(C)
        & comm_ring_1(C) )
     => ! [S: set(A),Sa: set(B),F2: fun(A,fun(B,C))] :
          ( ! [I3: A] :
              ( member(A,I3,S)
             => topolo81223032696312382ous_on(B,C,Sa,aa(A,fun(B,C),F2,I3)) )
         => topolo81223032696312382ous_on(B,C,Sa,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_ais(set(A),fun(fun(A,fun(B,C)),fun(B,C)),S),F2)) ) ) ).

% continuous_on_prod
tff(fact_7725_continuous__on__prod_H,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4987421752381908075d_mult(C) )
     => ! [I5: set(A),S: set(B),F2: fun(A,fun(B,C))] :
          ( ! [I3: A] :
              ( member(A,I3,I5)
             => topolo81223032696312382ous_on(B,C,S,aa(A,fun(B,C),F2,I3)) )
         => topolo81223032696312382ous_on(B,C,S,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_aiu(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2)) ) ) ).

% continuous_on_prod'
tff(fact_7726_continuous__on__sum,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo5987344860129210374id_add(C) )
     => ! [I5: set(A),S: set(B),F2: fun(A,fun(B,C))] :
          ( ! [I3: A] :
              ( member(A,I3,I5)
             => topolo81223032696312382ous_on(B,C,S,aa(A,fun(B,C),F2,I3)) )
         => topolo81223032696312382ous_on(B,C,S,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_aiw(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2)) ) ) ).

% continuous_on_sum
tff(fact_7727_continuous__on__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [Sa: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,Sa,F2)
         => topolo81223032696312382ous_on(A,B,Sa,aTP_Lamp_aix(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_on_minus
tff(fact_7728_continuous__on__min,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [A2: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,A2,F2)
         => ( topolo81223032696312382ous_on(A,B,A2,G)
           => topolo81223032696312382ous_on(A,B,A2,aa(fun(A,B),fun(A,B),aTP_Lamp_aiy(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_on_min
tff(fact_7729_open__Collect__neq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topological_t2_space(B) )
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,top_top(set(A)),F2)
         => ( topolo81223032696312382ous_on(A,B,top_top(set(A)),G)
           => topolo1002775350975398744n_open(A,collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_aiz(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G))) ) ) ) ).

% open_Collect_neq
tff(fact_7730_continuous__on__open__Un,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Sa: set(A),Ta: set(A),F2: fun(A,B)] :
          ( topolo1002775350975398744n_open(A,Sa)
         => ( topolo1002775350975398744n_open(A,Ta)
           => ( topolo81223032696312382ous_on(A,B,Sa,F2)
             => ( topolo81223032696312382ous_on(A,B,Ta,F2)
               => topolo81223032696312382ous_on(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Sa),Ta),F2) ) ) ) ) ) ).

% continuous_on_open_Un
tff(fact_7731_bounded__linear_Ocontinuous__on,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),Sa: set(C),G: fun(C,A)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( topolo81223032696312382ous_on(C,A,Sa,G)
           => topolo81223032696312382ous_on(C,B,Sa,aa(fun(C,A),fun(C,B),aTP_Lamp_aja(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G)) ) ) ) ).

% bounded_linear.continuous_on
tff(fact_7732_continuous__on__tendsto__compose,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(A) )
     => ! [Sa: set(A),F2: fun(A,B),G: fun(C,A),L: A,F3: filter(C)] :
          ( topolo81223032696312382ous_on(A,B,Sa,F2)
         => ( filterlim(C,A,G,topolo7230453075368039082e_nhds(A,L),F3)
           => ( member(A,L,Sa)
             => ( eventually(C,aa(fun(C,A),fun(C,$o),aTP_Lamp_ajb(set(A),fun(fun(C,A),fun(C,$o)),Sa),G),F3)
               => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_vc(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G),topolo7230453075368039082e_nhds(B,aa(A,B,F2,L)),F3) ) ) ) ) ) ).

% continuous_on_tendsto_compose
tff(fact_7733_open__Collect__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,top_top(set(A)),F2)
         => ( topolo81223032696312382ous_on(A,B,top_top(set(A)),G)
           => topolo1002775350975398744n_open(A,collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_ajc(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G))) ) ) ) ).

% open_Collect_less
tff(fact_7734_continuous__on__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Sa: set(A),F2: fun(A,A)] :
          ( topolo81223032696312382ous_on(A,A,Sa,F2)
         => ( ! [X4: A] :
                ( member(A,X4,Sa)
               => ( sin(A,aa(A,A,F2,X4)) != zero_zero(A) ) )
           => topolo81223032696312382ous_on(A,A,Sa,aTP_Lamp_wh(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_on_cot
tff(fact_7735_continuous__on__tanh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [A2: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,A2,F2)
         => ( ! [X4: A] :
                ( member(A,X4,A2)
               => ( cosh(B,aa(A,B,F2,X4)) != zero_zero(B) ) )
           => topolo81223032696312382ous_on(A,B,A2,aTP_Lamp_ajd(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_on_tanh
tff(fact_7736_ivl__disj__un__one_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$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)),set_ord_atLeast(A,U)) = set_ord_atLeast(A,L) ) ) ) ).

% ivl_disj_un_one(8)
tff(fact_7737_Ici__subset__Ioi__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(set(A),$o,ord_less_eq(set(A),set_ord_atLeast(A,A3)),set_ord_greaterThan(A,B3))
        <=> aa(A,$o,ord_less(A,B3),A3) ) ) ).

% Ici_subset_Ioi_iff
tff(fact_7738_continuous__on__arcosh_H,axiom,
    ! [A2: set(real),F2: fun(real,real)] :
      ( topolo81223032696312382ous_on(real,real,A2,F2)
     => ( ! [X4: real] :
            ( member(real,X4,A2)
           => aa(real,$o,ord_less_eq(real,one_one(real)),aa(real,real,F2,X4)) )
       => topolo81223032696312382ous_on(real,real,A2,aTP_Lamp_aje(fun(real,real),fun(real,real),F2)) ) ) ).

% continuous_on_arcosh'
tff(fact_7739_continuous__on__powr_H,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Sa: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,Sa,F2)
         => ( topolo81223032696312382ous_on(A,real,Sa,G)
           => ( ! [X4: A] :
                  ( member(A,X4,Sa)
                 => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(A,real,F2,X4))
                    & ( ( aa(A,real,F2,X4) = zero_zero(real) )
                     => aa(real,$o,ord_less(real,zero_zero(real)),aa(A,real,G,X4)) ) ) )
             => topolo81223032696312382ous_on(A,real,Sa,aa(fun(A,real),fun(A,real),aTP_Lamp_ahr(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ).

% continuous_on_powr'
tff(fact_7740_continuous__on__log,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Sa: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,Sa,F2)
         => ( topolo81223032696312382ous_on(A,real,Sa,G)
           => ( ! [X4: A] :
                  ( member(A,X4,Sa)
                 => aa(real,$o,ord_less(real,zero_zero(real)),aa(A,real,F2,X4)) )
             => ( ! [X4: A] :
                    ( member(A,X4,Sa)
                   => ( aa(A,real,F2,X4) != one_one(real) ) )
               => ( ! [X4: A] :
                      ( member(A,X4,Sa)
                     => aa(real,$o,ord_less(real,zero_zero(real)),aa(A,real,G,X4)) )
                 => topolo81223032696312382ous_on(A,real,Sa,aa(fun(A,real),fun(A,real),aTP_Lamp_ajf(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% continuous_on_log
tff(fact_7741_continuous__on__arccos_H,axiom,
    topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real)),arccos) ).

% continuous_on_arccos'
tff(fact_7742_continuous__on__arcsin_H,axiom,
    topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real)),arcsin) ).

% continuous_on_arcsin'
tff(fact_7743_continuous__on__arccos,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Sa: set(A),F2: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,Sa,F2)
         => ( ! [X4: A] :
                ( member(A,X4,Sa)
               => ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,F2,X4))
                  & aa(real,$o,ord_less_eq(real,aa(A,real,F2,X4)),one_one(real)) ) )
           => topolo81223032696312382ous_on(A,real,Sa,aTP_Lamp_ajg(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_on_arccos
tff(fact_7744_continuous__on__arcsin,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Sa: set(A),F2: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,Sa,F2)
         => ( ! [X4: A] :
                ( member(A,X4,Sa)
               => ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,F2,X4))
                  & aa(real,$o,ord_less_eq(real,aa(A,real,F2,X4)),one_one(real)) ) )
           => topolo81223032696312382ous_on(A,real,Sa,aTP_Lamp_ajh(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_on_arcsin
tff(fact_7745_continuous__on__artanh,axiom,
    ! [A2: set(real)] :
      ( aa(set(real),$o,ord_less_eq(set(real),A2),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real)))
     => topolo81223032696312382ous_on(real,real,A2,artanh(real)) ) ).

% continuous_on_artanh
tff(fact_7746_ivl__disj__un__singleton_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),insert(A,L),bot_bot(set(A)))),set_ord_greaterThan(A,L)) = set_ord_atLeast(A,L) ) ).

% ivl_disj_un_singleton(1)
tff(fact_7747_ivl__disj__un__one_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,ord_less_eq(A,L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,U)),set_ord_greaterThan(A,U)) = set_ord_atLeast(A,L) ) ) ) ).

% ivl_disj_un_one(7)
tff(fact_7748_ivl__disj__un__one_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,ord_less(A,L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,U)),set_ord_atLeast(A,U)) = set_ord_greaterThan(A,L) ) ) ) ).

% ivl_disj_un_one(6)
tff(fact_7749_continuous__on__artanh_H,axiom,
    ! [A2: set(real),F2: fun(real,real)] :
      ( topolo81223032696312382ous_on(real,real,A2,F2)
     => ( ! [X4: real] :
            ( member(real,X4,A2)
           => member(real,aa(real,real,F2,X4),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real))) )
       => topolo81223032696312382ous_on(real,real,A2,aTP_Lamp_aji(fun(real,real),fun(real,real),F2)) ) ) ).

% continuous_on_artanh'
tff(fact_7750_atLeast__Suc,axiom,
    ! [K: nat] : set_ord_atLeast(nat,aa(nat,nat,suc,K)) = aa(set(nat),set(nat),minus_minus(set(nat),set_ord_atLeast(nat,K)),aa(set(nat),set(nat),insert(nat,K),bot_bot(set(nat)))) ).

% atLeast_Suc
tff(fact_7751_Rolle__deriv,axiom,
    ! [A3: real,B3: real,F2: fun(real,real),F7: fun(real,fun(real,real))] :
      ( aa(real,$o,ord_less(real,A3),B3)
     => ( ( aa(real,real,F2,A3) = aa(real,real,F2,B3) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A3,B3),F2)
         => ( ! [X4: real] :
                ( aa(real,$o,ord_less(real,A3),X4)
               => ( aa(real,$o,ord_less(real,X4),B3)
                 => has_derivative(real,real,F2,aa(real,fun(real,real),F7,X4),topolo174197925503356063within(real,X4,top_top(set(real)))) ) )
           => ? [Z2: real] :
                ( aa(real,$o,ord_less(real,A3),Z2)
                & aa(real,$o,ord_less(real,Z2),B3)
                & ! [X2: real] : aa(real,real,aa(real,fun(real,real),F7,Z2),X2) = zero_zero(real) ) ) ) ) ) ).

% Rolle_deriv
tff(fact_7752_mvt,axiom,
    ! [A3: real,B3: real,F2: fun(real,real),F7: fun(real,fun(real,real))] :
      ( aa(real,$o,ord_less(real,A3),B3)
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A3,B3),F2)
       => ( ! [X4: real] :
              ( aa(real,$o,ord_less(real,A3),X4)
             => ( aa(real,$o,ord_less(real,X4),B3)
               => has_derivative(real,real,F2,aa(real,fun(real,real),F7,X4),topolo174197925503356063within(real,X4,top_top(set(real)))) ) )
         => ~ ! [Xi: real] :
                ( aa(real,$o,ord_less(real,A3),Xi)
               => ( aa(real,$o,ord_less(real,Xi),B3)
                 => ( aa(real,real,minus_minus(real,aa(real,real,F2,B3)),aa(real,real,F2,A3)) != aa(real,real,aa(real,fun(real,real),F7,Xi),aa(real,real,minus_minus(real,B3),A3)) ) ) ) ) ) ) ).

% mvt
tff(fact_7753_continuous__on__of__int__floor,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A)
        & ring_1(B)
        & topolo4958980785337419405_space(B) )
     => topolo81223032696312382ous_on(A,B,aa(set(A),set(A),minus_minus(set(A),top_top(set(A))),ring_1_Ints(A)),aTP_Lamp_ajj(A,B)) ) ).

% continuous_on_of_int_floor
tff(fact_7754_continuous__on__of__int__ceiling,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A)
        & ring_1(B)
        & topolo4958980785337419405_space(B) )
     => topolo81223032696312382ous_on(A,B,aa(set(A),set(A),minus_minus(set(A),top_top(set(A))),ring_1_Ints(A)),aTP_Lamp_ajk(A,B)) ) ).

% continuous_on_of_int_ceiling
tff(fact_7755_continuous__on__Icc__at__leftD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A3: A,B3: A,F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A3,B3),F2)
         => ( aa(A,$o,ord_less(A,A3),B3)
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,B3)),topolo174197925503356063within(A,B3,set_ord_lessThan(A,B3))) ) ) ) ).

% continuous_on_Icc_at_leftD
tff(fact_7756_continuous__on__Icc__at__rightD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A3: A,B3: A,F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A3,B3),F2)
         => ( aa(A,$o,ord_less(A,A3),B3)
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,A3)),topolo174197925503356063within(A,A3,set_ord_greaterThan(A,A3))) ) ) ) ).

% continuous_on_Icc_at_rightD
tff(fact_7757_DERIV__pos__imp__increasing__open,axiom,
    ! [A3: real,B3: real,F2: fun(real,real)] :
      ( aa(real,$o,ord_less(real,A3),B3)
     => ( ! [X4: real] :
            ( aa(real,$o,ord_less(real,A3),X4)
           => ( aa(real,$o,ord_less(real,X4),B3)
             => ? [Y4: real] :
                  ( has_field_derivative(real,F2,Y4,topolo174197925503356063within(real,X4,top_top(set(real))))
                  & aa(real,$o,ord_less(real,zero_zero(real)),Y4) ) ) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A3,B3),F2)
         => aa(real,$o,ord_less(real,aa(real,real,F2,A3)),aa(real,real,F2,B3)) ) ) ) ).

% DERIV_pos_imp_increasing_open
tff(fact_7758_DERIV__neg__imp__decreasing__open,axiom,
    ! [A3: real,B3: real,F2: fun(real,real)] :
      ( aa(real,$o,ord_less(real,A3),B3)
     => ( ! [X4: real] :
            ( aa(real,$o,ord_less(real,A3),X4)
           => ( aa(real,$o,ord_less(real,X4),B3)
             => ? [Y4: real] :
                  ( has_field_derivative(real,F2,Y4,topolo174197925503356063within(real,X4,top_top(set(real))))
                  & aa(real,$o,ord_less(real,Y4),zero_zero(real)) ) ) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A3,B3),F2)
         => aa(real,$o,ord_less(real,aa(real,real,F2,B3)),aa(real,real,F2,A3)) ) ) ) ).

% DERIV_neg_imp_decreasing_open
tff(fact_7759_DERIV__isconst__end,axiom,
    ! [A3: real,B3: real,F2: fun(real,real)] :
      ( aa(real,$o,ord_less(real,A3),B3)
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A3,B3),F2)
       => ( ! [X4: real] :
              ( aa(real,$o,ord_less(real,A3),X4)
             => ( aa(real,$o,ord_less(real,X4),B3)
               => has_field_derivative(real,F2,zero_zero(real),topolo174197925503356063within(real,X4,top_top(set(real)))) ) )
         => ( aa(real,real,F2,B3) = aa(real,real,F2,A3) ) ) ) ) ).

% DERIV_isconst_end
tff(fact_7760_complete__interval,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [A3: A,B3: A,P: fun(A,$o)] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(A,$o,P,A3)
           => ( ~ aa(A,$o,P,B3)
             => ? [C5: A] :
                  ( aa(A,$o,ord_less_eq(A,A3),C5)
                  & aa(A,$o,ord_less_eq(A,C5),B3)
                  & ! [X2: A] :
                      ( ( aa(A,$o,ord_less_eq(A,A3),X2)
                        & aa(A,$o,ord_less(A,X2),C5) )
                     => aa(A,$o,P,X2) )
                  & ! [D5: A] :
                      ( ! [X4: A] :
                          ( ( aa(A,$o,ord_less_eq(A,A3),X4)
                            & aa(A,$o,ord_less(A,X4),D5) )
                         => aa(A,$o,P,X4) )
                     => aa(A,$o,ord_less_eq(A,D5),C5) ) ) ) ) ) ) ).

% complete_interval
tff(fact_7761_DERIV__isconst2,axiom,
    ! [A3: real,B3: real,F2: fun(real,real),X: real] :
      ( aa(real,$o,ord_less(real,A3),B3)
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A3,B3),F2)
       => ( ! [X4: real] :
              ( aa(real,$o,ord_less(real,A3),X4)
             => ( aa(real,$o,ord_less(real,X4),B3)
               => has_field_derivative(real,F2,zero_zero(real),topolo174197925503356063within(real,X4,top_top(set(real)))) ) )
         => ( aa(real,$o,ord_less_eq(real,A3),X)
           => ( aa(real,$o,ord_less_eq(real,X),B3)
             => ( aa(real,real,F2,X) = aa(real,real,F2,A3) ) ) ) ) ) ) ).

% DERIV_isconst2
tff(fact_7762_continuous__on__IccI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),A3: A,B3: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,A3)),topolo174197925503356063within(A,A3,set_ord_greaterThan(A,A3)))
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,B3)),topolo174197925503356063within(A,B3,set_ord_lessThan(A,B3)))
           => ( ! [X4: A] :
                  ( aa(A,$o,ord_less(A,A3),X4)
                 => ( aa(A,$o,ord_less(A,X4),B3)
                   => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,X4)),topolo174197925503356063within(A,X4,top_top(set(A)))) ) )
             => ( aa(A,$o,ord_less(A,A3),B3)
               => topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A3,B3),F2) ) ) ) ) ) ).

% continuous_on_IccI
tff(fact_7763_Rolle,axiom,
    ! [A3: real,B3: real,F2: fun(real,real)] :
      ( aa(real,$o,ord_less(real,A3),B3)
     => ( ( aa(real,real,F2,A3) = aa(real,real,F2,B3) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A3,B3),F2)
         => ( ! [X4: real] :
                ( aa(real,$o,ord_less(real,A3),X4)
               => ( aa(real,$o,ord_less(real,X4),B3)
                 => differentiable(real,real,F2,topolo174197925503356063within(real,X4,top_top(set(real)))) ) )
           => ? [Z2: real] :
                ( aa(real,$o,ord_less(real,A3),Z2)
                & aa(real,$o,ord_less(real,Z2),B3)
                & has_field_derivative(real,F2,zero_zero(real),topolo174197925503356063within(real,Z2,top_top(set(real)))) ) ) ) ) ) ).

% Rolle
tff(fact_7764_int__of__integer__code,axiom,
    ! [K: code_integer] :
      code_int_of_integer(K) = $ite(
        aa(code_integer,$o,ord_less(code_integer,K),zero_zero(code_integer)),
        aa(int,int,uminus_uminus(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,product_case_prod(code_integer,code_integer,int,aTP_Lamp_ajl(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_7765_csqrt_Osimps_I1_J,axiom,
    ! [Z: complex] : re(csqrt(Z)) = aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,plus_plus(real,real_V7770717601297561774m_norm(complex,Z)),re(Z))),aa(num,real,numeral_numeral(real),bit0(one2)))) ).

% csqrt.simps(1)
tff(fact_7766_int__of__integer__of__nat,axiom,
    ! [Na: nat] : 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_7767_complex__Re__of__nat,axiom,
    ! [Na: nat] : re(aa(nat,complex,semiring_1_of_nat(complex),Na)) = aa(nat,real,semiring_1_of_nat(real),Na) ).

% complex_Re_of_nat
tff(fact_7768_continuous__on__of__real__o__iff,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),G: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,complex,S,aTP_Lamp_ajm(fun(A,real),fun(A,complex),G))
        <=> topolo81223032696312382ous_on(A,real,S,G) ) ) ).

% continuous_on_of_real_o_iff
tff(fact_7769_zero__integer_Orep__eq,axiom,
    code_int_of_integer(zero_zero(code_integer)) = zero_zero(int) ).

% zero_integer.rep_eq
tff(fact_7770_int__of__integer__numeral,axiom,
    ! [K: num] : 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_7771_one__integer_Orep__eq,axiom,
    code_int_of_integer(one_one(code_integer)) = one_one(int) ).

% one_integer.rep_eq
tff(fact_7772_complex__Re__numeral,axiom,
    ! [V2: num] : re(aa(num,complex,numeral_numeral(complex),V2)) = aa(num,real,numeral_numeral(real),V2) ).

% complex_Re_numeral
tff(fact_7773_Re__sum,axiom,
    ! [A: $tType,F2: fun(A,complex),Sa: set(A)] : re(aa(set(A),complex,groups7311177749621191930dd_sum(A,complex,F2),Sa)) = aa(set(A),real,groups7311177749621191930dd_sum(A,real,aTP_Lamp_ajn(fun(A,complex),fun(A,real),F2)),Sa) ).

% Re_sum
tff(fact_7774_Re__divide__of__nat,axiom,
    ! [Z: complex,Na: nat] : re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Z),aa(nat,complex,semiring_1_of_nat(complex),Na))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),re(Z)),aa(nat,real,semiring_1_of_nat(real),Na)) ).

% Re_divide_of_nat
tff(fact_7775_Re__divide__numeral,axiom,
    ! [Z: complex,W: num] : re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Z),aa(num,complex,numeral_numeral(complex),W))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),re(Z)),aa(num,real,numeral_numeral(real),W)) ).

% Re_divide_numeral
tff(fact_7776_cos__Arg__i__mult__zero,axiom,
    ! [Y: complex] :
      ( ( Y != zero_zero(complex) )
     => ( ( re(Y) = zero_zero(real) )
       => ( cos(real,arg(Y)) = zero_zero(real) ) ) ) ).

% cos_Arg_i_mult_zero
tff(fact_7777_continuous__on__cis,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A2: set(A),F2: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,A2,F2)
         => topolo81223032696312382ous_on(A,complex,A2,aTP_Lamp_ajo(fun(A,real),fun(A,complex),F2)) ) ) ).

% continuous_on_cis
tff(fact_7778_continuous__on__Re,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Sa: set(A),G: fun(A,complex)] :
          ( topolo81223032696312382ous_on(A,complex,Sa,G)
         => topolo81223032696312382ous_on(A,real,Sa,aTP_Lamp_ajp(fun(A,complex),fun(A,real),G)) ) ) ).

% continuous_on_Re
tff(fact_7779_summable__Re,axiom,
    ! [F2: fun(nat,complex)] :
      ( summable(complex,F2)
     => summable(real,aTP_Lamp_ajq(fun(nat,complex),fun(nat,real),F2)) ) ).

% summable_Re
tff(fact_7780_continuous__Re,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),G: fun(A,complex)] :
          ( topolo3448309680560233919inuous(A,complex,F3,G)
         => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_ajr(fun(A,complex),fun(A,real),G)) ) ) ).

% continuous_Re
tff(fact_7781_zero__complex_Osimps_I1_J,axiom,
    re(zero_zero(complex)) = zero_zero(real) ).

% zero_complex.simps(1)
tff(fact_7782_one__complex_Osimps_I1_J,axiom,
    re(one_one(complex)) = one_one(real) ).

% one_complex.simps(1)
tff(fact_7783_imaginary__unit_Osimps_I1_J,axiom,
    re(imaginary_unit) = zero_zero(real) ).

% imaginary_unit.simps(1)
tff(fact_7784_sums__Re,axiom,
    ! [X5: fun(nat,complex),A3: complex] :
      ( aa(complex,$o,sums(complex,X5),A3)
     => aa(real,$o,sums(real,aTP_Lamp_ajq(fun(nat,complex),fun(nat,real),X5)),re(A3)) ) ).

% sums_Re
tff(fact_7785_Cauchy__Re,axiom,
    ! [X5: fun(nat,complex)] :
      ( topolo3814608138187158403Cauchy(complex,X5)
     => topolo3814608138187158403Cauchy(real,aTP_Lamp_ajq(fun(nat,complex),fun(nat,real),X5)) ) ).

% Cauchy_Re
tff(fact_7786_has__derivative__Re,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,complex),G4: fun(A,complex),F3: filter(A)] :
          ( has_derivative(A,complex,G,G4,F3)
         => has_derivative(A,real,aTP_Lamp_ajs(fun(A,complex),fun(A,real),G),aTP_Lamp_ajs(fun(A,complex),fun(A,real),G4),F3) ) ) ).

% has_derivative_Re
tff(fact_7787_less__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] :
      ( aa(code_integer,$o,ord_less(code_integer,X),Xa2)
    <=> aa(int,$o,ord_less(int,code_int_of_integer(X)),code_int_of_integer(Xa2)) ) ).

% less_integer.rep_eq
tff(fact_7788_integer__less__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( aa(code_integer,$o,ord_less(code_integer,K),L)
    <=> aa(int,$o,ord_less(int,code_int_of_integer(K)),code_int_of_integer(L)) ) ).

% integer_less_iff
tff(fact_7789_int__of__integer__less__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa(int,$o,ord_less(int,code_int_of_integer(X)),code_int_of_integer(Y))
    <=> aa(code_integer,$o,ord_less(code_integer,X),Y) ) ).

% int_of_integer_less_iff
tff(fact_7790_Re__csqrt,axiom,
    ! [Z: complex] : aa(real,$o,ord_less_eq(real,zero_zero(real)),re(csqrt(Z))) ).

% Re_csqrt
tff(fact_7791_isCont__Re,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A3: A,G: fun(A,complex)] :
          ( topolo3448309680560233919inuous(A,complex,topolo174197925503356063within(A,A3,top_top(set(A))),G)
         => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,top_top(set(A))),aTP_Lamp_ajr(fun(A,complex),fun(A,real),G)) ) ) ).

% isCont_Re
tff(fact_7792_cmod__plus__Re__le__0__iff,axiom,
    ! [Z: complex] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,plus_plus(real,real_V7770717601297561774m_norm(complex,Z)),re(Z))),zero_zero(real))
    <=> ( re(Z) = aa(real,real,uminus_uminus(real),real_V7770717601297561774m_norm(complex,Z)) ) ) ).

% cmod_plus_Re_le_0_iff
tff(fact_7793_bin__last__integer_Orep__eq,axiom,
    ! [X: code_integer] :
      ( bits_b8758750999018896077nteger(X)
    <=> ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),code_int_of_integer(X)) ) ).

% bin_last_integer.rep_eq
tff(fact_7794_bin__rest__integer_Orep__eq,axiom,
    ! [X: code_integer] : code_int_of_integer(bits_b2549910563261871055nteger(X)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),code_int_of_integer(X)),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% bin_rest_integer.rep_eq
tff(fact_7795_tendsto__Re,axiom,
    ! [A: $tType,G: fun(A,complex),A3: complex,F3: filter(A)] :
      ( filterlim(A,complex,G,topolo7230453075368039082e_nhds(complex,A3),F3)
     => filterlim(A,real,aTP_Lamp_ajn(fun(A,complex),fun(A,real),G),topolo7230453075368039082e_nhds(real,re(A3)),F3) ) ).

% tendsto_Re
tff(fact_7796_cos__n__Re__cis__pow__n,axiom,
    ! [Na: nat,A3: real] : cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),A3)) = re(aa(nat,complex,power_power(complex,cis(A3)),Na)) ).

% cos_n_Re_cis_pow_n
tff(fact_7797_Bit__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: $o] : code_int_of_integer(bits_Bit_integer(X,(Xa2))) = aa(int,int,plus_plus(int,aa($o,int,zero_neq_one_of_bool(int),(Xa2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),code_int_of_integer(X))) ).

% Bit_integer.rep_eq
tff(fact_7798_num__of__integer__code,axiom,
    ! [K: code_integer] :
      code_num_of_integer(K) = $ite(aa(code_integer,$o,ord_less_eq(code_integer,K),one_one(code_integer)),one2,aa(product_prod(code_integer,code_integer),num,product_case_prod(code_integer,code_integer,num,aTP_Lamp_ajt(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_7799_csqrt_Ocode,axiom,
    ! [Z: complex] :
      csqrt(Z) = complex2(aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,plus_plus(real,real_V7770717601297561774m_norm(complex,Z)),re(Z))),aa(num,real,numeral_numeral(real),bit0(one2)))),
        aa(real,real,
          aa(real,fun(real,real),times_times(real),
            $ite(im(Z) = zero_zero(real),one_one(real),aa(real,real,sgn_sgn(real),im(Z)))),
          aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,real_V7770717601297561774m_norm(complex,Z)),re(Z))),aa(num,real,numeral_numeral(real),bit0(one2)))))) ).

% csqrt.code
tff(fact_7800_complex__Im__fact,axiom,
    ! [Na: nat] : im(semiring_char_0_fact(complex,Na)) = zero_zero(real) ).

% complex_Im_fact
tff(fact_7801_complex__Im__of__int,axiom,
    ! [Z: int] : im(aa(int,complex,ring_1_of_int(complex),Z)) = zero_zero(real) ).

% complex_Im_of_int
tff(fact_7802_Im__complex__of__real,axiom,
    ! [Z: real] : im(aa(real,complex,real_Vector_of_real(complex),Z)) = zero_zero(real) ).

% Im_complex_of_real
tff(fact_7803_Im__power__real,axiom,
    ! [X: complex,Na: nat] :
      ( ( im(X) = zero_zero(real) )
     => ( im(aa(nat,complex,power_power(complex,X),Na)) = zero_zero(real) ) ) ).

% Im_power_real
tff(fact_7804_complex__Im__numeral,axiom,
    ! [V2: num] : im(aa(num,complex,numeral_numeral(complex),V2)) = zero_zero(real) ).

% complex_Im_numeral
tff(fact_7805_complex__Im__of__nat,axiom,
    ! [Na: nat] : im(aa(nat,complex,semiring_1_of_nat(complex),Na)) = zero_zero(real) ).

% complex_Im_of_nat
tff(fact_7806_Im__sum,axiom,
    ! [A: $tType,F2: fun(A,complex),Sa: set(A)] : im(aa(set(A),complex,groups7311177749621191930dd_sum(A,complex,F2),Sa)) = aa(set(A),real,groups7311177749621191930dd_sum(A,real,aTP_Lamp_aju(fun(A,complex),fun(A,real),F2)),Sa) ).

% Im_sum
tff(fact_7807_Re__power__real,axiom,
    ! [X: complex,Na: nat] :
      ( ( im(X) = zero_zero(real) )
     => ( re(aa(nat,complex,power_power(complex,X),Na)) = aa(nat,real,power_power(real,re(X)),Na) ) ) ).

% Re_power_real
tff(fact_7808_Im__divide__numeral,axiom,
    ! [Z: complex,W: num] : im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Z),aa(num,complex,numeral_numeral(complex),W))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),im(Z)),aa(num,real,numeral_numeral(real),W)) ).

% Im_divide_numeral
tff(fact_7809_Im__divide__of__nat,axiom,
    ! [Z: complex,Na: nat] : im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Z),aa(nat,complex,semiring_1_of_nat(complex),Na))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),im(Z)),aa(nat,real,semiring_1_of_nat(real),Na)) ).

% Im_divide_of_nat
tff(fact_7810_csqrt__of__real__nonneg,axiom,
    ! [X: complex] :
      ( ( im(X) = zero_zero(real) )
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),re(X))
       => ( csqrt(X) = aa(real,complex,real_Vector_of_real(complex),aa(real,real,sqrt,re(X))) ) ) ) ).

% csqrt_of_real_nonneg
tff(fact_7811_csqrt__minus,axiom,
    ! [X: complex] :
      ( ( aa(real,$o,ord_less(real,im(X)),zero_zero(real))
        | ( ( im(X) = zero_zero(real) )
          & aa(real,$o,ord_less_eq(real,zero_zero(real)),re(X)) ) )
     => ( csqrt(aa(complex,complex,uminus_uminus(complex),X)) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),csqrt(X)) ) ) ).

% csqrt_minus
tff(fact_7812_csqrt__of__real__nonpos,axiom,
    ! [X: complex] :
      ( ( im(X) = zero_zero(real) )
     => ( aa(real,$o,ord_less_eq(real,re(X)),zero_zero(real))
       => ( csqrt(X) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),aa(real,complex,real_Vector_of_real(complex),aa(real,real,sqrt,aa(real,real,abs_abs(real),re(X))))) ) ) ) ).

% csqrt_of_real_nonpos
tff(fact_7813_continuous__on__Im,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Sa: set(A),G: fun(A,complex)] :
          ( topolo81223032696312382ous_on(A,complex,Sa,G)
         => topolo81223032696312382ous_on(A,real,Sa,aTP_Lamp_ajv(fun(A,complex),fun(A,real),G)) ) ) ).

% continuous_on_Im
tff(fact_7814_continuous__complex__iff,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,complex)] :
          ( topolo3448309680560233919inuous(A,complex,F3,F2)
        <=> ( topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_ajr(fun(A,complex),fun(A,real),F2))
            & topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_ajw(fun(A,complex),fun(A,real),F2)) ) ) ) ).

% continuous_complex_iff
tff(fact_7815_complex__is__Int__iff,axiom,
    ! [Z: complex] :
      ( member(complex,Z,ring_1_Ints(complex))
    <=> ( ( im(Z) = zero_zero(real) )
        & ? [I4: int] : re(Z) = aa(int,real,ring_1_of_int(real),I4) ) ) ).

% complex_is_Int_iff
tff(fact_7816_sums__complex__iff,axiom,
    ! [F2: fun(nat,complex),X: complex] :
      ( aa(complex,$o,sums(complex,F2),X)
    <=> ( aa(real,$o,sums(real,aTP_Lamp_ajq(fun(nat,complex),fun(nat,real),F2)),re(X))
        & aa(real,$o,sums(real,aTP_Lamp_ajx(fun(nat,complex),fun(nat,real),F2)),im(X)) ) ) ).

% sums_complex_iff
tff(fact_7817_has__derivative__Im,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,complex),G4: fun(A,complex),F3: filter(A)] :
          ( has_derivative(A,complex,G,G4,F3)
         => has_derivative(A,real,aTP_Lamp_ajy(fun(A,complex),fun(A,real),G),aTP_Lamp_ajy(fun(A,complex),fun(A,real),G4),F3) ) ) ).

% has_derivative_Im
tff(fact_7818_sums__Im,axiom,
    ! [X5: fun(nat,complex),A3: complex] :
      ( aa(complex,$o,sums(complex,X5),A3)
     => aa(real,$o,sums(real,aTP_Lamp_ajx(fun(nat,complex),fun(nat,real),X5)),im(A3)) ) ).

% sums_Im
tff(fact_7819_Cauchy__Im,axiom,
    ! [X5: fun(nat,complex)] :
      ( topolo3814608138187158403Cauchy(complex,X5)
     => topolo3814608138187158403Cauchy(real,aTP_Lamp_ajx(fun(nat,complex),fun(nat,real),X5)) ) ).

% Cauchy_Im
tff(fact_7820_zero__complex_Osimps_I2_J,axiom,
    im(zero_zero(complex)) = zero_zero(real) ).

% zero_complex.simps(2)
tff(fact_7821_imaginary__unit_Osimps_I2_J,axiom,
    im(imaginary_unit) = one_one(real) ).

% imaginary_unit.simps(2)
tff(fact_7822_one__complex_Osimps_I2_J,axiom,
    im(one_one(complex)) = zero_zero(real) ).

% one_complex.simps(2)
tff(fact_7823_continuous__Im,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),G: fun(A,complex)] :
          ( topolo3448309680560233919inuous(A,complex,F3,G)
         => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_ajw(fun(A,complex),fun(A,real),G)) ) ) ).

% continuous_Im
tff(fact_7824_summable__Im,axiom,
    ! [F2: fun(nat,complex)] :
      ( summable(complex,F2)
     => summable(real,aTP_Lamp_ajx(fun(nat,complex),fun(nat,real),F2)) ) ).

% summable_Im
tff(fact_7825_isCont__Im,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A3: A,G: fun(A,complex)] :
          ( topolo3448309680560233919inuous(A,complex,topolo174197925503356063within(A,A3,top_top(set(A))),G)
         => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,top_top(set(A))),aTP_Lamp_ajw(fun(A,complex),fun(A,real),G)) ) ) ).

% isCont_Im
tff(fact_7826_summable__complex__iff,axiom,
    ! [F2: fun(nat,complex)] :
      ( summable(complex,F2)
    <=> ( summable(real,aTP_Lamp_ajq(fun(nat,complex),fun(nat,real),F2))
        & summable(real,aTP_Lamp_ajx(fun(nat,complex),fun(nat,real),F2)) ) ) ).

% summable_complex_iff
tff(fact_7827_Im__eq__0,axiom,
    ! [Z: complex] :
      ( ( aa(real,real,abs_abs(real),re(Z)) = real_V7770717601297561774m_norm(complex,Z) )
     => ( im(Z) = zero_zero(real) ) ) ).

% Im_eq_0
tff(fact_7828_cmod__eq__Im,axiom,
    ! [Z: complex] :
      ( ( re(Z) = zero_zero(real) )
     => ( real_V7770717601297561774m_norm(complex,Z) = aa(real,real,abs_abs(real),im(Z)) ) ) ).

% cmod_eq_Im
tff(fact_7829_cmod__eq__Re,axiom,
    ! [Z: complex] :
      ( ( im(Z) = zero_zero(real) )
     => ( real_V7770717601297561774m_norm(complex,Z) = aa(real,real,abs_abs(real),re(Z)) ) ) ).

% cmod_eq_Re
tff(fact_7830_csqrt__principal,axiom,
    ! [Z: complex] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),re(csqrt(Z)))
      | ( ( re(csqrt(Z)) = zero_zero(real) )
        & aa(real,$o,ord_less_eq(real,zero_zero(real)),im(csqrt(Z))) ) ) ).

% csqrt_principal
tff(fact_7831_tendsto__Im,axiom,
    ! [A: $tType,G: fun(A,complex),A3: complex,F3: filter(A)] :
      ( filterlim(A,complex,G,topolo7230453075368039082e_nhds(complex,A3),F3)
     => filterlim(A,real,aTP_Lamp_aju(fun(A,complex),fun(A,real),G),topolo7230453075368039082e_nhds(real,im(A3)),F3) ) ).

% tendsto_Im
tff(fact_7832_sin__n__Im__cis__pow__n,axiom,
    ! [Na: nat,A3: real] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),A3)) = im(aa(nat,complex,power_power(complex,cis(A3)),Na)) ).

% sin_n_Im_cis_pow_n
tff(fact_7833_tendsto__complex__iff,axiom,
    ! [A: $tType,F2: fun(A,complex),X: complex,F3: filter(A)] :
      ( filterlim(A,complex,F2,topolo7230453075368039082e_nhds(complex,X),F3)
    <=> ( filterlim(A,real,aTP_Lamp_ajn(fun(A,complex),fun(A,real),F2),topolo7230453075368039082e_nhds(real,re(X)),F3)
        & filterlim(A,real,aTP_Lamp_aju(fun(A,complex),fun(A,real),F2),topolo7230453075368039082e_nhds(real,im(X)),F3) ) ) ).

% tendsto_complex_iff
tff(fact_7834_cmod__power2,axiom,
    ! [Z: complex] : aa(nat,real,power_power(real,real_V7770717601297561774m_norm(complex,Z)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(real,real,plus_plus(real,aa(nat,real,power_power(real,re(Z)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,im(Z)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% cmod_power2
tff(fact_7835_Im__power2,axiom,
    ! [X: complex] : im(aa(nat,complex,power_power(complex,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = 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))),re(X))),im(X)) ).

% Im_power2
tff(fact_7836_Re__power2,axiom,
    ! [X: complex] : re(aa(nat,complex,power_power(complex,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(real,real,minus_minus(real,aa(nat,real,power_power(real,re(X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,im(X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% Re_power2
tff(fact_7837_complex__eq__0,axiom,
    ! [Z: complex] :
      ( ( Z = zero_zero(complex) )
    <=> ( aa(real,real,plus_plus(real,aa(nat,real,power_power(real,re(Z)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,im(Z)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = zero_zero(real) ) ) ).

% complex_eq_0
tff(fact_7838_norm__complex__def,axiom,
    ! [Z: complex] : real_V7770717601297561774m_norm(complex,Z) = aa(real,real,sqrt,aa(real,real,plus_plus(real,aa(nat,real,power_power(real,re(Z)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,im(Z)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% norm_complex_def
tff(fact_7839_inverse__complex_Osimps_I1_J,axiom,
    ! [X: complex] : re(aa(complex,complex,inverse_inverse(complex),X)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),re(X)),aa(real,real,plus_plus(real,aa(nat,real,power_power(real,re(X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,im(X)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% inverse_complex.simps(1)
tff(fact_7840_complex__neq__0,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero(complex) )
    <=> aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,plus_plus(real,aa(nat,real,power_power(real,re(Z)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,im(Z)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% complex_neq_0
tff(fact_7841_Re__divide,axiom,
    ! [X: complex,Y: complex] : re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),X),Y)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,plus_plus(real,aa(real,real,aa(real,fun(real,real),times_times(real),re(X)),re(Y))),aa(real,real,aa(real,fun(real,real),times_times(real),im(X)),im(Y)))),aa(real,real,plus_plus(real,aa(nat,real,power_power(real,re(Y)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,im(Y)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% Re_divide
tff(fact_7842_csqrt__unique,axiom,
    ! [W: complex,Z: complex] :
      ( ( aa(nat,complex,power_power(complex,W),aa(num,nat,numeral_numeral(nat),bit0(one2))) = Z )
     => ( ( aa(real,$o,ord_less(real,zero_zero(real)),re(W))
          | ( ( re(W) = zero_zero(real) )
            & aa(real,$o,ord_less_eq(real,zero_zero(real)),im(W)) ) )
       => ( csqrt(Z) = W ) ) ) ).

% csqrt_unique
tff(fact_7843_csqrt__square,axiom,
    ! [B3: complex] :
      ( ( aa(real,$o,ord_less(real,zero_zero(real)),re(B3))
        | ( ( re(B3) = zero_zero(real) )
          & aa(real,$o,ord_less_eq(real,zero_zero(real)),im(B3)) ) )
     => ( csqrt(aa(nat,complex,power_power(complex,B3),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = B3 ) ) ).

% csqrt_square
tff(fact_7844_inverse__complex_Osimps_I2_J,axiom,
    ! [X: complex] : im(aa(complex,complex,inverse_inverse(complex),X)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),im(X))),aa(real,real,plus_plus(real,aa(nat,real,power_power(real,re(X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,im(X)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% inverse_complex.simps(2)
tff(fact_7845_Im__divide,axiom,
    ! [X: complex,Y: complex] : im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),X),Y)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),times_times(real),im(X)),re(Y))),aa(real,real,aa(real,fun(real,real),times_times(real),re(X)),im(Y)))),aa(real,real,plus_plus(real,aa(nat,real,power_power(real,re(Y)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,im(Y)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% Im_divide
tff(fact_7846_complex__abs__le__norm,axiom,
    ! [Z: complex] : aa(real,$o,ord_less_eq(real,aa(real,real,plus_plus(real,aa(real,real,abs_abs(real),re(Z))),aa(real,real,abs_abs(real),im(Z)))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2)))),real_V7770717601297561774m_norm(complex,Z))) ).

% complex_abs_le_norm
tff(fact_7847_complex__unit__circle,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero(complex) )
     => ( aa(real,real,plus_plus(real,aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),re(Z)),real_V7770717601297561774m_norm(complex,Z))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),im(Z)),real_V7770717601297561774m_norm(complex,Z))),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(real) ) ) ).

% complex_unit_circle
tff(fact_7848_inverse__complex_Ocode,axiom,
    ! [X: complex] : aa(complex,complex,inverse_inverse(complex),X) = complex2(aa(real,real,aa(real,fun(real,real),divide_divide(real),re(X)),aa(real,real,plus_plus(real,aa(nat,real,power_power(real,re(X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,im(X)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),im(X))),aa(real,real,plus_plus(real,aa(nat,real,power_power(real,re(X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,im(X)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% inverse_complex.code
tff(fact_7849_Complex__divide,axiom,
    ! [X: complex,Y: complex] : aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),X),Y) = complex2(aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,plus_plus(real,aa(real,real,aa(real,fun(real,real),times_times(real),re(X)),re(Y))),aa(real,real,aa(real,fun(real,real),times_times(real),im(X)),im(Y)))),aa(real,real,plus_plus(real,aa(nat,real,power_power(real,re(Y)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,im(Y)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),times_times(real),im(X)),re(Y))),aa(real,real,aa(real,fun(real,real),times_times(real),re(X)),im(Y)))),aa(real,real,plus_plus(real,aa(nat,real,power_power(real,re(Y)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,im(Y)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% Complex_divide
tff(fact_7850_csqrt_Osimps_I2_J,axiom,
    ! [Z: complex] :
      im(csqrt(Z)) = aa(real,real,
        aa(real,fun(real,real),times_times(real),
          $ite(im(Z) = zero_zero(real),one_one(real),aa(real,real,sgn_sgn(real),im(Z)))),
        aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,real_V7770717601297561774m_norm(complex,Z)),re(Z))),aa(num,real,numeral_numeral(real),bit0(one2))))) ).

% csqrt.simps(2)
tff(fact_7851_Im__Reals__divide,axiom,
    ! [R3: complex,Z: complex] :
      ( member(complex,R3,real_Vector_Reals(complex))
     => ( im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),R3),Z)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,uminus_uminus(real),re(R3))),im(Z))),aa(nat,real,power_power(real,real_V7770717601297561774m_norm(complex,Z)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% Im_Reals_divide
tff(fact_7852_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      code_nat_of_integer(K) = $ite(aa(code_integer,$o,ord_less_eq(code_integer,K),zero_zero(code_integer)),zero_zero(nat),aa(product_prod(code_integer,code_integer),nat,product_case_prod(code_integer,code_integer,nat,aTP_Lamp_ajz(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_7853_nat__of__integer__numeral,axiom,
    ! [Na: num] : code_nat_of_integer(aa(num,code_integer,numeral_numeral(code_integer),Na)) = aa(num,nat,numeral_numeral(nat),Na) ).

% nat_of_integer_numeral
tff(fact_7854_nat__of__integer__code__post_I3_J,axiom,
    ! [K: num] : code_nat_of_integer(aa(num,code_integer,numeral_numeral(code_integer),K)) = aa(num,nat,numeral_numeral(nat),K) ).

% nat_of_integer_code_post(3)
tff(fact_7855_nat__of__integer__1,axiom,
    code_nat_of_integer(one_one(code_integer)) = one_one(nat) ).

% nat_of_integer_1
tff(fact_7856_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_7857_nat__of__integer__non__positive,axiom,
    ! [K: code_integer] :
      ( aa(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_7858_real__eq__imaginary__iff,axiom,
    ! [Y: complex,X: complex] :
      ( member(complex,Y,real_Vector_Reals(complex))
     => ( member(complex,X,real_Vector_Reals(complex))
       => ( ( X = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),Y) )
        <=> ( ( X = zero_zero(complex) )
            & ( Y = zero_zero(complex) ) ) ) ) ) ).

% real_eq_imaginary_iff
tff(fact_7859_imaginary__eq__real__iff,axiom,
    ! [Y: complex,X: complex] :
      ( member(complex,Y,real_Vector_Reals(complex))
     => ( member(complex,X,real_Vector_Reals(complex))
       => ( ( aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),Y) = X )
        <=> ( ( X = zero_zero(complex) )
            & ( Y = zero_zero(complex) ) ) ) ) ) ).

% imaginary_eq_real_iff
tff(fact_7860_complex__is__Real__iff,axiom,
    ! [Z: complex] :
      ( member(complex,Z,real_Vector_Reals(complex))
    <=> ( im(Z) = zero_zero(real) ) ) ).

% complex_is_Real_iff
tff(fact_7861_Reals__of__nat,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Na: nat] : member(A,aa(nat,A,semiring_1_of_nat(A),Na),real_Vector_Reals(A)) ) ).

% Reals_of_nat
tff(fact_7862_nonzero__Reals__inverse,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [A3: A] :
          ( member(A,A3,real_Vector_Reals(A))
         => ( ( A3 != zero_zero(A) )
           => member(A,aa(A,A,inverse_inverse(A),A3),real_Vector_Reals(A)) ) ) ) ).

% nonzero_Reals_inverse
tff(fact_7863_Complex__in__Reals,axiom,
    ! [X: real] : member(complex,complex2(X,zero_zero(real)),real_Vector_Reals(complex)) ).

% Complex_in_Reals
tff(fact_7864_nonzero__Reals__divide,axiom,
    ! [A: $tType] :
      ( real_V7773925162809079976_field(A)
     => ! [A3: A,B3: A] :
          ( member(A,A3,real_Vector_Reals(A))
         => ( member(A,B3,real_Vector_Reals(A))
           => ( ( B3 != zero_zero(A) )
             => member(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3),real_Vector_Reals(A)) ) ) ) ) ).

% nonzero_Reals_divide
tff(fact_7865_Reals__1,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => member(A,one_one(A),real_Vector_Reals(A)) ) ).

% Reals_1
tff(fact_7866_Reals__power,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [A3: A,Na: nat] :
          ( member(A,A3,real_Vector_Reals(A))
         => member(A,aa(nat,A,power_power(A,A3),Na),real_Vector_Reals(A)) ) ) ).

% Reals_power
tff(fact_7867_Reals__numeral,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [W: num] : member(A,aa(num,A,numeral_numeral(A),W),real_Vector_Reals(A)) ) ).

% Reals_numeral
tff(fact_7868_Reals__0,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => member(A,zero_zero(A),real_Vector_Reals(A)) ) ).

% Reals_0
tff(fact_7869_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_7870_Re__prod__Reals,axiom,
    ! [A: $tType,A2: set(A),F2: fun(A,complex)] :
      ( ! [X4: A] :
          ( member(A,X4,A2)
         => member(complex,aa(A,complex,F2,X4),real_Vector_Reals(complex)) )
     => ( re(groups7121269368397514597t_prod(A,complex,F2,A2)) = groups7121269368397514597t_prod(A,real,aTP_Lamp_ajn(fun(A,complex),fun(A,real),F2),A2) ) ) ).

% Re_prod_Reals
tff(fact_7871_nat__of__integer__less__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa(code_integer,$o,ord_less_eq(code_integer,zero_zero(code_integer)),X)
     => ( aa(code_integer,$o,ord_less_eq(code_integer,zero_zero(code_integer)),Y)
       => ( aa(nat,$o,ord_less(nat,code_nat_of_integer(X)),code_nat_of_integer(Y))
        <=> aa(code_integer,$o,ord_less(code_integer,X),Y) ) ) ) ).

% nat_of_integer_less_iff
tff(fact_7872_image__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] :
      ( aa(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_7873_series__comparison__complex,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [G: fun(nat,complex),N2: nat,F2: fun(nat,A)] :
          ( summable(complex,G)
         => ( ! [N: nat] : member(complex,aa(nat,complex,G,N),real_Vector_Reals(complex))
           => ( ! [N: nat] : aa(real,$o,ord_less_eq(real,zero_zero(real)),re(aa(nat,complex,G,N)))
             => ( ! [N: nat] :
                    ( aa(nat,$o,ord_less_eq(nat,N2),N)
                   => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(nat,A,F2,N))),real_V7770717601297561774m_norm(complex,aa(nat,complex,G,N))) )
               => summable(A,F2) ) ) ) ) ) ).

% series_comparison_complex
tff(fact_7874_Re__Reals__divide,axiom,
    ! [R3: complex,Z: complex] :
      ( member(complex,R3,real_Vector_Reals(complex))
     => ( re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),R3),Z)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),re(R3)),re(Z))),aa(nat,real,power_power(real,real_V7770717601297561774m_norm(complex,Z)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% Re_Reals_divide
tff(fact_7875_integer__set__bit__code,axiom,
    ! [X: code_integer,Na: code_integer,B3: $o] :
      bits_integer_set_bit(X,Na,(B3)) = $ite(
        aa(code_integer,$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)))),X),Na),(B3)),
        $ite((B3),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se1065995026697491101ons_or(code_integer),X),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),X),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_7876_uint32__shiftr__def,axiom,
    ! [X: uint32,Na: code_integer] :
      uint32_shiftr(X,Na) = $ite(
        ( aa(code_integer,$o,ord_less(code_integer,Na),zero_zero(code_integer))
        | aa(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)),X),Na),
        aa(uint32,uint32,aa(nat,fun(uint32,uint32),bit_se4197421643247451524op_bit(uint32),code_nat_of_integer(Na)),X) ) ).

% uint32_shiftr_def
tff(fact_7877_integer__set__bit__def,axiom,
    ! [X: code_integer,Na: code_integer,B3: $o] :
      bits_integer_set_bit(X,Na,(B3)) = $ite(aa(code_integer,$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)))),X),Na),(B3)),aa($o,code_integer,aa(nat,fun($o,code_integer),aa(code_integer,fun(nat,fun($o,code_integer)),generi7602027413899671122et_bit(code_integer),X),code_nat_of_integer(Na)),(B3))) ).

% integer_set_bit_def
tff(fact_7878_uint32__shiftl__def,axiom,
    ! [X: uint32,Na: code_integer] :
      uint32_shiftl(X,Na) = $ite(
        ( aa(code_integer,$o,ord_less(code_integer,Na),zero_zero(code_integer))
        | aa(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)),X),Na),
        aa(uint32,uint32,aa(nat,fun(uint32,uint32),bit_se4730199178511100633sh_bit(uint32),code_nat_of_integer(Na)),X) ) ).

% uint32_shiftl_def
tff(fact_7879_uint32__set__bit__def,axiom,
    ! [X: uint32,Na: code_integer,B3: $o] :
      uint32_set_bit(X,Na,(B3)) = $ite(
        ( aa(code_integer,$o,ord_less(code_integer,Na),zero_zero(code_integer))
        | aa(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)),X),Na),(B3)),
        aa($o,uint32,aa(nat,fun($o,uint32),aa(uint32,fun(nat,fun($o,uint32)),generi7602027413899671122et_bit(uint32),X),code_nat_of_integer(Na)),(B3)) ) ).

% uint32_set_bit_def
tff(fact_7880_uint32__test__bit__def,axiom,
    ! [X: uint32,Na: code_integer] :
      ( uint32_test_bit(X,Na)
    <=> $ite(
          ( aa(code_integer,$o,ord_less(code_integer,Na),zero_zero(code_integer))
          | aa(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,$o)))),bit_se5641148757651400278ts_bit(uint32)),X),Na),
          aa(nat,$o,aa(uint32,fun(nat,$o),bit_se5641148757651400278ts_bit(uint32),X),code_nat_of_integer(Na)) ) ) ).

% uint32_test_bit_def
tff(fact_7881_uint32__shiftr__code,axiom,
    ! [W: uint32,Na: code_integer] :
      rep_uint322(uint32_shiftr(W,Na)) = $ite(
        ( aa(code_integer,$o,ord_less(code_integer,Na),zero_zero(code_integer))
        | aa(code_integer,$o,ord_less_eq(code_integer,aa(num,code_integer,numeral_numeral(code_integer),bit0(bit0(bit0(bit0(bit0(one2))))))),Na) ),
        rep_uint322(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)),W),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_uint322(W)) ) ).

% uint32_shiftr_code
tff(fact_7882_uint32_Oless__iff__word__of,axiom,
    ! [P2: uint32,Q3: uint32] :
      ( aa(uint32,$o,ord_less(uint32,P2),Q3)
    <=> aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o,ord_less(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322(P2)),rep_uint322(Q3)) ) ).

% uint32.less_iff_word_of
tff(fact_7883_less__uint32_Orep__eq,axiom,
    ! [X: uint32,Xa2: uint32] :
      ( aa(uint32,$o,ord_less(uint32,X),Xa2)
    <=> aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o,ord_less(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322(X)),rep_uint322(Xa2)) ) ).

% less_uint32.rep_eq
tff(fact_7884_zero__uint32_Orep__eq,axiom,
    rep_uint322(zero_zero(uint32)) = zero_zero(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))) ).

% zero_uint32.rep_eq
tff(fact_7885_one__uint32_Orep__eq,axiom,
    rep_uint322(one_one(uint32)) = one_one(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))) ).

% one_uint32.rep_eq
tff(fact_7886_uint32_Oeven__iff__word__of,axiom,
    ! [P2: uint32] :
      ( aa(uint32,$o,dvd_dvd(uint32,aa(num,uint32,numeral_numeral(uint32),bit0(one2))),P2)
    <=> aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o,dvd_dvd(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),aa(num,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),numeral_numeral(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),bit0(one2))),rep_uint322(P2)) ) ).

% uint32.even_iff_word_of
tff(fact_7887_uint32__test__bit__code,axiom,
    ! [W: uint32,Na: code_integer] :
      ( uint32_test_bit(W,Na)
    <=> $ite(
          ( aa(code_integer,$o,ord_less(code_integer,Na),zero_zero(code_integer))
          | aa(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,$o)))),bit_se5641148757651400278ts_bit(uint32)),W),Na),
          aa(nat,$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_uint322(W)),code_nat_of_integer(Na)) ) ) ).

% uint32_test_bit_code
tff(fact_7888_uint32__set__bit__code,axiom,
    ! [W: uint32,Na: code_integer,B3: $o] :
      rep_uint322(uint32_set_bit(W,Na,(B3))) = $ite(
        ( aa(code_integer,$o,ord_less(code_integer,Na),zero_zero(code_integer))
        | aa(code_integer,$o,ord_less(code_integer,aa(num,code_integer,numeral_numeral(code_integer),bit1(bit1(bit1(bit1(one2)))))),Na) ),
        rep_uint322(aa($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)),W),Na),(B3))),
        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_uint322(W)),code_nat_of_integer(Na)),(B3)) ) ).

% uint32_set_bit_code
tff(fact_7889_uint32__shiftl__code,axiom,
    ! [W: uint32,Na: code_integer] :
      rep_uint322(uint32_shiftl(W,Na)) = $ite(
        ( aa(code_integer,$o,ord_less(code_integer,Na),zero_zero(code_integer))
        | aa(code_integer,$o,ord_less_eq(code_integer,aa(num,code_integer,numeral_numeral(code_integer),bit0(bit0(bit0(bit0(bit0(one2))))))),Na) ),
        rep_uint322(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)),W),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_uint322(W)) ) ).

% uint32_shiftl_code
tff(fact_7890_Uint32__signed__code,axiom,
    ! [I: code_integer] :
      rep_uint322(uint32_signed(I)) = $ite(
        ( aa(code_integer,$o,ord_less(code_integer,I),aa(code_integer,code_integer,uminus_uminus(code_integer),aa(num,code_integer,numeral_numeral(code_integer),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))))))))))))))))))))))))))))))))))
        | aa(code_integer,$o,ord_less_eq(code_integer,aa(num,code_integer,numeral_numeral(code_integer),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))))))))))))))))))))))))))))))))),I) ),
        rep_uint322(aa(code_integer,uint32,aa(fun(code_integer,uint32),fun(code_integer,uint32),undefined(fun(fun(code_integer,uint32),fun(code_integer,uint32))),uint322),I)),
        aa(int,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),ring_1_of_int(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),code_I935103866777955880mbolic(I)) ) ).

% Uint32_signed_code
tff(fact_7891_integer__shiftl__def,axiom,
    ! [X: code_integer,Na: code_integer] :
      bits_integer_shiftl(X,Na) = $ite(aa(code_integer,$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))),X),Na),aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se4730199178511100633sh_bit(code_integer),code_nat_of_integer(Na)),X)) ).

% integer_shiftl_def
tff(fact_7892_one__uint32_Orsp,axiom,
    one_one(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))) = one_one(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))) ).

% one_uint32.rsp
tff(fact_7893_zero__uint32_Orsp,axiom,
    zero_zero(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))) = zero_zero(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))) ).

% zero_uint32.rsp
tff(fact_7894_integer__shiftl__code_I2_J,axiom,
    ! [X: code_integer] : bits_integer_shiftl(X,zero_zero(code_integer)) = X ).

% integer_shiftl_code(2)
tff(fact_7895_int__of__integer__symbolic__aux__code_I1_J,axiom,
    code_I935103866777955880mbolic(zero_zero(code_integer)) = zero_zero(int) ).

% int_of_integer_symbolic_aux_code(1)
tff(fact_7896_uint32_Oset__bits__aux__code,axiom,
    ! [F2: fun(nat,$o),Na: nat,W: uint32] :
      set_bits_aux_uint32(F2,Na,W) = $ite(
        Na = zero_zero(nat),
        W,
        $let(
          n2: nat,
          n2:= aa(nat,nat,minus_minus(nat,Na),one_one(nat)),
          set_bits_aux_uint32(F2,n2,
            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)),W)),
              $ite(aa(nat,$o,F2,n2),one_one(uint32),zero_zero(uint32)))) ) ) ).

% uint32.set_bits_aux_code
tff(fact_7897_shiftr__uint32__code,axiom,
    ! [Na: nat,X: uint32] :
      aa(uint32,uint32,aa(nat,fun(uint32,uint32),bit_se4197421643247451524op_bit(uint32),Na),X) = $ite(aa(nat,$o,ord_less(nat,Na),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(one2))))))),uint32_shiftr(X,code_integer_of_nat(Na)),zero_zero(uint32)) ).

% shiftr_uint32_code
tff(fact_7898_shiftl__uint32__code,axiom,
    ! [Na: nat,X: uint32] :
      aa(uint32,uint32,aa(nat,fun(uint32,uint32),bit_se4730199178511100633sh_bit(uint32),Na),X) = $ite(aa(nat,$o,ord_less(nat,Na),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(one2))))))),uint32_shiftl(X,code_integer_of_nat(Na)),zero_zero(uint32)) ).

% shiftl_uint32_code
tff(fact_7899_integer__of__nat_Orep__eq,axiom,
    ! [X: nat] : code_int_of_integer(code_integer_of_nat(X)) = aa(nat,int,semiring_1_of_nat(int),X) ).

% integer_of_nat.rep_eq
tff(fact_7900_int__of__integer__integer__of__nat,axiom,
    ! [Na: nat] : code_int_of_integer(code_integer_of_nat(Na)) = aa(nat,int,semiring_1_of_nat(int),Na) ).

% int_of_integer_integer_of_nat
tff(fact_7901_integer__of__nat_Oabs__eq,axiom,
    ! [X: nat] : code_integer_of_nat(X) = code_integer_of_int(aa(nat,int,semiring_1_of_nat(int),X)) ).

% integer_of_nat.abs_eq
tff(fact_7902_integer__of__nat__less__0__conv,axiom,
    ! [Na: nat] : ~ aa(code_integer,$o,ord_less(code_integer,code_integer_of_nat(Na)),zero_zero(code_integer)) ).

% integer_of_nat_less_0_conv
tff(fact_7903_integer__of__nat__numeral,axiom,
    ! [Na: num] : 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_7904_integer__of__nat__0,axiom,
    code_integer_of_nat(zero_zero(nat)) = zero_zero(code_integer) ).

% integer_of_nat_0
tff(fact_7905_integer__of__nat__1,axiom,
    code_integer_of_nat(one_one(nat)) = one_one(code_integer) ).

% integer_of_nat_1
tff(fact_7906_test__bit__uint32__code,axiom,
    ! [X: uint32,Na: nat] :
      ( aa(nat,$o,aa(uint32,fun(nat,$o),bit_se5641148757651400278ts_bit(uint32),X),Na)
    <=> ( aa(nat,$o,ord_less(nat,Na),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(one2)))))))
        & uint32_test_bit(X,code_integer_of_nat(Na)) ) ) ).

% test_bit_uint32_code
tff(fact_7907_set__bit__uint32__code,axiom,
    ! [X: uint32,Na: nat,B3: $o] :
      aa($o,uint32,aa(nat,fun($o,uint32),aa(uint32,fun(nat,fun($o,uint32)),generi7602027413899671122et_bit(uint32),X),Na),(B3)) = $ite(aa(nat,$o,ord_less(nat,Na),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(one2))))))),uint32_set_bit(X,code_integer_of_nat(Na),(B3)),X) ).

% set_bit_uint32_code
tff(fact_7908_uint32__divmod__code,axiom,
    ! [X: uint32,Y: uint32] :
      uint32_divmod(X,Y) = $ite(
        aa(uint32,$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,ord_less(uint32,X),Y),aa(uint32,product_prod(uint32,uint32),aa(uint32,fun(uint32,product_prod(uint32,uint32)),product_Pair(uint32,uint32),zero_zero(uint32)),X),aa(uint32,product_prod(uint32,uint32),aa(uint32,fun(uint32,product_prod(uint32,uint32)),product_Pair(uint32,uint32),one_one(uint32)),aa(uint32,uint32,minus_minus(uint32,X),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(X)),mod0_uint32(X)),
          $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)),X),Y)),
            $let(
              r: uint32,
              r:= aa(uint32,uint32,minus_minus(uint32,X),aa(uint32,uint32,aa(uint32,fun(uint32,uint32),times_times(uint32),q),Y)),
              $ite(aa(uint32,$o,ord_less_eq(uint32,Y),r),aa(uint32,product_prod(uint32,uint32),aa(uint32,fun(uint32,product_prod(uint32,uint32)),product_Pair(uint32,uint32),aa(uint32,uint32,plus_plus(uint32,q),one_one(uint32))),aa(uint32,uint32,minus_minus(uint32,r),Y)),aa(uint32,product_prod(uint32,uint32),aa(uint32,fun(uint32,product_prod(uint32,uint32)),product_Pair(uint32,uint32),q),r)) ) ) ) ) ).

% uint32_divmod_code
tff(fact_7909_sshiftr__uint32__code,axiom,
    ! [Na: nat,X: uint32] :
      aa(uint32,uint32,aa(nat,fun(uint32,uint32),signed489701013188660438uint32,Na),X) = $ite(
        aa(nat,$o,ord_less(nat,Na),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(one2))))))),
        uint32_sshiftr(X,code_integer_of_nat(Na)),
        $ite(aa(nat,$o,aa(uint32,fun(nat,$o),bit_se5641148757651400278ts_bit(uint32),X),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_7910_uint32__divmod__def,axiom,
    ! [X: uint32,Y: uint32] :
      uint32_divmod(X,Y) = $ite(Y = zero_zero(uint32),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),aa(fun(uint32,fun(uint32,uint32)),fun(uint32,fun(uint32,uint32)),undefined(fun(fun(uint32,fun(uint32,uint32)),fun(uint32,fun(uint32,uint32)))),divide_divide(uint32)),X),zero_zero(uint32))),aa(uint32,uint32,aa(uint32,fun(uint32,uint32),aa(fun(uint32,fun(uint32,uint32)),fun(uint32,fun(uint32,uint32)),undefined(fun(fun(uint32,fun(uint32,uint32)),fun(uint32,fun(uint32,uint32)))),modulo_modulo(uint32)),X),zero_zero(uint32))),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),divide_divide(uint32),X),Y)),aa(uint32,uint32,aa(uint32,fun(uint32,uint32),modulo_modulo(uint32),X),Y))) ).

% uint32_divmod_def
tff(fact_7911_mod0__uint32__def,axiom,
    ! [X: uint32] : mod0_uint32(X) = aa(uint32,uint32,aa(uint32,fun(uint32,uint32),aa(fun(uint32,fun(uint32,uint32)),fun(uint32,fun(uint32,uint32)),undefined(fun(fun(uint32,fun(uint32,uint32)),fun(uint32,fun(uint32,uint32)))),modulo_modulo(uint32)),X),zero_zero(uint32)) ).

% mod0_uint32_def
tff(fact_7912_div0__uint32__def,axiom,
    ! [X: uint32] : div0_uint32(X) = aa(uint32,uint32,aa(uint32,fun(uint32,uint32),aa(fun(uint32,fun(uint32,uint32)),fun(uint32,fun(uint32,uint32)),undefined(fun(fun(uint32,fun(uint32,uint32)),fun(uint32,fun(uint32,uint32)))),divide_divide(uint32)),X),zero_zero(uint32)) ).

% div0_uint32_def
tff(fact_7913_uint32__sdiv__code,axiom,
    ! [X: uint32,Y: uint32] :
      rep_uint322(uint32_sdiv(X,Y)) = $ite(Y = zero_zero(uint32),rep_uint322(aa(uint32,uint32,aa(uint32,fun(uint32,uint32),aa(fun(uint32,fun(uint32,uint32)),fun(uint32,fun(uint32,uint32)),undefined(fun(fun(uint32,fun(uint32,uint32)),fun(uint32,fun(uint32,uint32)))),divide_divide(uint32)),X),zero_zero(uint32))),signed7115095781618012415divide(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322(X),rep_uint322(Y))) ).

% uint32_sdiv_code
tff(fact_7914_uint32__sshiftr__def,axiom,
    ! [X: uint32,Na: code_integer] :
      uint32_sshiftr(X,Na) = $ite(
        ( aa(code_integer,$o,ord_less(code_integer,Na),zero_zero(code_integer))
        | aa(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),X),
        aa(uint32,uint32,aa(nat,fun(uint32,uint32),signed489701013188660438uint32,code_nat_of_integer(Na)),X) ) ).

% uint32_sshiftr_def
tff(fact_7915_uint32__sshiftr__code,axiom,
    ! [W: uint32,Na: code_integer] :
      rep_uint322(uint32_sshiftr(W,Na)) = $ite(
        ( aa(code_integer,$o,ord_less(code_integer,Na),zero_zero(code_integer))
        | aa(code_integer,$o,ord_less_eq(code_integer,aa(num,code_integer,numeral_numeral(code_integer),bit0(bit0(bit0(bit0(bit0(one2))))))),Na) ),
        rep_uint322(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),W)),
        signed_drop_bit(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))),code_nat_of_integer(Na),rep_uint322(W)) ) ).

% uint32_sshiftr_code
tff(fact_7916_integer__shiftr__def,axiom,
    ! [X: code_integer,Na: code_integer] :
      bits_integer_shiftr(X,Na) = $ite(aa(code_integer,$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))),X),Na),aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se4197421643247451524op_bit(code_integer),code_nat_of_integer(Na)),X)) ).

% integer_shiftr_def
tff(fact_7917_uint32_Oset__bits__code,axiom,
    ! [P: fun(nat,$o)] : bit_bi4170147762399595738t_bits(uint32,P) = set_bits_aux_uint32(P,aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(one2)))))),zero_zero(uint32)) ).

% uint32.set_bits_code
tff(fact_7918_integer__shiftr__code_I2_J,axiom,
    ! [X: code_integer] : bits_integer_shiftr(X,zero_zero(code_integer)) = X ).

% integer_shiftr_code(2)
tff(fact_7919_set__bits__False__eq,axiom,
    ! [A: $tType] :
      ( bit_bi6583157726757044596ension(A)
     => ( bit_bi4170147762399595738t_bits(A,aTP_Lamp_agi(nat,$o)) = zero_zero(A) ) ) ).

% set_bits_False_eq
tff(fact_7920_mod__uint32__code,axiom,
    ! [X: uint32,Y: uint32] :
      aa(uint32,uint32,aa(uint32,fun(uint32,uint32),modulo_modulo(uint32),X),Y) = $ite(Y = zero_zero(uint32),X,uint32_mod(X,Y)) ).

% mod_uint32_code
tff(fact_7921_int__set__bits__K__False,axiom,
    bit_bi4170147762399595738t_bits(int,aTP_Lamp_agi(nat,$o)) = zero_zero(int) ).

% int_set_bits_K_False
tff(fact_7922_set__bits__K__False,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( bit_bi4170147762399595738t_bits(word(A),aTP_Lamp_agi(nat,$o)) = zero_zero(word(A)) ) ) ).

% set_bits_K_False
tff(fact_7923_int__set__bits__K__True,axiom,
    bit_bi4170147762399595738t_bits(int,aTP_Lamp_aka(nat,$o)) = aa(int,int,uminus_uminus(int),one_one(int)) ).

% int_set_bits_K_True
tff(fact_7924_bit__set__bits__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(nat,$o),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),bit_bi4170147762399595738t_bits(word(A),P)),Na)
        <=> ( aa(nat,$o,ord_less(nat,Na),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,P,Na) ) ) ) ).

% bit_set_bits_word_iff
tff(fact_7925_word__of__int__conv__set__bits,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: int] : aa(int,word(A),ring_1_of_int(word(A)),I) = bit_bi4170147762399595738t_bits(word(A),aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),I)) ) ).

% word_of_int_conv_set_bits
tff(fact_7926_set__bits__conv__set__bits__aux,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F2: fun(nat,$o)] : bit_bi4170147762399595738t_bits(word(A),F2) = code_T2661198915054445665ts_aux(A,F2,type_len0_len_of(A,type2(A)),zero_zero(word(A))) ) ).

% set_bits_conv_set_bits_aux
tff(fact_7927_word__test__bit__set__bits,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F2: fun(nat,$o),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),bit_bi4170147762399595738t_bits(word(A),F2)),Na)
        <=> ( aa(nat,$o,ord_less(nat,Na),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,F2,Na) ) ) ) ).

% word_test_bit_set_bits
tff(fact_7928_div__uint32__code,axiom,
    ! [X: uint32,Y: uint32] :
      aa(uint32,uint32,aa(uint32,fun(uint32,uint32),divide_divide(uint32),X),Y) = $ite(Y = zero_zero(uint32),zero_zero(uint32),uint32_div(X,Y)) ).

% div_uint32_code
tff(fact_7929_bin__last__set__bits,axiom,
    ! [F2: fun(nat,$o)] :
      ( bit_wf_set_bits_int(F2)
     => ( ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),bit_bi4170147762399595738t_bits(int,F2))
      <=> aa(nat,$o,F2,zero_zero(nat)) ) ) ).

% bin_last_set_bits
tff(fact_7930_wf__set__bits__int__Suc,axiom,
    ! [F2: fun(nat,$o)] :
      ( bit_wf_set_bits_int(aTP_Lamp_age(fun(nat,$o),fun(nat,$o),F2))
    <=> bit_wf_set_bits_int(F2) ) ).

% wf_set_bits_int_Suc
tff(fact_7931_wf__set__bits__int__const,axiom,
    ! [B3: $o] : bit_wf_set_bits_int(aTP_Lamp_akb($o,fun(nat,$o),(B3))) ).

% wf_set_bits_int_const
tff(fact_7932_int__set__bits__unfold__BIT,axiom,
    ! [F2: fun(nat,$o)] :
      ( bit_wf_set_bits_int(F2)
     => ( bit_bi4170147762399595738t_bits(int,F2) = aa(int,int,plus_plus(int,aa($o,int,zero_neq_one_of_bool(int),aa(nat,$o,F2,zero_zero(nat)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),bit_bi4170147762399595738t_bits(int,comp(nat,$o,nat,F2,suc)))) ) ) ).

% int_set_bits_unfold_BIT
tff(fact_7933_msb__uint32__code,axiom,
    ! [X: uint32] :
      ( most_s684356279273892711sb_msb(uint32,X)
    <=> uint32_test_bit(X,aa(num,code_integer,numeral_numeral(code_integer),bit1(bit1(bit1(bit1(one2)))))) ) ).

% msb_uint32_code
tff(fact_7934_bin__rest__set__bits,axiom,
    ! [F2: fun(nat,$o)] :
      ( bit_wf_set_bits_int(F2)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),bit_bi4170147762399595738t_bits(int,F2)),aa(num,int,numeral_numeral(int),bit0(one2))) = bit_bi4170147762399595738t_bits(int,comp(nat,$o,nat,F2,suc)) ) ) ).

% bin_rest_set_bits
tff(fact_7935_summable__inverse__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,comp(A,A,nat,inverse_inverse(A),F2))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_akc(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ).

% summable_inverse_divide
tff(fact_7936_sum__comp__morphism,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( comm_monoid_add(B)
        & comm_monoid_add(A) )
     => ! [H: fun(B,A),G: fun(C,B),A2: set(C)] :
          ( ( aa(B,A,H,zero_zero(B)) = zero_zero(A) )
         => ( ! [X4: B,Y3: B] : aa(B,A,H,aa(B,B,plus_plus(B,X4),Y3)) = aa(A,A,plus_plus(A,aa(B,A,H,X4)),aa(B,A,H,Y3))
           => ( aa(set(C),A,groups7311177749621191930dd_sum(C,A,comp(B,A,C,H,G)),A2) = aa(B,A,H,aa(set(C),B,groups7311177749621191930dd_sum(C,B,G),A2)) ) ) ) ) ).

% sum_comp_morphism
tff(fact_7937_sum_Oreindex__nontrivial,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [A2: set(A),H: fun(A,B),G: fun(B,C)] :
          ( finite_finite2(A,A2)
         => ( ! [X4: A,Y3: A] :
                ( member(A,X4,A2)
               => ( member(A,Y3,A2)
                 => ( ( X4 != Y3 )
                   => ( ( aa(A,B,H,X4) = aa(A,B,H,Y3) )
                     => ( aa(B,C,G,aa(A,B,H,X4)) = zero_zero(C) ) ) ) ) )
           => ( aa(set(B),C,groups7311177749621191930dd_sum(B,C,G),aa(set(A),set(B),image(A,B,H),A2)) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,comp(B,C,A,G,H)),A2) ) ) ) ) ).

% sum.reindex_nontrivial
tff(fact_7938_prod_Oreindex__nontrivial,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [A2: set(A),H: fun(A,B),G: fun(B,C)] :
          ( finite_finite2(A,A2)
         => ( ! [X4: A,Y3: A] :
                ( member(A,X4,A2)
               => ( member(A,Y3,A2)
                 => ( ( X4 != Y3 )
                   => ( ( aa(A,B,H,X4) = aa(A,B,H,Y3) )
                     => ( aa(B,C,G,aa(A,B,H,X4)) = one_one(C) ) ) ) ) )
           => ( groups7121269368397514597t_prod(B,C,G,aa(set(A),set(B),image(A,B,H),A2)) = groups7121269368397514597t_prod(A,C,comp(B,C,A,G,H),A2) ) ) ) ) ).

% prod.reindex_nontrivial
tff(fact_7939_sum__image__le,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(A),G: fun(C,B),F2: fun(A,C)] :
          ( finite_finite2(A,I5)
         => ( ! [I3: A] :
                ( member(A,I3,I5)
               => aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(C,B,G,aa(A,C,F2,I3))) )
           => aa(B,$o,ord_less_eq(B,aa(set(C),B,groups7311177749621191930dd_sum(C,B,G),aa(set(A),set(C),image(A,C,F2),I5))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,comp(C,B,A,G,F2)),I5)) ) ) ) ).

% sum_image_le
tff(fact_7940_sum_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Na: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Na))) = aa(A,A,plus_plus(A,aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,comp(nat,A,nat,G,suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))) ) ).

% sum.atLeast0_lessThan_Suc_shift
tff(fact_7941_sum_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Na: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Na))) = aa(A,A,plus_plus(A,aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,comp(nat,A,nat,G,suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na))) ) ).

% sum.atLeast0_atMost_Suc_shift
tff(fact_7942_prod_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Na: nat] : groups7121269368397514597t_prod(nat,A,G,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,G,zero_zero(nat))),groups7121269368397514597t_prod(nat,A,comp(nat,A,nat,G,suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))) ) ).

% prod.atLeast0_lessThan_Suc_shift
tff(fact_7943_prod_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Na: nat] : groups7121269368397514597t_prod(nat,A,G,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,G,zero_zero(nat))),groups7121269368397514597t_prod(nat,A,comp(nat,A,nat,G,suc),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na))) ) ).

% prod.atLeast0_atMost_Suc_shift
tff(fact_7944_sum_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,Na)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,comp(nat,A,nat,G,plus_plus(nat,M))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,minus_minus(nat,Na),M))) ) ).

% sum.atLeastLessThan_shift_0
tff(fact_7945_prod_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,Na)) = groups7121269368397514597t_prod(nat,A,comp(nat,A,nat,G,plus_plus(nat,M)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,minus_minus(nat,Na),M))) ) ).

% prod.atLeastLessThan_shift_0
tff(fact_7946_tendsto__compose__at,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),Y: B,F3: filter(A),G: fun(B,C),Z: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Y),F3)
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,Z),topolo174197925503356063within(B,Y,top_top(set(B))))
           => ( eventually(A,aa(C,fun(A,$o),aa(fun(B,C),fun(C,fun(A,$o)),aa(B,fun(fun(B,C),fun(C,fun(A,$o))),aTP_Lamp_akd(fun(A,B),fun(B,fun(fun(B,C),fun(C,fun(A,$o)))),F2),Y),G),Z),F3)
             => filterlim(A,C,comp(B,C,A,G,F2),topolo7230453075368039082e_nhds(C,Z),F3) ) ) ) ) ).

% tendsto_compose_at
tff(fact_7947_sum_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,comp(nat,A,nat,G,aTP_Lamp_ake(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Na))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,Na)) ) ).

% sum.atLeast_lessThan_pred_shift
tff(fact_7948_sum_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,comp(nat,A,nat,G,aTP_Lamp_ake(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Na))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% sum.atLeast_atMost_pred_shift
tff(fact_7949_prod_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] : groups7121269368397514597t_prod(nat,A,comp(nat,A,nat,G,aTP_Lamp_ake(nat,nat)),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Na))) = groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,Na)) ) ).

% prod.atLeast_lessThan_pred_shift
tff(fact_7950_prod_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] : groups7121269368397514597t_prod(nat,A,comp(nat,A,nat,G,aTP_Lamp_ake(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Na))) = groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% prod.atLeast_atMost_pred_shift
tff(fact_7951_sum_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,ord_less_eq(nat,M),Na)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,Na)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,comp(nat,A,nat,G,plus_plus(nat,M))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,minus_minus(nat,Na),M))) ) ) ) ).

% sum.atLeastAtMost_shift_0
tff(fact_7952_prod_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,ord_less_eq(nat,M),Na)
         => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,Na)) = groups7121269368397514597t_prod(nat,A,comp(nat,A,nat,G,plus_plus(nat,M)),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,minus_minus(nat,Na),M))) ) ) ) ).

% prod.atLeastAtMost_shift_0
tff(fact_7953_lim__at__infinity__0,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),L: A] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,L),at_infinity(A))
        <=> filterlim(A,A,comp(A,A,A,F2,inverse_inverse(A)),topolo7230453075368039082e_nhds(A,L),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% lim_at_infinity_0
tff(fact_7954_tendsto__at__iff__sequentially,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),A3: B,X: A,Sa: set(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),topolo174197925503356063within(A,X,Sa))
        <=> ! [X9: fun(nat,A)] :
              ( ! [I4: nat] : member(A,aa(nat,A,X9,I4),aa(set(A),set(A),minus_minus(set(A),Sa),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))))
             => ( filterlim(nat,A,X9,topolo7230453075368039082e_nhds(A,X),at_top(nat))
               => filterlim(nat,B,comp(A,B,nat,F2,X9),topolo7230453075368039082e_nhds(B,A3),at_top(nat)) ) ) ) ) ).

% tendsto_at_iff_sequentially
tff(fact_7955_uint32__msb__test__bit,axiom,
    ! [X: uint32] :
      ( most_s684356279273892711sb_msb(uint32,X)
    <=> aa(nat,$o,aa(uint32,fun(nat,$o),bit_se5641148757651400278ts_bit(uint32),X),aa(num,nat,numeral_numeral(nat),bit1(bit1(bit1(bit1(one2)))))) ) ).

% uint32_msb_test_bit
tff(fact_7956_map__filter__on__comp,axiom,
    ! [A: $tType,C: $tType,B: $tType,G: fun(B,A),Y7: set(B),X5: set(A),F3: filter(B),F2: fun(A,C)] :
      ( aa(set(A),$o,ord_less_eq(set(A),aa(set(B),set(A),image(B,A,G),Y7)),X5)
     => ( eventually(B,aTP_Lamp_akf(set(B),fun(B,$o),Y7),F3)
       => ( map_filter_on(A,C,X5,F2,map_filter_on(B,A,Y7,G,F3)) = map_filter_on(B,C,Y7,comp(A,C,B,F2,G),F3) ) ) ) ).

% map_filter_on_comp
tff(fact_7957_complex__mult__cnj,axiom,
    ! [Z: complex] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Z),cnj(Z)) = aa(real,complex,real_Vector_of_real(complex),aa(real,real,plus_plus(real,aa(nat,real,power_power(real,re(Z)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,im(Z)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% complex_mult_cnj
tff(fact_7958_complex__cnj__zero,axiom,
    cnj(zero_zero(complex)) = zero_zero(complex) ).

% complex_cnj_zero
tff(fact_7959_complex__cnj__zero__iff,axiom,
    ! [Z: complex] :
      ( ( cnj(Z) = zero_zero(complex) )
    <=> ( Z = zero_zero(complex) ) ) ).

% complex_cnj_zero_iff
tff(fact_7960_complex__cnj__one__iff,axiom,
    ! [Z: complex] :
      ( ( cnj(Z) = one_one(complex) )
    <=> ( Z = one_one(complex) ) ) ).

% complex_cnj_one_iff
tff(fact_7961_complex__cnj__one,axiom,
    cnj(one_one(complex)) = one_one(complex) ).

% complex_cnj_one
tff(fact_7962_cnj__sum,axiom,
    ! [A: $tType,F2: fun(A,complex),Sa: set(A)] : cnj(aa(set(A),complex,groups7311177749621191930dd_sum(A,complex,F2),Sa)) = aa(set(A),complex,groups7311177749621191930dd_sum(A,complex,aTP_Lamp_akg(fun(A,complex),fun(A,complex),F2)),Sa) ).

% cnj_sum
tff(fact_7963_cnj__prod,axiom,
    ! [A: $tType,F2: fun(A,complex),Sa: set(A)] : cnj(groups7121269368397514597t_prod(A,complex,F2,Sa)) = groups7121269368397514597t_prod(A,complex,aTP_Lamp_akg(fun(A,complex),fun(A,complex),F2),Sa) ).

% cnj_prod
tff(fact_7964_complex__In__mult__cnj__zero,axiom,
    ! [Z: complex] : im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Z),cnj(Z))) = zero_zero(real) ).

% complex_In_mult_cnj_zero
tff(fact_7965_eventually__map__filter__on,axiom,
    ! [B: $tType,A: $tType,X5: set(A),F3: filter(A),P: fun(B,$o),F2: fun(A,B)] :
      ( eventually(A,aTP_Lamp_aa(set(A),fun(A,$o),X5),F3)
     => ( eventually(B,P,map_filter_on(A,B,X5,F2,F3))
      <=> eventually(A,aa(fun(A,B),fun(A,$o),aa(fun(B,$o),fun(fun(A,B),fun(A,$o)),aTP_Lamp_akh(set(A),fun(fun(B,$o),fun(fun(A,B),fun(A,$o))),X5),P),F2),F3) ) ) ).

% eventually_map_filter_on
tff(fact_7966_continuous__on__cnj,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Sa: set(A),G: fun(A,complex)] :
          ( topolo81223032696312382ous_on(A,complex,Sa,G)
         => topolo81223032696312382ous_on(A,complex,Sa,aTP_Lamp_aki(fun(A,complex),fun(A,complex),G)) ) ) ).

% continuous_on_cnj
tff(fact_7967_has__derivative__cnj,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,complex),G4: fun(A,complex),F3: filter(A)] :
          ( has_derivative(A,complex,G,G4,F3)
         => has_derivative(A,complex,aTP_Lamp_akj(fun(A,complex),fun(A,complex),G),aTP_Lamp_akj(fun(A,complex),fun(A,complex),G4),F3) ) ) ).

% has_derivative_cnj
tff(fact_7968_sums__cnj,axiom,
    ! [F2: fun(nat,complex),L: complex] :
      ( aa(complex,$o,sums(complex,aTP_Lamp_akk(fun(nat,complex),fun(nat,complex),F2)),cnj(L))
    <=> aa(complex,$o,sums(complex,F2),L) ) ).

% sums_cnj
tff(fact_7969_continuous__cnj,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),G: fun(A,complex)] :
          ( topolo3448309680560233919inuous(A,complex,F3,G)
         => topolo3448309680560233919inuous(A,complex,F3,aTP_Lamp_akl(fun(A,complex),fun(A,complex),G)) ) ) ).

% continuous_cnj
tff(fact_7970_differentiable__cnj__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(A,complex),X: A,A2: set(A)] :
          ( differentiable(A,complex,aTP_Lamp_akj(fun(A,complex),fun(A,complex),F2),topolo174197925503356063within(A,X,A2))
        <=> differentiable(A,complex,F2,topolo174197925503356063within(A,X,A2)) ) ) ).

% differentiable_cnj_iff
tff(fact_7971_tendsto__cnj,axiom,
    ! [A: $tType,G: fun(A,complex),A3: complex,F3: filter(A)] :
      ( filterlim(A,complex,G,topolo7230453075368039082e_nhds(complex,A3),F3)
     => filterlim(A,complex,aTP_Lamp_akg(fun(A,complex),fun(A,complex),G),topolo7230453075368039082e_nhds(complex,cnj(A3)),F3) ) ).

% tendsto_cnj
tff(fact_7972_lim__cnj,axiom,
    ! [A: $tType,F2: fun(A,complex),L: complex,F3: filter(A)] :
      ( filterlim(A,complex,aTP_Lamp_akg(fun(A,complex),fun(A,complex),F2),topolo7230453075368039082e_nhds(complex,cnj(L)),F3)
    <=> filterlim(A,complex,F2,topolo7230453075368039082e_nhds(complex,L),F3) ) ).

% lim_cnj
tff(fact_7973_isCont__cnj,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A3: A,G: fun(A,complex)] :
          ( topolo3448309680560233919inuous(A,complex,topolo174197925503356063within(A,A3,top_top(set(A))),G)
         => topolo3448309680560233919inuous(A,complex,topolo174197925503356063within(A,A3,top_top(set(A))),aTP_Lamp_akl(fun(A,complex),fun(A,complex),G)) ) ) ).

% isCont_cnj
tff(fact_7974_Re__complex__div__eq__0,axiom,
    ! [A3: complex,B3: complex] :
      ( ( re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A3),B3)) = zero_zero(real) )
    <=> ( re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B3))) = zero_zero(real) ) ) ).

% Re_complex_div_eq_0
tff(fact_7975_Im__complex__div__eq__0,axiom,
    ! [A3: complex,B3: complex] :
      ( ( im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A3),B3)) = zero_zero(real) )
    <=> ( im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B3))) = zero_zero(real) ) ) ).

% Im_complex_div_eq_0
tff(fact_7976_Re__complex__div__lt__0,axiom,
    ! [A3: complex,B3: complex] :
      ( aa(real,$o,ord_less(real,re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A3),B3))),zero_zero(real))
    <=> aa(real,$o,ord_less(real,re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B3)))),zero_zero(real)) ) ).

% Re_complex_div_lt_0
tff(fact_7977_Re__complex__div__gt__0,axiom,
    ! [A3: complex,B3: complex] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A3),B3)))
    <=> aa(real,$o,ord_less(real,zero_zero(real)),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B3)))) ) ).

% Re_complex_div_gt_0
tff(fact_7978_Re__complex__div__ge__0,axiom,
    ! [A3: complex,B3: complex] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A3),B3)))
    <=> aa(real,$o,ord_less_eq(real,zero_zero(real)),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B3)))) ) ).

% Re_complex_div_ge_0
tff(fact_7979_Re__complex__div__le__0,axiom,
    ! [A3: complex,B3: complex] :
      ( aa(real,$o,ord_less_eq(real,re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A3),B3))),zero_zero(real))
    <=> aa(real,$o,ord_less_eq(real,re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B3)))),zero_zero(real)) ) ).

% Re_complex_div_le_0
tff(fact_7980_Im__complex__div__gt__0,axiom,
    ! [A3: complex,B3: complex] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A3),B3)))
    <=> aa(real,$o,ord_less(real,zero_zero(real)),im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B3)))) ) ).

% Im_complex_div_gt_0
tff(fact_7981_Im__complex__div__lt__0,axiom,
    ! [A3: complex,B3: complex] :
      ( aa(real,$o,ord_less(real,im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A3),B3))),zero_zero(real))
    <=> aa(real,$o,ord_less(real,im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B3)))),zero_zero(real)) ) ).

% Im_complex_div_lt_0
tff(fact_7982_Im__complex__div__le__0,axiom,
    ! [A3: complex,B3: complex] :
      ( aa(real,$o,ord_less_eq(real,im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A3),B3))),zero_zero(real))
    <=> aa(real,$o,ord_less_eq(real,im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B3)))),zero_zero(real)) ) ).

% Im_complex_div_le_0
tff(fact_7983_Im__complex__div__ge__0,axiom,
    ! [A3: complex,B3: complex] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A3),B3)))
    <=> aa(real,$o,ord_less_eq(real,zero_zero(real)),im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B3)))) ) ).

% Im_complex_div_ge_0
tff(fact_7984_summable__reindex,axiom,
    ! [F2: fun(nat,real),G: fun(nat,nat)] :
      ( summable(real,F2)
     => ( inj_on(nat,nat,G,top_top(set(nat)))
       => ( ! [X4: nat] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(nat,real,F2,X4))
         => summable(real,comp(nat,real,nat,F2,G)) ) ) ) ).

% summable_reindex
tff(fact_7985_sum_OatLeast__int__atMost__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(int,A),M: nat,Na: nat] : aa(set(int),A,groups7311177749621191930dd_sum(int,A,G),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),M),aa(nat,int,semiring_1_of_nat(int),Na))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,comp(int,A,nat,G,semiring_1_of_nat(int))),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% sum.atLeast_int_atMost_int_shift
tff(fact_7986_sum_OatLeast__int__lessThan__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(int,A),M: nat,Na: nat] : aa(set(int),A,groups7311177749621191930dd_sum(int,A,G),set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),M),aa(nat,int,semiring_1_of_nat(int),Na))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,comp(int,A,nat,G,semiring_1_of_nat(int))),set_or7035219750837199246ssThan(nat,M,Na)) ) ).

% sum.atLeast_int_lessThan_int_shift
tff(fact_7987_prod_OatLeast__int__atMost__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(int,A),M: nat,Na: nat] : groups7121269368397514597t_prod(int,A,G,set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),M),aa(nat,int,semiring_1_of_nat(int),Na))) = groups7121269368397514597t_prod(nat,A,comp(int,A,nat,G,semiring_1_of_nat(int)),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% prod.atLeast_int_atMost_int_shift
tff(fact_7988_prod_OatLeast__int__lessThan__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(int,A),M: nat,Na: nat] : groups7121269368397514597t_prod(int,A,G,set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),M),aa(nat,int,semiring_1_of_nat(int),Na))) = groups7121269368397514597t_prod(nat,A,comp(int,A,nat,G,semiring_1_of_nat(int)),set_or7035219750837199246ssThan(nat,M,Na)) ) ).

% prod.atLeast_int_lessThan_int_shift
tff(fact_7989_complex__mod__mult__cnj,axiom,
    ! [Z: complex] : real_V7770717601297561774m_norm(complex,aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Z),cnj(Z))) = aa(nat,real,power_power(real,real_V7770717601297561774m_norm(complex,Z)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% complex_mod_mult_cnj
tff(fact_7990_complex__div__gt__0,axiom,
    ! [A3: complex,B3: complex] :
      ( ( aa(real,$o,ord_less(real,zero_zero(real)),re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A3),B3)))
      <=> aa(real,$o,ord_less(real,zero_zero(real)),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B3)))) )
      & ( aa(real,$o,ord_less(real,zero_zero(real)),im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A3),B3)))
      <=> aa(real,$o,ord_less(real,zero_zero(real)),im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B3)))) ) ) ).

% complex_div_gt_0
tff(fact_7991_suminf__reindex__mono,axiom,
    ! [F2: fun(nat,real),G: fun(nat,nat)] :
      ( summable(real,F2)
     => ( inj_on(nat,nat,G,top_top(set(nat)))
       => ( ! [X4: nat] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(nat,real,F2,X4))
         => aa(real,$o,ord_less_eq(real,suminf(real,comp(nat,real,nat,F2,G))),suminf(real,F2)) ) ) ) ).

% suminf_reindex_mono
tff(fact_7992_complex__norm__square,axiom,
    ! [Z: complex] : aa(real,complex,real_Vector_of_real(complex),aa(nat,real,power_power(real,real_V7770717601297561774m_norm(complex,Z)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Z),cnj(Z)) ).

% complex_norm_square
tff(fact_7993_complex__add__cnj,axiom,
    ! [Z: complex] : aa(complex,complex,plus_plus(complex,Z),cnj(Z)) = aa(real,complex,real_Vector_of_real(complex),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),re(Z))) ).

% complex_add_cnj
tff(fact_7994_filterlim__at__top__iff__inverse__0,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( eventually(A,aTP_Lamp_afz(fun(A,real),fun(A,$o),F2),F3)
     => ( filterlim(A,real,F2,at_top(real),F3)
      <=> filterlim(A,real,comp(real,real,A,inverse_inverse(real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% filterlim_at_top_iff_inverse_0
tff(fact_7995_suminf__reindex,axiom,
    ! [F2: fun(nat,real),G: fun(nat,nat)] :
      ( summable(real,F2)
     => ( inj_on(nat,nat,G,top_top(set(nat)))
       => ( ! [X4: nat] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(nat,real,F2,X4))
         => ( ! [X4: nat] :
                ( ~ member(nat,X4,aa(set(nat),set(nat),image(nat,nat,G),top_top(set(nat))))
               => ( aa(nat,real,F2,X4) = zero_zero(real) ) )
           => ( suminf(real,comp(nat,real,nat,F2,G)) = suminf(real,F2) ) ) ) ) ) ).

% suminf_reindex
tff(fact_7996_complex__diff__cnj,axiom,
    ! [Z: complex] : aa(complex,complex,minus_minus(complex,Z),cnj(Z)) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(real,complex,real_Vector_of_real(complex),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),im(Z)))),imaginary_unit) ).

% complex_diff_cnj
tff(fact_7997_complex__div__cnj,axiom,
    ! [A3: complex,B3: complex] : aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A3),B3) = aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B3))),aa(real,complex,real_Vector_of_real(complex),aa(nat,real,power_power(real,real_V7770717601297561774m_norm(complex,B3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% complex_div_cnj
tff(fact_7998_cnj__add__mult__eq__Re,axiom,
    ! [Z: complex,W: complex] : aa(complex,complex,plus_plus(complex,aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Z),cnj(W))),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),cnj(Z)),W)) = aa(real,complex,real_Vector_of_real(complex),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Z),cnj(W))))) ).

% cnj_add_mult_eq_Re
tff(fact_7999_Code__Target__Int_Onegative__def,axiom,
    code_Target_negative = comp(int,int,num,uminus_uminus(int),numeral_numeral(int)) ).

% Code_Target_Int.negative_def
tff(fact_8000_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)),comp(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),code_integer,comp(fun(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),code_integer,product_apsnd(code_integer,code_integer,code_integer),times_times(code_integer)),sgn_sgn(code_integer)),L),
            $ite(aa(code_integer,code_integer,sgn_sgn(code_integer),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),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer),aTP_Lamp_akm(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_8001_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_8002_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_8003_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,ord_less(code_integer,zero_zero(code_integer)),L),
          $ite(aa(code_integer,$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),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer),aTP_Lamp_akn(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,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),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer),aTP_Lamp_ako(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_8004_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),product_case_prod(code_integer,code_integer,product_prod(code_integer,$o),aTP_Lamp_akp(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,$o))),K)),code_divmod_abs(K,aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))))) ).

% bit_cut_integer_code
tff(fact_8005_integer__of__uint32__code,axiom,
    ! [Na: uint32] :
      aa(uint32,code_integer,integer_of_uint32,Na) = $ite(aa(nat,$o,aa(uint32,fun(nat,$o),bit_se5641148757651400278ts_bit(uint32),Na),aa(num,nat,numeral_numeral(nat),bit1(bit1(bit1(bit1(one2)))))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se1065995026697491101ons_or(code_integer),intege5370686899274169573signed(aa(uint32,uint32,aa(uint32,fun(uint32,uint32),bit_se5824344872417868541ns_and(uint32),Na),aa(num,uint32,numeral_numeral(uint32),bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(one2)))))))))))))))))))))))))))))))))),aa(num,code_integer,numeral_numeral(code_integer),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))))))))))))))))))))))))))))))))),intege5370686899274169573signed(Na)) ).

% integer_of_uint32_code
tff(fact_8006_integer__of__uint32__signed__def,axiom,
    ! [Na: uint32] :
      intege5370686899274169573signed(Na) = $ite(aa(nat,$o,aa(uint32,fun(nat,$o),bit_se5641148757651400278ts_bit(uint32),Na),aa(num,nat,numeral_numeral(nat),bit1(bit1(bit1(bit1(one2)))))),aa(uint32,code_integer,aa(fun(uint32,code_integer),fun(uint32,code_integer),undefined(fun(fun(uint32,code_integer),fun(uint32,code_integer))),integer_of_uint32),Na),aa(uint32,code_integer,integer_of_uint32,Na)) ).

% integer_of_uint32_signed_def
tff(fact_8007_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,dvd_dvd(code_integer,aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),K)) ).

% bit_cut_integer_def
tff(fact_8008_integer__of__uint32__signed__code,axiom,
    ! [Na: uint32] :
      intege5370686899274169573signed(Na) = $ite(aa(nat,$o,aa(uint32,fun(nat,$o),bit_se5641148757651400278ts_bit(uint32),Na),aa(num,nat,numeral_numeral(nat),bit1(bit1(bit1(bit1(one2)))))),aa(uint32,code_integer,aa(fun(uint32,code_integer),fun(uint32,code_integer),undefined(fun(fun(uint32,code_integer),fun(uint32,code_integer))),integer_of_uint32),Na),code_integer_of_int(aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),int,semiring_1_unsigned(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))),int),rep_uint32(Na)))) ).

% integer_of_uint32_signed_code
tff(fact_8009_uint32__sdiv__def,axiom,
    ! [X: uint32,Y: uint32] :
      uint32_sdiv(X,Y) = $ite(Y = zero_zero(uint32),aa(uint32,uint32,aa(uint32,fun(uint32,uint32),aa(fun(uint32,fun(uint32,uint32)),fun(uint32,fun(uint32,uint32)),undefined(fun(fun(uint32,fun(uint32,uint32)),fun(uint32,fun(uint32,uint32)))),divide_divide(uint32)),X),zero_zero(uint32)),abs_uint32(signed7115095781618012415divide(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322(X),rep_uint322(Y)))) ).

% uint32_sdiv_def
tff(fact_8010_one__uint32__def,axiom,
    one_one(uint32) = abs_uint32(one_one(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))))) ).

% one_uint32_def
tff(fact_8011_zero__uint32__def,axiom,
    zero_zero(uint32) = abs_uint32(zero_zero(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))))) ).

% zero_uint32_def
tff(fact_8012_less__uint32_Oabs__eq,axiom,
    ! [Xa2: word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),X: word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))] :
      ( aa(uint32,$o,ord_less(uint32,abs_uint32(Xa2)),abs_uint32(X))
    <=> aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o,ord_less(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),Xa2),X) ) ).

% less_uint32.abs_eq
tff(fact_8013_Rep__uint32_H__code,axiom,
    ! [X: uint32] : rep_uint32(X) = bit_bi4170147762399595738t_bits(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),aa(uint32,fun(nat,$o),bit_se5641148757651400278ts_bit(uint32),X)) ).

% Rep_uint32'_code
tff(fact_8014_empty__natural,axiom,
    ! [C: $tType,B: $tType,D: $tType,A: $tType,F2: fun(A,C),G: fun(D,B)] : comp(C,set(B),A,aTP_Lamp_akq(C,set(B)),F2) = comp(set(D),set(B),A,image(D,B,G),aTP_Lamp_akr(A,set(D))) ).

% empty_natural
tff(fact_8015_even__sum__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_parity(B)
     => ! [A2: set(A),F2: fun(A,B)] :
          ( finite_finite2(A,A2)
         => ( aa(B,$o,dvd_dvd(B,aa(num,B,numeral_numeral(B),bit0(one2))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A2))
          <=> aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),finite_card(A,collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_aks(set(A),fun(fun(A,B),fun(A,$o)),A2),F2)))) ) ) ) ).

% even_sum_iff
tff(fact_8016_card__UNIV__unit,axiom,
    finite_card(product_unit,top_top(set(product_unit))) = one_one(nat) ).

% card_UNIV_unit
tff(fact_8017_card__Collect__less__nat,axiom,
    ! [Na: nat] : finite_card(nat,collect(nat,aTP_Lamp_gc(nat,fun(nat,$o),Na))) = Na ).

% card_Collect_less_nat
tff(fact_8018_card__Collect__le__nat,axiom,
    ! [Na: nat] : finite_card(nat,collect(nat,aTP_Lamp_ni(nat,fun(nat,$o),Na))) = aa(nat,nat,suc,Na) ).

% card_Collect_le_nat
tff(fact_8019_card__UNIV__bool,axiom,
    finite_card($o,top_top(set($o))) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% card_UNIV_bool
tff(fact_8020_card_Oempty,axiom,
    ! [A: $tType] : finite_card(A,bot_bot(set(A))) = zero_zero(nat) ).

% card.empty
tff(fact_8021_card_Oinfinite,axiom,
    ! [A: $tType,A2: set(A)] :
      ( ~ finite_finite2(A,A2)
     => ( finite_card(A,A2) = zero_zero(nat) ) ) ).

% card.infinite
tff(fact_8022_prod__constant,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Y: A,A2: set(B)] : groups7121269368397514597t_prod(B,A,aTP_Lamp_akt(A,fun(B,A),Y),A2) = aa(nat,A,power_power(A,Y),finite_card(B,A2)) ) ).

% prod_constant
tff(fact_8023_card__0__eq,axiom,
    ! [A: $tType,A2: set(A)] :
      ( finite_finite2(A,A2)
     => ( ( finite_card(A,A2) = zero_zero(nat) )
      <=> ( A2 = bot_bot(set(A)) ) ) ) ).

% card_0_eq
tff(fact_8024_card__insert__disjoint,axiom,
    ! [A: $tType,A2: set(A),X: A] :
      ( finite_finite2(A,A2)
     => ( ~ member(A,X,A2)
       => ( finite_card(A,aa(set(A),set(A),insert(A,X),A2)) = aa(nat,nat,suc,finite_card(A,A2)) ) ) ) ).

% card_insert_disjoint
tff(fact_8025_sum__constant,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Y: A,A2: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_aku(A,fun(B,A),Y)),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),finite_card(B,A2))),Y) ) ).

% sum_constant
tff(fact_8026_card__Diff__insert,axiom,
    ! [A: $tType,A3: A,A2: set(A),B2: set(A)] :
      ( member(A,A3,A2)
     => ( ~ member(A,A3,B2)
       => ( finite_card(A,aa(set(A),set(A),minus_minus(set(A),A2),aa(set(A),set(A),insert(A,A3),B2))) = aa(nat,nat,minus_minus(nat,finite_card(A,aa(set(A),set(A),minus_minus(set(A),A2),B2))),one_one(nat)) ) ) ) ).

% card_Diff_insert
tff(fact_8027_card__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : finite_card(int,set_or1337092689740270186AtMost(int,L,U)) = nat2(aa(int,int,plus_plus(int,aa(int,int,minus_minus(int,U),L)),one_one(int))) ).

% card_atLeastAtMost_int
tff(fact_8028_card__doubleton__eq__2__iff,axiom,
    ! [A: $tType,A3: A,B3: A] :
      ( ( finite_card(A,aa(set(A),set(A),insert(A,A3),aa(set(A),set(A),insert(A,B3),bot_bot(set(A))))) = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
    <=> ( A3 != B3 ) ) ).

% card_doubleton_eq_2_iff
tff(fact_8029_card__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : finite_card(int,set_or5935395276787703475ssThan(int,L,U)) = nat2(aa(int,int,minus_minus(int,U),aa(int,int,plus_plus(int,L),one_one(int)))) ).

% card_greaterThanLessThan_int
tff(fact_8030_card__eq__sum,axiom,
    ! [A: $tType,A2: set(A)] : finite_card(A,A2) = aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aTP_Lamp_akv(A,nat)),A2) ).

% card_eq_sum
tff(fact_8031_sum__multicount__gen,axiom,
    ! [A: $tType,B: $tType,Sa: set(A),Ta: set(B),R: fun(A,fun(B,$o)),K: fun(B,nat)] :
      ( finite_finite2(A,Sa)
     => ( finite_finite2(B,Ta)
       => ( ! [X4: B] :
              ( member(B,X4,Ta)
             => ( finite_card(A,collect(A,aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_mc(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Sa),R),X4))) = aa(B,nat,K,X4) ) )
         => ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aa(fun(A,fun(B,$o)),fun(A,nat),aTP_Lamp_akw(set(B),fun(fun(A,fun(B,$o)),fun(A,nat)),Ta),R)),Sa) = aa(set(B),nat,groups7311177749621191930dd_sum(B,nat,K),Ta) ) ) ) ) ).

% sum_multicount_gen
tff(fact_8032_card__less__sym__Diff,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] :
      ( finite_finite2(A,A2)
     => ( finite_finite2(A,B2)
       => ( aa(nat,$o,ord_less(nat,finite_card(A,A2)),finite_card(A,B2))
         => aa(nat,$o,ord_less(nat,finite_card(A,aa(set(A),set(A),minus_minus(set(A),A2),B2))),finite_card(A,aa(set(A),set(A),minus_minus(set(A),B2),A2))) ) ) ) ).

% card_less_sym_Diff
tff(fact_8033_card__map__elide,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,ord_less_eq(nat,Na),finite_card(word(A),top_top(set(word(A)))))
         => ( 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))) = finite_card(nat,set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)) ) ) ) ).

% card_map_elide
tff(fact_8034_card__lists__length__eq,axiom,
    ! [A: $tType,A2: set(A),Na: nat] :
      ( finite_finite2(A,A2)
     => ( finite_card(list(A),collect(list(A),aa(nat,fun(list(A),$o),aTP_Lamp_ly(set(A),fun(nat,fun(list(A),$o)),A2),Na))) = aa(nat,nat,power_power(nat,finite_card(A,A2)),Na) ) ) ).

% card_lists_length_eq
tff(fact_8035_n__subsets,axiom,
    ! [A: $tType,A2: set(A),K: nat] :
      ( finite_finite2(A,A2)
     => ( finite_card(set(A),collect(set(A),aa(nat,fun(set(A),$o),aTP_Lamp_akx(set(A),fun(nat,fun(set(A),$o)),A2),K))) = aa(nat,nat,binomial(finite_card(A,A2)),K) ) ) ).

% n_subsets
tff(fact_8036_card__Un__le,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] : aa(nat,$o,ord_less_eq(nat,finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),B2))),aa(nat,nat,plus_plus(nat,finite_card(A,A2)),finite_card(A,B2))) ).

% card_Un_le
tff(fact_8037_card__insert__le,axiom,
    ! [A: $tType,A2: set(A),X: A] : aa(nat,$o,ord_less_eq(nat,finite_card(A,A2)),finite_card(A,aa(set(A),set(A),insert(A,X),A2))) ).

% card_insert_le
tff(fact_8038_card__length,axiom,
    ! [A: $tType,Xsa: list(A)] : aa(nat,$o,ord_less_eq(nat,finite_card(A,aa(list(A),set(A),set2(A),Xsa))),size_size(list(A),Xsa)) ).

% card_length
tff(fact_8039_card__insert__if,axiom,
    ! [A: $tType,A2: set(A),X: A] :
      ( finite_finite2(A,A2)
     => ( finite_card(A,aa(set(A),set(A),insert(A,X),A2)) = $ite(member(A,X,A2),finite_card(A,A2),aa(nat,nat,suc,finite_card(A,A2))) ) ) ).

% card_insert_if
tff(fact_8040_card__Suc__eq__finite,axiom,
    ! [A: $tType,A2: set(A),K: nat] :
      ( ( finite_card(A,A2) = aa(nat,nat,suc,K) )
    <=> ? [B9: A,B10: set(A)] :
          ( ( A2 = aa(set(A),set(A),insert(A,B9),B10) )
          & ~ member(A,B9,B10)
          & ( finite_card(A,B10) = K )
          & finite_finite2(A,B10) ) ) ).

% card_Suc_eq_finite
tff(fact_8041_card__atLeastZeroLessThan__int,axiom,
    ! [U: int] : finite_card(int,set_or7035219750837199246ssThan(int,zero_zero(int),U)) = nat2(U) ).

% card_atLeastZeroLessThan_int
tff(fact_8042_is__singleton__altdef,axiom,
    ! [A: $tType,A2: set(A)] :
      ( is_singleton(A,A2)
    <=> ( finite_card(A,A2) = one_one(nat) ) ) ).

% is_singleton_altdef
tff(fact_8043_card__2__iff_H,axiom,
    ! [A: $tType,S: set(A)] :
      ( ( finite_card(A,S) = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
    <=> ? [X3: A] :
          ( member(A,X3,S)
          & ? [Xa4: A] :
              ( member(A,Xa4,S)
              & ( X3 != Xa4 )
              & ! [Xb4: A] :
                  ( member(A,Xb4,S)
                 => ( ( Xb4 = X3 )
                    | ( Xb4 = Xa4 ) ) ) ) ) ) ).

% card_2_iff'
tff(fact_8044_card__1__singletonE,axiom,
    ! [A: $tType,A2: set(A)] :
      ( ( finite_card(A,A2) = one_one(nat) )
     => ~ ! [X4: A] : A2 != aa(set(A),set(A),insert(A,X4),bot_bot(set(A))) ) ).

% card_1_singletonE
tff(fact_8045_psubset__card__mono,axiom,
    ! [A: $tType,B2: set(A),A2: set(A)] :
      ( finite_finite2(A,B2)
     => ( aa(set(A),$o,ord_less(set(A),A2),B2)
       => aa(nat,$o,ord_less(nat,finite_card(A,A2)),finite_card(A,B2)) ) ) ).

% psubset_card_mono
tff(fact_8046_card__eq__0__iff,axiom,
    ! [A: $tType,A2: set(A)] :
      ( ( finite_card(A,A2) = zero_zero(nat) )
    <=> ( ( A2 = bot_bot(set(A)) )
        | ~ finite_finite2(A,A2) ) ) ).

% card_eq_0_iff
tff(fact_8047_card__ge__0__finite,axiom,
    ! [A: $tType,A2: set(A)] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),finite_card(A,A2))
     => finite_finite2(A,A2) ) ).

% card_ge_0_finite
tff(fact_8048_card__less,axiom,
    ! [M7: set(nat),I: nat] :
      ( member(nat,zero_zero(nat),M7)
     => ( finite_card(nat,collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_aky(set(nat),fun(nat,fun(nat,$o)),M7),I))) != zero_zero(nat) ) ) ).

% card_less
tff(fact_8049_card__less__Suc,axiom,
    ! [M7: set(nat),I: nat] :
      ( member(nat,zero_zero(nat),M7)
     => ( aa(nat,nat,suc,finite_card(nat,collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_akz(set(nat),fun(nat,fun(nat,$o)),M7),I)))) = finite_card(nat,collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_aky(set(nat),fun(nat,fun(nat,$o)),M7),I))) ) ) ).

% card_less_Suc
tff(fact_8050_card__less__Suc2,axiom,
    ! [M7: set(nat),I: nat] :
      ( ~ member(nat,zero_zero(nat),M7)
     => ( finite_card(nat,collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_akz(set(nat),fun(nat,fun(nat,$o)),M7),I))) = finite_card(nat,collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_aky(set(nat),fun(nat,fun(nat,$o)),M7),I))) ) ) ).

% card_less_Suc2
tff(fact_8051_pigeonhole,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A2: set(B)] :
      ( aa(nat,$o,ord_less(nat,finite_card(A,aa(set(B),set(A),image(B,A,F2),A2))),finite_card(B,A2))
     => ~ inj_on(B,A,F2,A2) ) ).

% pigeonhole
tff(fact_8052_sum__Suc,axiom,
    ! [A: $tType,F2: fun(A,nat),A2: set(A)] : aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aTP_Lamp_ala(fun(A,nat),fun(A,nat),F2)),A2) = aa(nat,nat,plus_plus(nat,aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A2)),finite_card(A,A2)) ).

% sum_Suc
tff(fact_8053_sum__constant__scaleR,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Y: A,A2: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_alb(A,fun(B,A),Y)),A2) = aa(A,A,real_V8093663219630862766scaleR(A,aa(nat,real,semiring_1_of_nat(real),finite_card(B,A2))),Y) ) ).

% sum_constant_scaleR
tff(fact_8054_sum__multicount,axiom,
    ! [A: $tType,B: $tType,S: set(A),T3: set(B),R: fun(A,fun(B,$o)),K: nat] :
      ( finite_finite2(A,S)
     => ( finite_finite2(B,T3)
       => ( ! [X4: B] :
              ( member(B,X4,T3)
             => ( finite_card(A,collect(A,aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_mc(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),S),R),X4))) = K ) )
         => ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aa(fun(A,fun(B,$o)),fun(A,nat),aTP_Lamp_akw(set(B),fun(fun(A,fun(B,$o)),fun(A,nat)),T3),R)),S) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),finite_card(B,T3)) ) ) ) ) ).

% sum_multicount
tff(fact_8055_real__of__card,axiom,
    ! [A: $tType,A2: set(A)] : aa(nat,real,semiring_1_of_nat(real),finite_card(A,A2)) = aa(set(A),real,groups7311177749621191930dd_sum(A,real,aTP_Lamp_alc(A,real)),A2) ).

% real_of_card
tff(fact_8056_sum__bounded__above,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [A2: set(A),F2: fun(A,B),K4: B] :
          ( ! [I3: A] :
              ( member(A,I3,A2)
             => aa(B,$o,ord_less_eq(B,aa(A,B,F2,I3)),K4) )
         => aa(B,$o,ord_less_eq(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A2)),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),finite_card(A,A2))),K4)) ) ) ).

% sum_bounded_above
tff(fact_8057_sum__bounded__below,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [A2: set(A),K4: B,F2: fun(A,B)] :
          ( ! [I3: A] :
              ( member(A,I3,A2)
             => aa(B,$o,ord_less_eq(B,K4),aa(A,B,F2,I3)) )
         => aa(B,$o,ord_less_eq(B,aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),finite_card(A,A2))),K4)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A2)) ) ) ).

% sum_bounded_below
tff(fact_8058_card__gt__0__iff,axiom,
    ! [A: $tType,A2: set(A)] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),finite_card(A,A2))
    <=> ( ( A2 != bot_bot(set(A)) )
        & finite_finite2(A,A2) ) ) ).

% card_gt_0_iff
tff(fact_8059_card__le__Suc0__iff__eq,axiom,
    ! [A: $tType,A2: set(A)] :
      ( finite_finite2(A,A2)
     => ( aa(nat,$o,ord_less_eq(nat,finite_card(A,A2)),aa(nat,nat,suc,zero_zero(nat)))
      <=> ! [X3: A] :
            ( member(A,X3,A2)
           => ! [Xa4: A] :
                ( member(A,Xa4,A2)
               => ( X3 = Xa4 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_8060_finite__UNIV__card__ge__0,axiom,
    ! [A: $tType] :
      ( finite_finite2(A,top_top(set(A)))
     => aa(nat,$o,ord_less(nat,zero_zero(nat)),finite_card(A,top_top(set(A)))) ) ).

% finite_UNIV_card_ge_0
tff(fact_8061_card__1__singleton__iff,axiom,
    ! [A: $tType,A2: set(A)] :
      ( ( finite_card(A,A2) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ? [X3: A] : A2 = aa(set(A),set(A),insert(A,X3),bot_bot(set(A))) ) ).

% card_1_singleton_iff
tff(fact_8062_card__eq__SucD,axiom,
    ! [A: $tType,A2: set(A),K: nat] :
      ( ( finite_card(A,A2) = aa(nat,nat,suc,K) )
     => ? [B5: A,B4: set(A)] :
          ( ( A2 = aa(set(A),set(A),insert(A,B5),B4) )
          & ~ member(A,B5,B4)
          & ( finite_card(A,B4) = K )
          & ( ( K = zero_zero(nat) )
           => ( B4 = bot_bot(set(A)) ) ) ) ) ).

% card_eq_SucD
tff(fact_8063_card__Suc__eq,axiom,
    ! [A: $tType,A2: set(A),K: nat] :
      ( ( finite_card(A,A2) = aa(nat,nat,suc,K) )
    <=> ? [B9: A,B10: set(A)] :
          ( ( A2 = aa(set(A),set(A),insert(A,B9),B10) )
          & ~ member(A,B9,B10)
          & ( finite_card(A,B10) = K )
          & ( ( K = zero_zero(nat) )
           => ( B10 = bot_bot(set(A)) ) ) ) ) ).

% card_Suc_eq
tff(fact_8064_card__le__Suc__iff,axiom,
    ! [A: $tType,Na: nat,A2: set(A)] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,Na)),finite_card(A,A2))
    <=> ? [A10: A,B10: set(A)] :
          ( ( A2 = aa(set(A),set(A),insert(A,A10),B10) )
          & ~ member(A,A10,B10)
          & aa(nat,$o,ord_less_eq(nat,Na),finite_card(A,B10))
          & finite_finite2(A,B10) ) ) ).

% card_le_Suc_iff
tff(fact_8065_card__1__singletonI,axiom,
    ! [A: $tType,S: set(A),X: A] :
      ( finite_finite2(A,S)
     => ( ( finite_card(A,S) = one_one(nat) )
       => ( member(A,X,S)
         => ( S = aa(set(A),set(A),insert(A,X),bot_bot(set(A))) ) ) ) ) ).

% card_1_singletonI
tff(fact_8066_card__Diff1__le,axiom,
    ! [A: $tType,A2: set(A),X: A] : aa(nat,$o,ord_less_eq(nat,finite_card(A,aa(set(A),set(A),minus_minus(set(A),A2),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))))),finite_card(A,A2)) ).

% card_Diff1_le
tff(fact_8067_card__psubset,axiom,
    ! [A: $tType,B2: set(A),A2: set(A)] :
      ( finite_finite2(A,B2)
     => ( aa(set(A),$o,ord_less_eq(set(A),A2),B2)
       => ( aa(nat,$o,ord_less(nat,finite_card(A,A2)),finite_card(A,B2))
         => aa(set(A),$o,ord_less(set(A),A2),B2) ) ) ) ).

% card_psubset
tff(fact_8068_card__lists__length__le,axiom,
    ! [A: $tType,A2: set(A),Na: nat] :
      ( finite_finite2(A,A2)
     => ( finite_card(list(A),collect(list(A),aa(nat,fun(list(A),$o),aTP_Lamp_mk(set(A),fun(nat,fun(list(A),$o)),A2),Na))) = aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,power_power(nat,finite_card(A,A2))),set_ord_atMost(nat,Na)) ) ) ).

% card_lists_length_le
tff(fact_8069_ex__bij__betw__nat__finite,axiom,
    ! [A: $tType,M7: set(A)] :
      ( finite_finite2(A,M7)
     => ? [H2: fun(nat,A)] : bij_betw(nat,A,H2,set_or7035219750837199246ssThan(nat,zero_zero(nat),finite_card(A,M7)),M7) ) ).

% ex_bij_betw_nat_finite
tff(fact_8070_ex__bij__betw__nat__finite__1,axiom,
    ! [A: $tType,M7: set(A)] :
      ( finite_finite2(A,M7)
     => ? [H2: fun(nat,A)] : bij_betw(nat,A,H2,set_or1337092689740270186AtMost(nat,one_one(nat),finite_card(A,M7)),M7) ) ).

% ex_bij_betw_nat_finite_1
tff(fact_8071_card__roots__unity,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Na: nat] :
          ( aa(nat,$o,ord_less_eq(nat,one_one(nat)),Na)
         => aa(nat,$o,ord_less_eq(nat,finite_card(A,collect(A,aTP_Lamp_nb(nat,fun(A,$o),Na)))),Na) ) ) ).

% card_roots_unity
tff(fact_8072_ex__bij__betw__finite__nat,axiom,
    ! [A: $tType,M7: set(A)] :
      ( finite_finite2(A,M7)
     => ? [H2: fun(A,nat)] : bij_betw(A,nat,H2,M7,set_or7035219750837199246ssThan(nat,zero_zero(nat),finite_card(A,M7))) ) ).

% ex_bij_betw_finite_nat
tff(fact_8073_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N2: set(nat),Na: nat] :
      ( aa(set(nat),$o,ord_less_eq(set(nat),N2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))
     => aa(nat,$o,ord_less_eq(nat,finite_card(nat,N2)),Na) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_8074_card__sum__le__nat__sum,axiom,
    ! [S: set(nat)] : aa(nat,$o,ord_less_eq(nat,aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_ik(nat,nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),finite_card(nat,S)))),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_ik(nat,nat)),S)) ).

% card_sum_le_nat_sum
tff(fact_8075_conj__subset__def,axiom,
    ! [A: $tType,A2: set(A),P: fun(A,$o),Q: fun(A,$o)] :
      ( aa(set(A),$o,ord_less_eq(set(A),A2),collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_nj(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)))
    <=> ( aa(set(A),$o,ord_less_eq(set(A),A2),collect(A,P))
        & aa(set(A),$o,ord_less_eq(set(A),A2),collect(A,Q)) ) ) ).

% conj_subset_def
tff(fact_8076_card__nth__roots,axiom,
    ! [C2: complex,Na: nat] :
      ( ( C2 != zero_zero(complex) )
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
       => ( finite_card(complex,collect(complex,aa(nat,fun(complex,$o),aTP_Lamp_jw(complex,fun(nat,fun(complex,$o)),C2),Na))) = Na ) ) ) ).

% card_nth_roots
tff(fact_8077_card__roots__unity__eq,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( finite_card(complex,collect(complex,aTP_Lamp_gr(nat,fun(complex,$o),Na))) = Na ) ) ).

% card_roots_unity_eq
tff(fact_8078_card__2__iff,axiom,
    ! [A: $tType,S: set(A)] :
      ( ( finite_card(A,S) = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
    <=> ? [X3: A,Y6: A] :
          ( ( S = aa(set(A),set(A),insert(A,X3),aa(set(A),set(A),insert(A,Y6),bot_bot(set(A)))) )
          & ( X3 != Y6 ) ) ) ).

% card_2_iff
tff(fact_8079_card__3__iff,axiom,
    ! [A: $tType,S: set(A)] :
      ( ( finite_card(A,S) = aa(num,nat,numeral_numeral(nat),bit1(one2)) )
    <=> ? [X3: A,Y6: A,Z6: A] :
          ( ( S = aa(set(A),set(A),insert(A,X3),aa(set(A),set(A),insert(A,Y6),aa(set(A),set(A),insert(A,Z6),bot_bot(set(A))))) )
          & ( X3 != Y6 )
          & ( Y6 != Z6 )
          & ( X3 != Z6 ) ) ) ).

% card_3_iff
tff(fact_8080_card__range__greater__zero,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A)] :
      ( finite_finite2(A,aa(set(B),set(A),image(B,A,F2),top_top(set(B))))
     => aa(nat,$o,ord_less(nat,zero_zero(nat)),finite_card(A,aa(set(B),set(A),image(B,A,F2),top_top(set(B))))) ) ).

% card_range_greater_zero
tff(fact_8081_odd__card__imp__not__empty,axiom,
    ! [A: $tType,A2: set(A)] :
      ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),finite_card(A,A2))
     => ( A2 != bot_bot(set(A)) ) ) ).

% odd_card_imp_not_empty
tff(fact_8082_card__insert__disjoint_H,axiom,
    ! [A: $tType,A2: set(A),X: A] :
      ( finite_finite2(A,A2)
     => ( ~ member(A,X,A2)
       => ( aa(nat,nat,minus_minus(nat,finite_card(A,aa(set(A),set(A),insert(A,X),A2))),aa(nat,nat,suc,zero_zero(nat))) = finite_card(A,A2) ) ) ) ).

% card_insert_disjoint'
tff(fact_8083_card__Suc__Diff1,axiom,
    ! [A: $tType,A2: set(A),X: A] :
      ( finite_finite2(A,A2)
     => ( member(A,X,A2)
       => ( aa(nat,nat,suc,finite_card(A,aa(set(A),set(A),minus_minus(set(A),A2),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))))) = finite_card(A,A2) ) ) ) ).

% card_Suc_Diff1
tff(fact_8084_card_Oinsert__remove,axiom,
    ! [A: $tType,A2: set(A),X: A] :
      ( finite_finite2(A,A2)
     => ( finite_card(A,aa(set(A),set(A),insert(A,X),A2)) = aa(nat,nat,suc,finite_card(A,aa(set(A),set(A),minus_minus(set(A),A2),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))))) ) ) ).

% card.insert_remove
tff(fact_8085_card_Oremove,axiom,
    ! [A: $tType,A2: set(A),X: A] :
      ( finite_finite2(A,A2)
     => ( member(A,X,A2)
       => ( finite_card(A,A2) = aa(nat,nat,suc,finite_card(A,aa(set(A),set(A),minus_minus(set(A),A2),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))))) ) ) ) ).

% card.remove
tff(fact_8086_card__Diff1__less__iff,axiom,
    ! [A: $tType,A2: set(A),X: A] :
      ( aa(nat,$o,ord_less(nat,finite_card(A,aa(set(A),set(A),minus_minus(set(A),A2),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))))),finite_card(A,A2))
    <=> ( finite_finite2(A,A2)
        & member(A,X,A2) ) ) ).

% card_Diff1_less_iff
tff(fact_8087_card__Diff2__less,axiom,
    ! [A: $tType,A2: set(A),X: A,Y: A] :
      ( finite_finite2(A,A2)
     => ( member(A,X,A2)
       => ( member(A,Y,A2)
         => aa(nat,$o,ord_less(nat,finite_card(A,aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),minus_minus(set(A),A2),aa(set(A),set(A),insert(A,X),bot_bot(set(A))))),aa(set(A),set(A),insert(A,Y),bot_bot(set(A)))))),finite_card(A,A2)) ) ) ) ).

% card_Diff2_less
tff(fact_8088_card__Diff1__less,axiom,
    ! [A: $tType,A2: set(A),X: A] :
      ( finite_finite2(A,A2)
     => ( member(A,X,A2)
       => aa(nat,$o,ord_less(nat,finite_card(A,aa(set(A),set(A),minus_minus(set(A),A2),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))))),finite_card(A,A2)) ) ) ).

% card_Diff1_less
tff(fact_8089_card__Diff__singleton__if,axiom,
    ! [A: $tType,A2: set(A),X: A] :
      finite_card(A,aa(set(A),set(A),minus_minus(set(A),A2),aa(set(A),set(A),insert(A,X),bot_bot(set(A))))) = $ite(member(A,X,A2),aa(nat,nat,minus_minus(nat,finite_card(A,A2)),one_one(nat)),finite_card(A,A2)) ).

% card_Diff_singleton_if
tff(fact_8090_card__Diff__singleton,axiom,
    ! [A: $tType,X: A,A2: set(A)] :
      ( member(A,X,A2)
     => ( finite_card(A,aa(set(A),set(A),minus_minus(set(A),A2),aa(set(A),set(A),insert(A,X),bot_bot(set(A))))) = aa(nat,nat,minus_minus(nat,finite_card(A,A2)),one_one(nat)) ) ) ).

% card_Diff_singleton
tff(fact_8091_sum__norm__bound,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [S: set(A),F2: fun(A,B),K4: real] :
          ( ! [X4: A] :
              ( member(A,X4,S)
             => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(A,B,F2,X4))),K4) )
         => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),S))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),finite_card(A,S))),K4)) ) ) ).

% sum_norm_bound
tff(fact_8092_conj__comp__iff,axiom,
    ! [B: $tType,A: $tType,P: fun(B,$o),Q: fun(B,$o),G: fun(A,B),X2: A] :
      ( aa(A,$o,comp(B,$o,A,aa(fun(B,$o),fun(B,$o),aTP_Lamp_ald(fun(B,$o),fun(fun(B,$o),fun(B,$o)),P),Q),G),X2)
    <=> ( aa(A,$o,comp(B,$o,A,P,G),X2)
        & aa(A,$o,comp(B,$o,A,Q,G),X2) ) ) ).

% conj_comp_iff
tff(fact_8093_prod__le__power,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(B)
     => ! [A2: set(A),F2: fun(A,B),Na: B,K: nat] :
          ( ! [I3: A] :
              ( member(A,I3,A2)
             => ( aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,F2,I3))
                & aa(B,$o,ord_less_eq(B,aa(A,B,F2,I3)),Na) ) )
         => ( aa(nat,$o,ord_less_eq(nat,finite_card(A,A2)),K)
           => ( aa(B,$o,ord_less_eq(B,one_one(B)),Na)
             => aa(B,$o,ord_less_eq(B,groups7121269368397514597t_prod(A,B,F2,A2)),aa(nat,B,power_power(B,Na),K)) ) ) ) ) ).

% prod_le_power
tff(fact_8094_sum__bounded__above__strict,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere8940638589300402666id_add(B)
        & semiring_1(B) )
     => ! [A2: set(A),F2: fun(A,B),K4: B] :
          ( ! [I3: A] :
              ( member(A,I3,A2)
             => aa(B,$o,ord_less(B,aa(A,B,F2,I3)),K4) )
         => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),finite_card(A,A2))
           => aa(B,$o,ord_less(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A2)),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),finite_card(A,A2))),K4)) ) ) ) ).

% sum_bounded_above_strict
tff(fact_8095_sum__bounded__above__divide,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_field(B)
     => ! [A2: set(A),F2: fun(A,B),K4: B] :
          ( ! [I3: A] :
              ( member(A,I3,A2)
             => aa(B,$o,ord_less_eq(B,aa(A,B,F2,I3)),aa(B,B,aa(B,fun(B,B),divide_divide(B),K4),aa(nat,B,semiring_1_of_nat(B),finite_card(A,A2)))) )
         => ( finite_finite2(A,A2)
           => ( ( A2 != bot_bot(set(A)) )
             => aa(B,$o,ord_less_eq(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A2)),K4) ) ) ) ) ).

% sum_bounded_above_divide
tff(fact_8096_card__insert__le__m1,axiom,
    ! [A: $tType,Na: nat,Y: set(A),X: A] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Na)
     => ( aa(nat,$o,ord_less_eq(nat,finite_card(A,Y)),aa(nat,nat,minus_minus(nat,Na),one_one(nat)))
       => aa(nat,$o,ord_less_eq(nat,finite_card(A,aa(set(A),set(A),insert(A,X),Y))),Na) ) ) ).

% card_insert_le_m1
tff(fact_8097_card__word__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : finite_card(word(A),top_top(set(word(A)))) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),size_size(word(A),X)) ) ).

% card_word_size
tff(fact_8098_sum__fun__comp,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [S: set(A),R: set(B),G: fun(A,B),F2: fun(B,C)] :
          ( finite_finite2(A,S)
         => ( finite_finite2(B,R)
           => ( aa(set(B),$o,ord_less_eq(set(B),aa(set(A),set(B),image(A,B,G),S)),R)
             => ( aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ale(fun(A,B),fun(fun(B,C),fun(A,C)),G),F2)),S) = aa(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_alf(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),S),G),F2)),R) ) ) ) ) ) ).

% sum_fun_comp
tff(fact_8099_prod__gen__delta,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),A3: A,B3: fun(A,B),C2: B] :
          ( finite_finite2(A,S)
         => ( groups7121269368397514597t_prod(A,B,aa(B,fun(A,B),aa(fun(A,B),fun(B,fun(A,B)),aTP_Lamp_alg(A,fun(fun(A,B),fun(B,fun(A,B))),A3),B3),C2),S) = $ite(member(A,A3,S),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,B3,A3)),aa(nat,B,power_power(B,C2),aa(nat,nat,minus_minus(nat,finite_card(A,S)),one_one(nat)))),aa(nat,B,power_power(B,C2),finite_card(A,S))) ) ) ) ).

% prod_gen_delta
tff(fact_8100_polyfun__roots__card,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),K: nat,Na: nat] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( aa(nat,$o,ord_less_eq(nat,K),Na)
           => aa(nat,$o,ord_less_eq(nat,finite_card(A,collect(A,aa(nat,fun(A,$o),aTP_Lamp_nh(fun(nat,A),fun(nat,fun(A,$o)),C2),Na)))),Na) ) ) ) ).

% polyfun_roots_card
tff(fact_8101_finite__fun__UNIVD1,axiom,
    ! [B: $tType,A: $tType] :
      ( finite_finite2(fun(A,B),top_top(set(fun(A,B))))
     => ( ( finite_card(B,top_top(set(B))) != aa(nat,nat,suc,zero_zero(nat)) )
       => finite_finite2(A,top_top(set(A))) ) ) ).

% finite_fun_UNIVD1
tff(fact_8102_card__map__elide2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,ord_less_eq(nat,Na),finite_card(word(A),top_top(set(word(A)))))
         => ( 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_8103_card__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( finite_card(word(A),top_top(set(word(A)))) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))) ) ) ).

% card_word
tff(fact_8104_polyfun__rootbound,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),K: nat,Na: nat] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( aa(nat,$o,ord_less_eq(nat,K),Na)
           => ( finite_finite2(A,collect(A,aa(nat,fun(A,$o),aTP_Lamp_nh(fun(nat,A),fun(nat,fun(A,$o)),C2),Na)))
              & aa(nat,$o,ord_less_eq(nat,finite_card(A,collect(A,aa(nat,fun(A,$o),aTP_Lamp_nh(fun(nat,A),fun(nat,fun(A,$o)),C2),Na)))),Na) ) ) ) ) ).

% polyfun_rootbound
tff(fact_8105_Cardinality_Ocard__set,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( finite_card(set(A),top_top(set(set(A)))) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),finite_card(A,top_top(set(A)))) ) ) ).

% Cardinality.card_set
tff(fact_8106_card__bit1,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( 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))),finite_card(A,top_top(set(A))))) ) ) ).

% card_bit1
tff(fact_8107_card__num0,axiom,
    finite_card(numeral_num0,top_top(set(numeral_num0))) = zero_zero(nat) ).

% card_num0
tff(fact_8108_card__num1,axiom,
    finite_card(numeral_num1,top_top(set(numeral_num1))) = one_one(nat) ).

% card_num1
tff(fact_8109_card__nat,axiom,
    finite_card(nat,top_top(set(nat))) = zero_zero(nat) ).

% card_nat
tff(fact_8110_card__bit0,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( finite_card(numeral_bit0(A),top_top(set(numeral_bit0(A)))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),finite_card(A,top_top(set(A)))) ) ) ).

% card_bit0
tff(fact_8111_card__literal,axiom,
    finite_card(literal,top_top(set(literal))) = zero_zero(nat) ).

% card_literal
tff(fact_8112_UNIV__bool,axiom,
    top_top(set($o)) = aa(set($o),set($o),insert($o,$false),aa(set($o),set($o),insert($o,$true),bot_bot(set($o)))) ).

% UNIV_bool
tff(fact_8113_bit0_OCARD__eq,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( finite_card(numeral_bit0(A),top_top(set(numeral_bit0(A)))) = nat2(aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit0(A),top_top(set(numeral_bit0(A)))))) ) ) ).

% bit0.CARD_eq
tff(fact_8114_bit1_OCARD__eq,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A)))) = nat2(aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A)))))) ) ) ).

% bit1.CARD_eq
tff(fact_8115_CARD__1,axiom,
    ! [A: $tType] :
      ( cARD_1(A)
     => ( finite_card(A,top_top(set(A))) = one_one(nat) ) ) ).

% CARD_1
tff(fact_8116_bit1_Osize0,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => aa(int,$o,ord_less(int,zero_zero(int)),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A)))))) ) ).

% bit1.size0
tff(fact_8117_bit0_Osize0,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => aa(int,$o,ord_less(int,zero_zero(int)),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit0(A),top_top(set(numeral_bit0(A)))))) ) ).

% bit0.size0
tff(fact_8118_bit0_Osize1,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => aa(int,$o,ord_less(int,one_one(int)),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit0(A),top_top(set(numeral_bit0(A)))))) ) ).

% bit0.size1
tff(fact_8119_bit1_Osize1,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => aa(int,$o,ord_less(int,one_one(int)),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A)))))) ) ).

% bit1.size1
tff(fact_8120_zero__less__card__finite,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => aa(nat,$o,ord_less(nat,zero_zero(nat)),finite_card(A,top_top(set(A)))) ) ).

% zero_less_card_finite
tff(fact_8121_card__UNIV__sum,axiom,
    ! [B: $tType,A: $tType] :
      finite_card(sum_sum(A,B),top_top(set(sum_sum(A,B)))) = $ite(
        ( ( finite_card(A,top_top(set(A))) != zero_zero(nat) )
        & ( finite_card(B,top_top(set(B))) != zero_zero(nat) ) ),
        aa(nat,nat,plus_plus(nat,finite_card(A,top_top(set(A)))),finite_card(B,top_top(set(B)))),
        zero_zero(nat) ) ).

% card_UNIV_sum
tff(fact_8122_bit0__cases,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [X: numeral_bit0(A)] :
          ~ ! [Z2: int] :
              ( ( X = aa(int,numeral_bit0(A),ring_1_of_int(numeral_bit0(A)),Z2) )
             => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z2)
               => ~ aa(int,$o,ord_less(int,Z2),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit0(A),top_top(set(numeral_bit0(A)))))) ) ) ) ).

% bit0_cases
tff(fact_8123_bit1__cases,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [X: numeral_bit1(A)] :
          ~ ! [Z2: int] :
              ( ( X = aa(int,numeral_bit1(A),ring_1_of_int(numeral_bit1(A)),Z2) )
             => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z2)
               => ~ aa(int,$o,ord_less(int,Z2),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A)))))) ) ) ) ).

% bit1_cases
tff(fact_8124_bit0__induct,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [P: fun(numeral_bit0(A),$o),X: numeral_bit0(A)] :
          ( ! [Z2: int] :
              ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z2)
             => ( aa(int,$o,ord_less(int,Z2),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit0(A),top_top(set(numeral_bit0(A))))))
               => aa(numeral_bit0(A),$o,P,aa(int,numeral_bit0(A),ring_1_of_int(numeral_bit0(A)),Z2)) ) )
         => aa(numeral_bit0(A),$o,P,X) ) ) ).

% bit0_induct
tff(fact_8125_bit1__induct,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [P: fun(numeral_bit1(A),$o),X: numeral_bit1(A)] :
          ( ! [Z2: int] :
              ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z2)
             => ( aa(int,$o,ord_less(int,Z2),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A))))))
               => aa(numeral_bit1(A),$o,P,aa(int,numeral_bit1(A),ring_1_of_int(numeral_bit1(A)),Z2)) ) )
         => aa(numeral_bit1(A),$o,P,X) ) ) ).

% bit1_induct
tff(fact_8126_one__less__card,axiom,
    ! [A: $tType] :
      ( card2(A)
     => aa(nat,$o,ord_less(nat,aa(nat,nat,suc,zero_zero(nat))),finite_card(A,top_top(set(A)))) ) ).

% one_less_card
tff(fact_8127_one__le__card__finite,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,zero_zero(nat))),finite_card(A,top_top(set(A)))) ) ).

% one_le_card_finite
tff(fact_8128_card__UNIV__option,axiom,
    ! [A: $tType] :
      finite_card(option(A),top_top(set(option(A)))) = $ite(finite_card(A,top_top(set(A))) = zero_zero(nat),zero_zero(nat),aa(nat,nat,plus_plus(nat,finite_card(A,top_top(set(A)))),one_one(nat))) ).

% card_UNIV_option
tff(fact_8129_one__less__int__card,axiom,
    ! [A: $tType] :
      ( card2(A)
     => aa(int,$o,ord_less(int,one_one(int)),aa(nat,int,semiring_1_of_nat(int),finite_card(A,top_top(set(A))))) ) ).

% one_less_int_card
tff(fact_8130_two__le__card,axiom,
    ! [A: $tType] :
      ( card2(A)
     => aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),finite_card(A,top_top(set(A)))) ) ).

% two_le_card
tff(fact_8131_card__fun,axiom,
    ! [A: $tType,B: $tType] :
      finite_card(fun(A,B),top_top(set(fun(A,B)))) = $ite(
        ( ( ( finite_card(A,top_top(set(A))) != zero_zero(nat) )
          & ( finite_card(B,top_top(set(B))) != zero_zero(nat) ) )
        | ( finite_card(B,top_top(set(B))) = one_one(nat) ) ),
        aa(nat,nat,power_power(nat,finite_card(B,top_top(set(B)))),finite_card(A,top_top(set(A)))),
        zero_zero(nat) ) ).

% card_fun
tff(fact_8132_finite__UNIV__fun,axiom,
    ! [A: $tType,B: $tType] :
      ( finite_finite2(fun(A,B),top_top(set(fun(A,B))))
    <=> ( ( finite_finite2(A,top_top(set(A)))
          & finite_finite2(B,top_top(set(B))) )
        | ( finite_card(B,top_top(set(B))) = one_one(nat) ) ) ) ).

% finite_UNIV_fun
tff(fact_8133_card__UNIV__set,axiom,
    ! [A: $tType] :
      finite_card(set(A),top_top(set(set(A)))) = $ite(finite_card(A,top_top(set(A))) = zero_zero(nat),zero_zero(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),finite_card(A,top_top(set(A))))) ).

% card_UNIV_set
tff(fact_8134_inj__on__Abs__bit1,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => inj_on(int,numeral_bit1(A),numeral_Abs_bit12(A),set_or7035219750837199246ssThan(int,zero_zero(int),aa(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(nat,int,semiring_1_of_nat(int),finite_card(A,top_top(set(A)))))))) ) ).

% inj_on_Abs_bit1
tff(fact_8135_Abs__bit1__inject,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [X: int,Y: int] :
          ( member(int,X,set_or7035219750837199246ssThan(int,zero_zero(int),aa(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(nat,int,semiring_1_of_nat(int),finite_card(A,top_top(set(A))))))))
         => ( member(int,Y,set_or7035219750837199246ssThan(int,zero_zero(int),aa(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(nat,int,semiring_1_of_nat(int),finite_card(A,top_top(set(A))))))))
           => ( ( aa(int,numeral_bit1(A),numeral_Abs_bit12(A),X) = aa(int,numeral_bit1(A),numeral_Abs_bit12(A),Y) )
            <=> ( X = Y ) ) ) ) ) ).

% Abs_bit1_inject
tff(fact_8136_one__bit1__def,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( one_one(numeral_bit1(A)) = aa(int,numeral_bit1(A),numeral_Abs_bit12(A),one_one(int)) ) ) ).

% one_bit1_def
tff(fact_8137_zero__bit1__def,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( zero_zero(numeral_bit1(A)) = aa(int,numeral_bit1(A),numeral_Abs_bit12(A),zero_zero(int)) ) ) ).

% zero_bit1_def
tff(fact_8138_bit1_Oof__nat__eq,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [K: nat] : aa(nat,numeral_bit1(A),semiring_1_of_nat(numeral_bit1(A)),K) = aa(int,numeral_bit1(A),numeral_Abs_bit12(A),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(nat,int,semiring_1_of_nat(int),K)),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A))))))) ) ).

% bit1.of_nat_eq
tff(fact_8139_bit1_Oof__int__eq,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Z: int] : aa(int,numeral_bit1(A),ring_1_of_int(numeral_bit1(A)),Z) = aa(int,numeral_bit1(A),numeral_Abs_bit12(A),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Z),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A))))))) ) ).

% bit1.of_int_eq
tff(fact_8140_bit1_OUNIV__eq,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( top_top(set(numeral_bit1(A))) = aa(set(int),set(numeral_bit1(A)),image(int,numeral_bit1(A),numeral_Abs_bit12(A)),set_or7035219750837199246ssThan(int,zero_zero(int),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A))))))) ) ) ).

% bit1.UNIV_eq
tff(fact_8141_Abs__bit1__cases,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [X: numeral_bit1(A)] :
          ~ ! [Y3: int] :
              ( ( X = aa(int,numeral_bit1(A),numeral_Abs_bit12(A),Y3) )
             => ~ member(int,Y3,set_or7035219750837199246ssThan(int,zero_zero(int),aa(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(nat,int,semiring_1_of_nat(int),finite_card(A,top_top(set(A)))))))) ) ) ).

% Abs_bit1_cases
tff(fact_8142_Abs__bit1__induct,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [P: fun(numeral_bit1(A),$o),X: numeral_bit1(A)] :
          ( ! [Y3: int] :
              ( member(int,Y3,set_or7035219750837199246ssThan(int,zero_zero(int),aa(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(nat,int,semiring_1_of_nat(int),finite_card(A,top_top(set(A))))))))
             => aa(numeral_bit1(A),$o,P,aa(int,numeral_bit1(A),numeral_Abs_bit12(A),Y3)) )
         => aa(numeral_bit1(A),$o,P,X) ) ) ).

% Abs_bit1_induct
tff(fact_8143_Abs__bit1__inverse,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Y: int] :
          ( member(int,Y,set_or7035219750837199246ssThan(int,zero_zero(int),aa(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(nat,int,semiring_1_of_nat(int),finite_card(A,top_top(set(A))))))))
         => ( aa(numeral_bit1(A),int,numeral_Rep_bit1(A),aa(int,numeral_bit1(A),numeral_Abs_bit12(A),Y)) = Y ) ) ) ).

% Abs_bit1_inverse
tff(fact_8144_Abs__bit1_H__def,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [X: int] : numeral_Abs_bit1(A,X) = aa(int,numeral_bit1(A),numeral_Abs_bit12(A),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),X),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A))))))) ) ).

% Abs_bit1'_def
tff(fact_8145_Abs__bit1_H__code,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [X: int] : aa(numeral_bit1(A),int,numeral_Rep_bit1(A),numeral_Abs_bit1(A,X)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),X),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A)))))) ) ).

% Abs_bit1'_code
tff(fact_8146_bit1_ORep__Abs__1,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( aa(numeral_bit1(A),int,numeral_Rep_bit1(A),aa(int,numeral_bit1(A),numeral_Abs_bit12(A),one_one(int))) = one_one(int) ) ) ).

% bit1.Rep_Abs_1
tff(fact_8147_bit1_ORep__Abs__0,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( aa(numeral_bit1(A),int,numeral_Rep_bit1(A),aa(int,numeral_bit1(A),numeral_Abs_bit12(A),zero_zero(int))) = zero_zero(int) ) ) ).

% bit1.Rep_Abs_0
tff(fact_8148_less__bit1__def,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [A3: numeral_bit1(A),B3: numeral_bit1(A)] :
          ( aa(numeral_bit1(A),$o,ord_less(numeral_bit1(A),A3),B3)
        <=> aa(int,$o,ord_less(int,aa(numeral_bit1(A),int,numeral_Rep_bit1(A),A3)),aa(numeral_bit1(A),int,numeral_Rep_bit1(A),B3)) ) ) ).

% less_bit1_def
tff(fact_8149_bit1_ORep__0,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( aa(numeral_bit1(A),int,numeral_Rep_bit1(A),zero_zero(numeral_bit1(A))) = zero_zero(int) ) ) ).

% bit1.Rep_0
tff(fact_8150_bit1_ORep__1,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( aa(numeral_bit1(A),int,numeral_Rep_bit1(A),one_one(numeral_bit1(A))) = one_one(int) ) ) ).

% bit1.Rep_1
tff(fact_8151_bit1_ORep__le__n,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [X: numeral_bit1(A)] : aa(int,$o,ord_less_eq(int,aa(numeral_bit1(A),int,numeral_Rep_bit1(A),X)),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A)))))) ) ).

% bit1.Rep_le_n
tff(fact_8152_bit1_ORep__less__n,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [X: numeral_bit1(A)] : aa(int,$o,ord_less(int,aa(numeral_bit1(A),int,numeral_Rep_bit1(A),X)),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A)))))) ) ).

% bit1.Rep_less_n
tff(fact_8153_bit1_ORep__mod,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [X: numeral_bit1(A)] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(numeral_bit1(A),int,numeral_Rep_bit1(A),X)),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A)))))) = aa(numeral_bit1(A),int,numeral_Rep_bit1(A),X) ) ).

% bit1.Rep_mod
tff(fact_8154_bit1_ORep__numeral,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [W: num] : aa(numeral_bit1(A),int,numeral_Rep_bit1(A),aa(num,numeral_bit1(A),numeral_numeral(numeral_bit1(A)),W)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(num,int,numeral_numeral(int),W)),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A)))))) ) ).

% bit1.Rep_numeral
tff(fact_8155_bit1_ORep__Abs__mod,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [M: int] : aa(numeral_bit1(A),int,numeral_Rep_bit1(A),aa(int,numeral_bit1(A),numeral_Abs_bit12(A),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),M),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A)))))))) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),M),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A)))))) ) ).

% bit1.Rep_Abs_mod
tff(fact_8156_bit1_Ominus__def,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [X: numeral_bit1(A)] : aa(numeral_bit1(A),numeral_bit1(A),uminus_uminus(numeral_bit1(A)),X) = aa(int,numeral_bit1(A),numeral_Abs_bit12(A),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),aa(numeral_bit1(A),int,numeral_Rep_bit1(A),X))),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A))))))) ) ).

% bit1.minus_def
tff(fact_8157_bit1_Oadd__def,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [X: numeral_bit1(A),Y: numeral_bit1(A)] : aa(numeral_bit1(A),numeral_bit1(A),plus_plus(numeral_bit1(A),X),Y) = aa(int,numeral_bit1(A),numeral_Abs_bit12(A),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,plus_plus(int,aa(numeral_bit1(A),int,numeral_Rep_bit1(A),X)),aa(numeral_bit1(A),int,numeral_Rep_bit1(A),Y))),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A))))))) ) ).

% bit1.add_def
tff(fact_8158_bit1_Omult__def,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [X: numeral_bit1(A),Y: numeral_bit1(A)] : aa(numeral_bit1(A),numeral_bit1(A),aa(numeral_bit1(A),fun(numeral_bit1(A),numeral_bit1(A)),times_times(numeral_bit1(A)),X),Y) = aa(int,numeral_bit1(A),numeral_Abs_bit12(A),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(numeral_bit1(A),int,numeral_Rep_bit1(A),X)),aa(numeral_bit1(A),int,numeral_Rep_bit1(A),Y))),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A))))))) ) ).

% bit1.mult_def
tff(fact_8159_bit1_Odiff__def,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [X: numeral_bit1(A),Y: numeral_bit1(A)] : aa(numeral_bit1(A),numeral_bit1(A),minus_minus(numeral_bit1(A),X),Y) = aa(int,numeral_bit1(A),numeral_Abs_bit12(A),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,minus_minus(int,aa(numeral_bit1(A),int,numeral_Rep_bit1(A),X)),aa(numeral_bit1(A),int,numeral_Rep_bit1(A),Y))),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A))))))) ) ).

% bit1.diff_def
tff(fact_8160_bit1_OAbs__inverse,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [M: int] :
          ( member(int,M,set_or7035219750837199246ssThan(int,zero_zero(int),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A)))))))
         => ( aa(numeral_bit1(A),int,numeral_Rep_bit1(A),aa(int,numeral_bit1(A),numeral_Abs_bit12(A),M)) = M ) ) ) ).

% bit1.Abs_inverse
tff(fact_8161_Rep__bit1__induct,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Y: int,P: fun(int,$o)] :
          ( member(int,Y,set_or7035219750837199246ssThan(int,zero_zero(int),aa(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(nat,int,semiring_1_of_nat(int),finite_card(A,top_top(set(A))))))))
         => ( ! [X4: numeral_bit1(A)] : aa(int,$o,P,aa(numeral_bit1(A),int,numeral_Rep_bit1(A),X4))
           => aa(int,$o,P,Y) ) ) ) ).

% Rep_bit1_induct
tff(fact_8162_Rep__bit1__cases,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Y: int] :
          ( member(int,Y,set_or7035219750837199246ssThan(int,zero_zero(int),aa(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(nat,int,semiring_1_of_nat(int),finite_card(A,top_top(set(A))))))))
         => ~ ! [X4: numeral_bit1(A)] : Y != aa(numeral_bit1(A),int,numeral_Rep_bit1(A),X4) ) ) ).

% Rep_bit1_cases
tff(fact_8163_Rep__bit1,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [X: numeral_bit1(A)] : member(int,aa(numeral_bit1(A),int,numeral_Rep_bit1(A),X),set_or7035219750837199246ssThan(int,zero_zero(int),aa(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(nat,int,semiring_1_of_nat(int),finite_card(A,top_top(set(A)))))))) ) ).

% Rep_bit1
tff(fact_8164_type__definition__bit1,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => type_definition(numeral_bit1(A),int,numeral_Rep_bit1(A),numeral_Abs_bit12(A),set_or7035219750837199246ssThan(int,zero_zero(int),aa(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(nat,int,semiring_1_of_nat(int),finite_card(A,top_top(set(A)))))))) ) ).

% type_definition_bit1
tff(fact_8165_card__lists__distinct__length__eq,axiom,
    ! [A: $tType,A2: set(A),K: nat] :
      ( finite_finite2(A,A2)
     => ( aa(nat,$o,ord_less_eq(nat,K),finite_card(A,A2))
       => ( finite_card(list(A),collect(list(A),aa(nat,fun(list(A),$o),aTP_Lamp_alh(set(A),fun(nat,fun(list(A),$o)),A2),K))) = groups7121269368397514597t_prod(nat,nat,aTP_Lamp_ik(nat,nat),set_or1337092689740270186AtMost(nat,aa(nat,nat,plus_plus(nat,aa(nat,nat,minus_minus(nat,finite_card(A,A2)),K)),one_one(nat)),finite_card(A,A2))) ) ) ) ).

% card_lists_distinct_length_eq
tff(fact_8166_distinct__union,axiom,
    ! [A: $tType,Xsa: list(A),Ys: list(A)] :
      ( distinct(A,union(A,Xsa,Ys))
    <=> distinct(A,Ys) ) ).

% distinct_union
tff(fact_8167_finite__lists__distinct__length__eq,axiom,
    ! [A: $tType,A2: set(A),Na: nat] :
      ( finite_finite2(A,A2)
     => finite_finite2(list(A),collect(list(A),aa(nat,fun(list(A),$o),aTP_Lamp_alh(set(A),fun(nat,fun(list(A),$o)),A2),Na))) ) ).

% finite_lists_distinct_length_eq
tff(fact_8168_finite__set__image,axiom,
    ! [A: $tType,A2: set(list(A))] :
      ( finite_finite2(set(A),aa(set(list(A)),set(set(A)),image(list(A),set(A),set2(A)),A2))
     => ( ! [Xs: list(A)] :
            ( member(list(A),Xs,A2)
           => distinct(A,Xs) )
       => finite_finite2(list(A),A2) ) ) ).

% finite_set_image
tff(fact_8169_distinct__length__le,axiom,
    ! [A: $tType,Ys: list(A),Xsa: list(A)] :
      ( distinct(A,Ys)
     => ( ( aa(list(A),set(A),set2(A),Ys) = aa(list(A),set(A),set2(A),Xsa) )
       => aa(nat,$o,ord_less_eq(nat,size_size(list(A),Ys)),size_size(list(A),Xsa)) ) ) ).

% distinct_length_le
tff(fact_8170_finite__distinct__list,axiom,
    ! [A: $tType,A2: set(A)] :
      ( finite_finite2(A,A2)
     => ? [Xs: list(A)] :
          ( ( aa(list(A),set(A),set2(A),Xs) = A2 )
          & distinct(A,Xs) ) ) ).

% finite_distinct_list
tff(fact_8171_distinct__finite__set,axiom,
    ! [A: $tType,X: set(A)] : finite_finite2(list(A),collect(list(A),aTP_Lamp_ali(set(A),fun(list(A),$o),X))) ).

% distinct_finite_set
tff(fact_8172_distinct__length__2__or__more,axiom,
    ! [A: $tType,A3: A,B3: A,Xsa: list(A)] :
      ( distinct(A,aa(list(A),list(A),cons(A,A3),aa(list(A),list(A),cons(A,B3),Xsa)))
    <=> ( ( A3 != B3 )
        & distinct(A,aa(list(A),list(A),cons(A,A3),Xsa))
        & distinct(A,aa(list(A),list(A),cons(A,B3),Xsa)) ) ) ).

% distinct_length_2_or_more
tff(fact_8173_distinct_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Xsa: list(A)] :
      ( distinct(A,aa(list(A),list(A),cons(A,X),Xsa))
    <=> ( ~ member(A,X,aa(list(A),set(A),set2(A),Xsa))
        & distinct(A,Xsa) ) ) ).

% distinct.simps(2)
tff(fact_8174_distinct__product,axiom,
    ! [A: $tType,B: $tType,Xsa: list(A),Ys: list(B)] :
      ( distinct(A,Xsa)
     => ( distinct(B,Ys)
       => distinct(product_prod(A,B),product(A,B,Xsa,Ys)) ) ) ).

% distinct_product
tff(fact_8175_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xsa: list(A)] :
          ( distinct(A,Xsa)
         => distinct(A,Xsa) ) ) ).

% sorted_list_of_set.distinct_if_distinct_map
tff(fact_8176_nth__eq__iff__index__eq,axiom,
    ! [A: $tType,Xsa: list(A),I: nat,J: nat] :
      ( distinct(A,Xsa)
     => ( aa(nat,$o,ord_less(nat,I),size_size(list(A),Xsa))
       => ( aa(nat,$o,ord_less(nat,J),size_size(list(A),Xsa))
         => ( ( aa(nat,A,nth(A,Xsa),I) = aa(nat,A,nth(A,Xsa),J) )
          <=> ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
tff(fact_8177_distinct__conv__nth,axiom,
    ! [A: $tType,Xsa: list(A)] :
      ( distinct(A,Xsa)
    <=> ! [I4: nat] :
          ( aa(nat,$o,ord_less(nat,I4),size_size(list(A),Xsa))
         => ! [J3: nat] :
              ( aa(nat,$o,ord_less(nat,J3),size_size(list(A),Xsa))
             => ( ( I4 != J3 )
               => ( aa(nat,A,nth(A,Xsa),I4) != aa(nat,A,nth(A,Xsa),J3) ) ) ) ) ) ).

% distinct_conv_nth
tff(fact_8178_ATP_Olambda__1,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_bv(nat,real),Uu) = 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,power_power(real,aa(real,real,uminus_uminus(real),one_one(real))),Uu)),one_one(real))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uu),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat)))) ).

% ATP.lambda_1
tff(fact_8179_ATP_Olambda__2,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A] : aa(A,A,aTP_Lamp_yx(A,A),Uu) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,exp(A),Uu)),one_one(A))),Uu) ) ).

% ATP.lambda_2
tff(fact_8180_ATP_Olambda__3,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_cm(nat,real),Uu) = aa(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,Uu)) ).

% ATP.lambda_3
tff(fact_8181_ATP_Olambda__4,axiom,
    ! [Uu: real] :
      ( aa(real,$o,aTP_Lamp_de(real,$o),Uu)
    <=> ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Uu)
        & aa(real,$o,ord_less_eq(real,Uu),aa(num,real,numeral_numeral(real),bit0(one2)))
        & ( cos(real,Uu) = zero_zero(real) ) ) ) ).

% ATP.lambda_4
tff(fact_8182_ATP_Olambda__5,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_aas(nat,A),Uu) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Uu))),aa(nat,A,semiring_1_of_nat(A),Uu)) ) ).

% ATP.lambda_5
tff(fact_8183_ATP_Olambda__6,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_hd(nat,real),Uu) = aa(real,real,aa(real,fun(real,real),times_times(real),cos_coeff(Uu)),aa(nat,real,power_power(real,zero_zero(real)),Uu)) ).

% ATP.lambda_6
tff(fact_8184_ATP_Olambda__7,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_aat(nat,A),Uu) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),Uu)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Uu))) ) ).

% ATP.lambda_7
tff(fact_8185_ATP_Olambda__8,axiom,
    ! [Uu: real] : aa(real,real,aTP_Lamp_zb(real,real),Uu) = aa(real,real,aa(real,fun(real,real),divide_divide(real),cos(real,Uu)),sin(real,Uu)) ).

% ATP.lambda_8
tff(fact_8186_ATP_Olambda__9,axiom,
    ! [Uu: real] : aa(real,real,aTP_Lamp_acc(real,real),Uu) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),Uu)),Uu) ).

% ATP.lambda_9
tff(fact_8187_ATP_Olambda__10,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Uu: word(B)] :
          ( aa(word(B),$o,aTP_Lamp_pl(word(B),$o),Uu)
        <=> aa(word(B),$o,ord_less(word(B),Uu),aa(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_10
tff(fact_8188_ATP_Olambda__11,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_aaa(nat,real),Uu) = aa(real,real,root(Uu),aa(nat,real,semiring_1_of_nat(real),Uu)) ).

% ATP.lambda_11
tff(fact_8189_ATP_Olambda__12,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_ake(nat,nat),Uu) = aa(nat,nat,minus_minus(nat,Uu),aa(nat,nat,suc,zero_zero(nat))) ).

% ATP.lambda_12
tff(fact_8190_ATP_Olambda__13,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_ak(A,A),Uu) = aa(A,A,plus_plus(A,Uu),one_one(A)) ) ).

% ATP.lambda_13
tff(fact_8191_ATP_Olambda__14,axiom,
    ! [A: $tType,Uu: A] : aa(A,set(A),aTP_Lamp_qc(A,set(A)),Uu) = aa(set(A),set(A),insert(A,Uu),bot_bot(set(A))) ).

% ATP.lambda_14
tff(fact_8192_ATP_Olambda__15,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_aak(nat,real),Uu) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),Uu)) ).

% ATP.lambda_15
tff(fact_8193_ATP_Olambda__16,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_aar(nat,A),Uu) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),Uu)) ) ).

% ATP.lambda_16
tff(fact_8194_ATP_Olambda__17,axiom,
    ! [Uu: int] :
      ( aa(int,$o,aTP_Lamp_dk(int,$o),Uu)
    <=> ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Uu) ) ).

% ATP.lambda_17
tff(fact_8195_ATP_Olambda__18,axiom,
    ! [Uu: real] : aa(real,real,aTP_Lamp_rz(real,real),Uu) = suminf(real,aTP_Lamp_aw(real,fun(nat,real),Uu)) ).

% ATP.lambda_18
tff(fact_8196_ATP_Olambda__19,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B)] : aa(fun(A,B),set(product_prod(A,B)),aTP_Lamp_qa(fun(A,B),set(product_prod(A,B))),Uu) = collect(product_prod(A,B),product_case_prod(A,B,$o,aTP_Lamp_pz(fun(A,B),fun(A,fun(B,$o)),Uu))) ).

% ATP.lambda_19
tff(fact_8197_ATP_Olambda__20,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_aal(nat,real),Uu) = aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uu))) ).

% ATP.lambda_20
tff(fact_8198_ATP_Olambda__21,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_dl(nat,A),Uu) = aa(A,A,inverse_inverse(A),semiring_char_0_fact(A,Uu)) ) ).

% ATP.lambda_21
tff(fact_8199_ATP_Olambda__22,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_aad(nat,A),Uu) = aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Uu)) ) ).

% ATP.lambda_22
tff(fact_8200_ATP_Olambda__23,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A)
        & ring_1(B)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: A] : aa(A,B,aTP_Lamp_ajj(A,B),Uu) = aa(int,B,ring_1_of_int(B),archim6421214686448440834_floor(A,Uu)) ) ).

% ATP.lambda_23
tff(fact_8201_ATP_Olambda__24,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A)
        & ring_1(B)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: A] : aa(A,B,aTP_Lamp_ajk(A,B),Uu) = aa(int,B,ring_1_of_int(B),archimedean_ceiling(A,Uu)) ) ).

% ATP.lambda_24
tff(fact_8202_ATP_Olambda__25,axiom,
    ! [Uu: int] : aa(int,fun(int,product_prod(int,int)),aTP_Lamp_la(int,fun(int,product_prod(int,int))),Uu) = aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),Uu)) ).

% ATP.lambda_25
tff(fact_8203_ATP_Olambda__26,axiom,
    ! [Uu: int] : aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ld(int,fun(int,product_prod(int,int))),Uu) = aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,abs_abs(int),Uu)) ).

% ATP.lambda_26
tff(fact_8204_ATP_Olambda__27,axiom,
    ! [Uu: nat] : aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_ki(nat,fun(nat,product_prod(nat,nat))),Uu) = aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,suc,Uu)) ).

% ATP.lambda_27
tff(fact_8205_ATP_Olambda__28,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] :
          aa(nat,A,aTP_Lamp_ea(A,fun(nat,A),Uu),Uua) = $ite(aa(nat,$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,power_power(A,Uu),Uua)),zero_zero(A)) ) ).

% ATP.lambda_28
tff(fact_8206_ATP_Olambda__29,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] :
          aa(nat,A,aTP_Lamp_ga(nat,fun(nat,A),Uu),Uua) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua)),zero_zero(A)) ) ).

% ATP.lambda_29
tff(fact_8207_ATP_Olambda__30,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] :
          aa(nat,A,aTP_Lamp_dz(A,fun(nat,A),Uu),Uua) = $ite(aa(nat,$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,power_power(A,Uu),Uua))) ) ).

% ATP.lambda_30
tff(fact_8208_ATP_Olambda__31,axiom,
    ! [Uu: fun(nat,real),Uua: nat] :
      aa(nat,real,aTP_Lamp_cn(fun(nat,real),fun(nat,real),Uu),Uua) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua),zero_zero(real),aa(nat,real,Uu,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,Uua),one_one(nat))),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% ATP.lambda_31
tff(fact_8209_ATP_Olambda__32,axiom,
    ! [Uu: int,Uua: int] :
      aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_kz(int,fun(int,product_prod(int,int))),Uu),Uua) = $ite(Uu = zero_zero(int),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),Uu)),Uua)),aa(int,int,abs_abs(int),Uu))) ).

% ATP.lambda_32
tff(fact_8210_ATP_Olambda__33,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] :
          aa(nat,A,aTP_Lamp_gb(nat,fun(nat,A),Uu),Uua) = $ite(~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua)),zero_zero(A)) ) ).

% ATP.lambda_33
tff(fact_8211_ATP_Olambda__34,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(nat,A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_acp(fun(nat,A),fun(A,$o),Uu),Uua)
        <=> filterlim(nat,A,Uu,topolo7230453075368039082e_nhds(A,Uua),at_top(nat)) ) ) ).

% ATP.lambda_34
tff(fact_8212_ATP_Olambda__35,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_kp(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ko(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_35
tff(fact_8213_ATP_Olambda__36,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_kn(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_km(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_36
tff(fact_8214_ATP_Olambda__37,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_bu(real,fun(nat,real),Uu),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,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,plus_plus(nat,Uua),one_one(nat)))))),aa(nat,real,power_power(real,aa(real,real,minus_minus(real,Uu),one_one(real))),aa(nat,nat,suc,Uua))) ).

% ATP.lambda_37
tff(fact_8215_ATP_Olambda__38,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_ce(real,fun(nat,real),Uu),Uua) = 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,power_power(real,aa(real,real,uminus_uminus(real),one_one(real))),Uua)),semiring_char_0_fact(real,aa(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,real,power_power(real,Uu),aa(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_38
tff(fact_8216_ATP_Olambda__39,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_cl(real,fun(nat,real),Uu),Uua) = 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,power_power(real,aa(real,real,uminus_uminus(real),one_one(real))),Uua)),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)))),aa(nat,real,power_power(real,Uu),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))) ).

% ATP.lambda_39
tff(fact_8217_ATP_Olambda__40,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_fu(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),Uua)),aa(nat,A,semiring_1_of_nat(A),Uua))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua))) ) ).

% ATP.lambda_40
tff(fact_8218_ATP_Olambda__41,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_fv(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,plus_plus(nat,Uu),Uua))),Uua)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Uua)) ) ).

% ATP.lambda_41
tff(fact_8219_ATP_Olambda__42,axiom,
    ! [Uu: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_df(real,fun(real,$o),Uu),Uua)
    <=> ( aa(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,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) = Uu ) ) ) ).

% ATP.lambda_42
tff(fact_8220_ATP_Olambda__43,axiom,
    ! [Uu: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_cz(real,fun(real,$o),Uu),Uua)
    <=> ( aa(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,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) = Uu ) ) ) ).

% ATP.lambda_43
tff(fact_8221_ATP_Olambda__44,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa(code_integer,int,aa(code_integer,fun(code_integer,int),aTP_Lamp_ajl(code_integer,fun(code_integer,int)),Uu),Uua) = $let(
        l2: int,
        l2:= aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),code_int_of_integer(Uu)),
        $ite(Uua = zero_zero(code_integer),l2,aa(int,int,plus_plus(int,l2),one_one(int))) ) ).

% ATP.lambda_44
tff(fact_8222_ATP_Olambda__45,axiom,
    ! [Uu: nat,Uua: nat] :
      aa(nat,a,aa(nat,fun(nat,a),aTP_Lamp_ku(nat,fun(nat,a)),Uu),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),Uu)),
        $ite(Uua = zero_zero(nat),m2,aa(a,a,plus_plus(a,m2),one_one(a))) ) ).

% ATP.lambda_45
tff(fact_8223_ATP_Olambda__46,axiom,
    ! [Uu: complex,Uua: real] :
      ( aa(real,$o,aTP_Lamp_jx(complex,fun(real,$o),Uu),Uua)
    <=> ( ( aa(complex,complex,sgn_sgn(complex),Uu) = cis(Uua) )
        & aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),pi)),Uua)
        & aa(real,$o,ord_less_eq(real,Uua),pi) ) ) ).

% ATP.lambda_46
tff(fact_8224_ATP_Olambda__47,axiom,
    ! [Uu: real,Uua: int] :
      ( aa(int,$o,aTP_Lamp_dh(real,fun(int,$o),Uu),Uua)
    <=> ( aa(real,$o,ord_less_eq(real,aa(int,real,ring_1_of_int(real),Uua)),Uu)
        & aa(real,$o,ord_less(real,Uu),aa(int,real,ring_1_of_int(real),aa(int,int,plus_plus(int,Uua),one_one(int)))) ) ) ).

% ATP.lambda_47
tff(fact_8225_ATP_Olambda__48,axiom,
    ! [Uu: rat,Uua: int] :
      ( aa(int,$o,aTP_Lamp_di(rat,fun(int,$o),Uu),Uua)
    <=> ( aa(rat,$o,ord_less_eq(rat,aa(int,rat,ring_1_of_int(rat),Uua)),Uu)
        & aa(rat,$o,ord_less(rat,Uu),aa(int,rat,ring_1_of_int(rat),aa(int,int,plus_plus(int,Uua),one_one(int)))) ) ) ).

% ATP.lambda_48
tff(fact_8226_ATP_Olambda__49,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_ali(set(A),fun(list(A),$o),Uu),Uua)
    <=> ( ( aa(list(A),set(A),set2(A),Uua) = Uu )
        & distinct(A,Uua) ) ) ).

% ATP.lambda_49
tff(fact_8227_ATP_Olambda__50,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_aw(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(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,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,power_power(real,Uu),aa(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_50
tff(fact_8228_ATP_Olambda__51,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_sa(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,power_power(real,aa(real,real,uminus_uminus(real),one_one(real))),Uua)),aa(nat,real,power_power(real,Uu),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% ATP.lambda_51
tff(fact_8229_ATP_Olambda__52,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_zh(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,power_power(real,aa(real,real,uminus_uminus(real),one_one(real))),Uua)),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_52
tff(fact_8230_ATP_Olambda__53,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_fy(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),Uua)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua))) ) ).

% ATP.lambda_53
tff(fact_8231_ATP_Olambda__54,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_eu(A,fun(nat,A),Uu),Uua) = aa(nat,A,gbinomial(A,aa(A,A,plus_plus(A,Uu),aa(nat,A,semiring_1_of_nat(A),Uua))),Uua) ) ).

% ATP.lambda_54
tff(fact_8232_ATP_Olambda__55,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_fz(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,Uu),Uua)),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Uu),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_55
tff(fact_8233_ATP_Olambda__56,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_fg(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,Uu),Uua)),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),Uua)) ) ).

% ATP.lambda_56
tff(fact_8234_ATP_Olambda__57,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_fl(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,power_power(nat,aa(nat,nat,binomial(Uu),Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% ATP.lambda_57
tff(fact_8235_ATP_Olambda__58,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_ht(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,power_power(real,Uu),Uua)),semiring_char_0_fact(real,Uua)) ).

% ATP.lambda_58
tff(fact_8236_ATP_Olambda__59,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_ace(nat,fun(real,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,power_power(real,Uua),Uu)),aa(real,real,exp(real),Uua)) ).

% ATP.lambda_59
tff(fact_8237_ATP_Olambda__60,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_eq(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,binomial(aa(nat,nat,plus_plus(nat,Uu),Uua)),Uu) ).

% ATP.lambda_60
tff(fact_8238_ATP_Olambda__61,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_er(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,binomial(aa(nat,nat,plus_plus(nat,Uu),Uua)),Uua) ).

% ATP.lambda_61
tff(fact_8239_ATP_Olambda__62,axiom,
    ! [Uu: nat,Uua: complex] :
      ( aa(complex,$o,aTP_Lamp_gr(nat,fun(complex,$o),Uu),Uua)
    <=> ( aa(nat,complex,power_power(complex,Uua),Uu) = one_one(complex) ) ) ).

% ATP.lambda_62
tff(fact_8240_ATP_Olambda__63,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu: nat,Uua: A] :
          ( aa(A,$o,aTP_Lamp_nb(nat,fun(A,$o),Uu),Uua)
        <=> ( aa(nat,A,power_power(A,Uua),Uu) = one_one(A) ) ) ) ).

% ATP.lambda_63
tff(fact_8241_ATP_Olambda__64,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_mz(A,fun(A,$o),Uu),Uua)
        <=> ( member(A,Uua,ring_1_Ints(A))
            & aa(A,$o,ord_less_eq(A,aa(A,A,abs_abs(A),Uua)),Uu) ) ) ) ).

% ATP.lambda_64
tff(fact_8242_ATP_Olambda__65,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_abb(real,fun(nat,real),Uu),Uua) = aa(nat,real,power_power(real,aa(real,real,plus_plus(real,one_one(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),Uu),aa(nat,real,semiring_1_of_nat(real),Uua)))),Uua) ).

% ATP.lambda_65
tff(fact_8243_ATP_Olambda__66,axiom,
    ! [Uu: real,Uua: real] : aa(real,real,aTP_Lamp_ada(real,fun(real,real),Uu),Uua) = powr(real,aa(real,real,plus_plus(real,one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),Uu),Uua)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),Uua)) ).

% ATP.lambda_66
tff(fact_8244_ATP_Olambda__67,axiom,
    ! [Uu: real,Uua: real] : aa(real,real,aTP_Lamp_acf(real,fun(real,real),Uu),Uua) = powr(real,aa(real,real,plus_plus(real,one_one(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),Uu),Uua)),Uua) ).

% ATP.lambda_67
tff(fact_8245_ATP_Olambda__68,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_av(real,fun(nat,real),Uu),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,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,power_power(real,Uu),aa(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_68
tff(fact_8246_ATP_Olambda__69,axiom,
    ! [Uu: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_dd(real,fun(real,$o),Uu),Uua)
    <=> ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Uua)
        & aa(real,$o,ord_less_eq(real,Uua),pi)
        & ( cos(real,Uua) = Uu ) ) ) ).

% ATP.lambda_69
tff(fact_8247_ATP_Olambda__70,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_jr(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))),aa(nat,A,Uu,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)))) ) ).

% ATP.lambda_70
tff(fact_8248_ATP_Olambda__71,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fc(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,plus_plus(A,aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))),aa(nat,A,Uu,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)))) ) ).

% ATP.lambda_71
tff(fact_8249_ATP_Olambda__72,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_aag(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,aa(nat,A,Uu,aa(nat,nat,suc,Uua))),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_72
tff(fact_8250_ATP_Olambda__73,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_hh(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,aa(nat,A,Uu,aa(nat,nat,suc,Uua))),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_73
tff(fact_8251_ATP_Olambda__74,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_nw(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),Uu,Uua),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_74
tff(fact_8252_ATP_Olambda__75,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_nu(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),Uu,Uua)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_75
tff(fact_8253_ATP_Olambda__76,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bm(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)),aa(nat,A,power_power(A,zero_zero(A)),Uua)) ) ).

% ATP.lambda_76
tff(fact_8254_ATP_Olambda__77,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bs(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)),aa(nat,A,power_power(A,zero_zero(A)),Uua)) ) ).

% ATP.lambda_77
tff(fact_8255_ATP_Olambda__78,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_lw(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)),aa(nat,A,power_power(A,zero_zero(A)),Uua)) ) ).

% ATP.lambda_78
tff(fact_8256_ATP_Olambda__79,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bn(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)),aa(nat,A,power_power(A,zero_zero(A)),Uua)) ) ).

% ATP.lambda_79
tff(fact_8257_ATP_Olambda__80,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ii(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,minus_minus(nat,Uua),one_one(nat)))) ) ).

% ATP.lambda_80
tff(fact_8258_ATP_Olambda__81,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ih(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,plus_plus(nat,Uua),one_one(nat)))) ) ).

% ATP.lambda_81
tff(fact_8259_ATP_Olambda__82,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_aah(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_82
tff(fact_8260_ATP_Olambda__83,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_eo(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_83
tff(fact_8261_ATP_Olambda__84,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_ahj(fun(A,$o),fun(A,$o),Uu),Uua)
        <=> ( aa(A,$o,Uu,Uua)
            & ! [Y6: A] :
                ( aa(A,$o,Uu,Y6)
               => aa(A,$o,ord_less_eq(A,Y6),Uua) ) ) ) ) ).

% ATP.lambda_84
tff(fact_8262_ATP_Olambda__85,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_jk(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),Uu,Uua),set_ord_lessThan(nat,Uua)) ) ).

% ATP.lambda_85
tff(fact_8263_ATP_Olambda__86,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_id(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),Uu,Uua)),set_ord_lessThan(nat,Uua)) ) ).

% ATP.lambda_86
tff(fact_8264_ATP_Olambda__87,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,product_prod(B,A),aTP_Lamp_qf(fun(A,B),fun(A,product_prod(B,A)),Uu),Uua) = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),aa(A,B,Uu,Uua)),Uua) ).

% ATP.lambda_87
tff(fact_8265_ATP_Olambda__88,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_aga(fun(A,real),fun(A,$o),Uu),Uua)
    <=> aa(real,$o,ord_less(real,aa(A,real,Uu,Uua)),zero_zero(real)) ) ).

% ATP.lambda_88
tff(fact_8266_ATP_Olambda__89,axiom,
    ! [B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(B,real),Uua: B] : aa(B,real,aTP_Lamp_uk(fun(B,real),fun(B,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(B,real,Uu,Uua)),zero_zero(real)) ) ).

% ATP.lambda_89
tff(fact_8267_ATP_Olambda__90,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,complex,aTP_Lamp_gl(fun(A,real),fun(A,complex),Uu),Uua) = complex2(aa(A,real,Uu,Uua),zero_zero(real)) ).

% ATP.lambda_90
tff(fact_8268_ATP_Olambda__91,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(A,B),Uua: A] :
          ( aa(A,$o,aTP_Lamp_mw(fun(A,B),fun(A,$o),Uu),Uua)
        <=> ( aa(A,B,Uu,Uua) = zero_zero(B) ) ) ) ).

% ATP.lambda_91
tff(fact_8269_ATP_Olambda__92,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,B),Uua: A] :
          ( aa(A,$o,aTP_Lamp_my(fun(A,B),fun(A,$o),Uu),Uua)
        <=> ( aa(A,B,Uu,Uua) = one_one(B) ) ) ) ).

% ATP.lambda_92
tff(fact_8270_ATP_Olambda__93,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_abl(fun(nat,real),fun(nat,real),Uu),Uua) = aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_zh(fun(nat,real),fun(nat,real),Uu)),set_ord_lessThan(nat,aa(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_93
tff(fact_8271_ATP_Olambda__94,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_abk(fun(nat,real),fun(nat,real),Uu),Uua) = aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_zh(fun(nat,real),fun(nat,real),Uu)),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_94
tff(fact_8272_ATP_Olambda__95,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_acj(fun(A,B),fun(A,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uua))),real_V7770717601297561774m_norm(A,Uua)) ) ).

% ATP.lambda_95
tff(fact_8273_ATP_Olambda__96,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_dn(A,fun(nat,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,aa(nat,nat,plus_plus(nat,Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,A,power_power(A,Uu),aa(nat,nat,plus_plus(nat,Uua),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% ATP.lambda_96
tff(fact_8274_ATP_Olambda__97,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_dy(A,fun(nat,A),Uu),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,power_power(A,Uu),aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_97
tff(fact_8275_ATP_Olambda__98,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_dq(A,fun(nat,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uua))),aa(nat,A,power_power(A,Uu),Uua)) ) ).

% ATP.lambda_98
tff(fact_8276_ATP_Olambda__99,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_dm(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),semiring_char_0_fact(A,Uua))),aa(nat,A,power_power(A,Uu),Uua)) ) ).

% ATP.lambda_99
tff(fact_8277_ATP_Olambda__100,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_cp(A,fun(nat,A),Uu),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,power_power(A,Uu),Uua)) ) ).

% ATP.lambda_100
tff(fact_8278_ATP_Olambda__101,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_dx(A,fun(nat,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,cos_coeff(Uua)),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),Uu)),Uua)) ) ).

% ATP.lambda_101
tff(fact_8279_ATP_Olambda__102,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_po(nat,fun(real,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),Uua)),aa(nat,real,power_power(real,aa(real,real,abs_abs(real),Uua)),Uu)) ).

% ATP.lambda_102
tff(fact_8280_ATP_Olambda__103,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_dr(A,fun(nat,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,sin_coeff(Uua)),aa(nat,A,power_power(A,Uu),Uua)) ) ).

% ATP.lambda_103
tff(fact_8281_ATP_Olambda__104,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ds(A,fun(nat,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,cos_coeff(Uua)),aa(nat,A,power_power(A,Uu),Uua)) ) ).

% ATP.lambda_104
tff(fact_8282_ATP_Olambda__105,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_abg(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),Uua)),aa(nat,A,power_power(A,Uu),Uua)) ) ).

% ATP.lambda_105
tff(fact_8283_ATP_Olambda__106,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_abf(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uua)),aa(nat,A,power_power(A,Uu),Uua)) ) ).

% ATP.lambda_106
tff(fact_8284_ATP_Olambda__107,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_hc(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),sin_coeff(Uua)),aa(nat,real,power_power(real,Uu),Uua)) ).

% ATP.lambda_107
tff(fact_8285_ATP_Olambda__108,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_hb(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),cos_coeff(Uua)),aa(nat,real,power_power(real,Uu),Uua)) ).

% ATP.lambda_108
tff(fact_8286_ATP_Olambda__109,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa(code_integer,num,aa(code_integer,fun(code_integer,num),aTP_Lamp_ajt(code_integer,fun(code_integer,num)),Uu),Uua) = $let(
        l2: num,
        l2:= code_num_of_integer(Uu),
        $let(
          l3: num,
          l3:= aa(num,num,plus_plus(num,l2),l2),
          $ite(Uua = zero_zero(code_integer),l3,aa(num,num,plus_plus(num,l3),one2)) ) ) ).

% ATP.lambda_109
tff(fact_8287_ATP_Olambda__110,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa(code_integer,nat,aa(code_integer,fun(code_integer,nat),aTP_Lamp_ajz(code_integer,fun(code_integer,nat)),Uu),Uua) = $let(
        l2: nat,
        l2:= code_nat_of_integer(Uu),
        $let(
          l3: nat,
          l3:= aa(nat,nat,plus_plus(nat,l2),l2),
          $ite(Uua = zero_zero(code_integer),l3,aa(nat,nat,plus_plus(nat,l3),one_one(nat))) ) ) ).

% ATP.lambda_110
tff(fact_8288_ATP_Olambda__111,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_ij(nat,fun(nat,A),Uu),Uua) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Uua)),Uu) ) ).

% ATP.lambda_111
tff(fact_8289_ATP_Olambda__112,axiom,
    ! [Uu: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_da(real,fun(real,$o),Uu),Uua)
    <=> ( aa(real,real,exp(real),Uua) = Uu ) ) ).

% ATP.lambda_112
tff(fact_8290_ATP_Olambda__113,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Uu: A,Uua: A] : aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_kb(A,fun(A,product_prod(A,A))),Uu),Uua) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),aa(A,A,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_113
tff(fact_8291_ATP_Olambda__114,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Uu: A,Uua: A] : aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_kc(A,fun(A,product_prod(A,A))),Uu),Uua) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)) ) ).

% ATP.lambda_114
tff(fact_8292_ATP_Olambda__115,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_iq(A,fun(nat,A),Uu),Uua) = aa(A,A,plus_plus(A,Uu),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),Uua)),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% ATP.lambda_115
tff(fact_8293_ATP_Olambda__116,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_fx(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(nat,nat,binomial(Uu),Uua)) ).

% ATP.lambda_116
tff(fact_8294_ATP_Olambda__117,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_dc(set(A),fun(A,$o),Uu),Uua)
    <=> ( Uu = aa(set(A),set(A),insert(A,Uua),bot_bot(set(A))) ) ) ).

% ATP.lambda_117
tff(fact_8295_ATP_Olambda__118,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_abd(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),Uu),aa(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_118
tff(fact_8296_ATP_Olambda__119,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_aaw(fun(nat,A),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_or1337092689740270186AtMost(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_119
tff(fact_8297_ATP_Olambda__120,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,product_prod(A,B),aTP_Lamp_qe(fun(A,B),fun(A,product_prod(A,B)),Uu),Uua) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),aa(A,B,Uu,Uua)) ).

% ATP.lambda_120
tff(fact_8298_ATP_Olambda__121,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_oa(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Uua),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Uu),Uua))) ) ).

% ATP.lambda_121
tff(fact_8299_ATP_Olambda__122,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_aay(fun(nat,A),fun(nat,real),Uu),Uua) = aa(real,real,root(Uua),real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua))) ) ).

% ATP.lambda_122
tff(fact_8300_ATP_Olambda__123,axiom,
    ! [Uu: int,Uua: int] : aa(int,int,aa(int,fun(int,int),aTP_Lamp_ks(int,fun(int,int)),Uu),Uua) = aa(int,int,plus_plus(int,Uu),aa($o,int,zero_neq_one_of_bool(int),Uua != zero_zero(int))) ).

% ATP.lambda_123
tff(fact_8301_ATP_Olambda__124,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_aax(real,fun(nat,real),Uu),Uua) = aa(real,real,plus_plus(real,Uu),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_124
tff(fact_8302_ATP_Olambda__125,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_aan(real,fun(nat,real),Uu),Uua) = aa(real,real,plus_plus(real,Uu),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uua)))) ).

% ATP.lambda_125
tff(fact_8303_ATP_Olambda__126,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_aao(fun(nat,A),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_ord_lessThan(nat,Uua)) ) ).

% ATP.lambda_126
tff(fact_8304_ATP_Olambda__127,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_aaz(fun(nat,A),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_ord_lessThan(nat,Uua)) ) ).

% ATP.lambda_127
tff(fact_8305_ATP_Olambda__128,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_aap(fun(nat,A),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_128
tff(fact_8306_ATP_Olambda__129,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_aba(fun(nat,A),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_129
tff(fact_8307_ATP_Olambda__130,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_aab(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_130
tff(fact_8308_ATP_Olambda__131,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ju(A,fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_131
tff(fact_8309_ATP_Olambda__132,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_od(A,fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_132
tff(fact_8310_ATP_Olambda__133,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_jp(A,fun(nat,A),Uu),Uua) = aa(A,A,plus_plus(A,Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_133
tff(fact_8311_ATP_Olambda__134,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_aea(fun(A,$o),fun(A,$o),Uu),Uua)
        <=> eventually(A,Uu,topolo7230453075368039082e_nhds(A,Uua)) ) ) ).

% ATP.lambda_134
tff(fact_8312_ATP_Olambda__135,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat,Uua: A] : aa(A,A,aTP_Lamp_yv(nat,fun(A,A),Uu),Uua) = comm_s3205402744901411588hammer(A,Uua,Uu) ) ).

% ATP.lambda_135
tff(fact_8313_ATP_Olambda__136,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A,Uua: real] : aa(real,A,aTP_Lamp_abt(A,fun(real,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,Uua),Uu) ) ).

% ATP.lambda_136
tff(fact_8314_ATP_Olambda__137,axiom,
    ! [Uu: nat,Uua: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aTP_Lamp_a(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),Uu),Uua) = vEBT_vebt_inserti(Uua,Uu) ).

% ATP.lambda_137
tff(fact_8315_ATP_Olambda__138,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_no(set(A),fun(set(A),$o),Uu),Uua)
    <=> aa(set(A),$o,ord_less_eq(set(A),Uua),Uu) ) ).

% ATP.lambda_138
tff(fact_8316_ATP_Olambda__139,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_ni(nat,fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,ord_less_eq(nat,Uua),Uu) ) ).

% ATP.lambda_139
tff(fact_8317_ATP_Olambda__140,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_adm(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,ord_less_eq(A,Uua),Uu) ) ) ).

% ATP.lambda_140
tff(fact_8318_ATP_Olambda__141,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_ei(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,ord_less_eq(A,Uua),Uu) ) ) ).

% ATP.lambda_141
tff(fact_8319_ATP_Olambda__142,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_pn(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Uua),Uu) ).

% ATP.lambda_142
tff(fact_8320_ATP_Olambda__143,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_abv(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu) ) ).

% ATP.lambda_143
tff(fact_8321_ATP_Olambda__144,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ol(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu) ) ).

% ATP.lambda_144
tff(fact_8322_ATP_Olambda__145,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_pq(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu) ) ).

% ATP.lambda_145
tff(fact_8323_ATP_Olambda__146,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_gc(nat,fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,ord_less(nat,Uua),Uu) ) ).

% ATP.lambda_146
tff(fact_8324_ATP_Olambda__147,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_adk(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,ord_less(A,Uua),Uu) ) ) ).

% ATP.lambda_147
tff(fact_8325_ATP_Olambda__148,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_he(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,ord_less(A,Uua),Uu) ) ) ).

% ATP.lambda_148
tff(fact_8326_ATP_Olambda__149,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_zx(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),Uu) ).

% ATP.lambda_149
tff(fact_8327_ATP_Olambda__150,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_abu(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).

% ATP.lambda_150
tff(fact_8328_ATP_Olambda__151,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ar(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).

% ATP.lambda_151
tff(fact_8329_ATP_Olambda__152,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_oz(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).

% ATP.lambda_152
tff(fact_8330_ATP_Olambda__153,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_pg(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,minus_minus(nat,Uua),Uu) ).

% ATP.lambda_153
tff(fact_8331_ATP_Olambda__154,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ok(A,fun(A,A),Uu),Uua) = aa(A,A,minus_minus(A,Uua),Uu) ) ).

% ATP.lambda_154
tff(fact_8332_ATP_Olambda__155,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ou(A,fun(A,A),Uu),Uua) = aa(A,A,minus_minus(A,Uua),Uu) ) ).

% ATP.lambda_155
tff(fact_8333_ATP_Olambda__156,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_rk(nat,fun(real,real),Uu),Uua) = aa(nat,real,power_power(real,Uua),Uu) ).

% ATP.lambda_156
tff(fact_8334_ATP_Olambda__157,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa(code_integer,code_integer,aTP_Lamp_pf(code_integer,fun(code_integer,code_integer),Uu),Uua) = aa(code_integer,code_integer,plus_plus(code_integer,Uua),Uu) ).

% ATP.lambda_157
tff(fact_8335_ATP_Olambda__158,axiom,
    ! [Uu: int,Uua: int] : aa(int,int,aTP_Lamp_pe(int,fun(int,int),Uu),Uua) = aa(int,int,plus_plus(int,Uua),Uu) ).

% ATP.lambda_158
tff(fact_8336_ATP_Olambda__159,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_px(A,fun(A,A),Uu),Uua) = aa(A,A,plus_plus(A,Uua),Uu) ) ).

% ATP.lambda_159
tff(fact_8337_ATP_Olambda__160,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_oj(A,fun(A,A),Uu),Uua) = aa(A,A,plus_plus(A,Uua),Uu) ) ).

% ATP.lambda_160
tff(fact_8338_ATP_Olambda__161,axiom,
    ! [Uu: real,Uua: real] : aa(real,real,aTP_Lamp_rn(real,fun(real,real),Uu),Uua) = powr(real,Uua,Uu) ).

% ATP.lambda_161
tff(fact_8339_ATP_Olambda__162,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_mx(nat,fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,dvd_dvd(nat,Uua),Uu) ) ).

% ATP.lambda_162
tff(fact_8340_ATP_Olambda__163,axiom,
    ! [Uu: int,Uua: int] :
      ( aa(int,$o,aTP_Lamp_np(int,fun(int,$o),Uu),Uua)
    <=> aa(int,$o,dvd_dvd(int,Uua),Uu) ) ).

% ATP.lambda_163
tff(fact_8341_ATP_Olambda__164,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_ap(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,dvd_dvd(A,Uua),Uu) ) ) ).

% ATP.lambda_164
tff(fact_8342_ATP_Olambda__165,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,product_prod(B,A),aa(A,fun(B,product_prod(B,A)),aTP_Lamp_qd(A,fun(B,product_prod(B,A))),Uu),Uua) = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uua),Uu) ).

% ATP.lambda_165
tff(fact_8343_ATP_Olambda__166,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_en(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,binomial(Uua),Uu) ).

% ATP.lambda_166
tff(fact_8344_ATP_Olambda__167,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] : aa(A,list(A),aa(list(A),fun(A,list(A)),aTP_Lamp_qi(list(A),fun(A,list(A))),Uu),Uua) = aa(list(A),list(A),cons(A,Uua),Uu) ).

% ATP.lambda_167
tff(fact_8345_ATP_Olambda__168,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_aam(real,fun(nat,real),Uu),Uua) = aa(real,real,root(Uua),Uu) ).

% ATP.lambda_168
tff(fact_8346_ATP_Olambda__169,axiom,
    ! [B: $tType,Uu: set(B),Uua: B] :
      ( aa(B,$o,aTP_Lamp_akf(set(B),fun(B,$o),Uu),Uua)
    <=> member(B,Uua,Uu) ) ).

% ATP.lambda_169
tff(fact_8347_ATP_Olambda__170,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_ady(set(A),fun(A,$o),Uu),Uua)
        <=> member(A,Uua,Uu) ) ) ).

% ATP.lambda_170
tff(fact_8348_ATP_Olambda__171,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_aa(set(A),fun(A,$o),Uu),Uua)
    <=> member(A,Uua,Uu) ) ).

% ATP.lambda_171
tff(fact_8349_ATP_Olambda__172,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( aa(A,$o,aTP_Lamp_ad(A,fun(A,$o),Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_172
tff(fact_8350_ATP_Olambda__173,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_afy(fun(A,real),fun(A,$o),Uu),Uua)
    <=> aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_173
tff(fact_8351_ATP_Olambda__174,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] :
          ( aa(A,$o,aTP_Lamp_afp(fun(A,real),fun(A,$o),Uu),Uua)
        <=> aa(real,$o,ord_less_eq(real,one_one(real)),aa(A,real,Uu,Uua)) ) ) ).

% ATP.lambda_174
tff(fact_8352_ATP_Olambda__175,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_aft(fun(A,real),fun(A,$o),Uu),Uua)
    <=> aa(real,$o,ord_less_eq(real,one_one(real)),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_175
tff(fact_8353_ATP_Olambda__176,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_afz(fun(A,real),fun(A,$o),Uu),Uua)
    <=> aa(real,$o,ord_less(real,zero_zero(real)),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_176
tff(fact_8354_ATP_Olambda__177,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] :
      ( aa(A,$o,aTP_Lamp_mp(fun(A,nat),fun(A,$o),Uu),Uua)
    <=> aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(A,nat,Uu,Uua)) ) ).

% ATP.lambda_177
tff(fact_8355_ATP_Olambda__178,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_hs(fun(nat,real),fun(nat,real),Uu),Uua) = aa(nat,real,Uu,aa(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_178
tff(fact_8356_ATP_Olambda__179,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_hr(fun(nat,real),fun(nat,real),Uu),Uua) = aa(nat,real,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)) ).

% ATP.lambda_179
tff(fact_8357_ATP_Olambda__180,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_agz(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,Uu,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),Uua)) ) ).

% ATP.lambda_180
tff(fact_8358_ATP_Olambda__181,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(set(nat),A),Uua: nat] : aa(nat,A,aTP_Lamp_zy(fun(set(nat),A),fun(nat,A),Uu),Uua) = aa(set(nat),A,Uu,set_ord_lessThan(nat,Uua)) ) ).

% ATP.lambda_181
tff(fact_8359_ATP_Olambda__182,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(set(nat),A),Uua: nat] : aa(nat,A,aTP_Lamp_zz(fun(set(nat),A),fun(nat,A),Uu),Uua) = aa(set(nat),A,Uu,set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_182
tff(fact_8360_ATP_Olambda__183,axiom,
    ! [Uu: fun(real,$o),Uua: real] :
      ( aa(real,$o,aTP_Lamp_afs(fun(real,$o),fun(real,$o),Uu),Uua)
    <=> aa(real,$o,Uu,aa(real,real,inverse_inverse(real),Uua)) ) ).

% ATP.lambda_183
tff(fact_8361_ATP_Olambda__184,axiom,
    ! [A: $tType,Uu: fun(real,A),Uua: real] : aa(real,A,aTP_Lamp_adf(fun(real,A),fun(real,A),Uu),Uua) = aa(real,A,Uu,aa(real,real,inverse_inverse(real),Uua)) ).

% ATP.lambda_184
tff(fact_8362_ATP_Olambda__185,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(real,A),Uua: nat] : aa(nat,A,aTP_Lamp_agx(fun(real,A),fun(nat,A),Uu),Uua) = aa(real,A,Uu,aa(nat,real,semiring_1_of_nat(real),Uua)) ) ).

% ATP.lambda_185
tff(fact_8363_ATP_Olambda__186,axiom,
    ! [A: $tType,Uu: fun(int,A),Uua: nat] : aa(nat,A,aTP_Lamp_aby(fun(int,A),fun(nat,A),Uu),Uua) = aa(int,A,Uu,aa(nat,int,semiring_1_of_nat(int),Uua)) ).

% ATP.lambda_186
tff(fact_8364_ATP_Olambda__187,axiom,
    ! [Uu: fun(real,real),Uua: real] : aa(real,real,aTP_Lamp_re(fun(real,real),fun(real,real),Uu),Uua) = aa(real,real,Uu,aa(real,real,uminus_uminus(real),Uua)) ).

% ATP.lambda_187
tff(fact_8365_ATP_Olambda__188,axiom,
    ! [Uu: fun(real,$o),Uua: real] :
      ( aa(real,$o,aTP_Lamp_afo(fun(real,$o),fun(real,$o),Uu),Uua)
    <=> aa(real,$o,Uu,aa(real,real,uminus_uminus(real),Uua)) ) ).

% ATP.lambda_188
tff(fact_8366_ATP_Olambda__189,axiom,
    ! [A: $tType,Uu: fun(real,A),Uua: real] : aa(real,A,aTP_Lamp_adc(fun(real,A),fun(real,A),Uu),Uua) = aa(real,A,Uu,aa(real,real,uminus_uminus(real),Uua)) ).

% ATP.lambda_189
tff(fact_8367_ATP_Olambda__190,axiom,
    ! [Uu: fun(nat,$o),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_age(fun(nat,$o),fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_190
tff(fact_8368_ATP_Olambda__191,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bw(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_191
tff(fact_8369_ATP_Olambda__192,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_zq(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_192
tff(fact_8370_ATP_Olambda__193,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ci(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_193
tff(fact_8371_ATP_Olambda__194,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_jf(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_194
tff(fact_8372_ATP_Olambda__195,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ep(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_195
tff(fact_8373_ATP_Olambda__196,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_zo(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ).

% ATP.lambda_196
tff(fact_8374_ATP_Olambda__197,axiom,
    ! [Uu: fun(nat,real),Uua: real] : aa(real,real,aTP_Lamp_sd(fun(nat,real),fun(real,real),Uu),Uua) = suminf(real,aa(real,fun(nat,real),aTP_Lamp_sc(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua)) ).

% ATP.lambda_197
tff(fact_8375_ATP_Olambda__198,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A] : aa(A,A,aTP_Lamp_rl(fun(nat,A),fun(A,A),Uu),Uua) = suminf(A,aa(A,fun(nat,A),aTP_Lamp_cv(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua)) ) ).

% ATP.lambda_198
tff(fact_8376_ATP_Olambda__199,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,complex,aTP_Lamp_hz(nat,fun(nat,complex),Uu),Uua) = cis(aa(real,real,aa(real,fun(real,real),divide_divide(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))),pi)),aa(nat,real,semiring_1_of_nat(real),Uua))),aa(nat,real,semiring_1_of_nat(real),Uu))) ).

% ATP.lambda_199
tff(fact_8377_ATP_Olambda__200,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ti(A,fun(A,A),Uu),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),Uu)),Uua)),aa(A,A,inverse_inverse(A),Uu))) ) ).

% ATP.lambda_200
tff(fact_8378_ATP_Olambda__201,axiom,
    ! [Uu: fun(real,real),Uua: real] :
      ( aa(real,$o,aTP_Lamp_adh(fun(real,real),fun(real,$o),Uu),Uua)
    <=> ( aa(real,real,Uu,Uua) != zero_zero(real) ) ) ).

% ATP.lambda_201
tff(fact_8379_ATP_Olambda__202,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,real,aTP_Lamp_dv(A,fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uua))),aa(nat,A,power_power(A,Uu),Uua))) ) ).

% ATP.lambda_202
tff(fact_8380_ATP_Olambda__203,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_bt(nat,fun(nat,$o),Uu),Uua)
    <=> ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))) ) ).

% ATP.lambda_203
tff(fact_8381_ATP_Olambda__204,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_dw(A,fun(nat,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,sin_coeff(Uua)),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),Uu)),Uua))) ) ).

% ATP.lambda_204
tff(fact_8382_ATP_Olambda__205,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,real,aTP_Lamp_dt(A,fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(A,A,real_V8093663219630862766scaleR(A,sin_coeff(Uua)),aa(nat,A,power_power(A,Uu),Uua))) ) ).

% ATP.lambda_205
tff(fact_8383_ATP_Olambda__206,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,real,aTP_Lamp_du(A,fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(A,A,real_V8093663219630862766scaleR(A,cos_coeff(Uua)),aa(nat,A,power_power(A,Uu),Uua))) ) ).

% ATP.lambda_206
tff(fact_8384_ATP_Olambda__207,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_cy(A,fun(nat,fun(A,A)),Uu),Uua) = aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Uu),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_207
tff(fact_8385_ATP_Olambda__208,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_bq(A,fun(nat,fun(A,A)),Uu),Uua) = aa(A,fun(A,A),times_times(A),aa(A,A,plus_plus(A,Uu),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_208
tff(fact_8386_ATP_Olambda__209,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_aav(real,fun(nat,real),Uu),Uua) = aa(real,real,inverse_inverse(real),aa(nat,real,power_power(real,Uu),Uua)) ).

% ATP.lambda_209
tff(fact_8387_ATP_Olambda__210,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_rg(A,fun(A,A),Uu),Uua) = cos(A,aa(A,A,plus_plus(A,Uua),Uu)) ) ).

% ATP.lambda_210
tff(fact_8388_ATP_Olambda__211,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_am(set(A),fun(A,$o),Uu),Uua)
    <=> ~ member(A,Uua,Uu) ) ).

% ATP.lambda_211
tff(fact_8389_ATP_Olambda__212,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_agj(A,fun(A,$o),Uu),Uua)
        <=> ( Uua != Uu ) ) ) ).

% ATP.lambda_212
tff(fact_8390_ATP_Olambda__213,axiom,
    ! [A: $tType] :
      ( topological_t1_space(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_aeb(A,fun(A,$o),Uu),Uua)
        <=> ( Uua != Uu ) ) ) ).

% ATP.lambda_213
tff(fact_8391_ATP_Olambda__214,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_adx(A,fun(A,$o),Uu),Uua)
        <=> ( Uua != Uu ) ) ) ).

% ATP.lambda_214
tff(fact_8392_ATP_Olambda__215,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_adl(A,fun(A,$o),Uu),Uua)
        <=> ( Uua != Uu ) ) ) ).

% ATP.lambda_215
tff(fact_8393_ATP_Olambda__216,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_cb(fun(nat,A),fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_216
tff(fact_8394_ATP_Olambda__217,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_ez(fun(nat,A),fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_217
tff(fact_8395_ATP_Olambda__218,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_ca(fun(nat,A),fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_218
tff(fact_8396_ATP_Olambda__219,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,real,aTP_Lamp_ej(fun(B,A),fun(B,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_219
tff(fact_8397_ATP_Olambda__220,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: fun(B,A),Uua: B] : aa(B,real,aTP_Lamp_jd(fun(B,A),fun(B,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_220
tff(fact_8398_ATP_Olambda__221,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_aih(fun(A,B),fun(A,real),Uu),Uua) = real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_221
tff(fact_8399_ATP_Olambda__222,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_vg(fun(A,B),fun(A,real),Uu),Uua) = real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_222
tff(fact_8400_ATP_Olambda__223,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & comm_semiring_1(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_je(fun(A,B),fun(A,real),Uu),Uua) = real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_223
tff(fact_8401_ATP_Olambda__224,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_wk(fun(A,B),fun(A,real),Uu),Uua) = real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_224
tff(fact_8402_ATP_Olambda__225,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_aaj(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,inverse_inverse(real),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_225
tff(fact_8403_ATP_Olambda__226,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_tg(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,inverse_inverse(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_226
tff(fact_8404_ATP_Olambda__227,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_ahm(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_227
tff(fact_8405_ATP_Olambda__228,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_ahp(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_228
tff(fact_8406_ATP_Olambda__229,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_qm(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,inverse_inverse(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_229
tff(fact_8407_ATP_Olambda__230,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_yt(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_230
tff(fact_8408_ATP_Olambda__231,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_abx(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,inverse_inverse(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_231
tff(fact_8409_ATP_Olambda__232,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_we(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_232
tff(fact_8410_ATP_Olambda__233,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: fun(B,nat),Uua: B] : aa(B,A,aTP_Lamp_is(fun(B,nat),fun(B,A),Uu),Uua) = aa(nat,A,semiring_1_of_nat(A),aa(B,nat,Uu,Uua)) ) ).

% ATP.lambda_233
tff(fact_8411_ATP_Olambda__234,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Uu: fun(B,nat),Uua: B] : aa(B,A,aTP_Lamp_ed(fun(B,nat),fun(B,A),Uu),Uua) = aa(nat,A,semiring_1_of_nat(A),aa(B,nat,Uu,Uua)) ) ).

% ATP.lambda_234
tff(fact_8412_ATP_Olambda__235,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,real,aTP_Lamp_abz(fun(A,nat),fun(A,real),Uu),Uua) = aa(nat,real,semiring_1_of_nat(real),aa(A,nat,Uu,Uua)) ).

% ATP.lambda_235
tff(fact_8413_ATP_Olambda__236,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,int,aTP_Lamp_gj(fun(A,nat),fun(A,int),Uu),Uua) = aa(nat,int,semiring_1_of_nat(int),aa(A,nat,Uu,Uua)) ).

% ATP.lambda_236
tff(fact_8414_ATP_Olambda__237,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_tp(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_237
tff(fact_8415_ATP_Olambda__238,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ahs(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_238
tff(fact_8416_ATP_Olambda__239,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_zw(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_239
tff(fact_8417_ATP_Olambda__240,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ng(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_240
tff(fact_8418_ATP_Olambda__241,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,complex,aTP_Lamp_bd(fun(nat,real),fun(nat,complex),Uu),Uua) = aa(real,complex,real_Vector_of_real(complex),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_241
tff(fact_8419_ATP_Olambda__242,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,real),Uua: nat] : aa(nat,A,aTP_Lamp_cf(fun(nat,real),fun(nat,A),Uu),Uua) = aa(real,A,real_Vector_of_real(A),aa(nat,real,Uu,Uua)) ) ).

% ATP.lambda_242
tff(fact_8420_ATP_Olambda__243,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(nat,real),Uua: nat] : aa(nat,A,aTP_Lamp_bl(fun(nat,real),fun(nat,A),Uu),Uua) = aa(real,A,real_Vector_of_real(A),aa(nat,real,Uu,Uua)) ) ).

% ATP.lambda_243
tff(fact_8421_ATP_Olambda__244,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Uu: fun(nat,real),Uua: nat] : aa(nat,A,aTP_Lamp_ct(fun(nat,real),fun(nat,A),Uu),Uua) = aa(real,A,real_Vector_of_real(A),aa(nat,real,Uu,Uua)) ) ).

% ATP.lambda_244
tff(fact_8422_ATP_Olambda__245,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Uu: fun(B,real),Uua: B] : aa(B,A,aTP_Lamp_ef(fun(B,real),fun(B,A),Uu),Uua) = aa(real,A,real_Vector_of_real(A),aa(B,real,Uu,Uua)) ) ).

% ATP.lambda_245
tff(fact_8423_ATP_Olambda__246,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2191834092415804123ebra_1(A) )
     => ! [Uu: fun(B,real),Uua: B] : aa(B,A,aTP_Lamp_iu(fun(B,real),fun(B,A),Uu),Uua) = aa(real,A,real_Vector_of_real(A),aa(B,real,Uu,Uua)) ) ).

% ATP.lambda_246
tff(fact_8424_ATP_Olambda__247,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,real),Uua: A] : aa(A,B,aTP_Lamp_sn(fun(A,real),fun(A,B),Uu),Uua) = aa(real,B,real_Vector_of_real(B),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_247
tff(fact_8425_ATP_Olambda__248,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,complex,aTP_Lamp_ajm(fun(A,real),fun(A,complex),Uu),Uua) = aa(real,complex,real_Vector_of_real(complex),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_248
tff(fact_8426_ATP_Olambda__249,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,real),Uua: A] : aa(A,B,aTP_Lamp_aii(fun(A,real),fun(A,B),Uu),Uua) = aa(real,B,real_Vector_of_real(B),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_249
tff(fact_8427_ATP_Olambda__250,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,real),Uua: A] : aa(A,B,aTP_Lamp_vh(fun(A,real),fun(A,B),Uu),Uua) = aa(real,B,real_Vector_of_real(B),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_250
tff(fact_8428_ATP_Olambda__251,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,B,aTP_Lamp_xn(fun(A,real),fun(A,B),Uu),Uua) = aa(real,B,real_Vector_of_real(B),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_251
tff(fact_8429_ATP_Olambda__252,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,real),Uua: A] : aa(A,B,aTP_Lamp_xm(fun(A,real),fun(A,B),Uu),Uua) = aa(real,B,real_Vector_of_real(B),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_252
tff(fact_8430_ATP_Olambda__253,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bj(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_253
tff(fact_8431_ATP_Olambda__254,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_dj(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_254
tff(fact_8432_ATP_Olambda__255,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cu(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_255
tff(fact_8433_ATP_Olambda__256,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_gz(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_256
tff(fact_8434_ATP_Olambda__257,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_sv(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_257
tff(fact_8435_ATP_Olambda__258,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_aix(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_258
tff(fact_8436_ATP_Olambda__259,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_qj(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_259
tff(fact_8437_ATP_Olambda__260,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_ys(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_260
tff(fact_8438_ATP_Olambda__261,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_vp(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_261
tff(fact_8439_ATP_Olambda__262,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_adb(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,uminus_uminus(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_262
tff(fact_8440_ATP_Olambda__263,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_vq(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_263
tff(fact_8441_ATP_Olambda__264,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_jn(fun(nat,A),fun(nat,fun(A,A)),Uu),Uua) = aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_264
tff(fact_8442_ATP_Olambda__265,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: fun(B,int),Uua: B] : aa(B,A,aTP_Lamp_it(fun(B,int),fun(B,A),Uu),Uua) = aa(int,A,ring_1_of_int(A),aa(B,int,Uu,Uua)) ) ).

% ATP.lambda_265
tff(fact_8443_ATP_Olambda__266,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(A)
     => ! [Uu: fun(B,int),Uua: B] : aa(B,A,aTP_Lamp_ee(fun(B,int),fun(B,A),Uu),Uua) = aa(int,A,ring_1_of_int(A),aa(B,int,Uu,Uua)) ) ).

% ATP.lambda_266
tff(fact_8444_ATP_Olambda__267,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_ig(fun(nat,A),fun(nat,fun(A,A)),Uu),Uua) = plus_plus(A,aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_267
tff(fact_8445_ATP_Olambda__268,axiom,
    ! [Uu: fun(real,real),Uua: real] : aa(real,real,aTP_Lamp_aji(fun(real,real),fun(real,real),Uu),Uua) = aa(real,real,artanh(real),aa(real,real,Uu,Uua)) ).

% ATP.lambda_268
tff(fact_8446_ATP_Olambda__269,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_acs(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,artanh(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_269
tff(fact_8447_ATP_Olambda__270,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_yn(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,artanh(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_270
tff(fact_8448_ATP_Olambda__271,axiom,
    ! [Uu: fun(real,real),Uua: real] : aa(real,real,aTP_Lamp_aid(fun(real,real),fun(real,real),Uu),Uua) = aa(real,real,arsinh(real),aa(real,real,Uu,Uua)) ).

% ATP.lambda_271
tff(fact_8449_ATP_Olambda__272,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_zm(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arsinh(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_272
tff(fact_8450_ATP_Olambda__273,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_xq(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arsinh(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_273
tff(fact_8451_ATP_Olambda__274,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ub(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arctan,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_274
tff(fact_8452_ATP_Olambda__275,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_aif(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arctan,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_275
tff(fact_8453_ATP_Olambda__276,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_zp(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arctan,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_276
tff(fact_8454_ATP_Olambda__277,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_xl(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arctan,aa(A,real,Uu,Uua)) ).

% ATP.lambda_277
tff(fact_8455_ATP_Olambda__278,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_se(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcsin,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_278
tff(fact_8456_ATP_Olambda__279,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ajh(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcsin,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_279
tff(fact_8457_ATP_Olambda__280,axiom,
    ! [Uu: fun(real,real),Uua: real] : aa(real,real,aTP_Lamp_aje(fun(real,real),fun(real,real),Uu),Uua) = aa(real,real,arcosh(real),aa(real,real,Uu,Uua)) ).

% ATP.lambda_280
tff(fact_8458_ATP_Olambda__281,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_acr(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcosh(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_281
tff(fact_8459_ATP_Olambda__282,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_vo(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcosh(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_282
tff(fact_8460_ATP_Olambda__283,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_uh(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arccos,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_283
tff(fact_8461_ATP_Olambda__284,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ajg(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arccos,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_284
tff(fact_8462_ATP_Olambda__285,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_ahq(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,sgn_sgn(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_285
tff(fact_8463_ATP_Olambda__286,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_yu(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,sgn_sgn(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_286
tff(fact_8464_ATP_Olambda__287,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_wl(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,sgn_sgn(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_287
tff(fact_8465_ATP_Olambda__288,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_bk(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,abs_abs(real),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_288
tff(fact_8466_ATP_Olambda__289,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_gh(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,abs_abs(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_289
tff(fact_8467_ATP_Olambda__290,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_field(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_iy(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,abs_abs(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_290
tff(fact_8468_ATP_Olambda__291,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_aib(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,abs_abs(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_291
tff(fact_8469_ATP_Olambda__292,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_zs(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,abs_abs(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_292
tff(fact_8470_ATP_Olambda__293,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_wf(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,abs_abs(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_293
tff(fact_8471_ATP_Olambda__294,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_ajd(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,tanh(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_294
tff(fact_8472_ATP_Olambda__295,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_yz(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,tanh(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_295
tff(fact_8473_ATP_Olambda__296,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_rr(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,tanh(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_296
tff(fact_8474_ATP_Olambda__297,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_wg(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,tanh(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_297
tff(fact_8475_ATP_Olambda__298,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_ain(fun(A,B),fun(A,B),Uu),Uua) = sinh(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_298
tff(fact_8476_ATP_Olambda__299,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_xb(fun(A,B),fun(A,B),Uu),Uua) = sinh(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_299
tff(fact_8477_ATP_Olambda__300,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_qt(fun(A,A),fun(A,A),Uu),Uua) = sinh(A,aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_300
tff(fact_8478_ATP_Olambda__301,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_xd(fun(A,B),fun(A,B),Uu),Uua) = sinh(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_301
tff(fact_8479_ATP_Olambda__302,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_aim(fun(A,B),fun(A,B),Uu),Uua) = cosh(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_302
tff(fact_8480_ATP_Olambda__303,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_xc(fun(A,B),fun(A,B),Uu),Uua) = cosh(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_303
tff(fact_8481_ATP_Olambda__304,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_qs(fun(A,A),fun(A,A),Uu),Uua) = cosh(A,aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_304
tff(fact_8482_ATP_Olambda__305,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_xe(fun(A,B),fun(A,B),Uu),Uua) = cosh(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_305
tff(fact_8483_ATP_Olambda__306,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ud(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,tan(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_306
tff(fact_8484_ATP_Olambda__307,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_wi(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,tan(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_307
tff(fact_8485_ATP_Olambda__308,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_tc(fun(A,real),fun(A,real),Uu),Uua) = sin(real,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_308
tff(fact_8486_ATP_Olambda__309,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_xo(fun(A,B),fun(A,B),Uu),Uua) = sin(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_309
tff(fact_8487_ATP_Olambda__310,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_rb(fun(A,A),fun(A,A),Uu),Uua) = sin(A,aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_310
tff(fact_8488_ATP_Olambda__311,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ta(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,exp(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_311
tff(fact_8489_ATP_Olambda__312,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_aik(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,exp(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_312
tff(fact_8490_ATP_Olambda__313,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_xg(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,exp(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_313
tff(fact_8491_ATP_Olambda__314,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_ra(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,exp(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_314
tff(fact_8492_ATP_Olambda__315,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_xh(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,exp(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_315
tff(fact_8493_ATP_Olambda__316,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_Vector_banach(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_na(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,exp(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_316
tff(fact_8494_ATP_Olambda__317,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_wh(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,cot(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_317
tff(fact_8495_ATP_Olambda__318,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_abj(fun(nat,real),fun(nat,real),Uu),Uua) = cos(real,aa(nat,real,Uu,Uua)) ).

% ATP.lambda_318
tff(fact_8496_ATP_Olambda__319,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_tl(fun(A,real),fun(A,real),Uu),Uua) = cos(real,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_319
tff(fact_8497_ATP_Olambda__320,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_xi(fun(A,B),fun(A,B),Uu),Uua) = cos(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_320
tff(fact_8498_ATP_Olambda__321,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_rf(fun(A,A),fun(A,A),Uu),Uua) = cos(A,aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_321
tff(fact_8499_ATP_Olambda__322,axiom,
    ! [Uu: fun(nat,complex),Uua: nat] : aa(nat,real,aTP_Lamp_ajq(fun(nat,complex),fun(nat,real),Uu),Uua) = re(aa(nat,complex,Uu,Uua)) ).

% ATP.lambda_322
tff(fact_8500_ATP_Olambda__323,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,complex),Uua: A] : aa(A,real,aTP_Lamp_ajs(fun(A,complex),fun(A,real),Uu),Uua) = re(aa(A,complex,Uu,Uua)) ) ).

% ATP.lambda_323
tff(fact_8501_ATP_Olambda__324,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,complex),Uua: A] : aa(A,real,aTP_Lamp_ajp(fun(A,complex),fun(A,real),Uu),Uua) = re(aa(A,complex,Uu,Uua)) ) ).

% ATP.lambda_324
tff(fact_8502_ATP_Olambda__325,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,complex),Uua: A] : aa(A,real,aTP_Lamp_ajr(fun(A,complex),fun(A,real),Uu),Uua) = re(aa(A,complex,Uu,Uua)) ) ).

% ATP.lambda_325
tff(fact_8503_ATP_Olambda__326,axiom,
    ! [A: $tType,Uu: fun(A,complex),Uua: A] : aa(A,real,aTP_Lamp_ajn(fun(A,complex),fun(A,real),Uu),Uua) = re(aa(A,complex,Uu,Uua)) ).

% ATP.lambda_326
tff(fact_8504_ATP_Olambda__327,axiom,
    ! [Uu: fun(nat,complex),Uua: nat] : aa(nat,real,aTP_Lamp_ajx(fun(nat,complex),fun(nat,real),Uu),Uua) = im(aa(nat,complex,Uu,Uua)) ).

% ATP.lambda_327
tff(fact_8505_ATP_Olambda__328,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,complex),Uua: A] : aa(A,real,aTP_Lamp_ajy(fun(A,complex),fun(A,real),Uu),Uua) = im(aa(A,complex,Uu,Uua)) ) ).

% ATP.lambda_328
tff(fact_8506_ATP_Olambda__329,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,complex),Uua: A] : aa(A,real,aTP_Lamp_ajv(fun(A,complex),fun(A,real),Uu),Uua) = im(aa(A,complex,Uu,Uua)) ) ).

% ATP.lambda_329
tff(fact_8507_ATP_Olambda__330,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,complex),Uua: A] : aa(A,real,aTP_Lamp_ajw(fun(A,complex),fun(A,real),Uu),Uua) = im(aa(A,complex,Uu,Uua)) ) ).

% ATP.lambda_330
tff(fact_8508_ATP_Olambda__331,axiom,
    ! [A: $tType,Uu: fun(A,complex),Uua: A] : aa(A,real,aTP_Lamp_aju(fun(A,complex),fun(A,real),Uu),Uua) = im(aa(A,complex,Uu,Uua)) ).

% ATP.lambda_331
tff(fact_8509_ATP_Olambda__332,axiom,
    ! [Uu: fun(real,fun(nat,real)),Uua: real] : aa(real,real,aTP_Lamp_ug(fun(real,fun(nat,real)),fun(real,real),Uu),Uua) = suminf(real,aa(real,fun(nat,real),Uu,Uua)) ).

% ATP.lambda_332
tff(fact_8510_ATP_Olambda__333,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_Vector_banach(B) )
     => ! [Uu: fun(A,fun(nat,B)),Uua: A] : aa(A,B,aTP_Lamp_za(fun(A,fun(nat,B)),fun(A,B),Uu),Uua) = suminf(B,aa(A,fun(nat,B),Uu,Uua)) ) ).

% ATP.lambda_333
tff(fact_8511_ATP_Olambda__334,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [Uu: fun(A,fun(nat,B)),Uua: A] : aa(A,B,aTP_Lamp_ex(fun(A,fun(nat,B)),fun(A,B),Uu),Uua) = suminf(B,aa(A,fun(nat,B),Uu,Uua)) ) ).

% ATP.lambda_334
tff(fact_8512_ATP_Olambda__335,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_tz(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,sqrt,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_335
tff(fact_8513_ATP_Olambda__336,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_aig(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,sqrt,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_336
tff(fact_8514_ATP_Olambda__337,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_zn(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,sqrt,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_337
tff(fact_8515_ATP_Olambda__338,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_xj(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,sqrt,aa(A,real,Uu,Uua)) ).

% ATP.lambda_338
tff(fact_8516_ATP_Olambda__339,axiom,
    ! [Uu: fun(nat,complex),Uua: nat] : aa(nat,complex,aTP_Lamp_akk(fun(nat,complex),fun(nat,complex),Uu),Uua) = cnj(aa(nat,complex,Uu,Uua)) ).

% ATP.lambda_339
tff(fact_8517_ATP_Olambda__340,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,complex),Uua: A] : aa(A,complex,aTP_Lamp_akj(fun(A,complex),fun(A,complex),Uu),Uua) = cnj(aa(A,complex,Uu,Uua)) ) ).

% ATP.lambda_340
tff(fact_8518_ATP_Olambda__341,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,complex),Uua: A] : aa(A,complex,aTP_Lamp_aki(fun(A,complex),fun(A,complex),Uu),Uua) = cnj(aa(A,complex,Uu,Uua)) ) ).

% ATP.lambda_341
tff(fact_8519_ATP_Olambda__342,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,complex),Uua: A] : aa(A,complex,aTP_Lamp_akl(fun(A,complex),fun(A,complex),Uu),Uua) = cnj(aa(A,complex,Uu,Uua)) ) ).

% ATP.lambda_342
tff(fact_8520_ATP_Olambda__343,axiom,
    ! [A: $tType,Uu: fun(A,complex),Uua: A] : aa(A,complex,aTP_Lamp_akg(fun(A,complex),fun(A,complex),Uu),Uua) = cnj(aa(A,complex,Uu,Uua)) ).

% ATP.lambda_343
tff(fact_8521_ATP_Olambda__344,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,complex,aTP_Lamp_ajo(fun(A,real),fun(A,complex),Uu),Uua) = cis(aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_344
tff(fact_8522_ATP_Olambda__345,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,nat,aTP_Lamp_ala(fun(A,nat),fun(A,nat),Uu),Uua) = aa(nat,nat,suc,aa(A,nat,Uu,Uua)) ).

% ATP.lambda_345
tff(fact_8523_ATP_Olambda__346,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_uy(fun(A,$o),fun(A,$o),Uu),Uua)
        <=> ~ aa(A,$o,Uu,Uua) ) ) ).

% ATP.lambda_346
tff(fact_8524_ATP_Olambda__347,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A] :
      ( aa(A,$o,aTP_Lamp_al(fun(A,$o),fun(A,$o),Uu),Uua)
    <=> ~ aa(A,$o,Uu,Uua) ) ).

% ATP.lambda_347
tff(fact_8525_ATP_Olambda__348,axiom,
    ! [B: $tType,A: $tType] :
      ( finite_finite(A)
     => ! [Uu: fun(B,fun(A,$o)),Uua: B] :
          ( aa(B,$o,aTP_Lamp_ahe(fun(B,fun(A,$o)),fun(B,$o),Uu),Uua)
        <=> ! [X_1: A] : aa(A,$o,aa(B,fun(A,$o),Uu,Uua),X_1) ) ) ).

% ATP.lambda_348
tff(fact_8526_ATP_Olambda__349,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A) )
     => ! [Uu: fun(real,A),Uua: real] : aa(real,real,aTP_Lamp_zd(fun(real,A),fun(real,real),Uu),Uua) = aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(A,aa(real,A,Uu,Uua))) ) ).

% ATP.lambda_349
tff(fact_8527_ATP_Olambda__350,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B)
        & ring_1(C)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,C,aTP_Lamp_xv(fun(A,B),fun(A,C),Uu),Uua) = aa(int,C,ring_1_of_int(C),archim6421214686448440834_floor(B,aa(A,B,Uu,Uua))) ) ).

% ATP.lambda_350
tff(fact_8528_ATP_Olambda__351,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B)
        & ring_1(C)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,C,aTP_Lamp_xu(fun(A,B),fun(A,C),Uu),Uua) = aa(int,C,ring_1_of_int(C),archimedean_ceiling(B,aa(A,B,Uu,Uua))) ) ).

% ATP.lambda_351
tff(fact_8529_ATP_Olambda__352,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_ahg(fun(A,$o),fun(A,$o),Uu),Uua)
        <=> ! [Y6: A] :
              ( aa(A,$o,ord_less_eq(A,Uua),Y6)
             => aa(A,$o,Uu,Y6) ) ) ) ).

% ATP.lambda_352
tff(fact_8530_ATP_Olambda__353,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,real),Uub: nat] :
      aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_co(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Uu),Uua),Uub) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub),aa(nat,real,Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,Uu,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,Uub),one_one(nat))),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% ATP.lambda_353
tff(fact_8531_ATP_Olambda__354,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,real),Uub: nat] :
      aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_hq(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Uu),Uua),Uub) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub),aa(nat,real,Uu,Uub),aa(nat,real,Uua,Uub)) ).

% ATP.lambda_354
tff(fact_8532_ATP_Olambda__355,axiom,
    ! [Uu: num,Uua: code_integer,Uub: code_integer] :
      aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_kg(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = $ite(aa(code_integer,$o,ord_less_eq(code_integer,aa(num,code_integer,numeral_numeral(code_integer),Uu)),Uub),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,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,minus_minus(code_integer,Uub),aa(num,code_integer,numeral_numeral(code_integer),Uu))),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),Uua)),Uub)) ).

% ATP.lambda_355
tff(fact_8533_ATP_Olambda__356,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] :
      aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_ke(num,fun(nat,fun(nat,product_prod(nat,nat))),Uu),Uua),Uub) = $ite(aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),Uu)),Uub),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,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,minus_minus(nat,Uub),aa(num,nat,numeral_numeral(nat),Uu))),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),Uub)) ).

% ATP.lambda_356
tff(fact_8534_ATP_Olambda__357,axiom,
    ! [Uu: num,Uua: int,Uub: int] :
      aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_kf(num,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = $ite(aa(int,$o,ord_less_eq(int,aa(num,int,numeral_numeral(int),Uu)),Uub),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,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,minus_minus(int,Uub),aa(num,int,numeral_numeral(int),Uu))),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Uua)),Uub)) ).

% ATP.lambda_357
tff(fact_8535_ATP_Olambda__358,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Uu: num,Uua: A,Uub: A] :
          aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_kh(num,fun(A,fun(A,product_prod(A,A))),Uu),Uua),Uub) = $ite(aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),Uu)),Uub),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,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,minus_minus(A,Uub),aa(num,A,numeral_numeral(A),Uu))),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)),Uub)) ) ).

% ATP.lambda_358
tff(fact_8536_ATP_Olambda__359,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: set(nat),Uua: fun(nat,A),Uub: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_lu(set(nat),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = $ite(member(nat,Uub,Uu),aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_359
tff(fact_8537_ATP_Olambda__360,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_lr(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uu = Uub,aa(A,B,Uua,Uub),zero_zero(B)) ) ).

% ATP.lambda_360
tff(fact_8538_ATP_Olambda__361,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ls(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uu = Uub,aa(A,B,Uua,Uub),one_one(B)) ) ).

% ATP.lambda_361
tff(fact_8539_ATP_Olambda__362,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: nat,Uua: fun(nat,A),Uub: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bb(nat,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = $ite(Uub = Uu,aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_362
tff(fact_8540_ATP_Olambda__363,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_lq(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uub = Uu,aa(A,B,Uua,Uub),zero_zero(B)) ) ).

% ATP.lambda_363
tff(fact_8541_ATP_Olambda__364,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_lt(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uub = Uu,aa(A,B,Uua,Uub),one_one(B)) ) ).

% ATP.lambda_364
tff(fact_8542_ATP_Olambda__365,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_ako(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = $ite(Uub = zero_zero(code_integer),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,minus_minus(code_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,minus_minus(code_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),Uu)),Uub))) ).

% ATP.lambda_365
tff(fact_8543_ATP_Olambda__366,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_akm(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = $ite(Uub = zero_zero(code_integer),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,minus_minus(code_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,minus_minus(code_integer,aa(code_integer,code_integer,abs_abs(code_integer),Uu)),Uub))) ).

% ATP.lambda_366
tff(fact_8544_ATP_Olambda__367,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_akn(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = $ite(Uub = zero_zero(code_integer),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,minus_minus(code_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,minus_minus(code_integer,Uu),Uub))) ).

% ATP.lambda_367
tff(fact_8545_ATP_Olambda__368,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,$o),Uua: fun(nat,A),Uub: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_lv(fun(nat,$o),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = $ite(aa(nat,$o,Uu,Uub),aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_368
tff(fact_8546_ATP_Olambda__369,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(A,B),Uua: fun(A,$o),Uub: A] :
          aa(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_mr(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu),Uua),Uub) = $ite(aa(A,$o,Uua,Uub),aa(A,B,Uu,Uub),zero_zero(B)) ) ).

% ATP.lambda_369
tff(fact_8547_ATP_Olambda__370,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,B),Uua: fun(A,$o),Uub: A] :
          aa(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_ms(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu),Uua),Uub) = $ite(aa(A,$o,Uua,Uub),aa(A,B,Uu,Uub),one_one(B)) ) ).

% ATP.lambda_370
tff(fact_8548_ATP_Olambda__371,axiom,
    ! [Uu: fun(real,real),Uua: fun(real,real),Uub: real] :
      ( aa(real,$o,aa(fun(real,real),fun(real,$o),aTP_Lamp_adi(fun(real,real),fun(fun(real,real),fun(real,$o)),Uu),Uua),Uub)
    <=> has_field_derivative(real,Uu,aa(real,real,Uua,Uub),topolo174197925503356063within(real,Uub,top_top(set(real)))) ) ).

% ATP.lambda_371
tff(fact_8549_ATP_Olambda__372,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: filter(B),Uua: fun(B,A),Uub: A] :
          ( aa(A,$o,aa(fun(B,A),fun(A,$o),aTP_Lamp_aco(filter(B),fun(fun(B,A),fun(A,$o)),Uu),Uua),Uub)
        <=> filterlim(B,A,Uua,topolo7230453075368039082e_nhds(A,Uub),Uu) ) ) ).

% ATP.lambda_372
tff(fact_8550_ATP_Olambda__373,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ko(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),aa(nat,nat,minus_minus(nat,Uua),Uub)) ) ).

% ATP.lambda_373
tff(fact_8551_ATP_Olambda__374,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_km(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),aa(nat,nat,minus_minus(nat,Uua),Uub)) ) ).

% ATP.lambda_374
tff(fact_8552_ATP_Olambda__375,axiom,
    ! [Uu: fun(real,fun(nat,real)),Uua: nat,Uub: real] : aa(real,real,aa(nat,fun(real,real),aTP_Lamp_uf(fun(real,fun(nat,real)),fun(nat,fun(real,real)),Uu),Uua),Uub) = aa(nat,real,aa(real,fun(nat,real),Uu,Uub),Uua) ).

% ATP.lambda_375
tff(fact_8553_ATP_Olambda__376,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jl(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),Uua) ) ).

% ATP.lambda_376
tff(fact_8554_ATP_Olambda__377,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ie(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),Uua) ) ).

% ATP.lambda_377
tff(fact_8555_ATP_Olambda__378,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu: fun(B,fun(A,B)),Uua: A,Uub: B] : aa(B,B,aa(A,fun(B,B),aTP_Lamp_qk(fun(B,fun(A,B)),fun(A,fun(B,B)),Uu),Uua),Uub) = aa(A,B,aa(B,fun(A,B),Uu,Uub),Uua) ) ).

% ATP.lambda_378
tff(fact_8556_ATP_Olambda__379,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: C,Uub: B] : aa(B,A,aa(C,fun(B,A),aTP_Lamp_ja(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uua),Uub) = aa(C,A,aa(B,fun(C,A),Uu,Uub),Uua) ) ).

% ATP.lambda_379
tff(fact_8557_ATP_Olambda__380,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: C,Uub: B] : aa(B,A,aa(C,fun(B,A),aTP_Lamp_gp(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uua),Uub) = aa(C,A,aa(B,fun(C,A),Uu,Uub),Uua) ) ).

% ATP.lambda_380
tff(fact_8558_ATP_Olambda__381,axiom,
    ! [B: $tType,A: $tType] :
      ( finite_finite(A)
     => ! [Uu: fun(B,fun(A,$o)),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_ahd(fun(B,fun(A,$o)),fun(A,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,aa(B,fun(A,$o),Uu,Uub),Uua) ) ) ).

% ATP.lambda_381
tff(fact_8559_ATP_Olambda__382,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [Uu: fun(B,fun(A,C)),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_wm(fun(B,fun(A,C)),fun(A,fun(B,C)),Uu),Uua),Uub) = aa(A,C,aa(B,fun(A,C),Uu,Uub),Uua) ) ).

% ATP.lambda_382
tff(fact_8560_ATP_Olambda__383,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [Uu: fun(B,fun(A,C)),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_vv(fun(B,fun(A,C)),fun(A,fun(B,C)),Uu),Uua),Uub) = aa(A,C,aa(B,fun(A,C),Uu,Uub),Uua) ) ).

% ATP.lambda_383
tff(fact_8561_ATP_Olambda__384,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [Uu: fun(A,fun(nat,B)),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_ek(fun(A,fun(nat,B)),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(A,fun(nat,B),Uu,Uub),Uua) ) ).

% ATP.lambda_384
tff(fact_8562_ATP_Olambda__385,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_st(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_385
tff(fact_8563_ATP_Olambda__386,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(C) )
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_tt(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_386
tff(fact_8564_ATP_Olambda__387,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & real_V4412858255891104859lgebra(C)
        & comm_ring_1(C) )
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_air(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_387
tff(fact_8565_ATP_Olambda__388,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4987421752381908075d_mult(C) )
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_ait(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_388
tff(fact_8566_ATP_Olambda__389,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo5987344860129210374id_add(C) )
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_aiv(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_389
tff(fact_8567_ATP_Olambda__390,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & real_V4412858255891104859lgebra(C)
        & comm_ring_1(C) )
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_wq(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_390
tff(fact_8568_ATP_Olambda__391,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & topolo4987421752381908075d_mult(C) )
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_wo(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_391
tff(fact_8569_ATP_Olambda__392,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & topolo5987344860129210374id_add(C) )
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_vr(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_392
tff(fact_8570_ATP_Olambda__393,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(C)
        & comm_ring_1(C) )
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_wu(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_393
tff(fact_8571_ATP_Olambda__394,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_ws(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_394
tff(fact_8572_ATP_Olambda__395,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_vt(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_395
tff(fact_8573_ATP_Olambda__396,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_mb(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_396
tff(fact_8574_ATP_Olambda__397,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_mf(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_397
tff(fact_8575_ATP_Olambda__398,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_gg(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_gf(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_398
tff(fact_8576_ATP_Olambda__399,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ge(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_gd(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_399
tff(fact_8577_ATP_Olambda__400,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_fb(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(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))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_400
tff(fact_8578_ATP_Olambda__401,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ec(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_eb(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_401
tff(fact_8579_ATP_Olambda__402,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: nat,Uua: A,Uub: nat] :
          aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ck(nat,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              $ite(Uub = Uu,one_one(A),zero_zero(A))),
            aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_402
tff(fact_8580_ATP_Olambda__403,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa(code_integer,product_prod(code_integer,$o),aa(code_integer,fun(code_integer,product_prod(code_integer,$o)),aTP_Lamp_akp(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,$o))),Uu),Uua),Uub) = aa($o,product_prod(code_integer,$o),
        aa(code_integer,fun($o,product_prod(code_integer,$o)),product_Pair(code_integer,$o),
          $ite(aa(code_integer,$o,ord_less(code_integer,zero_zero(code_integer)),Uu),Uua,aa(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_403
tff(fact_8581_ATP_Olambda__404,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,$o)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_lp(fun(A,fun(A,$o)),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,aa(A,fun(A,$o),Uu,Uua),Uub)
        | ( Uua = Uub ) ) ) ).

% ATP.lambda_404
tff(fact_8582_ATP_Olambda__405,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jm(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jl(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_405
tff(fact_8583_ATP_Olambda__406,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_if(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ie(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_406
tff(fact_8584_ATP_Olambda__407,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: A] : aa(A,A,aa(nat,fun(A,A),aTP_Lamp_zc(fun(nat,A),fun(nat,fun(A,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_bp(fun(nat,A),fun(A,fun(nat,A)),Uu),Uub)),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_407
tff(fact_8585_ATP_Olambda__408,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ln(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_lm(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_408
tff(fact_8586_ATP_Olambda__409,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ll(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_lk(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_409
tff(fact_8587_ATP_Olambda__410,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_lj(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_li(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_410
tff(fact_8588_ATP_Olambda__411,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_lh(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_lg(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_411
tff(fact_8589_ATP_Olambda__412,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_lf(rat,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> aa(product_prod(int,int),$o,product_case_prod(int,int,$o,aa(int,fun(int,fun(int,$o)),aTP_Lamp_le(int,fun(int,fun(int,fun(int,$o))),Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_412
tff(fact_8590_ATP_Olambda__413,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_lc(rat,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> aa(product_prod(int,int),$o,product_case_prod(int,int,$o,aa(int,fun(int,fun(int,$o)),aTP_Lamp_lb(int,fun(int,fun(int,fun(int,$o))),Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_413
tff(fact_8591_ATP_Olambda__414,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4987421752381908075d_mult(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_aiu(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_ait(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub),Uu) ) ).

% ATP.lambda_414
tff(fact_8592_ATP_Olambda__415,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(C)
        & comm_ring_1(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_ais(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_air(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub),Uu) ) ).

% ATP.lambda_415
tff(fact_8593_ATP_Olambda__416,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(C)
        & comm_ring_1(C) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_wv(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_wu(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub),Uu) ) ).

% ATP.lambda_416
tff(fact_8594_ATP_Olambda__417,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_wt(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_ws(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub),Uu) ) ).

% ATP.lambda_417
tff(fact_8595_ATP_Olambda__418,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(C)
        & comm_ring_1(C)
        & topological_t2_space(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_wr(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_wq(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub),Uu) ) ).

% ATP.lambda_418
tff(fact_8596_ATP_Olambda__419,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4987421752381908075d_mult(C)
        & topological_t2_space(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_wp(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_wo(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub),Uu) ) ).

% ATP.lambda_419
tff(fact_8597_ATP_Olambda__420,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_tu(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_tt(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub),Uu) ) ).

% ATP.lambda_420
tff(fact_8598_ATP_Olambda__421,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(B),Uub: C] : aa(C,A,aa(set(B),fun(C,A),aTP_Lamp_jb(fun(B,fun(C,A)),fun(set(B),fun(C,A)),Uu),Uua),Uub) = groups7121269368397514597t_prod(B,A,aa(C,fun(B,A),aTP_Lamp_ja(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uub),Uua) ) ).

% ATP.lambda_421
tff(fact_8599_ATP_Olambda__422,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_aiw(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(B,fun(A,C),aTP_Lamp_aiv(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub)),Uu) ) ).

% ATP.lambda_422
tff(fact_8600_ATP_Olambda__423,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_vu(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(B,fun(A,C),aTP_Lamp_vt(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub)),Uu) ) ).

% ATP.lambda_423
tff(fact_8601_ATP_Olambda__424,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(C)
        & topological_t2_space(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_vs(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(B,fun(A,C),aTP_Lamp_vr(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub)),Uu) ) ).

% ATP.lambda_424
tff(fact_8602_ATP_Olambda__425,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_su(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(B,fun(A,C),aTP_Lamp_st(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub)),Uu) ) ).

% ATP.lambda_425
tff(fact_8603_ATP_Olambda__426,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(B),Uub: C] : aa(C,A,aa(set(B),fun(C,A),aTP_Lamp_gq(fun(B,fun(C,A)),fun(set(B),fun(C,A)),Uu),Uua),Uub) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(C,fun(B,A),aTP_Lamp_gp(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uub)),Uua) ) ).

% ATP.lambda_426
tff(fact_8604_ATP_Olambda__427,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(nat,B)),Uub: nat] : aa(nat,B,aa(fun(A,fun(nat,B)),fun(nat,B),aTP_Lamp_el(set(A),fun(fun(A,fun(nat,B)),fun(nat,B)),Uu),Uua),Uub) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(nat,fun(A,B),aTP_Lamp_ek(fun(A,fun(nat,B)),fun(nat,fun(A,B)),Uua),Uub)),Uu) ) ).

% ATP.lambda_427
tff(fact_8605_ATP_Olambda__428,axiom,
    ! [Uu: fun(nat,fun(real,real)),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_rs(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Uu),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),Uu,Uub),zero_zero(real))),semiring_char_0_fact(real,Uub))),aa(nat,real,power_power(real,Uua),Uub)) ).

% ATP.lambda_428
tff(fact_8606_ATP_Olambda__429,axiom,
    ! [Uu: real,Uua: fun(nat,fun(real,real)),Uub: nat] : aa(nat,real,aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_rt(real,fun(fun(nat,fun(real,real)),fun(nat,real)),Uu),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,power_power(real,Uu),Uub)) ).

% ATP.lambda_429
tff(fact_8607_ATP_Olambda__430,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [Uu: real,Uua: fun(nat,fun(A,real)),Uub: nat] : aa(nat,real,aa(fun(nat,fun(A,real)),fun(nat,real),aTP_Lamp_ho(real,fun(fun(nat,fun(A,real)),fun(nat,real)),Uu),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,power_power(real,Uu),Uub)) ) ).

% ATP.lambda_430
tff(fact_8608_ATP_Olambda__431,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu: fun(nat,A),Uua: nat,Uub: A] :
          ( aa(A,$o,aa(nat,fun(A,$o),aTP_Lamp_nh(fun(nat,A),fun(nat,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_es(fun(nat,A),fun(A,fun(nat,A)),Uu),Uub)),set_ord_atMost(nat,Uua)) = zero_zero(A) ) ) ) ).

% ATP.lambda_431
tff(fact_8609_ATP_Olambda__432,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_yw(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,Uu,aa(A,A,plus_plus(A,Uua),Uub))),aa(A,A,Uu,Uua))),Uub) ) ).

% ATP.lambda_432
tff(fact_8610_ATP_Olambda__433,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_yo(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,Uu,aa(A,A,plus_plus(A,Uua),Uub))),aa(A,A,Uu,Uua))),Uub) ) ).

% ATP.lambda_433
tff(fact_8611_ATP_Olambda__434,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_lx(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,power_power(A,zero_zero(A)),Uub))),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_434
tff(fact_8612_ATP_Olambda__435,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_ut(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,Uu,Uub)),aa(A,A,Uu,Uua))),aa(A,A,minus_minus(A,Uub),Uua)) ) ).

% ATP.lambda_435
tff(fact_8613_ATP_Olambda__436,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_yp(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,Uu,Uub)),aa(A,A,Uu,Uua))),aa(A,A,minus_minus(A,Uub),Uua)) ) ).

% ATP.lambda_436
tff(fact_8614_ATP_Olambda__437,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_sb(fun(nat,real),fun(real,fun(nat,real)),Uu),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,Uu,Uub)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uub)))),aa(nat,real,power_power(real,Uua),Uub)) ).

% ATP.lambda_437
tff(fact_8615_ATP_Olambda__438,axiom,
    ! [Uu: real,Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_hp(real,fun(fun(nat,real),fun(nat,real)),Uu),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,power_power(real,Uu),Uub)) ).

% ATP.lambda_438
tff(fact_8616_ATP_Olambda__439,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & ring_1(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_cr(fun(nat,A),fun(A,fun(nat,A)),Uu),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,Uu,Uub))),aa(nat,A,power_power(A,Uua),aa(nat,nat,minus_minus(nat,Uub),aa(nat,nat,suc,zero_zero(nat))))) ) ).

% ATP.lambda_439
tff(fact_8617_ATP_Olambda__440,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_alh(set(A),fun(nat,fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( ( size_size(list(A),Uub) = Uua )
        & distinct(A,Uub)
        & aa(set(A),$o,ord_less_eq(set(A),aa(list(A),set(A),set2(A),Uub)),Uu) ) ) ).

% ATP.lambda_440
tff(fact_8618_ATP_Olambda__441,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_mk(set(A),fun(nat,fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,ord_less_eq(set(A),aa(list(A),set(A),set2(A),Uub)),Uu)
        & aa(nat,$o,ord_less_eq(nat,size_size(list(A),Uub)),Uua) ) ) ).

% ATP.lambda_441
tff(fact_8619_ATP_Olambda__442,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_ly(set(A),fun(nat,fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,ord_less_eq(set(A),aa(list(A),set(A),set2(A),Uub)),Uu)
        & ( size_size(list(A),Uub) = Uua ) ) ) ).

% ATP.lambda_442
tff(fact_8620_ATP_Olambda__443,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ro(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,diffs(A,Uu)),Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_443
tff(fact_8621_ATP_Olambda__444,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_akz(set(nat),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( member(nat,aa(nat,nat,suc,Uub),Uu)
        & aa(nat,$o,ord_less(nat,Uub),Uua) ) ) ).

% ATP.lambda_444
tff(fact_8622_ATP_Olambda__445,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_oc(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,Uu),aa(nat,A,semiring_1_of_nat(A),Uub))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,Uua),Uub))) ) ).

% ATP.lambda_445
tff(fact_8623_ATP_Olambda__446,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: set(A)] :
      ( aa(set(A),$o,aa(nat,fun(set(A),$o),aTP_Lamp_akx(set(A),fun(nat,fun(set(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,ord_less_eq(set(A),Uub),Uu)
        & ( finite_card(A,Uub) = Uua ) ) ) ).

% ATP.lambda_446
tff(fact_8624_ATP_Olambda__447,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_aky(set(nat),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( member(nat,Uub,Uu)
        & aa(nat,$o,ord_less(nat,Uub),aa(nat,nat,suc,Uua)) ) ) ).

% ATP.lambda_447
tff(fact_8625_ATP_Olambda__448,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & ring_1(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_cq(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,Uu),Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_448
tff(fact_8626_ATP_Olambda__449,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_cw(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,Uu),Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_449
tff(fact_8627_ATP_Olambda__450,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cx(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,Uua),Uub)),aa(nat,A,power_power(A,Uu),Uub)) ) ).

% ATP.lambda_450
tff(fact_8628_ATP_Olambda__451,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_ev(nat,fun(nat,fun(nat,nat)),Uu),Uua),Uub) = aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,Uua),Uub)),aa(nat,nat,minus_minus(nat,Uu),Uub)) ).

% ATP.lambda_451
tff(fact_8629_ATP_Olambda__452,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_ac(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,Uu)
        | member(A,Uub,Uua) ) ) ).

% ATP.lambda_452
tff(fact_8630_ATP_Olambda__453,axiom,
    ! [A: $tType,Uu: A,Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_ah(A,fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | member(A,Uub,Uua) ) ) ).

% ATP.lambda_453
tff(fact_8631_ATP_Olambda__454,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_kv(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,ord_less_eq(int,Uu),Uua)
        & aa(int,$o,ord_less(int,Uua),Uub) ) ) ).

% ATP.lambda_454
tff(fact_8632_ATP_Olambda__455,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: word(A),Uua: word(A),Uub: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),aTP_Lamp_an(word(A),fun(word(A),fun(word(A),$o)),Uu),Uua),Uub)
        <=> ( aa(word(A),$o,ord_less_eq(word(A),Uu),Uub)
            & aa(word(A),$o,ord_less_eq(word(A),Uub),Uua) ) ) ) ).

% ATP.lambda_455
tff(fact_8633_ATP_Olambda__456,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_nk(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,ord_less_eq(int,Uu),Uub)
        & aa(int,$o,ord_less_eq(int,Uub),Uua) ) ) ).

% ATP.lambda_456
tff(fact_8634_ATP_Olambda__457,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_kw(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,ord_less_eq(int,Uu),Uub)
        & aa(int,$o,ord_less(int,Uua),Uub) ) ) ).

% ATP.lambda_457
tff(fact_8635_ATP_Olambda__458,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_nn(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,ord_less_eq(int,Uu),Uub)
        & aa(int,$o,ord_less(int,Uub),Uua) ) ) ).

% ATP.lambda_458
tff(fact_8636_ATP_Olambda__459,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_nm(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,ord_less(int,Uu),Uub)
        & aa(int,$o,ord_less_eq(int,Uub),Uua) ) ) ).

% ATP.lambda_459
tff(fact_8637_ATP_Olambda__460,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_nl(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,ord_less(int,Uu),Uub)
        & aa(int,$o,ord_less(int,Uub),Uua) ) ) ).

% ATP.lambda_460
tff(fact_8638_ATP_Olambda__461,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_mj(set(A),fun(set(A),fun(set(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),set(A),minus_minus(set(A),Uub),Uu) = aa(set(A),set(A),minus_minus(set(A),Uua),Uu) ) ) ).

% ATP.lambda_461
tff(fact_8639_ATP_Olambda__462,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ao(set(A),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,Uu)
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_462
tff(fact_8640_ATP_Olambda__463,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_af(A,fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_463
tff(fact_8641_ATP_Olambda__464,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ae(A,fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_464
tff(fact_8642_ATP_Olambda__465,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_ml(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_465
tff(fact_8643_ATP_Olambda__466,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_nq(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_466
tff(fact_8644_ATP_Olambda__467,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_mn(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & ( aa(A,B,Uua,Uub) != one_one(B) ) ) ) ) ).

% ATP.lambda_467
tff(fact_8645_ATP_Olambda__468,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,A),Uua: set(B),Uub: B] :
          ( aa(B,$o,aa(set(B),fun(B,$o),aTP_Lamp_nr(fun(B,A),fun(set(B),fun(B,$o)),Uu),Uua),Uub)
        <=> ( member(B,Uub,Uua)
            & ( aa(B,A,Uu,Uub) != zero_zero(A) ) ) ) ) ).

% ATP.lambda_468
tff(fact_8646_ATP_Olambda__469,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_parity(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_aks(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & ~ aa(B,$o,dvd_dvd(B,aa(num,B,numeral_numeral(B),bit0(one2))),aa(A,B,Uua,Uub)) ) ) ) ).

% ATP.lambda_469
tff(fact_8647_ATP_Olambda__470,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_aq(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,Uu)
        & ~ member(A,Uub,Uua) ) ) ).

% ATP.lambda_470
tff(fact_8648_ATP_Olambda__471,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_kl(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,ord_less_eq(nat,aa(nat,nat,plus_plus(nat,Uua),Uub)),Uu) ) ).

% ATP.lambda_471
tff(fact_8649_ATP_Olambda__472,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: A,Uua: real,Uub: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aTP_Lamp_acu(A,fun(real,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_472
tff(fact_8650_ATP_Olambda__473,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_kq(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,ord_less(nat,aa(nat,nat,plus_plus(nat,Uua),Uub)),Uu) ) ).

% ATP.lambda_473
tff(fact_8651_ATP_Olambda__474,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_pd(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Uub),Uu)),Uua) ) ).

% ATP.lambda_474
tff(fact_8652_ATP_Olambda__475,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_pb(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),Uu),Uub)),Uua) ) ).

% ATP.lambda_475
tff(fact_8653_ATP_Olambda__476,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_pc(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Uub),Uu)),Uua) ) ).

% ATP.lambda_476
tff(fact_8654_ATP_Olambda__477,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_pa(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),Uu),Uub)),Uua) ) ).

% ATP.lambda_477
tff(fact_8655_ATP_Olambda__478,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: B,Uub: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_ka(set(product_prod(B,A)),fun(B,fun(A,$o)),Uu),Uua),Uub)
    <=> member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uua),Uub),Uu) ) ).

% ATP.lambda_478
tff(fact_8656_ATP_Olambda__479,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_jz(set(product_prod(A,B)),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub),Uu) ) ).

% ATP.lambda_479
tff(fact_8657_ATP_Olambda__480,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( aa(complex,$o,aa(complex,fun(complex,$o),aTP_Lamp_gn(nat,fun(complex,fun(complex,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,complex,power_power(complex,Uub),Uu) = Uua ) ) ).

% ATP.lambda_480
tff(fact_8658_ATP_Olambda__481,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( aa(complex,$o,aa(nat,fun(complex,$o),aTP_Lamp_jw(complex,fun(nat,fun(complex,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,complex,power_power(complex,Uub),Uua) = Uu ) ) ).

% ATP.lambda_481
tff(fact_8659_ATP_Olambda__482,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu: A,Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_mt(A,fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,ring_1_Ints(A))
            & aa(A,$o,ord_less_eq(A,Uu),Uub)
            & aa(A,$o,ord_less_eq(A,Uub),Uua) ) ) ) ).

% ATP.lambda_482
tff(fact_8660_ATP_Olambda__483,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_bz(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,suc,Uub))),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_483
tff(fact_8661_ATP_Olambda__484,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_by(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,suc,Uub))),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_484
tff(fact_8662_ATP_Olambda__485,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,int),Uub: nat] : aa(nat,real,aa(fun(nat,int),fun(nat,real),aTP_Lamp_abi(fun(nat,real),fun(fun(nat,int),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,minus_minus(real,aa(nat,real,Uu,Uub)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),aa(nat,int,Uua,Uub))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))) ).

% ATP.lambda_485
tff(fact_8663_ATP_Olambda__486,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_sc(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,Uu,Uub)),aa(nat,real,power_power(real,Uua),aa(nat,nat,suc,Uub))) ).

% ATP.lambda_486
tff(fact_8664_ATP_Olambda__487,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_az(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,Uu,Uub)),aa(nat,real,power_power(real,Uua),Uub)) ).

% ATP.lambda_487
tff(fact_8665_ATP_Olambda__488,axiom,
    ! [Uu: fun(nat,nat),Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_fh(fun(nat,nat),fun(nat,fun(nat,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,Uu,Uub)),aa(nat,nat,power_power(nat,Uua),Uub)) ).

% ATP.lambda_488
tff(fact_8666_ATP_Olambda__489,axiom,
    ! [B: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(nat,B),Uua: B,Uub: nat] : aa(nat,B,aa(B,fun(nat,B),aTP_Lamp_ze(fun(nat,B),fun(B,fun(nat,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,Uu,Uub)),aa(nat,B,power_power(B,Uua),Uub)) ) ).

% ATP.lambda_489
tff(fact_8667_ATP_Olambda__490,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_es(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_490
tff(fact_8668_ATP_Olambda__491,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_bp(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_491
tff(fact_8669_ATP_Olambda__492,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_bx(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_492
tff(fact_8670_ATP_Olambda__493,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_cv(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_493
tff(fact_8671_ATP_Olambda__494,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_mult(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_et(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_494
tff(fact_8672_ATP_Olambda__495,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_fd(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_495
tff(fact_8673_ATP_Olambda__496,axiom,
    ! [Uu: fun(nat,$o),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_mh(fun(nat,$o),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,Uu,Uub)
        & aa(nat,$o,ord_less(nat,Uub),Uua) ) ) ).

% ATP.lambda_496
tff(fact_8674_ATP_Olambda__497,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,real),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_sg(fun(A,real),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_497
tff(fact_8675_ATP_Olambda__498,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,real),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ail(fun(A,real),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_498
tff(fact_8676_ATP_Olambda__499,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,real),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vi(fun(A,real),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_499
tff(fact_8677_ATP_Olambda__500,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,real),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_xf(fun(A,real),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_500
tff(fact_8678_ATP_Olambda__501,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,real,aa(fun(A,B),fun(A,real),aTP_Lamp_aho(fun(A,B),fun(fun(A,B),fun(A,real)),Uu),Uua),Uub) = real_V557655796197034286t_dist(B,aa(A,B,Uu,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_501
tff(fact_8679_ATP_Olambda__502,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,real,aa(fun(A,B),fun(A,real),aTP_Lamp_acv(fun(A,B),fun(fun(A,B),fun(A,real)),Uu),Uua),Uub) = real_V557655796197034286t_dist(B,aa(A,B,Uu,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_502
tff(fact_8680_ATP_Olambda__503,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,real,aa(fun(A,B),fun(A,real),aTP_Lamp_acw(fun(A,B),fun(fun(A,B),fun(A,real)),Uu),Uua),Uub) = real_V557655796197034286t_dist(B,aa(A,B,Uu,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_503
tff(fact_8681_ATP_Olambda__504,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] :
      ( aa(A,$o,aa(fun(A,real),fun(A,$o),aTP_Lamp_aew(fun(A,real),fun(fun(A,real),fun(A,$o)),Uu),Uua),Uub)
    <=> aa(real,$o,ord_less_eq(real,aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_504
tff(fact_8682_ATP_Olambda__505,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_aei(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less_eq(B,aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_505
tff(fact_8683_ATP_Olambda__506,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_aen(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less_eq(B,aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_506
tff(fact_8684_ATP_Olambda__507,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_afj(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less_eq(B,aa(A,B,Uua,Uub)),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_507
tff(fact_8685_ATP_Olambda__508,axiom,
    ! [Uu: fun(real,real),Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_agb(fun(real,real),fun(fun(real,real),fun(real,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,Uu,Uub)),aa(real,real,Uua,Uub)) ).

% ATP.lambda_508
tff(fact_8686_ATP_Olambda__509,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ix(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_509
tff(fact_8687_ATP_Olambda__510,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tr(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_510
tff(fact_8688_ATP_Olambda__511,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_te(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_511
tff(fact_8689_ATP_Olambda__512,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aht(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_512
tff(fact_8690_ATP_Olambda__513,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_qy(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,Uu,Uub)),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_513
tff(fact_8691_ATP_Olambda__514,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yq(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_514
tff(fact_8692_ATP_Olambda__515,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_abw(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_515
tff(fact_8693_ATP_Olambda__516,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_agv(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_516
tff(fact_8694_ATP_Olambda__517,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_wa(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_517
tff(fact_8695_ATP_Olambda__518,axiom,
    ! [Uu: fun(real,real),Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_adj(fun(real,real),fun(fun(real,real),fun(real,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,Uua,Uub)),aa(real,real,Uu,Uub)) ).

% ATP.lambda_518
tff(fact_8696_ATP_Olambda__519,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_ajc(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less(B,aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_519
tff(fact_8697_ATP_Olambda__520,axiom,
    ! [Uu: fun(nat,nat),Uua: fun(nat,nat),Uub: nat] : aa(nat,nat,aa(fun(nat,nat),fun(nat,nat),aTP_Lamp_of(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,Uu,Uub)),aa(nat,nat,Uua,Uub)) ).

% ATP.lambda_520
tff(fact_8698_ATP_Olambda__521,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_agm(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_521
tff(fact_8699_ATP_Olambda__522,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_oe(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_522
tff(fact_8700_ATP_Olambda__523,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_jc(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_523
tff(fact_8701_ATP_Olambda__524,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_sw(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_524
tff(fact_8702_ATP_Olambda__525,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ahw(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_525
tff(fact_8703_ATP_Olambda__526,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4211221413907600880p_mult(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ahx(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_526
tff(fact_8704_ATP_Olambda__527,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_qw(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,Uu,Uub)),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_527
tff(fact_8705_ATP_Olambda__528,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ye(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_528
tff(fact_8706_ATP_Olambda__529,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4211221413907600880p_mult(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yd(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_529
tff(fact_8707_ATP_Olambda__530,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_abq(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_530
tff(fact_8708_ATP_Olambda__531,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_wd(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_531
tff(fact_8709_ATP_Olambda__532,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_agt(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_532
tff(fact_8710_ATP_Olambda__533,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ya(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_533
tff(fact_8711_ATP_Olambda__534,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_xx(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_534
tff(fact_8712_ATP_Olambda__535,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_aca(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uub)),aa(A,real,Uu,Uub)) ).

% ATP.lambda_535
tff(fact_8713_ATP_Olambda__536,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ha(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uub)),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_536
tff(fact_8714_ATP_Olambda__537,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_aai(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,minus_minus(real,aa(nat,real,Uu,Uub)),aa(nat,real,Uua,Uub)) ).

% ATP.lambda_537
tff(fact_8715_ATP_Olambda__538,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bi(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,minus_minus(A,aa(nat,A,Uu,Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_538
tff(fact_8716_ATP_Olambda__539,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_gx(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,minus_minus(A,aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_539
tff(fact_8717_ATP_Olambda__540,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_sk(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,minus_minus(B,aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_540
tff(fact_8718_ATP_Olambda__541,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ahv(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,minus_minus(B,aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_541
tff(fact_8719_ATP_Olambda__542,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_qv(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,minus_minus(A,aa(A,A,Uu,Uub)),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_542
tff(fact_8720_ATP_Olambda__543,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yr(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,minus_minus(B,aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_543
tff(fact_8721_ATP_Olambda__544,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yg(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,minus_minus(B,aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_544
tff(fact_8722_ATP_Olambda__545,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yk(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,minus_minus(B,aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_545
tff(fact_8723_ATP_Olambda__546,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yf(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,minus_minus(B,aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_546
tff(fact_8724_ATP_Olambda__547,axiom,
    ! [A: $tType] :
      ( ( topolo1287966508704411220up_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_nc(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,minus_minus(A,aa(nat,A,Uua,Uub)),aa(nat,A,Uu,Uub)) ) ).

% ATP.lambda_547
tff(fact_8725_ATP_Olambda__548,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: fun(A,nat),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_hg(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,minus_minus(nat,aa(A,nat,Uua,Uub)),aa(A,nat,Uu,Uub)) ) ).

% ATP.lambda_548
tff(fact_8726_ATP_Olambda__549,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_gk(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,minus_minus(nat,aa(A,nat,Uua,Uub)),aa(A,nat,Uu,Uub)) ).

% ATP.lambda_549
tff(fact_8727_ATP_Olambda__550,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ym(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,minus_minus(B,aa(A,B,Uua,Uub)),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_550
tff(fact_8728_ATP_Olambda__551,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1898628316856586783d_mult(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,nat),Uub: A] : aa(A,B,aa(fun(A,nat),fun(A,B),aTP_Lamp_aiq(fun(A,B),fun(fun(A,nat),fun(A,B)),Uu),Uua),Uub) = aa(nat,B,power_power(B,aa(A,B,Uu,Uub)),aa(A,nat,Uua,Uub)) ) ).

% ATP.lambda_551
tff(fact_8729_ATP_Olambda__552,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1898628316856586783d_mult(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,nat),Uub: A] : aa(A,B,aa(fun(A,nat),fun(A,B),aTP_Lamp_ww(fun(A,B),fun(fun(A,nat),fun(A,B)),Uu),Uua),Uub) = aa(nat,B,power_power(B,aa(A,B,Uu,Uub)),aa(A,nat,Uua,Uub)) ) ).

% ATP.lambda_552
tff(fact_8730_ATP_Olambda__553,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [Uu: fun(A,B),Uua: fun(A,nat),Uub: A] : aa(A,B,aa(fun(A,nat),fun(A,B),aTP_Lamp_wx(fun(A,B),fun(fun(A,nat),fun(A,B)),Uu),Uua),Uub) = aa(nat,B,power_power(B,aa(A,B,Uu,Uub)),aa(A,nat,Uua,Uub)) ) ).

% ATP.lambda_553
tff(fact_8731_ATP_Olambda__554,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aiy(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = ord_min(B,aa(A,B,Uu,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_554
tff(fact_8732_ATP_Olambda__555,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vn(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = ord_min(B,aa(A,B,Uu,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_555
tff(fact_8733_ATP_Olambda__556,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vm(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = ord_min(B,aa(A,B,Uu,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_556
tff(fact_8734_ATP_Olambda__557,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aic(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),ord_max(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_557
tff(fact_8735_ATP_Olambda__558,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_xt(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),ord_max(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_558
tff(fact_8736_ATP_Olambda__559,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_xs(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),ord_max(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_559
tff(fact_8737_ATP_Olambda__560,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,complex,aa(fun(A,real),fun(A,complex),aTP_Lamp_xp(fun(A,real),fun(fun(A,real),fun(A,complex)),Uu),Uua),Uub) = complex2(aa(A,real,Uu,Uub),aa(A,real,Uua,Uub)) ).

% ATP.lambda_560
tff(fact_8738_ATP_Olambda__561,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bf(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,plus_plus(A,aa(nat,A,Uu,Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_561
tff(fact_8739_ATP_Olambda__562,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_gs(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,plus_plus(A,aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_562
tff(fact_8740_ATP_Olambda__563,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_sj(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,plus_plus(B,aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_563
tff(fact_8741_ATP_Olambda__564,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ahu(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,plus_plus(B,aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_564
tff(fact_8742_ATP_Olambda__565,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_qx(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,plus_plus(A,aa(A,A,Uu,Uub)),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_565
tff(fact_8743_ATP_Olambda__566,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yh(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,plus_plus(B,aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_566
tff(fact_8744_ATP_Olambda__567,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_abr(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,plus_plus(real,aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_567
tff(fact_8745_ATP_Olambda__568,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_agr(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,plus_plus(B,aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_568
tff(fact_8746_ATP_Olambda__569,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yi(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,plus_plus(B,aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_569
tff(fact_8747_ATP_Olambda__570,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ns(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,plus_plus(B,aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_570
tff(fact_8748_ATP_Olambda__571,axiom,
    ! [Uu: fun(real,real),Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_rq(fun(real,real),fun(fun(real,real),fun(real,real)),Uu),Uua),Uub) = powr(real,aa(real,real,Uu,Uub),aa(real,real,Uua,Uub)) ).

% ATP.lambda_571
tff(fact_8749_ATP_Olambda__572,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_tx(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uu,Uub),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_572
tff(fact_8750_ATP_Olambda__573,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ahr(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uu,Uub),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_573
tff(fact_8751_ATP_Olambda__574,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_zv(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uu,Uub),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_574
tff(fact_8752_ATP_Olambda__575,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_wj(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uu,Uub),aa(A,real,Uua,Uub)) ).

% ATP.lambda_575
tff(fact_8753_ATP_Olambda__576,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ajf(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,log(aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_576
tff(fact_8754_ATP_Olambda__577,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_aaq(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,log(aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_577
tff(fact_8755_ATP_Olambda__578,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_vy(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,log(aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_578
tff(fact_8756_ATP_Olambda__579,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,C),Uub: A] : aa(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_aia(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),Uu),Uua),Uub) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu,Uub)),aa(A,C,Uua,Uub)) ) ).

% ATP.lambda_579
tff(fact_8757_ATP_Olambda__580,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,C),Uub: A] : aa(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_ux(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),Uu),Uua),Uub) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu,Uub)),aa(A,C,Uua,Uub)) ) ).

% ATP.lambda_580
tff(fact_8758_ATP_Olambda__581,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,C),Uub: A] : aa(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_xw(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),Uu),Uua),Uub) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu,Uub)),aa(A,C,Uua,Uub)) ) ).

% ATP.lambda_581
tff(fact_8759_ATP_Olambda__582,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_aj(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,Uu,Uub)
       => aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_582
tff(fact_8760_ATP_Olambda__583,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
          ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_acm(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,$o,Uu,Uub)
            | aa(A,$o,Uua,Uub) ) ) ) ).

% ATP.lambda_583
tff(fact_8761_ATP_Olambda__584,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ab(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,Uu,Uub)
        | aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_584
tff(fact_8762_ATP_Olambda__585,axiom,
    ! [B: $tType,Uu: fun(B,$o),Uua: fun(B,$o),Uub: B] :
      ( aa(B,$o,aa(fun(B,$o),fun(B,$o),aTP_Lamp_ald(fun(B,$o),fun(fun(B,$o),fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(B,$o,Uu,Uub)
        & aa(B,$o,Uua,Uub) ) ) ).

% ATP.lambda_585
tff(fact_8763_ATP_Olambda__586,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
          ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_acl(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,$o,Uu,Uub)
            & aa(A,$o,Uua,Uub) ) ) ) ).

% ATP.lambda_586
tff(fact_8764_ATP_Olambda__587,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_nj(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,Uu,Uub)
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_587
tff(fact_8765_ATP_Olambda__588,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_agh(fun(nat,A),fun(fun(nat,A),fun(nat,$o)),Uu),Uua),Uub)
        <=> ( aa(nat,A,Uu,Uub) = aa(nat,A,Uua,Uub) ) ) ) ).

% ATP.lambda_588
tff(fact_8766_ATP_Olambda__589,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(B,A),Uub: B] :
      ( aa(B,$o,aa(fun(B,A),fun(B,$o),aTP_Lamp_adr(fun(B,A),fun(fun(B,A),fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(B,A,Uu,Uub) = aa(B,A,Uua,Uub) ) ) ).

% ATP.lambda_589
tff(fact_8767_ATP_Olambda__590,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_aeh(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_590
tff(fact_8768_ATP_Olambda__591,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: fun(A,A),Uub: A] :
          ( aa(A,$o,aa(fun(A,A),fun(A,$o),aTP_Lamp_aec(fun(A,A),fun(fun(A,A),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,A,Uu,Uub) = aa(A,A,Uua,Uub) ) ) ) ).

% ATP.lambda_591
tff(fact_8769_ATP_Olambda__592,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_aff(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_592
tff(fact_8770_ATP_Olambda__593,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_aed(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,Uu,Uub)
      <=> aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_593
tff(fact_8771_ATP_Olambda__594,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_adp(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_594
tff(fact_8772_ATP_Olambda__595,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
      ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_adw(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,B,Uu,Uub) = aa(A,B,Uua,Uub) ) ) ).

% ATP.lambda_595
tff(fact_8773_ATP_Olambda__596,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_afx(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less(B,aa(A,B,Uu,Uub)),aa(int,B,ring_1_of_int(B),archimedean_ceiling(B,Uua))) ) ) ).

% ATP.lambda_596
tff(fact_8774_ATP_Olambda__597,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: nat,Uub: A] : aa(A,A,aa(nat,fun(A,A),aTP_Lamp_ri(fun(A,A),fun(nat,fun(A,A)),Uu),Uua),Uub) = aa(nat,A,power_power(A,aa(A,A,Uu,Uub)),aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_597
tff(fact_8775_ATP_Olambda__598,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_aij(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = comm_s3205402744901411588hammer(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_598
tff(fact_8776_ATP_Olambda__599,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,real),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_dp(fun(nat,real),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,aa(nat,real,Uu,Uub)),Uua) ) ).

% ATP.lambda_599
tff(fact_8777_ATP_Olambda__600,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: fun(B,real),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_eg(fun(B,real),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,aa(B,real,Uu,Uub)),Uua) ) ).

% ATP.lambda_600
tff(fact_8778_ATP_Olambda__601,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,real),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_sq(fun(A,real),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uu,Uub)),Uua) ) ).

% ATP.lambda_601
tff(fact_8779_ATP_Olambda__602,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(C),Uub: B] : aa(B,A,aa(set(C),fun(B,A),aTP_Lamp_iz(fun(B,fun(C,A)),fun(set(C),fun(B,A)),Uu),Uua),Uub) = groups7121269368397514597t_prod(C,A,aa(B,fun(C,A),Uu,Uub),Uua) ) ).

% ATP.lambda_602
tff(fact_8780_ATP_Olambda__603,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [Uu: set(A),Uua: fun(B,fun(A,C)),Uub: B] : aa(B,C,aa(fun(B,fun(A,C)),fun(B,C),aTP_Lamp_wn(set(A),fun(fun(B,fun(A,C)),fun(B,C)),Uu),Uua),Uub) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),Uua,Uub),Uu) ) ).

% ATP.lambda_603
tff(fact_8781_ATP_Olambda__604,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(C),Uub: B] : aa(B,A,aa(set(C),fun(B,A),aTP_Lamp_go(fun(B,fun(C,A)),fun(set(C),fun(B,A)),Uu),Uua),Uub) = aa(set(C),A,groups7311177749621191930dd_sum(C,A,aa(B,fun(C,A),Uu,Uub)),Uua) ) ).

% ATP.lambda_604
tff(fact_8782_ATP_Olambda__605,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu: set(A),Uua: fun(B,fun(A,B)),Uub: B] : aa(B,B,aa(fun(B,fun(A,B)),fun(B,B),aTP_Lamp_ql(set(A),fun(fun(B,fun(A,B)),fun(B,B)),Uu),Uua),Uub) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(B,fun(A,B),Uua,Uub)),Uu) ) ).

% ATP.lambda_605
tff(fact_8783_ATP_Olambda__606,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [Uu: set(A),Uua: fun(B,fun(A,C)),Uub: B] : aa(B,C,aa(fun(B,fun(A,C)),fun(B,C),aTP_Lamp_vw(set(A),fun(fun(B,fun(A,C)),fun(B,C)),Uu),Uua),Uub) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(B,fun(A,C),Uua,Uub)),Uu) ) ).

% ATP.lambda_606
tff(fact_8784_ATP_Olambda__607,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,real,aa(B,fun(A,real),aTP_Lamp_acx(fun(A,B),fun(B,fun(A,real)),Uu),Uua),Uub) = real_V557655796197034286t_dist(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_607
tff(fact_8785_ATP_Olambda__608,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [Uu: fun(nat,set(A)),Uua: set(A),Uub: nat] :
          ( aa(nat,$o,aa(set(A),fun(nat,$o),aTP_Lamp_agq(fun(nat,set(A)),fun(set(A),fun(nat,$o)),Uu),Uua),Uub)
        <=> aa(set(A),$o,ord_less_eq(set(A),aa(nat,set(A),Uu,Uub)),Uua) ) ) ).

% ATP.lambda_608
tff(fact_8786_ATP_Olambda__609,axiom,
    ! [Uu: fun(nat,nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_mi(fun(nat,nat),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,ord_less_eq(nat,aa(nat,nat,Uu,Uub)),Uua) ) ).

% ATP.lambda_609
tff(fact_8787_ATP_Olambda__610,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_afh(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less_eq(B,aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_610
tff(fact_8788_ATP_Olambda__611,axiom,
    ! [A: $tType,B: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_afg(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less_eq(B,aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_611
tff(fact_8789_ATP_Olambda__612,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_aez(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less_eq(B,aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_612
tff(fact_8790_ATP_Olambda__613,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_aeu(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less_eq(B,aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_613
tff(fact_8791_ATP_Olambda__614,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_em(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_614
tff(fact_8792_ATP_Olambda__615,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ax(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_615
tff(fact_8793_ATP_Olambda__616,axiom,
    ! [B: $tType,A: $tType] :
      ( field(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_gy(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_616
tff(fact_8794_ATP_Olambda__617,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_qu(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_617
tff(fact_8795_ATP_Olambda__618,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu: 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)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_618
tff(fact_8796_ATP_Olambda__619,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_zk(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,Uua,Uub)),Uu) ) ).

% ATP.lambda_619
tff(fact_8797_ATP_Olambda__620,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_aem(fun(B,A),fun(A,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,ord_less(A,aa(B,A,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_620
tff(fact_8798_ATP_Olambda__621,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_aej(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less(B,aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_621
tff(fact_8799_ATP_Olambda__622,axiom,
    ! [A: $tType,B: $tType] :
      ( ( dense_linorder(B)
        & no_bot(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_aev(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less(B,aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_622
tff(fact_8800_ATP_Olambda__623,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_bh(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_623
tff(fact_8801_ATP_Olambda__624,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_0(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_gv(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_624
tff(fact_8802_ATP_Olambda__625,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_sm(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_625
tff(fact_8803_ATP_Olambda__626,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_ahk(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_626
tff(fact_8804_ATP_Olambda__627,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_ahz(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_627
tff(fact_8805_ATP_Olambda__628,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_qr(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_628
tff(fact_8806_ATP_Olambda__629,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_yb(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_629
tff(fact_8807_ATP_Olambda__630,axiom,
    ! [A: $tType,Uu: fun(A,assn),Uua: assn,Uub: A] : aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_au(fun(A,assn),fun(assn,fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(A,assn,Uu,Uub)),Uua) ).

% ATP.lambda_630
tff(fact_8808_ATP_Olambda__631,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_wc(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_631
tff(fact_8809_ATP_Olambda__632,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_xy(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_632
tff(fact_8810_ATP_Olambda__633,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: real,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_tk(real,fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uub)),Uu) ) ).

% ATP.lambda_633
tff(fact_8811_ATP_Olambda__634,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cg(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uua,Uub)),Uu) ) ).

% ATP.lambda_634
tff(fact_8812_ATP_Olambda__635,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_zj(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uua,Uub)),Uu) ) ).

% ATP.lambda_635
tff(fact_8813_ATP_Olambda__636,axiom,
    ! [B: $tType,A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_up(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uua,Uub)),Uu) ) ).

% ATP.lambda_636
tff(fact_8814_ATP_Olambda__637,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_hi(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,minus_minus(A,aa(nat,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_637
tff(fact_8815_ATP_Olambda__638,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_yl(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,minus_minus(B,aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_638
tff(fact_8816_ATP_Olambda__639,axiom,
    ! [Uu: fun(real,real),Uua: nat,Uub: real] : aa(real,real,aa(nat,fun(real,real),aTP_Lamp_rm(fun(real,real),fun(nat,fun(real,real)),Uu),Uua),Uub) = aa(nat,real,power_power(real,aa(real,real,Uu,Uub)),Uua) ).

% ATP.lambda_639
tff(fact_8817_ATP_Olambda__640,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: fun(B,A),Uua: nat,Uub: B] : aa(B,A,aa(nat,fun(B,A),aTP_Lamp_iw(fun(B,A),fun(nat,fun(B,A)),Uu),Uua),Uub) = aa(nat,A,power_power(A,aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_640
tff(fact_8818_ATP_Olambda__641,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_tn(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,power_power(B,aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_641
tff(fact_8819_ATP_Olambda__642,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_aip(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,power_power(B,aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_642
tff(fact_8820_ATP_Olambda__643,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: nat,Uub: A] : aa(A,A,aa(nat,fun(A,A),aTP_Lamp_rj(fun(A,A),fun(nat,fun(A,A)),Uu),Uua),Uub) = aa(nat,A,power_power(A,aa(A,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_643
tff(fact_8821_ATP_Olambda__644,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_wz(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,power_power(B,aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_644
tff(fact_8822_ATP_Olambda__645,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_agu(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,power_power(B,aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_645
tff(fact_8823_ATP_Olambda__646,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_vx(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,power_power(B,aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_646
tff(fact_8824_ATP_Olambda__647,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_wy(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,power_power(B,aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_647
tff(fact_8825_ATP_Olambda__648,axiom,
    ! [Uu: nat,Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_acz(nat,fun(fun(real,real),fun(real,real)),Uu),Uua),Uub) = aa(nat,real,power_power(real,aa(real,real,Uua,Uub)),Uu) ).

% ATP.lambda_648
tff(fact_8826_ATP_Olambda__649,axiom,
    ! [A: $tType,Uu: nat,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ur(nat,fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(nat,real,power_power(real,aa(A,real,Uua,Uub)),Uu) ).

% ATP.lambda_649
tff(fact_8827_ATP_Olambda__650,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_agn(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,plus_plus(A,aa(nat,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_650
tff(fact_8828_ATP_Olambda__651,axiom,
    ! [Uu: fun(real,real),Uua: real,Uub: real] : aa(real,real,aa(real,fun(real,real),aTP_Lamp_rp(fun(real,real),fun(real,fun(real,real)),Uu),Uua),Uub) = powr(real,aa(real,real,Uu,Uub),Uua) ).

% ATP.lambda_651
tff(fact_8829_ATP_Olambda__652,axiom,
    ! [A: $tType,Uu: real,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_acb(real,fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uua,Uub),Uu) ).

% ATP.lambda_652
tff(fact_8830_ATP_Olambda__653,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(B,A),Uua: set(A),Uub: B] :
          ( aa(B,$o,aa(set(A),fun(B,$o),aTP_Lamp_aet(fun(B,A),fun(set(A),fun(B,$o)),Uu),Uua),Uub)
        <=> member(A,aa(B,A,Uu,Uub),Uua) ) ) ).

% ATP.lambda_653
tff(fact_8831_ATP_Olambda__654,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu: fun(A,B),Uua: set(B),Uub: A] :
          ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_aes(fun(A,B),fun(set(B),fun(A,$o)),Uu),Uua),Uub)
        <=> member(B,aa(A,B,Uu,Uub),Uua) ) ) ).

% ATP.lambda_654
tff(fact_8832_ATP_Olambda__655,axiom,
    ! [C: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: set(A),Uua: fun(C,A),Uub: C] :
          ( aa(C,$o,aa(fun(C,A),fun(C,$o),aTP_Lamp_ajb(set(A),fun(fun(C,A),fun(C,$o)),Uu),Uua),Uub)
        <=> member(A,aa(C,A,Uua,Uub),Uu) ) ) ).

% ATP.lambda_655
tff(fact_8833_ATP_Olambda__656,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,A),Uub: B] :
      ( aa(B,$o,aa(fun(B,A),fun(B,$o),aTP_Lamp_pt(set(A),fun(fun(B,A),fun(B,$o)),Uu),Uua),Uub)
    <=> member(A,aa(B,A,Uua,Uub),Uu) ) ).

% ATP.lambda_656
tff(fact_8834_ATP_Olambda__657,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: $o,Uub: A] :
      ( aa(A,$o,aa($o,fun(A,$o),aTP_Lamp_aee(fun(A,$o),fun($o,fun(A,$o)),Uu),(Uua)),Uub)
    <=> ( aa(A,$o,Uu,Uub)
        | (Uua) ) ) ).

% ATP.lambda_657
tff(fact_8835_ATP_Olambda__658,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_adt(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) = Uua ) ) ) ).

% ATP.lambda_658
tff(fact_8836_ATP_Olambda__659,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo8865339358273720382pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_adv(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) = Uua ) ) ) ).

% ATP.lambda_659
tff(fact_8837_ATP_Olambda__660,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ag(A,fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_660
tff(fact_8838_ATP_Olambda__661,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ob(nat,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,Uua),Uub))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,Uu),Uub))) ) ).

% ATP.lambda_661
tff(fact_8839_ATP_Olambda__662,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: real,Uua: fun(nat,A),Uub: nat] : aa(nat,real,aa(fun(nat,A),fun(nat,real),aTP_Lamp_cd(real,fun(fun(nat,A),fun(nat,real)),Uu),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,power_power(real,Uu),Uub)) ) ).

% ATP.lambda_662
tff(fact_8840_ATP_Olambda__663,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_ags(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uub))),aa(nat,real,Uua,Uub)) ) ) ).

% ATP.lambda_663
tff(fact_8841_ATP_Olambda__664,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,B),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,B),fun(nat,$o),aTP_Lamp_agg(fun(nat,A),fun(fun(nat,B),fun(nat,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uub))),real_V7770717601297561774m_norm(B,aa(nat,B,Uua,Uub))) ) ) ).

% ATP.lambda_664
tff(fact_8842_ATP_Olambda__665,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_afl(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ( archim6421214686448440834_floor(B,aa(A,B,Uu,Uub)) = archim6421214686448440834_floor(B,Uua) ) ) ) ).

% ATP.lambda_665
tff(fact_8843_ATP_Olambda__666,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_afm(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ( archimedean_ceiling(B,aa(A,B,Uu,Uub)) = archimedean_ceiling(B,Uua) ) ) ) ).

% ATP.lambda_666
tff(fact_8844_ATP_Olambda__667,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: real,Uub: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aTP_Lamp_agw(fun(A,B),fun(real,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uub))),Uua) ) ) ).

% ATP.lambda_667
tff(fact_8845_ATP_Olambda__668,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_hw(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,aa(nat,nat,plus_plus(nat,Uub),Uua)))),aa(nat,A,power_power(A,Uu),aa(nat,nat,plus_plus(nat,Uub),Uua))) ) ).

% ATP.lambda_668
tff(fact_8846_ATP_Olambda__669,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_afw(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less(B,aa(int,B,ring_1_of_int(B),archim6421214686448440834_floor(B,Uua))),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_669
tff(fact_8847_ATP_Olambda__670,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: nat,Uua: nat,Uub: word(A)] :
          ( aa(word(A),$o,aa(nat,fun(word(A),$o),aTP_Lamp_og(nat,fun(nat,fun(word(A),$o)),Uu),Uua),Uub)
        <=> aa(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),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Uu)),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Uua))) ) ) ).

% ATP.lambda_670
tff(fact_8848_ATP_Olambda__671,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_oh(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,ord_less(nat,Uub),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uu)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))) ) ).

% ATP.lambda_671
tff(fact_8849_ATP_Olambda__672,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ny(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = groups7121269368397514597t_prod(nat,A,Uu,set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua),aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uua))) ) ).

% ATP.lambda_672
tff(fact_8850_ATP_Olambda__673,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_nz(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua),aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uua))) ) ).

% ATP.lambda_673
tff(fact_8851_ATP_Olambda__674,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_nx(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua),aa(nat,nat,plus_plus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uua))) ) ).

% ATP.lambda_674
tff(fact_8852_ATP_Olambda__675,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_in(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,plus_plus(A,Uu),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uub)),Uua)) ) ).

% ATP.lambda_675
tff(fact_8853_ATP_Olambda__676,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_il(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,plus_plus(A,Uu),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uub)),Uua)) ) ).

% ATP.lambda_676
tff(fact_8854_ATP_Olambda__677,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_hn(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,power_power(A,Uu),aa(nat,nat,minus_minus(nat,Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_677
tff(fact_8855_ATP_Olambda__678,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_aau(real,fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),divide_divide(real),Uua),aa(nat,real,power_power(real,Uu),Uub)) ).

% ATP.lambda_678
tff(fact_8856_ATP_Olambda__679,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_fn(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,power_power(A,Uu),aa(nat,nat,plus_plus(nat,Uua),Uub)) ) ).

% ATP.lambda_679
tff(fact_8857_ATP_Olambda__680,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_im(nat,fun(nat,fun(nat,nat)),Uu),Uua),Uub) = aa(nat,nat,plus_plus(nat,Uu),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)) ).

% ATP.lambda_680
tff(fact_8858_ATP_Olambda__681,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( aa(real,$o,aa(real,fun(real,$o),aTP_Lamp_afq(real,fun(real,fun(real,$o)),Uu),Uua),Uub)
    <=> member(real,Uub,set_or5935395276787703475ssThan(real,Uu,Uua)) ) ).

% ATP.lambda_681
tff(fact_8859_ATP_Olambda__682,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: real,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_eh(real,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,Uu),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_682
tff(fact_8860_ATP_Olambda__683,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: real,Uub: nat] : aa(nat,A,aa(real,fun(nat,A),aTP_Lamp_do(fun(nat,A),fun(real,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,Uua),aa(nat,A,Uu,Uub)) ) ).

% ATP.lambda_683
tff(fact_8861_ATP_Olambda__684,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: real,Uub: A] : aa(A,B,aa(real,fun(A,B),aTP_Lamp_sp(fun(A,B),fun(real,fun(A,B)),Uu),Uua),Uub) = aa(B,B,real_V8093663219630862766scaleR(B,Uua),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_684
tff(fact_8862_ATP_Olambda__685,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: real,Uub: A] : aa(A,A,aa(real,fun(A,A),aTP_Lamp_qo(fun(A,A),fun(real,fun(A,A)),Uu),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,Uua),aa(A,A,Uu,Uub)) ) ).

% ATP.lambda_685
tff(fact_8863_ATP_Olambda__686,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: B,Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_aey(B,fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less_eq(B,Uu),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_686
tff(fact_8864_ATP_Olambda__687,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_afi(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less_eq(B,Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_687
tff(fact_8865_ATP_Olambda__688,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_afa(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less_eq(B,Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_688
tff(fact_8866_ATP_Olambda__689,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_aeo(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less_eq(B,Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_689
tff(fact_8867_ATP_Olambda__690,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_akc(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),aa(nat,A,Uu,Uub)) ) ).

% ATP.lambda_690
tff(fact_8868_ATP_Olambda__691,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_ael(fun(B,A),fun(A,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,ord_less(A,Uua),aa(B,A,Uu,Uub)) ) ) ).

% ATP.lambda_691
tff(fact_8869_ATP_Olambda__692,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_aep(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less(B,Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_692
tff(fact_8870_ATP_Olambda__693,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_aek(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less(B,Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_693
tff(fact_8871_ATP_Olambda__694,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: B,Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ahl(B,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uu),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_694
tff(fact_8872_ATP_Olambda__695,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ay(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_695
tff(fact_8873_ATP_Olambda__696,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_gw(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_696
tff(fact_8874_ATP_Olambda__697,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ch(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_697
tff(fact_8875_ATP_Olambda__698,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_zi(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_698
tff(fact_8876_ATP_Olambda__699,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_uq(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_699
tff(fact_8877_ATP_Olambda__700,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_bg(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),aa(nat,A,Uu,Uub)) ) ).

% ATP.lambda_700
tff(fact_8878_ATP_Olambda__701,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_sl(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_701
tff(fact_8879_ATP_Olambda__702,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_ahy(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_702
tff(fact_8880_ATP_Olambda__703,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_qq(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),aa(A,A,Uu,Uub)) ) ).

% ATP.lambda_703
tff(fact_8881_ATP_Olambda__704,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_yc(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_704
tff(fact_8882_ATP_Olambda__705,axiom,
    ! [A: $tType,Uu: fun(A,assn),Uua: assn,Uub: A] : aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_jy(fun(A,assn),fun(assn,fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Uua),aa(A,assn,Uu,Uub)) ).

% ATP.lambda_705
tff(fact_8883_ATP_Olambda__706,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_wb(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_706
tff(fact_8884_ATP_Olambda__707,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_xz(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_707
tff(fact_8885_ATP_Olambda__708,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linordered_field(B)
        & topolo1944317154257567458pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_ade(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_708
tff(fact_8886_ATP_Olambda__709,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A,Uua: fun(B,nat),Uub: B] : aa(B,A,aa(fun(B,nat),fun(B,A),aTP_Lamp_jg(A,fun(fun(B,nat),fun(B,A)),Uu),Uua),Uub) = aa(nat,A,power_power(A,Uu),aa(B,nat,Uua,Uub)) ) ).

% ATP.lambda_709
tff(fact_8887_ATP_Olambda__710,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [Uu: fun(A,nat),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_abc(fun(A,nat),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,power_power(B,Uua),aa(A,nat,Uu,Uub)) ) ).

% ATP.lambda_710
tff(fact_8888_ATP_Olambda__711,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [Uu: B,Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yj(B,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,plus_plus(B,Uu),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_711
tff(fact_8889_ATP_Olambda__712,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: nat,Uub: A] : aa(A,real,aa(nat,fun(A,real),aTP_Lamp_aio(fun(A,real),fun(nat,fun(A,real)),Uu),Uua),Uub) = aa(real,real,root(Uua),aa(A,real,Uu,Uub)) ) ).

% ATP.lambda_712
tff(fact_8890_ATP_Olambda__713,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: nat,Uub: A] : aa(A,real,aa(nat,fun(A,real),aTP_Lamp_zl(fun(A,real),fun(nat,fun(A,real)),Uu),Uua),Uub) = aa(real,real,root(Uua),aa(A,real,Uu,Uub)) ) ).

% ATP.lambda_713
tff(fact_8891_ATP_Olambda__714,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: nat,Uub: A] : aa(A,real,aa(nat,fun(A,real),aTP_Lamp_xa(fun(A,real),fun(nat,fun(A,real)),Uu),Uua),Uub) = aa(real,real,root(Uua),aa(A,real,Uu,Uub)) ).

% ATP.lambda_714
tff(fact_8892_ATP_Olambda__715,axiom,
    ! [A: $tType,Uu: $o,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_aeg($o,fun(fun(A,$o),fun(A,$o)),(Uu)),Uua),Uub)
    <=> ( (Uu)
       => aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_715
tff(fact_8893_ATP_Olambda__716,axiom,
    ! [A: $tType,Uu: $o,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_aef($o,fun(fun(A,$o),fun(A,$o)),(Uu)),Uua),Uub)
    <=> ( (Uu)
        | aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_716
tff(fact_8894_ATP_Olambda__717,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_pz(fun(A,B),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> ( Uub = aa(A,B,Uu,Uua) ) ) ).

% ATP.lambda_717
tff(fact_8895_ATP_Olambda__718,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jq(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,plus_plus(A,Uu),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,Uua),Uub))) ) ).

% ATP.lambda_718
tff(fact_8896_ATP_Olambda__719,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: real,Uub: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aTP_Lamp_agy(fun(A,B),fun(real,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,ord_less_eq(real,Uua),real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uub))) ) ) ).

% ATP.lambda_719
tff(fact_8897_ATP_Olambda__720,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_mu(fun(A,nat),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,ord_less(nat,zero_zero(nat)),
          $ite(Uub = Uua,aa(nat,nat,suc,aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uub))) ) ).

% ATP.lambda_720
tff(fact_8898_ATP_Olambda__721,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_mq(fun(A,nat),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,ord_less(nat,zero_zero(nat)),
          $ite(aa(A,$o,Uua,Uub),aa(A,nat,Uu,Uub),zero_zero(nat))) ) ).

% ATP.lambda_721
tff(fact_8899_ATP_Olambda__722,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: fun(A,nat),Uub: A] :
      ( aa(A,$o,aa(fun(A,nat),fun(A,$o),aTP_Lamp_mv(fun(A,nat),fun(fun(A,nat),fun(A,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(nat,nat,minus_minus(nat,aa(A,nat,Uu,Uub)),aa(A,nat,Uua,Uub))) ) ).

% ATP.lambda_722
tff(fact_8900_ATP_Olambda__723,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ji(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,minus_minus(nat,Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_723
tff(fact_8901_ATP_Olambda__724,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_hf(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,minus_minus(nat,Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_724
tff(fact_8902_ATP_Olambda__725,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aac(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)) ) ).

% ATP.lambda_725
tff(fact_8903_ATP_Olambda__726,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_zt(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,minus_minus(nat,Uub),Uua)) ) ).

% ATP.lambda_726
tff(fact_8904_ATP_Olambda__727,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: A,Uub: A] : aa(A,B,aa(A,fun(A,B),aTP_Lamp_uv(fun(A,B),fun(A,fun(A,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(A,A,plus_plus(A,Uua),Uub)) ) ).

% ATP.lambda_727
tff(fact_8905_ATP_Olambda__728,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_rh(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,Uu,aa(A,A,plus_plus(A,Uua),Uub)) ) ).

% ATP.lambda_728
tff(fact_8906_ATP_Olambda__729,axiom,
    ! [Uu: fun(real,$o),Uua: real,Uub: real] :
      ( aa(real,$o,aa(real,fun(real,$o),aTP_Lamp_afn(fun(real,$o),fun(real,fun(real,$o)),Uu),Uua),Uub)
    <=> aa(real,$o,Uu,aa(real,real,plus_plus(real,Uub),Uua)) ) ).

% ATP.lambda_729
tff(fact_8907_ATP_Olambda__730,axiom,
    ! [A: $tType,Uu: fun(real,A),Uua: real,Uub: real] : aa(real,A,aa(real,fun(real,A),aTP_Lamp_add(fun(real,A),fun(real,fun(real,A)),Uu),Uua),Uub) = aa(real,A,Uu,aa(real,real,plus_plus(real,Uub),Uua)) ).

% ATP.lambda_730
tff(fact_8908_ATP_Olambda__731,axiom,
    ! [Uu: fun(nat,$o),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_agf(fun(nat,$o),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,Uu,aa(nat,nat,plus_plus(nat,Uub),Uua)) ) ).

% ATP.lambda_731
tff(fact_8909_ATP_Olambda__732,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ba(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,plus_plus(nat,Uub),Uua)) ) ).

% ATP.lambda_732
tff(fact_8910_ATP_Olambda__733,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_zu(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,plus_plus(nat,Uub),Uua)) ) ).

% ATP.lambda_733
tff(fact_8911_ATP_Olambda__734,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ago(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,plus_plus(nat,Uub),Uua)) ) ).

% ATP.lambda_734
tff(fact_8912_ATP_Olambda__735,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jh(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,plus_plus(nat,Uub),Uua)) ) ).

% ATP.lambda_735
tff(fact_8913_ATP_Olambda__736,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ib(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,plus_plus(nat,Uub),Uua)) ) ).

% ATP.lambda_736
tff(fact_8914_ATP_Olambda__737,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,$o),Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_aex(fun(A,$o),fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,Uu,aa(A,A,plus_plus(A,Uub),Uua)) ) ) ).

% ATP.lambda_737
tff(fact_8915_ATP_Olambda__738,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: A,Uub: A] : aa(A,B,aa(A,fun(A,B),aTP_Lamp_uw(fun(A,B),fun(A,fun(A,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(A,A,plus_plus(A,Uub),Uua)) ) ).

% ATP.lambda_738
tff(fact_8916_ATP_Olambda__739,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,B),Uua: A,Uub: A] : aa(A,B,aa(A,fun(A,B),aTP_Lamp_yy(fun(A,B),fun(A,fun(A,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(A,A,plus_plus(A,Uub),Uua)) ) ).

% ATP.lambda_739
tff(fact_8917_ATP_Olambda__740,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_qz(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,Uu,aa(A,A,plus_plus(A,Uub),Uua)) ) ).

% ATP.lambda_740
tff(fact_8918_ATP_Olambda__741,axiom,
    ! [A: $tType,B: $tType,Uu: fun(product_prod(A,B),$o),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_kk(fun(product_prod(A,B),$o),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> aa(product_prod(A,B),$o,Uu,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub)) ) ).

% ATP.lambda_741
tff(fact_8919_ATP_Olambda__742,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: fun(product_prod(A,B),C),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_kt(fun(product_prod(A,B),C),fun(A,fun(B,C)),Uu),Uua),Uub) = aa(product_prod(A,B),C,Uu,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub)) ).

% ATP.lambda_742
tff(fact_8920_ATP_Olambda__743,axiom,
    ! [C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: B,Uua: fun(B,C),Uub: B] : aa(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_act(B,fun(fun(B,C),fun(B,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(B,B,plus_plus(B,Uu),Uub)) ) ).

% ATP.lambda_743
tff(fact_8921_ATP_Olambda__744,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: A,Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_uu(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(A,B,Uua,aa(A,A,plus_plus(A,Uu),Uub)) ) ).

% ATP.lambda_744
tff(fact_8922_ATP_Olambda__745,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cj(nat,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uua,aa(nat,nat,plus_plus(nat,Uub),Uu)) ) ).

% ATP.lambda_745
tff(fact_8923_ATP_Olambda__746,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(real,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_tj(fun(real,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,Uu,aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_746
tff(fact_8924_ATP_Olambda__747,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [Uu: fun(real,A),Uua: fun(nat,real),Uub: nat] : aa(nat,A,aa(fun(nat,real),fun(nat,A),aTP_Lamp_acd(fun(real,A),fun(fun(nat,real),fun(nat,A)),Uu),Uua),Uub) = aa(real,A,Uu,aa(nat,real,Uua,Uub)) ) ).

% ATP.lambda_747
tff(fact_8925_ATP_Olambda__748,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,nat),Uub: nat] : aa(nat,A,aa(fun(nat,nat),fun(nat,A),aTP_Lamp_agl(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,Uua,Uub)) ) ).

% ATP.lambda_748
tff(fact_8926_ATP_Olambda__749,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(C,A),Uua: fun(B,C),Uub: B] : aa(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_pj(fun(C,A),fun(fun(B,C),fun(B,A)),Uu),Uua),Uub) = aa(C,A,Uu,aa(B,C,Uua,Uub)) ).

% ATP.lambda_749
tff(fact_8927_ATP_Olambda__750,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(B,A),Uua: fun(C,B),Uub: C] : aa(C,A,aa(fun(C,B),fun(C,A),aTP_Lamp_op(fun(B,A),fun(fun(C,B),fun(C,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(C,B,Uua,Uub)) ).

% ATP.lambda_750
tff(fact_8928_ATP_Olambda__751,axiom,
    ! [A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [Uu: fun(A,$o),Uua: fun(nat,A),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_ahb(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,Uu,aa(nat,A,Uua,Uub)) ) ) ).

% ATP.lambda_751
tff(fact_8929_ATP_Olambda__752,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(nat,A),Uub: nat] : aa(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_ahh(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_752
tff(fact_8930_ATP_Olambda__753,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [Uu: fun(A,$o),Uua: fun(nat,A),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_ado(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,Uu,aa(nat,A,Uua,Uub)) ) ) ).

% ATP.lambda_753
tff(fact_8931_ATP_Olambda__754,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(nat,A),Uub: nat] : aa(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_aae(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_754
tff(fact_8932_ATP_Olambda__755,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(nat,A),Uub: nat] : aa(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_abs(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_755
tff(fact_8933_ATP_Olambda__756,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_sy(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_756
tff(fact_8934_ATP_Olambda__757,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_aja(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_757
tff(fact_8935_ATP_Olambda__758,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_abo(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_758
tff(fact_8936_ATP_Olambda__759,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_abp(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_759
tff(fact_8937_ATP_Olambda__760,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(nat,A),Uub: nat] : aa(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_aaf(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_760
tff(fact_8938_ATP_Olambda__761,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ahn(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_761
tff(fact_8939_ATP_Olambda__762,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_vc(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_762
tff(fact_8940_ATP_Olambda__763,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_rc(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,Uu,aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_763
tff(fact_8941_ATP_Olambda__764,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ve(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_764
tff(fact_8942_ATP_Olambda__765,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,$o),Uua: fun(B,A),Uub: B] :
      ( aa(B,$o,aa(fun(B,A),fun(B,$o),aTP_Lamp_adq(fun(A,$o),fun(fun(B,A),fun(B,$o)),Uu),Uua),Uub)
    <=> aa(A,$o,Uu,aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_765
tff(fact_8943_ATP_Olambda__766,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_xk(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ).

% ATP.lambda_766
tff(fact_8944_ATP_Olambda__767,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & topological_t2_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(C,A),Uua: fun(A,B),Uub: C] : aa(C,B,aa(fun(A,B),fun(C,B),aTP_Lamp_vf(fun(C,A),fun(fun(A,B),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uua,aa(C,A,Uu,Uub)) ) ).

% ATP.lambda_767
tff(fact_8945_ATP_Olambda__768,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(A,C),Uub: B] : aa(B,C,aa(fun(A,C),fun(B,C),aTP_Lamp_pk(fun(B,A),fun(fun(A,C),fun(B,C)),Uu),Uua),Uub) = aa(A,C,Uua,aa(B,A,Uu,Uub)) ).

% ATP.lambda_768
tff(fact_8946_ATP_Olambda__769,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_si(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_769
tff(fact_8947_ATP_Olambda__770,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_vk(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_770
tff(fact_8948_ATP_Olambda__771,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_afd(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_771
tff(fact_8949_ATP_Olambda__772,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_rd(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,Uua,aa(A,A,Uu,Uub)) ) ).

% ATP.lambda_772
tff(fact_8950_ATP_Olambda__773,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_acy(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_773
tff(fact_8951_ATP_Olambda__774,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topological_t2_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_va(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_774
tff(fact_8952_ATP_Olambda__775,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(B,$o),Uub: A] :
      ( aa(A,$o,aa(fun(B,$o),fun(A,$o),aTP_Lamp_ads(fun(A,B),fun(fun(B,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> aa(B,$o,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_775
tff(fact_8953_ATP_Olambda__776,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_iv(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_776
tff(fact_8954_ATP_Olambda__777,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ia(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_777
tff(fact_8955_ATP_Olambda__778,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( semiring_1(C)
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ale(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_778
tff(fact_8956_ATP_Olambda__779,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: C] : aa(C,B,aa(A,fun(C,B),aTP_Lamp_ps(fun(A,B),fun(A,fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,Uua) ).

% ATP.lambda_779
tff(fact_8957_ATP_Olambda__780,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: fun(nat,B),Uub: A] : aa(A,B,aa(fun(nat,B),fun(A,B),aTP_Lamp_zg(fun(A,B),fun(fun(nat,B),fun(A,B)),Uu),Uua),Uub) = suminf(B,aa(A,fun(nat,B),aa(fun(nat,B),fun(A,fun(nat,B)),aTP_Lamp_zf(fun(A,B),fun(fun(nat,B),fun(A,fun(nat,B))),Uu),Uua),Uub)) ) ).

% ATP.lambda_780
tff(fact_8958_ATP_Olambda__781,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_um(fun(nat,A),fun(A,fun(A,A)),Uu),Uua),Uub) = suminf(A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_ul(fun(nat,A),fun(A,fun(A,fun(nat,A))),Uu),Uua),Uub)) ) ).

% ATP.lambda_781
tff(fact_8959_ATP_Olambda__782,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,real,aa(A,fun(nat,real),aTP_Lamp_cc(fun(nat,A),fun(A,fun(nat,real)),Uu),Uua),Uub) = real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,power_power(A,Uua),Uub))) ) ).

% ATP.lambda_782
tff(fact_8960_ATP_Olambda__783,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,real,aa(fun(A,B),fun(A,real),aTP_Lamp_jo(fun(A,B),fun(fun(A,B),fun(A,real)),Uu),Uua),Uub) = real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub))) ) ).

% ATP.lambda_783
tff(fact_8961_ATP_Olambda__784,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topological_t2_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_aiz(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) != aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_784
tff(fact_8962_ATP_Olambda__785,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_abh(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua),Uub) = cos(real,aa(real,real,minus_minus(real,aa(nat,real,Uu,Uub)),Uua)) ).

% ATP.lambda_785
tff(fact_8963_ATP_Olambda__786,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_afc(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) != Uua ) ) ) ).

% ATP.lambda_786
tff(fact_8964_ATP_Olambda__787,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_agp(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) != Uua ) ) ) ).

% ATP.lambda_787
tff(fact_8965_ATP_Olambda__788,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_afb(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) != Uua ) ) ) ).

% ATP.lambda_788
tff(fact_8966_ATP_Olambda__789,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t1_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_adu(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) != Uua ) ) ) ).

% ATP.lambda_789
tff(fact_8967_ATP_Olambda__790,axiom,
    ! [C: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: A,Uua: fun(C,A),Uub: C] :
          ( aa(C,$o,aa(fun(C,A),fun(C,$o),aTP_Lamp_afe(A,fun(fun(C,A),fun(C,$o)),Uu),Uua),Uub)
        <=> ( aa(C,A,Uua,Uub) != Uu ) ) ) ).

% ATP.lambda_790
tff(fact_8968_ATP_Olambda__791,axiom,
    ! [B: $tType,A: $tType,Uu: set(B),Uua: fun(A,fun(B,$o)),Uub: A] : aa(A,nat,aa(fun(A,fun(B,$o)),fun(A,nat),aTP_Lamp_akw(set(B),fun(fun(A,fun(B,$o)),fun(A,nat)),Uu),Uua),Uub) = finite_card(B,collect(B,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_lz(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uua),Uub))) ).

% ATP.lambda_791
tff(fact_8969_ATP_Olambda__792,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A) )
     => ! [Uu: fun(B,real),Uua: fun(real,A),Uub: B] : aa(B,real,aa(fun(real,A),fun(B,real),aTP_Lamp_uj(fun(B,real),fun(fun(real,A),fun(B,real)),Uu),Uua),Uub) = aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(A,aa(real,A,Uua,aa(B,real,Uu,Uub)))) ) ).

% ATP.lambda_792
tff(fact_8970_ATP_Olambda__793,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_aha(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),Uu),Uua),Uub)
        <=> ! [N3: nat] :
              ( aa(nat,$o,ord_less_eq(nat,Uub),N3)
             => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_or7035219750837199246ssThan(nat,Uub,N3)))),aa(nat,real,Uua,Uub)) ) ) ) ).

% ATP.lambda_793
tff(fact_8971_ATP_Olambda__794,axiom,
    ! [A: $tType] :
      ( ( real_V8037385150606011577_space(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_ahi(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),Uu),Uua),Uub)
        <=> ! [A10: nat] :
              ( aa(nat,$o,ord_less_eq(nat,Uub),A10)
             => ! [B9: nat] :
                  ( aa(nat,$o,ord_less(nat,A10),B9)
                 => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_or3652927894154168847AtMost(nat,A10,B9)))),aa(nat,real,Uua,A10)) ) ) ) ) ).

% ATP.lambda_794
tff(fact_8972_ATP_Olambda__795,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,Uu: fun(B,fun(C,fun(D,A))),Uua: D,Uub: B,Uuc: C] : aa(C,A,aa(B,fun(C,A),aa(D,fun(B,fun(C,A)),aTP_Lamp_kd(fun(B,fun(C,fun(D,A))),fun(D,fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(D,A,aa(C,fun(D,A),aa(B,fun(C,fun(D,A)),Uu,Uub),Uuc),Uua) ).

% ATP.lambda_795
tff(fact_8973_ATP_Olambda__796,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] :
          aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_gd(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(
            ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub)
            & aa(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,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),aa(nat,nat,binomial(Uub),Uuc))))),semiring_char_0_fact(real,Uub))),aa(nat,A,power_power(A,Uu),Uuc))),aa(nat,A,power_power(A,Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))),
            zero_zero(A) ) ) ).

% ATP.lambda_796
tff(fact_8974_ATP_Olambda__797,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] :
          aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_eb(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(
            ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub)
            & ~ aa(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,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),aa(nat,nat,binomial(Uub),Uuc))))),semiring_char_0_fact(real,Uub)))),aa(nat,A,power_power(A,Uu),Uuc))),aa(nat,A,power_power(A,Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))),
            zero_zero(A) ) ) ).

% ATP.lambda_797
tff(fact_8975_ATP_Olambda__798,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] :
          aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_gf(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(aa(nat,$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,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),aa(nat,nat,binomial(Uub),Uuc))))),semiring_char_0_fact(real,Uub))),aa(nat,A,power_power(A,Uu),Uuc))),aa(nat,A,power_power(A,Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))),zero_zero(A)) ) ).

% ATP.lambda_798
tff(fact_8976_ATP_Olambda__799,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: set(A),Uub: A,Uuc: B] :
      aa(B,A,aa(A,fun(B,A),aa(set(A),fun(A,fun(B,A)),aTP_Lamp_qg(fun(A,B),fun(set(A),fun(A,fun(B,A))),Uu),Uua),Uub),Uuc) = $ite(member(B,Uuc,aa(set(A),set(B),image(A,B,Uu),Uua)),the_inv_into(A,B,Uua,Uu,Uuc),Uub) ).

% ATP.lambda_799
tff(fact_8977_ATP_Olambda__800,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_fo(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(
            aa(nat,$o,ord_less(nat,Uuc),Uu),
            aa(nat,A,Uua,Uuc),
            $ite(Uuc = Uu,zero_zero(A),aa(nat,A,Uub,aa(nat,nat,minus_minus(nat,Uuc),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).

% ATP.lambda_800
tff(fact_8978_ATP_Olambda__801,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_js(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(
            aa(nat,$o,ord_less(nat,Uuc),Uu),
            aa(nat,A,Uua,Uuc),
            $ite(Uuc = Uu,one_one(A),aa(nat,A,Uub,aa(nat,nat,minus_minus(nat,Uuc),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).

% ATP.lambda_801
tff(fact_8979_ATP_Olambda__802,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: A,Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_adg(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(aa(A,$o,ord_less_eq(A,Uuc),Uu),aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_802
tff(fact_8980_ATP_Olambda__803,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_jt(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(aa(nat,$o,ord_less(nat,Uuc),Uu),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_803
tff(fact_8981_ATP_Olambda__804,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_fp(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(aa(nat,$o,ord_less(nat,Uuc),Uu),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_804
tff(fact_8982_ATP_Olambda__805,axiom,
    ! [A: $tType] :
      ( ( topolo1287966508704411220up_add(A)
        & topological_t2_space(A) )
     => ! [Uu: 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_nd(fun(nat,A),fun(set(nat),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(member(nat,Uuc,Uua),aa(nat,A,Uub,Uuc),aa(nat,A,Uu,Uuc)) ) ).

% ATP.lambda_805
tff(fact_8983_ATP_Olambda__806,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_nf(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_806
tff(fact_8984_ATP_Olambda__807,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_ne(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_807
tff(fact_8985_ATP_Olambda__808,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: B,Uuc: A] :
          aa(A,B,aa(B,fun(A,B),aa(fun(A,B),fun(B,fun(A,B)),aTP_Lamp_alg(A,fun(fun(A,B),fun(B,fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa(A,B,Uua,Uuc),Uub) ) ).

% ATP.lambda_808
tff(fact_8986_ATP_Olambda__809,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,B),Uua: fun(A,$o),Uub: fun(A,B),Uuc: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,$o),fun(fun(A,B),fun(A,B)),aTP_Lamp_uz(fun(A,B),fun(fun(A,$o),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(aa(A,$o,Uua,Uuc),aa(A,B,Uu,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_809
tff(fact_8987_ATP_Olambda__810,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_hy(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_hx(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_lessThan(nat,aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,Uub),aa(nat,nat,suc,zero_zero(nat)))),Uuc))) ) ).

% ATP.lambda_810
tff(fact_8988_ATP_Olambda__811,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( semiring_0(A)
     => ! [Uu: fun(B,A),Uua: fun(C,A),Uub: set(C),Uuc: B] : aa(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_gu(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(set(C),A,groups7311177749621191930dd_sum(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_gt(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uuc)),Uub) ) ).

% ATP.lambda_811
tff(fact_8989_ATP_Olambda__812,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: nat,Uua: A,Uub: A,Uuc: nat] :
          aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_ft(nat,fun(A,fun(A,fun(nat,A))),Uu),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 = Uu,one_one(A),zero_zero(A)) )),
            aa(nat,A,power_power(A,Uua),Uuc)) ) ).

% ATP.lambda_812
tff(fact_8990_ATP_Olambda__813,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$o)),Uuc: B] : aa(B,C,aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_md(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),Uu),Uua),Uub),Uuc) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_mb(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uuc),collect(A,aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_mc(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_813
tff(fact_8991_ATP_Olambda__814,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$o)),Uuc: B] : aa(B,C,aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_mg(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(B,fun(A,C),aTP_Lamp_mf(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uuc)),collect(A,aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_mc(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_814
tff(fact_8992_ATP_Olambda__815,axiom,
    ! [Uu: 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_fj(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aa(fun(nat,nat),fun(nat,fun(nat,nat)),aTP_Lamp_fi(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),Uu),Uua),Uuc)),set_ord_atMost(nat,Uuc))),aa(nat,nat,power_power(nat,Uub),Uuc)) ).

% ATP.lambda_815
tff(fact_8993_ATP_Olambda__816,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: 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_ff(fun(nat,A),fun(fun(nat,A),fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aTP_Lamp_fe(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu),Uua),Uuc)),set_ord_atMost(nat,Uuc))),aa(nat,A,power_power(A,Uub),Uuc)) ) ).

% ATP.lambda_816
tff(fact_8994_ATP_Olambda__817,axiom,
    ! [Uu: 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_ry(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Uu),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),Uu,aa(nat,nat,plus_plus(nat,aa(nat,nat,suc,Uua)),Uuc)),zero_zero(real))),semiring_char_0_fact(real,Uuc))),aa(nat,real,power_power(real,Uub),Uuc)) ).

% ATP.lambda_817
tff(fact_8995_ATP_Olambda__818,axiom,
    ! [Uu: 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_rw(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Uu),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),Uu,aa(nat,nat,plus_plus(nat,Uua),Uuc)),zero_zero(real))),semiring_char_0_fact(real,Uuc))),aa(nat,real,power_power(real,Uub),Uuc)) ).

% ATP.lambda_818
tff(fact_8996_ATP_Olambda__819,axiom,
    ! [Uu: 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_rv(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Uu),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),Uu,Uuc),Uua)),semiring_char_0_fact(real,Uuc))),aa(nat,real,power_power(real,aa(real,real,minus_minus(real,Uub),Uua)),Uuc)) ).

% ATP.lambda_819
tff(fact_8997_ATP_Olambda__820,axiom,
    ! [Uu: 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_ru(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Uu),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),Uu,Uuc),Uub)),semiring_char_0_fact(real,Uuc))),aa(nat,real,power_power(real,aa(real,real,minus_minus(real,Uua),Uub)),Uuc)) ).

% ATP.lambda_820
tff(fact_8998_ATP_Olambda__821,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_hj(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,plus_plus(A,Uu),Uua)),aa(nat,nat,minus_minus(nat,Uub),Uuc))),aa(nat,A,power_power(A,Uu),Uuc))),aa(nat,A,power_power(A,Uu),Uub)) ) ).

% ATP.lambda_821
tff(fact_8999_ATP_Olambda__822,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_hx(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,plus_plus(A,Uua),Uu)),Uuc)),aa(nat,A,power_power(A,Uua),aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Uuc))) ) ).

% ATP.lambda_822
tff(fact_9000_ATP_Olambda__823,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: 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_abe(fun(A,A),fun(A,fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,Uu,aa(A,A,plus_plus(A,Uua),aa(nat,A,Uub,Uuc)))),aa(A,A,Uu,Uua))),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_823
tff(fact_9001_ATP_Olambda__824,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(B)
     => ! [Uu: fun(A,B),Uua: set(B),Uub: B,Uuc: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aa(set(B),fun(B,fun(A,$o)),aTP_Lamp_adn(fun(A,B),fun(set(B),fun(B,fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> ( member(B,aa(A,B,Uu,Uuc),Uua)
            & ( aa(A,B,Uu,Uuc) != Uub ) ) ) ) ).

% ATP.lambda_824
tff(fact_9002_ATP_Olambda__825,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: set(B),Uuc: A] :
          ( aa(A,$o,aa(set(B),fun(A,$o),aa(B,fun(set(B),fun(A,$o)),aTP_Lamp_aeq(fun(A,B),fun(B,fun(set(B),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> ( member(B,aa(A,B,Uu,Uuc),Uub)
            & ( aa(A,B,Uu,Uuc) != Uua ) ) ) ) ).

% ATP.lambda_825
tff(fact_9003_ATP_Olambda__826,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: real,Uuc: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_agd(fun(A,B),fun(B,fun(real,fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> aa(real,$o,ord_less_eq(real,real_V557655796197034286t_dist(B,aa(A,B,Uu,Uuc),Uua)),Uub) ) ) ).

% ATP.lambda_826
tff(fact_9004_ATP_Olambda__827,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: real,Uuc: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_afr(fun(A,B),fun(B,fun(real,fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> aa(real,$o,ord_less(real,real_V557655796197034286t_dist(B,aa(A,B,Uu,Uuc),Uua)),Uub) ) ) ).

% ATP.lambda_827
tff(fact_9005_ATP_Olambda__828,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_fm(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uub),Uuc))),comm_s3205402744901411588hammer(A,Uu,Uuc))),comm_s3205402744901411588hammer(A,Uua,aa(nat,nat,minus_minus(nat,Uub),Uuc))) ) ).

% ATP.lambda_828
tff(fact_9006_ATP_Olambda__829,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_ey(nat,fun(nat,fun(nat,fun(nat,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,semiring_1_of_nat(nat),aa(nat,nat,binomial(Uub),Uuc))),aa(nat,nat,power_power(nat,Uu),Uuc))),aa(nat,nat,power_power(nat,Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))) ).

% ATP.lambda_829
tff(fact_9007_ATP_Olambda__830,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_fk(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uub),Uuc))),aa(nat,A,power_power(A,Uu),Uuc))),aa(nat,A,power_power(A,Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))) ) ).

% ATP.lambda_830
tff(fact_9008_ATP_Olambda__831,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_fs(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uuc))),aa(nat,A,power_power(A,Uu),Uuc))),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,aa(nat,nat,minus_minus(nat,Uub),Uuc)))),aa(nat,A,power_power(A,Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc)))) ) ).

% ATP.lambda_831
tff(fact_9009_ATP_Olambda__832,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: nat,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_hm(A,fun(nat,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,Uub),aa(nat,nat,minus_minus(nat,Uua),aa(nat,nat,suc,Uuc)))),aa(nat,A,power_power(A,Uu),Uuc)) ) ).

% ATP.lambda_832
tff(fact_9010_ATP_Olambda__833,axiom,
    ! [A: $tType,Uu: $o,Uua: A,Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(A,fun(A,fun(A,$o)),aTP_Lamp_dg($o,fun(A,fun(A,fun(A,$o))),(Uu)),Uua),Uub),Uuc)
    <=> ( ( (Uu)
         => ( Uuc = Uua ) )
        & ( ~ (Uu)
         => ( Uuc = Uub ) ) ) ) ).

% ATP.lambda_833
tff(fact_9011_ATP_Olambda__834,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: set(B),Uub: fun(A,$o),Uuc: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(set(B),fun(fun(A,$o),fun(A,$o)),aTP_Lamp_on(fun(B,A),fun(set(B),fun(fun(A,$o),fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,aa(set(B),set(A),image(B,A,Uu),Uua))
        & aa(A,$o,Uub,Uuc) ) ) ).

% ATP.lambda_834
tff(fact_9012_ATP_Olambda__835,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: fun(A,fun(B,$o)),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_lz(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(B,Uuc,Uu)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uub),Uuc) ) ) ).

% ATP.lambda_835
tff(fact_9013_ATP_Olambda__836,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,fun(B,$o)),Uub: B,Uuc: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_mc(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,Uu)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uuc),Uub) ) ) ).

% ATP.lambda_836
tff(fact_9014_ATP_Olambda__837,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_hk(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,Uu),Uuc)),aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(A,A,plus_plus(A,Uu),Uua)),aa(nat,nat,minus_minus(nat,Uub),Uuc))),aa(nat,A,power_power(A,Uu),aa(nat,nat,minus_minus(nat,Uub),Uuc)))) ) ).

% ATP.lambda_837
tff(fact_9015_ATP_Olambda__838,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: set(A),Uua: fun(A,A),Uub: fun(A,A),Uuc: A] :
          ( aa(A,$o,aa(fun(A,A),fun(A,$o),aa(fun(A,A),fun(fun(A,A),fun(A,$o)),aTP_Lamp_aer(set(A),fun(fun(A,A),fun(fun(A,A),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> ( member(A,Uuc,Uu)
           => ( aa(A,A,Uua,Uuc) = aa(A,A,Uub,Uuc) ) ) ) ) ).

% ATP.lambda_838
tff(fact_9016_ATP_Olambda__839,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(A,B),Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,B),fun(A,fun(A,$o)),aTP_Lamp_om(set(A),fun(fun(A,B),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,Uu)
        & ( aa(A,B,Uua,Uuc) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_839
tff(fact_9017_ATP_Olambda__840,axiom,
    ! [B: $tType,A: $tType,Uu: set(B),Uua: fun(B,A),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(B,A),fun(A,fun(B,$o)),aTP_Lamp_pp(set(B),fun(fun(B,A),fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(B,Uuc,Uu)
        & ( aa(B,A,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_840
tff(fact_9018_ATP_Olambda__841,axiom,
    ! [A: $tType,C: $tType,Uu: set(A),Uua: fun(A,C),Uub: C,Uuc: A] :
      ( aa(A,$o,aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aTP_Lamp_os(set(A),fun(fun(A,C),fun(C,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,Uu)
        & ( aa(A,C,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_841
tff(fact_9019_ATP_Olambda__842,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,B),Uub: B,Uuc: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_ow(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,Uu)
        & ( aa(A,B,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_842
tff(fact_9020_ATP_Olambda__843,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,A),Uub: set(B),Uuc: B] :
      ( aa(B,$o,aa(set(B),fun(B,$o),aa(fun(B,A),fun(set(B),fun(B,$o)),aTP_Lamp_pw(set(A),fun(fun(B,A),fun(set(B),fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(B,Uuc,Uub)
        & member(A,aa(B,A,Uua,Uuc),Uu) ) ) ).

% ATP.lambda_843
tff(fact_9021_ATP_Olambda__844,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: nat,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_hl(A,fun(nat,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,Uu),Uuc)),aa(nat,A,power_power(A,Uub),aa(nat,nat,minus_minus(nat,Uua),Uuc))) ) ).

% ATP.lambda_844
tff(fact_9022_ATP_Olambda__845,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_ew(nat,fun(nat,fun(nat,fun(nat,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(Uu),Uuc)),aa(nat,nat,binomial(Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))) ).

% ATP.lambda_845
tff(fact_9023_ATP_Olambda__846,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_le(int,fun(int,fun(int,fun(int,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub)) ) ).

% ATP.lambda_846
tff(fact_9024_ATP_Olambda__847,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_lb(int,fun(int,fun(int,fun(int,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub)) ) ).

% ATP.lambda_847
tff(fact_9025_ATP_Olambda__848,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: B,Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(B,fun(A,fun(B,$o)),aTP_Lamp_kj(A,fun(B,fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( ( Uu = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_848
tff(fact_9026_ATP_Olambda__849,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: fun(A,$o),Uuc: B] :
      ( aa(B,$o,aa(fun(A,$o),fun(B,$o),aa(set(B),fun(fun(A,$o),fun(B,$o)),aTP_Lamp_oo(fun(B,A),fun(set(B),fun(fun(A,$o),fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(B,Uuc,Uua)
        & aa(A,$o,Uub,aa(B,A,Uu,Uuc)) ) ) ).

% ATP.lambda_849
tff(fact_9027_ATP_Olambda__850,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_mo(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> ( member(A,Uuc,Uu)
            & ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uuc)),aa(A,B,Uub,Uuc)) != one_one(B) ) ) ) ) ).

% ATP.lambda_850
tff(fact_9028_ATP_Olambda__851,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_mm(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> ( member(A,Uuc,Uu)
            & ( aa(B,B,plus_plus(B,aa(A,B,Uua,Uuc)),aa(A,B,Uub,Uuc)) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_851
tff(fact_9029_ATP_Olambda__852,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: 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_abm(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),Uu),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,minus_minus(A,Uuc),Uub)))),aa(B,B,minus_minus(B,aa(A,B,Uu,Uuc)),aa(B,B,plus_plus(B,aa(A,B,Uu,Uub)),aa(A,B,Uua,aa(A,A,minus_minus(A,Uuc),Uub))))) ) ).

% ATP.lambda_852
tff(fact_9030_ATP_Olambda__853,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_bo(fun(nat,A),fun(nat,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,plus_plus(nat,Uuc),Uua))),aa(nat,A,power_power(A,Uub),Uuc)) ) ).

% ATP.lambda_853
tff(fact_9031_ATP_Olambda__854,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: list(B),Uuc: nat] : aa(nat,A,aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_py(fun(B,A),fun(A,fun(list(B),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,aa(nat,B,nth(B,Uub),Uuc))),aa(nat,A,power_power(A,Uua),Uuc)) ) ).

% ATP.lambda_854
tff(fact_9032_ATP_Olambda__855,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,$o),Uub: fun(A,B),Uuc: A] :
      ( aa(A,$o,aa(fun(A,B),fun(A,$o),aa(fun(B,$o),fun(fun(A,B),fun(A,$o)),aTP_Lamp_akh(set(A),fun(fun(B,$o),fun(fun(A,B),fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(B,$o,Uua,aa(A,B,Uub,Uuc))
        & member(A,Uuc,Uu) ) ) ).

% ATP.lambda_855
tff(fact_9033_ATP_Olambda__856,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: 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_ul(fun(nat,A),fun(A,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uuc)),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(A,A,plus_plus(A,Uua),Uub)),Uuc)),aa(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,power_power(A,Uua),aa(nat,nat,minus_minus(nat,Uuc),aa(nat,nat,suc,zero_zero(nat))))))) ) ).

% ATP.lambda_856
tff(fact_9034_ATP_Olambda__857,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B)
        & topological_t2_space(A) )
     => ! [Uu: fun(A,B),Uua: fun(nat,B),Uub: A,Uuc: nat] : aa(nat,B,aa(A,fun(nat,B),aa(fun(nat,B),fun(A,fun(nat,B)),aTP_Lamp_zf(fun(A,B),fun(fun(nat,B),fun(A,fun(nat,B))),Uu),Uua),Uub),Uuc) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,Uua,Uuc)),aa(nat,B,power_power(B,aa(A,B,Uu,Uub)),Uuc)) ) ).

% ATP.lambda_857
tff(fact_9035_ATP_Olambda__858,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: 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_ua(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),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,Uu,Uua)))),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ).

% ATP.lambda_858
tff(fact_9036_ATP_Olambda__859,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: 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_io(fun(nat,A),fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uuc)),aa(nat,A,power_power(A,Uua),aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,Uuc),Uub)),one_one(nat)))) ) ).

% ATP.lambda_859
tff(fact_9037_ATP_Olambda__860,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: set(B),Uuc: A] :
          ( aa(A,$o,aa(set(B),fun(A,$o),aa(B,fun(set(B),fun(A,$o)),aTP_Lamp_afu(fun(A,B),fun(B,fun(set(B),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> member(B,aa(A,B,Uu,Uuc),aa(set(B),set(B),minus_minus(set(B),Uub),aa(set(B),set(B),insert(B,Uua),bot_bot(set(B))))) ) ) ).

% ATP.lambda_860
tff(fact_9038_ATP_Olambda__861,axiom,
    ! [Uu: 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_fi(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,Uu,Uuc)),aa(nat,nat,Uua,aa(nat,nat,minus_minus(nat,Uub),Uuc))) ).

% ATP.lambda_861
tff(fact_9039_ATP_Olambda__862,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: 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))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uuc)),aa(nat,A,Uua,aa(nat,nat,minus_minus(nat,Uub),Uuc))) ) ).

% ATP.lambda_862
tff(fact_9040_ATP_Olambda__863,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: 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_fe(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uuc)),aa(nat,A,Uua,aa(nat,nat,minus_minus(nat,Uub),Uuc))) ) ).

% ATP.lambda_863
tff(fact_9041_ATP_Olambda__864,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( semiring_0(A)
     => ! [Uu: fun(B,A),Uua: fun(C,A),Uub: B,Uuc: C] : aa(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_gt(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),aa(C,A,Uua,Uuc)) ) ).

% ATP.lambda_864
tff(fact_9042_ATP_Olambda__865,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: set(B),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$o)),Uuc: A] : aa(A,C,aa(fun(A,fun(B,$o)),fun(A,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C)),aTP_Lamp_ma(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),Uu),Uua),Uub),Uuc) = groups7121269368397514597t_prod(B,C,aa(A,fun(B,C),Uua,Uuc),collect(B,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_lz(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_865
tff(fact_9043_ATP_Olambda__866,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: set(B),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$o)),Uuc: A] : aa(A,C,aa(fun(A,fun(B,$o)),fun(A,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C)),aTP_Lamp_me(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),Uu),Uua),Uub),Uuc) = aa(set(B),C,groups7311177749621191930dd_sum(B,C,aa(A,fun(B,C),Uua,Uuc)),collect(B,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_lz(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_866
tff(fact_9044_ATP_Olambda__867,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: 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_ue(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),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(nat,real,power_power(real,cos(real,aa(A,real,Uu,Uua))),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% ATP.lambda_867
tff(fact_9045_ATP_Olambda__868,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: A] : aa(A,real,aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_uc(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uuc)),aa(real,real,inverse_inverse(real),aa(real,real,plus_plus(real,one_one(real)),aa(nat,real,power_power(real,aa(A,real,Uu,Uub)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ).

% ATP.lambda_868
tff(fact_9046_ATP_Olambda__869,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: A] : aa(A,real,aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_tb(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uuc)),aa(real,real,exp(real),aa(A,real,Uu,Uub))) ) ).

% ATP.lambda_869
tff(fact_9047_ATP_Olambda__870,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: A] : aa(A,real,aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_td(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uuc)),cos(real,aa(A,real,Uu,Uub))) ) ).

% ATP.lambda_870
tff(fact_9048_ATP_Olambda__871,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: 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_tq(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),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,Uu,Uua))) ) ).

% ATP.lambda_871
tff(fact_9049_ATP_Olambda__872,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: 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_sf(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),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,minus_minus(real,one_one(real)),aa(nat,real,power_power(real,aa(A,real,Uu,Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) ).

% ATP.lambda_872
tff(fact_9050_ATP_Olambda__873,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: A] : aa(A,real,aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_tm(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uuc)),aa(real,real,uminus_uminus(real),sin(real,aa(A,real,Uu,Uub)))) ) ).

% ATP.lambda_873
tff(fact_9051_ATP_Olambda__874,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: 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_ui(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),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,minus_minus(real,one_one(real)),aa(nat,real,power_power(real,aa(A,real,Uu,Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))) ) ).

% ATP.lambda_874
tff(fact_9052_ATP_Olambda__875,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: 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_abn(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(B,B,minus_minus(B,aa(A,B,Uu,aa(A,A,plus_plus(A,Uub),Uuc))),aa(A,B,Uu,Uub))),aa(A,B,Uua,Uuc)))),real_V7770717601297561774m_norm(A,Uuc)) ) ).

% ATP.lambda_875
tff(fact_9053_ATP_Olambda__876,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: 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_acg(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(B,B,minus_minus(B,aa(A,B,Uu,Uuc)),aa(A,B,Uu,Uub))),aa(A,B,Uua,aa(A,A,minus_minus(A,Uuc),Uub))))),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Uuc),Uub))) ) ).

% ATP.lambda_876
tff(fact_9054_ATP_Olambda__877,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,$o),Uua: A,Uub: set(A),Uuc: A] :
          ( aa(A,$o,aa(set(A),fun(A,$o),aa(A,fun(set(A),fun(A,$o)),aTP_Lamp_adz(fun(A,$o),fun(A,fun(set(A),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> ( ( Uuc != Uua )
           => ( member(A,Uuc,Uub)
             => aa(A,$o,Uu,Uuc) ) ) ) ) ).

% ATP.lambda_877
tff(fact_9055_ATP_Olambda__878,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Uu: fun(A,B),Uua: fun(A,C),Uub: real,Uuc: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aa(fun(A,C),fun(real,fun(A,$o)),aTP_Lamp_afv(fun(A,B),fun(fun(A,C),fun(real,fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> aa(real,$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,Uu,Uuc))),Uub)) ) ) ).

% ATP.lambda_878
tff(fact_9056_ATP_Olambda__879,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: 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_ack(fun(A,B),fun(fun(A,B),fun(filter(A),fun(A,B))),Uu),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Uuc),topolo3827282254853284352ce_Lim(A,A,Uub,aTP_Lamp_so(A,A)))))),aa(B,B,minus_minus(B,aa(B,B,minus_minus(B,aa(A,B,Uu,Uuc)),aa(A,B,Uu,topolo3827282254853284352ce_Lim(A,A,Uub,aTP_Lamp_so(A,A))))),aa(A,B,Uua,aa(A,A,minus_minus(A,Uuc),topolo3827282254853284352ce_Lim(A,A,Uub,aTP_Lamp_so(A,A)))))) ) ).

% ATP.lambda_879
tff(fact_9057_ATP_Olambda__880,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: 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_aci(fun(A,B),fun(A,fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Uuc),Uua)))),aa(B,B,minus_minus(B,aa(B,B,minus_minus(B,aa(A,B,Uub,Uuc)),aa(A,B,Uub,Uua))),aa(A,B,Uu,aa(A,A,minus_minus(A,Uuc),Uua)))) ) ).

% ATP.lambda_880
tff(fact_9058_ATP_Olambda__881,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: 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_ach(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),Uu),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Uuc),Uub)))),aa(B,B,minus_minus(B,aa(B,B,minus_minus(B,aa(A,B,Uu,Uuc)),aa(A,B,Uu,Uub))),aa(A,B,Uua,aa(A,A,minus_minus(A,Uuc),Uub)))) ) ).

% ATP.lambda_881
tff(fact_9059_ATP_Olambda__882,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(B,C),Uuc: B] : aa(B,C,aa(fun(B,C),fun(B,C),aa(fun(A,B),fun(fun(B,C),fun(B,C)),aTP_Lamp_alf(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(nat,C,semiring_1_of_nat(C),finite_card(A,collect(A,aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_ow(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uuc))))),aa(B,C,Uub,Uuc)) ) ).

% ATP.lambda_882
tff(fact_9060_ATP_Olambda__883,axiom,
    ! [A: $tType,B: $tType,Uu: $o,Uua: fun(A,fun(B,$o)),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_kx($o,fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),(Uu)),Uua),Uub),Uuc)
    <=> ( (Uu)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uub),Uuc) ) ) ).

% ATP.lambda_883
tff(fact_9061_ATP_Olambda__884,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: C] : aa(C,B,aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_ov(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),Uu),Uua),Uub),Uuc) = groups7121269368397514597t_prod(A,B,Uua,collect(A,aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aTP_Lamp_os(set(A),fun(fun(A,C),fun(C,fun(A,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_884
tff(fact_9062_ATP_Olambda__885,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: B] : aa(B,C,aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_oy(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),Uu),Uua),Uub),Uuc) = groups7121269368397514597t_prod(A,C,Uub,collect(A,aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_ow(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uuc))) ) ).

% ATP.lambda_885
tff(fact_9063_ATP_Olambda__886,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: C] : aa(C,B,aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_ot(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),Uu),Uua),Uub),Uuc) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,Uua),collect(A,aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aTP_Lamp_os(set(A),fun(fun(A,C),fun(C,fun(A,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_886
tff(fact_9064_ATP_Olambda__887,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: B] : aa(B,C,aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_ox(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,Uub),collect(A,aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_ow(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uuc))) ) ).

% ATP.lambda_887
tff(fact_9065_ATP_Olambda__888,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: real,Uuc: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aa(nat,fun(real,fun(A,$o)),aTP_Lamp_agc(fun(nat,A),fun(nat,fun(real,fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> aa(real,$o,ord_less_eq(real,Uub),real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_bp(fun(nat,A),fun(A,fun(nat,A)),Uu),Uuc)),set_ord_atMost(nat,Uua)))) ) ) ).

% ATP.lambda_888
tff(fact_9066_ATP_Olambda__889,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType,Uu: fun(B,A),Uua: fun(C,fun(D,B)),Uub: C,Uuc: D] : aa(D,A,aa(C,fun(D,A),aa(fun(C,fun(D,B)),fun(C,fun(D,A)),aTP_Lamp_kr(fun(B,A),fun(fun(C,fun(D,B)),fun(C,fun(D,A))),Uu),Uua),Uub),Uuc) = aa(B,A,Uu,aa(D,B,aa(C,fun(D,B),Uua,Uub),Uuc)) ).

% ATP.lambda_889
tff(fact_9067_ATP_Olambda__890,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_nv(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,minus_minus(nat,aa(nat,nat,plus_plus(nat,Uub),Uua)),aa(nat,nat,suc,Uuc))) ) ).

% ATP.lambda_890
tff(fact_9068_ATP_Olambda__891,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_nt(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,minus_minus(nat,aa(nat,nat,plus_plus(nat,Uub),Uua)),aa(nat,nat,suc,Uuc))) ) ).

% ATP.lambda_891
tff(fact_9069_ATP_Olambda__892,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_jj(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,minus_minus(nat,aa(nat,nat,plus_plus(nat,Uub),Uua)),Uuc)) ) ).

% ATP.lambda_892
tff(fact_9070_ATP_Olambda__893,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_ic(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,minus_minus(nat,aa(nat,nat,plus_plus(nat,Uub),Uua)),Uuc)) ) ).

% ATP.lambda_893
tff(fact_9071_ATP_Olambda__894,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_lm(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uub),Uua))),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uuc))) ).

% ATP.lambda_894
tff(fact_9072_ATP_Olambda__895,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_lg(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,plus_plus(int,aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uub),Uua))),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uuc))) ).

% ATP.lambda_895
tff(fact_9073_ATP_Olambda__896,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: 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_th(fun(B,A),fun(B,fun(fun(B,A),fun(B,A))),Uu),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,Uu,Uua))),aa(B,A,Uub,Uuc))),aa(A,A,inverse_inverse(A),aa(B,A,Uu,Uua)))) ) ).

% ATP.lambda_896
tff(fact_9074_ATP_Olambda__897,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_lk(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uub)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uuc))) ).

% ATP.lambda_897
tff(fact_9075_ATP_Olambda__898,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_li(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub))) ).

% ATP.lambda_898
tff(fact_9076_ATP_Olambda__899,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: set(B),Uub: B,Uuc: fun(A,B)] :
      ( aa(fun(A,B),$o,aa(B,fun(fun(A,B),$o),aa(set(B),fun(B,fun(fun(A,B),$o)),aTP_Lamp_ahf(set(A),fun(set(B),fun(B,fun(fun(A,B),$o))),Uu),Uua),Uub),Uuc)
    <=> ! [X3: A] :
          ( ( member(A,X3,Uu)
           => member(B,aa(A,B,Uuc,X3),Uua) )
          & ( ~ member(A,X3,Uu)
           => ( aa(A,B,Uuc,X3) = Uub ) ) ) ) ).

% ATP.lambda_899
tff(fact_9077_ATP_Olambda__900,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: 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_sz(fun(A,fun(A,B)),fun(fun(C,A),fun(C,fun(fun(C,A),fun(C,B)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,B,aa(A,fun(A,B),Uu,aa(C,A,Uua,Uub)),aa(C,A,Uuc,Uud)) ) ).

% ATP.lambda_900
tff(fact_9078_ATP_Olambda__901,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu: 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_tw(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,C)))),Uu),Uua),Uub),Uuc),Uud) = aa(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_tv(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C))))),Uu),Uua),Uub),Uuc),Uud)),Uu) ) ).

% ATP.lambda_901
tff(fact_9079_ATP_Olambda__902,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: 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_hv(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(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_hu(fun(nat,A),fun(A,fun(A,fun(nat,fun(nat,A)))),Uua),Uub),Uuc),Uud)),set_ord_lessThan(nat,aa(nat,nat,minus_minus(nat,Uu),Uud))) ) ).

% ATP.lambda_902
tff(fact_9080_ATP_Olambda__903,axiom,
    ! [Uu: 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($o,assn))),aTP_Lamp_cs(nat,fun(set(nat),fun(vEBT_VEBTi,fun(nat,fun($o,assn)))),Uu),Uua),Uub),Uuc),(Uud)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,vEBT_Intf_vebt_assn(Uu,Uua),Uub)),pure_assn((Uud) = member(nat,Uuc,Uua))) ).

% ATP.lambda_903
tff(fact_9081_ATP_Olambda__904,axiom,
    ! [Uu: 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_rx(nat,fun(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real)))),Uu),Uua),Uub),Uuc),Uud) = aa(real,real,minus_minus(real,aa(real,real,aa(nat,fun(real,real),Uua,Uuc),Uud)),aa(real,real,plus_plus(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_rw(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Uua),Uuc),Uud)),set_ord_lessThan(nat,aa(nat,nat,minus_minus(nat,Uu),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,power_power(real,Uud),aa(nat,nat,minus_minus(nat,Uu),Uuc))),semiring_char_0_fact(real,aa(nat,nat,minus_minus(nat,Uu),Uuc)))))) ).

% ATP.lambda_904
tff(fact_9082_ATP_Olambda__905,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: 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_ip(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_io(fun(nat,A),fun(A,fun(nat,fun(nat,A))),Uua),Uuc),Uud)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uud),Uu))),aa(nat,A,power_power(A,Uub),Uud)) ) ).

% ATP.lambda_905
tff(fact_9083_ATP_Olambda__906,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: A,Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_fw(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,aa(A,A,plus_plus(A,aa(nat,A,semiring_1_of_nat(A),Uud)),Uua)),one_one(A))),Uud)),aa(nat,A,power_power(A,Uub),Uud))),aa(nat,A,power_power(A,aa(A,A,plus_plus(A,Uub),Uuc)),aa(nat,nat,minus_minus(nat,Uu),Uud))) ) ).

% ATP.lambda_906
tff(fact_9084_ATP_Olambda__907,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: A,Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_fq(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,aa(A,A,plus_plus(A,aa(nat,A,semiring_1_of_nat(A),Uu)),Uua)),Uud)),aa(nat,A,power_power(A,Uub),Uud))),aa(nat,A,power_power(A,Uuc),aa(nat,nat,minus_minus(nat,Uu),Uud))) ) ).

% ATP.lambda_907
tff(fact_9085_ATP_Olambda__908,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: A,Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_fr(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,aa(A,A,uminus_uminus(A),Uua)),Uud)),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),Uub)),Uud))),aa(nat,A,power_power(A,aa(A,A,plus_plus(A,Uub),Uuc)),aa(nat,nat,minus_minus(nat,Uu),Uud))) ) ).

% ATP.lambda_908
tff(fact_9086_ATP_Olambda__909,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: 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_hu(fun(nat,A),fun(A,fun(A,fun(nat,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,plus_plus(nat,aa(nat,nat,plus_plus(nat,Uuc),Uud)),one_one(nat)))),aa(nat,A,power_power(A,Uub),Uud))),aa(nat,A,power_power(A,Uua),Uuc)) ) ).

% ATP.lambda_909
tff(fact_9087_ATP_Olambda__910,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( real_V7819770556892013058_space(B)
        & real_V7819770556892013058_space(C) )
     => ! [Uu: fun(A,B),Uua: B,Uub: fun(A,C),Uuc: C,Uud: A] :
          ( aa(A,$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,$o))),aTP_Lamp_afk(fun(A,B),fun(B,fun(fun(A,C),fun(C,fun(A,$o)))),Uu),Uua),Uub),Uuc),Uud)
        <=> aa(real,$o,ord_less_eq(real,real_V557655796197034286t_dist(C,aa(A,C,Uub,Uud),Uuc)),real_V557655796197034286t_dist(B,aa(A,B,Uu,Uud),Uua)) ) ) ).

% ATP.lambda_910
tff(fact_9088_ATP_Olambda__911,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: fun(B,C),Uuc: C,Uud: A] :
          ( aa(A,$o,aa(C,fun(A,$o),aa(fun(B,C),fun(C,fun(A,$o)),aa(B,fun(fun(B,C),fun(C,fun(A,$o))),aTP_Lamp_akd(fun(A,B),fun(B,fun(fun(B,C),fun(C,fun(A,$o)))),Uu),Uua),Uub),Uuc),Uud)
        <=> ( ( aa(A,B,Uu,Uud) = Uua )
           => ( aa(B,C,Uub,Uua) = Uuc ) ) ) ) ).

% ATP.lambda_911
tff(fact_9089_ATP_Olambda__912,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: nat,Uud: A] : aa(A,B,aa(nat,fun(A,B),aa(A,fun(nat,fun(A,B)),aa(fun(A,B),fun(A,fun(nat,fun(A,B))),aTP_Lamp_to(fun(A,B),fun(fun(A,B),fun(A,fun(nat,fun(A,B)))),Uu),Uua),Uub),Uuc),Uud) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),Uuc)),aa(A,B,Uua,Uud))),aa(nat,B,power_power(B,aa(A,B,Uu,Uub)),aa(nat,nat,minus_minus(nat,Uuc),one_one(nat)))) ) ).

% ATP.lambda_912
tff(fact_9090_ATP_Olambda__913,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu: 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_tv(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C))))),Uu),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)),groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_tt(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uuc),aa(set(A),set(A),minus_minus(set(A),Uu),aa(set(A),set(A),insert(A,Uue),bot_bot(set(A)))))) ) ).

% ATP.lambda_913
tff(fact_9091_ATP_Olambda__914,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: 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_tf(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(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,Uu,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_914
tff(fact_9092_ATP_Olambda__915,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: 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_ty(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,aa(A,real,Uu,Uub),aa(A,real,Uuc,Uub))),aa(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,Uu,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,Uu,Uub)))) ) ).

% ATP.lambda_915
tff(fact_9093_ATP_Olambda__916,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,real),Uua: fun(A,real),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,real),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_sh(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(B,B,plus_plus(B,aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uu,Uub)),aa(A,B,Uud,Uue))),aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uua,Uue)),aa(A,B,Uuc,Uub))) ) ).

% ATP.lambda_916
tff(fact_9094_ATP_Olambda__917,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: 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_sx(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(B,B,plus_plus(B,aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uud,Uue))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uue)),aa(A,B,Uuc,Uub))) ) ).

% ATP.lambda_917
tff(fact_9095_ATP_Olambda__918,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu: 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_ts(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(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,Uu,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_918
tff(fact_9096_ATP_Olambda__919,axiom,
    ! [B: $tType,A: $tType,Uu: $o,Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_pm($o,fun(A,fun(B,$o)),(Uu)),Uua),Uub)
    <=> (Uu) ) ).

% ATP.lambda_919
tff(fact_9097_ATP_Olambda__920,axiom,
    ! [Uu: $o,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_akb($o,fun(nat,$o),(Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_920
tff(fact_9098_ATP_Olambda__921,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: $o,Uua: A] :
          ( aa(A,$o,aTP_Lamp_acn($o,fun(A,$o),(Uu)),Uua)
        <=> (Uu) ) ) ).

% ATP.lambda_921
tff(fact_9099_ATP_Olambda__922,axiom,
    ! [A: $tType,Uu: $o,Uua: A] :
      ( aa(A,$o,aTP_Lamp_ph($o,fun(A,$o),(Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_922
tff(fact_9100_ATP_Olambda__923,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_sr(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_923
tff(fact_9101_ATP_Olambda__924,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_aie(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_924
tff(fact_9102_ATP_Olambda__925,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topological_t2_space(A) )
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_xr(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_925
tff(fact_9103_ATP_Olambda__926,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_un(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_926
tff(fact_9104_ATP_Olambda__927,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_agk(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_927
tff(fact_9105_ATP_Olambda__928,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & topolo8386298272705272623_space(A) )
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_vb(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_928
tff(fact_9106_ATP_Olambda__929,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_vl(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_929
tff(fact_9107_ATP_Olambda__930,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,B,aTP_Lamp_oq(B,fun(A,B),Uu),Uua) = Uu ).

% ATP.lambda_930
tff(fact_9108_ATP_Olambda__931,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_be(A,fun(nat,A),Uu),Uua) = Uu ) ).

% ATP.lambda_931
tff(fact_9109_ATP_Olambda__932,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_qn(A,fun(A,A),Uu),Uua) = Uu ) ).

% ATP.lambda_932
tff(fact_9110_ATP_Olambda__933,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_alb(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_933
tff(fact_9111_ATP_Olambda__934,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_zr(A,fun(nat,A),Uu),Uua) = Uu ) ).

% ATP.lambda_934
tff(fact_9112_ATP_Olambda__935,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo8386298272705272623_space(B) )
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_vd(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_935
tff(fact_9113_ATP_Olambda__936,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_akt(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_936
tff(fact_9114_ATP_Olambda__937,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_aku(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_937
tff(fact_9115_ATP_Olambda__938,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ahc(A,fun(nat,A),Uu),Uua) = Uu ) ).

% ATP.lambda_938
tff(fact_9116_ATP_Olambda__939,axiom,
    ! [B: $tType,A: $tType] :
      ( ( zero(A)
        & topological_t2_space(A)
        & topolo8386298272705272623_space(B) )
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_vj(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_939
tff(fact_9117_ATP_Olambda__940,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,A,aTP_Lamp_or(A,fun(B,A),Uu),Uua) = Uu ).

% ATP.lambda_940
tff(fact_9118_ATP_Olambda__941,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_ky(A,fun(B,$o)),Uu),Uua)
    <=> $true ) ).

% ATP.lambda_941
tff(fact_9119_ATP_Olambda__942,axiom,
    ! [Uu: complex] : aa(complex,complex,aTP_Lamp_gm(complex,complex),Uu) = Uu ).

% ATP.lambda_942
tff(fact_9120_ATP_Olambda__943,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: word(A)] : aa(word(A),word(A),aTP_Lamp_qb(word(A),word(A)),Uu) = Uu ) ).

% ATP.lambda_943
tff(fact_9121_ATP_Olambda__944,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_ik(nat,nat),Uu) = Uu ).

% ATP.lambda_944
tff(fact_9122_ATP_Olambda__945,axiom,
    ! [Uu: int] : aa(int,int,aTP_Lamp_jv(int,int),Uu) = Uu ).

% ATP.lambda_945
tff(fact_9123_ATP_Olambda__946,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: B] : aa(B,B,aTP_Lamp_pv(B,B),Uu) = Uu ) ).

% ATP.lambda_946
tff(fact_9124_ATP_Olambda__947,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: B] : aa(B,B,aTP_Lamp_pu(B,B),Uu) = Uu ) ).

% ATP.lambda_947
tff(fact_9125_ATP_Olambda__948,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_so(A,A),Uu) = Uu ) ).

% ATP.lambda_948
tff(fact_9126_ATP_Olambda__949,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_uo(A,A),Uu) = Uu ) ).

% ATP.lambda_949
tff(fact_9127_ATP_Olambda__950,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_qp(A,A),Uu) = Uu ) ).

% ATP.lambda_950
tff(fact_9128_ATP_Olambda__951,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_us(A,A),Uu) = Uu ) ).

% ATP.lambda_951
tff(fact_9129_ATP_Olambda__952,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A] : aa(A,A,aTP_Lamp_acq(A,A),Uu) = Uu ) ).

% ATP.lambda_952
tff(fact_9130_ATP_Olambda__953,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_pr(A,A),Uu) = Uu ) ).

% ATP.lambda_953
tff(fact_9131_ATP_Olambda__954,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_at(A,A),Uu) = Uu ) ).

% ATP.lambda_954
tff(fact_9132_ATP_Olambda__955,axiom,
    ! [A: $tType,Uu: A] : aa(A,A,aTP_Lamp_oi(A,A),Uu) = Uu ).

% ATP.lambda_955
tff(fact_9133_ATP_Olambda__956,axiom,
    ! [C: $tType,B: $tType,Uu: C] : aa(C,set(B),aTP_Lamp_akq(C,set(B)),Uu) = bot_bot(set(B)) ).

% ATP.lambda_956
tff(fact_9134_ATP_Olambda__957,axiom,
    ! [A: $tType,D: $tType,Uu: A] : aa(A,set(D),aTP_Lamp_akr(A,set(D)),Uu) = bot_bot(set(D)) ).

% ATP.lambda_957
tff(fact_9135_ATP_Olambda__958,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_bc(nat,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_958
tff(fact_9136_ATP_Olambda__959,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_br(nat,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_959
tff(fact_9137_ATP_Olambda__960,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_gi(B,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_960
tff(fact_9138_ATP_Olambda__961,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: A] : aa(A,B,aTP_Lamp_ss(A,B),Uu) = zero_zero(B) ) ).

% ATP.lambda_961
tff(fact_9139_ATP_Olambda__962,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_as(A,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_962
tff(fact_9140_ATP_Olambda__963,axiom,
    ! [A: $tType,B: $tType] :
      ( zero(B)
     => ! [Uu: A] : aa(A,B,aTP_Lamp_lo(A,B),Uu) = zero_zero(B) ) ).

% ATP.lambda_963
tff(fact_9141_ATP_Olambda__964,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_ir(B,A),Uu) = one_one(A) ) ).

% ATP.lambda_964
tff(fact_9142_ATP_Olambda__965,axiom,
    ! [A: $tType,Uu: A] : aa(A,real,aTP_Lamp_alc(A,real),Uu) = one_one(real) ).

% ATP.lambda_965
tff(fact_9143_ATP_Olambda__966,axiom,
    ! [A: $tType,Uu: A] : aa(A,nat,aTP_Lamp_akv(A,nat),Uu) = one_one(nat) ).

% ATP.lambda_966
tff(fact_9144_ATP_Olambda__967,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,B,aTP_Lamp_qh(A,B),Uu) = undefined(B) ).

% ATP.lambda_967
tff(fact_9145_ATP_Olambda__968,axiom,
    ! [Uu: real] :
      ( aa(real,$o,aTP_Lamp_db(real,$o),Uu)
    <=> $false ) ).

% ATP.lambda_968
tff(fact_9146_ATP_Olambda__969,axiom,
    ! [Uu: nat] :
      ( aa(nat,$o,aTP_Lamp_agi(nat,$o),Uu)
    <=> $false ) ).

% ATP.lambda_969
tff(fact_9147_ATP_Olambda__970,axiom,
    ! [A: $tType,Uu: A] :
      ( aa(A,$o,aTP_Lamp_ai(A,$o),Uu)
    <=> $false ) ).

% ATP.lambda_970
tff(fact_9148_ATP_Olambda__971,axiom,
    ! [Uu: nat] :
      ( aa(nat,$o,aTP_Lamp_aka(nat,$o),Uu)
    <=> $true ) ).

% ATP.lambda_971
tff(fact_9149_ATP_Olambda__972,axiom,
    ! [A: $tType,Uu: A] :
      ( aa(A,$o,aTP_Lamp_pi(A,$o),Uu)
    <=> $true ) ).

% ATP.lambda_972

% Type constructors (1270)
tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder,axiom,
    comple5582772986160207858norder(product_unit) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__linorder_1,axiom,
    comple5582772986160207858norder(extended_enat) ).

tff(tcon_Option_Ooption___Complete__Lattices_Ocomplete__linorder_2,axiom,
    ! [A11: $tType] :
      ( comple5582772986160207858norder(A11)
     => comple5582772986160207858norder(option(A11)) ) ).

tff(tcon_Enum_Ofinite__3___Complete__Lattices_Ocomplete__linorder_3,axiom,
    comple5582772986160207858norder(finite_3) ).

tff(tcon_Enum_Ofinite__2___Complete__Lattices_Ocomplete__linorder_4,axiom,
    comple5582772986160207858norder(finite_2) ).

tff(tcon_Enum_Ofinite__1___Complete__Lattices_Ocomplete__linorder_5,axiom,
    comple5582772986160207858norder(finite_1) ).

tff(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
    ! [A11: $tType,A13: $tType] :
      ( bounded_lattice(A13)
     => bounde4967611905675639751up_bot(fun(A11,A13)) ) ).

tff(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
    ! [A11: $tType,A13: $tType] :
      ( boolea8198339166811842893lgebra(A13)
     => boolea8198339166811842893lgebra(fun(A11,A13)) ) ).

tff(tcon_fun___Lattices_Obounded__lattice__top,axiom,
    ! [A11: $tType,A13: $tType] :
      ( bounded_lattice(A13)
     => bounded_lattice_top(fun(A11,A13)) ) ).

tff(tcon_fun___Lattices_Osemilattice__sup,axiom,
    ! [A11: $tType,A13: $tType] :
      ( semilattice_sup(A13)
     => semilattice_sup(fun(A11,A13)) ) ).

tff(tcon_fun___Lattices_Obounded__lattice,axiom,
    ! [A11: $tType,A13: $tType] :
      ( bounded_lattice(A13)
     => bounded_lattice(fun(A11,A13)) ) ).

tff(tcon_fun___Orderings_Oorder__top,axiom,
    ! [A11: $tType,A13: $tType] :
      ( order_top(A13)
     => order_top(fun(A11,A13)) ) ).

tff(tcon_fun___Orderings_Oorder__bot,axiom,
    ! [A11: $tType,A13: $tType] :
      ( order_bot(A13)
     => order_bot(fun(A11,A13)) ) ).

tff(tcon_fun___Orderings_Opreorder,axiom,
    ! [A11: $tType,A13: $tType] :
      ( preorder(A13)
     => preorder(fun(A11,A13)) ) ).

tff(tcon_fun___Finite__Set_Ofinite,axiom,
    ! [A11: $tType,A13: $tType] :
      ( ( finite_finite(A11)
        & finite_finite(A13) )
     => finite_finite(fun(A11,A13)) ) ).

tff(tcon_fun___Lattices_Olattice,axiom,
    ! [A11: $tType,A13: $tType] :
      ( lattice(A13)
     => lattice(fun(A11,A13)) ) ).

tff(tcon_fun___Orderings_Oorder,axiom,
    ! [A11: $tType,A13: $tType] :
      ( order(A13)
     => order(fun(A11,A13)) ) ).

tff(tcon_fun___Orderings_Otop,axiom,
    ! [A11: $tType,A13: $tType] :
      ( top(A13)
     => top(fun(A11,A13)) ) ).

tff(tcon_fun___Orderings_Oord,axiom,
    ! [A11: $tType,A13: $tType] :
      ( ord(A13)
     => ord(fun(A11,A13)) ) ).

tff(tcon_fun___Orderings_Obot,axiom,
    ! [A11: $tType,A13: $tType] :
      ( bot(A13)
     => bot(fun(A11,A13)) ) ).

tff(tcon_fun___Groups_Ouminus,axiom,
    ! [A11: $tType,A13: $tType] :
      ( uminus(A13)
     => uminus(fun(A11,A13)) ) ).

tff(tcon_fun___Groups_Ominus,axiom,
    ! [A11: $tType,A13: $tType] :
      ( minus(A13)
     => minus(fun(A11,A13)) ) ).

tff(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
    condit6923001295902523014norder(int) ).

tff(tcon_Int_Oint___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,axiom,
    bit_un5681908812861735899ations(int) ).

tff(tcon_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
    semiri1453513574482234551roduct(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring__with__nat,axiom,
    euclid5411537665997757685th_nat(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__ring__with__nat,axiom,
    euclid8789492081693882211th_nat(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
    ordere1937475149494474687imp_le(int) ).

tff(tcon_Int_Oint___Euclidean__Division_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___Topological__Spaces_Odiscrete__topology,axiom,
    topolo8865339358273720382pology(int) ).

tff(tcon_Int_Oint___Limits_Otopological__comm__monoid__mult,axiom,
    topolo4987421752381908075d_mult(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___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___Limits_Otopological__monoid__mult,axiom,
    topolo1898628316856586783d_mult(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___Rings_Ocomm__semiring__1__cancel,axiom,
    comm_s4317794764714335236cancel(int) ).

tff(tcon_Int_Oint___Bit__Operations_Osemiring__bits,axiom,
    bit_semiring_bits(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Ot2__space,axiom,
    topological_t2_space(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Ot1__space,axiom,
    topological_t1_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___Least__significant__bit_Olsb,axiom,
    least_6119777620449941438nt_lsb(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_6,axiom,
    semilattice_sup(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___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___Groups_Osemigroup__mult,axiom,
    semigroup_mult(int) ).

tff(tcon_Int_Oint___Rings_Osemidom__modulo,axiom,
    semidom_modulo(int) ).

tff(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
    semidom_divide(int) ).

tff(tcon_Int_Oint___Num_Osemiring__numeral,axiom,
    semiring_numeral(int) ).

tff(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
    semigroup_add(int) ).

tff(tcon_Int_Oint___Rings_Ozero__less__one,axiom,
    zero_less_one(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring,axiom,
    comm_semiring(int) ).

tff(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
    semiring_char_0(int) ).

tff(tcon_Int_Oint___Groups_Oab__group__add,axiom,
    ab_group_add(int) ).

tff(tcon_Int_Oint___Rings_Ozero__neq__one,axiom,
    zero_neq_one(int) ).

tff(tcon_Int_Oint___Rings_Oordered__ring,axiom,
    ordered_ring(int) ).

tff(tcon_Int_Oint___Rings_Oidom__abs__sgn,axiom,
    idom_abs_sgn(int) ).

tff(tcon_Int_Oint___Parity_Oring__parity,axiom,
    ring_parity(int) ).

tff(tcon_Int_Oint___Orderings_Opreorder_7,axiom,
    preorder(int) ).

tff(tcon_Int_Oint___Orderings_Olinorder,axiom,
    linorder(int) ).

tff(tcon_Int_Oint___Groups_Omonoid__mult,axiom,
    monoid_mult(int) ).

tff(tcon_Int_Oint___Rings_Oidom__modulo,axiom,
    idom_modulo(int) ).

tff(tcon_Int_Oint___Rings_Oidom__divide,axiom,
    idom_divide(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__ring__1,axiom,
    comm_ring_1(int) ).

tff(tcon_Int_Oint___Groups_Omonoid__add,axiom,
    monoid_add(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__1,axiom,
    semiring_1(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__0,axiom,
    semiring_0(int) ).

tff(tcon_Int_Oint___Orderings_Ono__top,axiom,
    no_top(int) ).

tff(tcon_Int_Oint___Orderings_Ono__bot,axiom,
    no_bot(int) ).

tff(tcon_Int_Oint___Lattices_Olattice_8,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___Rings_Omult__zero,axiom,
    mult_zero(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__ring,axiom,
    comm_ring(int) ).

tff(tcon_Int_Oint___Orderings_Oorder_9,axiom,
    order(int) ).

tff(tcon_Int_Oint___Num_Oneg__numeral,axiom,
    neg_numeral(int) ).

tff(tcon_Int_Oint___Nat_Oring__char__0,axiom,
    ring_char_0(int) ).

tff(tcon_Int_Oint___Rings_Osemiring,axiom,
    semiring(int) ).

tff(tcon_Int_Oint___Rings_Osemidom,axiom,
    semidom(int) ).

tff(tcon_Int_Oint___Orderings_Oord_10,axiom,
    ord(int) ).

tff(tcon_Int_Oint___Groups_Ouminus_11,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_Ominus_12,axiom,
    minus(int) ).

tff(tcon_Int_Oint___Power_Opower,axiom,
    power(int) ).

tff(tcon_Int_Oint___Num_Onumeral,axiom,
    numeral(int) ).

tff(tcon_Int_Oint___Groups_Ozero,axiom,
    zero(int) ).

tff(tcon_Int_Oint___Groups_Oplus,axiom,
    plus(int) ).

tff(tcon_Int_Oint___Rings_Oring,axiom,
    ring(int) ).

tff(tcon_Int_Oint___Rings_Oidom,axiom,
    idom(int) ).

tff(tcon_Int_Oint___Groups_Oone,axiom,
    one(int) ).

tff(tcon_Int_Oint___Rings_Odvd,axiom,
    dvd(int) ).

tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_13,axiom,
    condit6923001295902523014norder(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_14,axiom,
    bit_un5681908812861735899ations(nat) ).

tff(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_15,axiom,
    semiri1453513574482234551roduct(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_16,axiom,
    euclid5411537665997757685th_nat(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_17,axiom,
    ordere1937475149494474687imp_le(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_18,axiom,
    euclid4440199948858584721cancel(nat) ).

tff(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_19,axiom,
    unique1627219031080169319umeral(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_20,axiom,
    semiri6575147826004484403cancel(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_21,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_22,axiom,
    ordere580206878836729694up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_23,axiom,
    ordere2412721322843649153imp_le(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_24,axiom,
    bit_se359711467146920520ations(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_25,axiom,
    linord2810124833399127020strict(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_26,axiom,
    strict7427464778891057005id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_27,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_28,axiom,
    euclid3725896446679973847miring(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Otopological__space_29,axiom,
    topolo4958980785337419405_space(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_30,axiom,
    topolo1944317154257567458pology(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Odiscrete__topology_31,axiom,
    topolo8865339358273720382pology(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__comm__monoid__mult_32,axiom,
    topolo4987421752381908075d_mult(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__comm__monoid__add_33,axiom,
    topolo5987344860129210374id_add(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Oorder__topology_34,axiom,
    topolo2564578578187576103pology(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_35,axiom,
    semiri2026040879449505780visors(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_36,axiom,
    linord181362715937106298miring(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__semigroup__mult_37,axiom,
    topolo4211221413907600880p_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_38,axiom,
    linord8928482502909563296strict(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_39,axiom,
    semiri3467727345109120633visors(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_40,axiom,
    ordere6658533253407199908up_add(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__monoid__mult_41,axiom,
    topolo1898628316856586783d_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_42,axiom,
    ordere6911136660526730532id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_43,axiom,
    cancel2418104881723323429up_add(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__monoid__add_44,axiom,
    topolo6943815403480290642id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_45,axiom,
    cancel1802427076303600483id_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_46,axiom,
    comm_s4317794764714335236cancel(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_47,axiom,
    bit_semiring_bits(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Ot2__space_48,axiom,
    topological_t2_space(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Ot1__space_49,axiom,
    topological_t1_space(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__comm__semiring_50,axiom,
    ordere2520102378445227354miring(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_51,axiom,
    cancel_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring_52,axiom,
    linordered_semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring__0_53,axiom,
    ordered_semiring_0(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semidom_54,axiom,
    linordered_semidom(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__sup_55,axiom,
    semilattice_sup(nat) ).

tff(tcon_Nat_Onat___Groups_Oab__semigroup__mult_56,axiom,
    ab_semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__cancel_57,axiom,
    semiring_1_cancel(nat) ).

tff(tcon_Nat_Onat___Rings_Oalgebraic__semidom_58,axiom,
    algebraic_semidom(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_59,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_60,axiom,
    ab_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring_61,axiom,
    ordered_semiring(nat) ).

tff(tcon_Nat_Onat___Parity_Osemiring__parity_62,axiom,
    semiring_parity(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__add_63,axiom,
    comm_monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__modulo_64,axiom,
    semiring_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1_65,axiom,
    comm_semiring_1(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__mult_66,axiom,
    semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__modulo_67,axiom,
    semidom_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__divide_68,axiom,
    semidom_divide(nat) ).

tff(tcon_Nat_Onat___Num_Osemiring__numeral_69,axiom,
    semiring_numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__add_70,axiom,
    semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__less__one_71,axiom,
    zero_less_one(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring_72,axiom,
    comm_semiring(nat) ).

tff(tcon_Nat_Onat___Orderings_Owellorder,axiom,
    wellorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder__bot_73,axiom,
    order_bot(nat) ).

tff(tcon_Nat_Onat___Nat_Osemiring__char__0_74,axiom,
    semiring_char_0(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__neq__one_75,axiom,
    zero_neq_one(nat) ).

tff(tcon_Nat_Onat___Orderings_Opreorder_76,axiom,
    preorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Olinorder_77,axiom,
    linorder(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__mult_78,axiom,
    monoid_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__add_79,axiom,
    monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1_80,axiom,
    semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__0_81,axiom,
    semiring_0(nat) ).

tff(tcon_Nat_Onat___Orderings_Ono__top_82,axiom,
    no_top(nat) ).

tff(tcon_Nat_Onat___Lattices_Olattice_83,axiom,
    lattice(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__gcd_84,axiom,
    semiring_gcd(nat) ).

tff(tcon_Nat_Onat___Rings_Omult__zero_85,axiom,
    mult_zero(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder_86,axiom,
    order(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring_87,axiom,
    semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom_88,axiom,
    semidom(nat) ).

tff(tcon_Nat_Onat___Orderings_Oord_89,axiom,
    ord(nat) ).

tff(tcon_Nat_Onat___Orderings_Obot_90,axiom,
    bot(nat) ).

tff(tcon_Nat_Onat___Groups_Ominus_91,axiom,
    minus(nat) ).

tff(tcon_Nat_Onat___Power_Opower_92,axiom,
    power(nat) ).

tff(tcon_Nat_Onat___Num_Onumeral_93,axiom,
    numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Ozero_94,axiom,
    zero(nat) ).

tff(tcon_Nat_Onat___Groups_Oplus_95,axiom,
    plus(nat) ).

tff(tcon_Nat_Onat___Groups_Oone_96,axiom,
    one(nat) ).

tff(tcon_Nat_Onat___Rings_Odvd_97,axiom,
    dvd(nat) ).

tff(tcon_Nat_Onat___Nat_Osize,axiom,
    size(nat) ).

tff(tcon_Num_Onum___Orderings_Opreorder_98,axiom,
    preorder(num) ).

tff(tcon_Num_Onum___Orderings_Olinorder_99,axiom,
    linorder(num) ).

tff(tcon_Num_Onum___Orderings_Oorder_100,axiom,
    order(num) ).

tff(tcon_Num_Onum___Orderings_Oord_101,axiom,
    ord(num) ).

tff(tcon_Num_Onum___Groups_Oplus_102,axiom,
    plus(num) ).

tff(tcon_Num_Onum___Nat_Osize_103,axiom,
    size(num) ).

tff(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_104,axiom,
    semiri1453513574482234551roduct(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_105,axiom,
    ordere1937475149494474687imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_106,axiom,
    semiri6575147826004484403cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_107,axiom,
    strict9044650504122735259up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_108,axiom,
    ordere580206878836729694up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_109,axiom,
    ordere2412721322843649153imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_110,axiom,
    linord2810124833399127020strict(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_111,axiom,
    strict7427464778891057005id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_112,axiom,
    ordere8940638589300402666id_add(rat) ).

tff(tcon_Rat_Orat___Archimedean__Field_Oarchimedean__field,axiom,
    archim462609752435547400_field(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1__strict_113,axiom,
    linord715952674999750819strict(rat) ).

tff(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
    unboun7993243217541854897norder(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_114,axiom,
    semiri2026040879449505780visors(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_115,axiom,
    linord181362715937106298miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_116,axiom,
    linord8928482502909563296strict(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_117,axiom,
    semiri3467727345109120633visors(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_118,axiom,
    ordere6658533253407199908up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_119,axiom,
    ordere166539214618696060dd_abs(rat) ).

tff(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
    archim2362893244070406136eiling(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_120,axiom,
    ordere6911136660526730532id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_121,axiom,
    linord5086331880401160121up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_122,axiom,
    cancel2418104881723323429up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_123,axiom,
    ring_15535105094025558882visors(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_124,axiom,
    cancel1802427076303600483id_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring__strict_125,axiom,
    linord4710134922213307826strict(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_126,axiom,
    comm_s4317794764714335236cancel(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__comm__semiring_127,axiom,
    ordere2520102378445227354miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1_128,axiom,
    linord6961819062388156250ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add_129,axiom,
    ordered_ab_group_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_130,axiom,
    cancel_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring_131,axiom,
    linordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring__0_132,axiom,
    ordered_semiring_0(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semidom_133,axiom,
    linordered_semidom(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
    dense_linorder(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__sup_134,axiom,
    semilattice_sup(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__mult_135,axiom,
    ab_semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__cancel_136,axiom,
    semiring_1_cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_137,axiom,
    comm_monoid_mult(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__add_138,axiom,
    ab_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
    linordered_field(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring_139,axiom,
    ordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring__abs_140,axiom,
    ordered_ring_abs(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__add_141,axiom,
    comm_monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring_142,axiom,
    linordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__idom_143,axiom,
    linordered_idom(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1_144,axiom,
    comm_semiring_1(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__order,axiom,
    dense_order(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__mult_145,axiom,
    semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom__divide_146,axiom,
    semidom_divide(rat) ).

tff(tcon_Rat_Orat___Num_Osemiring__numeral_147,axiom,
    semiring_numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__add_148,axiom,
    semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
    division_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Ozero__less__one_149,axiom,
    zero_less_one(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring_150,axiom,
    comm_semiring(rat) ).

tff(tcon_Rat_Orat___Nat_Osemiring__char__0_151,axiom,
    semiring_char_0(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__group__add_152,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_153,axiom,
    zero_neq_one(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring_154,axiom,
    ordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__abs__sgn_155,axiom,
    idom_abs_sgn(rat) ).

tff(tcon_Rat_Orat___Orderings_Opreorder_156,axiom,
    preorder(rat) ).

tff(tcon_Rat_Orat___Orderings_Olinorder_157,axiom,
    linorder(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__mult_158,axiom,
    monoid_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__divide_159,axiom,
    idom_divide(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring__1_160,axiom,
    comm_ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__add_161,axiom,
    monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1_162,axiom,
    semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__0_163,axiom,
    semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__top_164,axiom,
    no_top(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__bot_165,axiom,
    no_bot(rat) ).

tff(tcon_Rat_Orat___Lattices_Olattice_166,axiom,
    lattice(rat) ).

tff(tcon_Rat_Orat___Groups_Ogroup__add_167,axiom,
    group_add(rat) ).

tff(tcon_Rat_Orat___Rings_Omult__zero_168,axiom,
    mult_zero(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring_169,axiom,
    comm_ring(rat) ).

tff(tcon_Rat_Orat___Orderings_Oorder_170,axiom,
    order(rat) ).

tff(tcon_Rat_Orat___Num_Oneg__numeral_171,axiom,
    neg_numeral(rat) ).

tff(tcon_Rat_Orat___Nat_Oring__char__0_172,axiom,
    ring_char_0(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring_173,axiom,
    semiring(rat) ).

tff(tcon_Rat_Orat___Fields_Oinverse,axiom,
    inverse(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom_174,axiom,
    semidom(rat) ).

tff(tcon_Rat_Orat___Orderings_Oord_175,axiom,
    ord(rat) ).

tff(tcon_Rat_Orat___Groups_Ouminus_176,axiom,
    uminus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1_177,axiom,
    ring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Oabs__if_178,axiom,
    abs_if(rat) ).

tff(tcon_Rat_Orat___Groups_Ominus_179,axiom,
    minus(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield,axiom,
    field(rat) ).

tff(tcon_Rat_Orat___Power_Opower_180,axiom,
    power(rat) ).

tff(tcon_Rat_Orat___Num_Onumeral_181,axiom,
    numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Ozero_182,axiom,
    zero(rat) ).

tff(tcon_Rat_Orat___Groups_Oplus_183,axiom,
    plus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring_184,axiom,
    ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom_185,axiom,
    idom(rat) ).

tff(tcon_Rat_Orat___Groups_Oone_186,axiom,
    one(rat) ).

tff(tcon_Rat_Orat___Rings_Odvd_187,axiom,
    dvd(rat) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_188,axiom,
    ! [A11: $tType] : bounde4967611905675639751up_bot(set(A11)) ).

tff(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_189,axiom,
    ! [A11: $tType] : boolea8198339166811842893lgebra(set(A11)) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice__top_190,axiom,
    ! [A11: $tType] : bounded_lattice_top(set(A11)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__sup_191,axiom,
    ! [A11: $tType] : semilattice_sup(set(A11)) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice_192,axiom,
    ! [A11: $tType] : bounded_lattice(set(A11)) ).

tff(tcon_Set_Oset___Orderings_Oorder__top_193,axiom,
    ! [A11: $tType] : order_top(set(A11)) ).

tff(tcon_Set_Oset___Orderings_Oorder__bot_194,axiom,
    ! [A11: $tType] : order_bot(set(A11)) ).

tff(tcon_Set_Oset___Orderings_Opreorder_195,axiom,
    ! [A11: $tType] : preorder(set(A11)) ).

tff(tcon_Set_Oset___Finite__Set_Ofinite_196,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => finite_finite(set(A11)) ) ).

tff(tcon_Set_Oset___Lattices_Olattice_197,axiom,
    ! [A11: $tType] : lattice(set(A11)) ).

tff(tcon_Set_Oset___Orderings_Oorder_198,axiom,
    ! [A11: $tType] : order(set(A11)) ).

tff(tcon_Set_Oset___Orderings_Otop_199,axiom,
    ! [A11: $tType] : top(set(A11)) ).

tff(tcon_Set_Oset___Orderings_Oord_200,axiom,
    ! [A11: $tType] : ord(set(A11)) ).

tff(tcon_Set_Oset___Orderings_Obot_201,axiom,
    ! [A11: $tType] : bot(set(A11)) ).

tff(tcon_Set_Oset___Groups_Ouminus_202,axiom,
    ! [A11: $tType] : uminus(set(A11)) ).

tff(tcon_Set_Oset___Groups_Ominus_203,axiom,
    ! [A11: $tType] : minus(set(A11)) ).

tff(tcon_HOL_Obool___Topological__Spaces_Otopological__space_204,axiom,
    topolo4958980785337419405_space($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_205,axiom,
    topolo1944317154257567458pology($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Odiscrete__topology_206,axiom,
    topolo8865339358273720382pology($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_207,axiom,
    bounde4967611905675639751up_bot($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_208,axiom,
    topolo2564578578187576103pology($o) ).

tff(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_209,axiom,
    boolea8198339166811842893lgebra($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice__top_210,axiom,
    bounded_lattice_top($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Ot2__space_211,axiom,
    topological_t2_space($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Ot1__space_212,axiom,
    topological_t1_space($o) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__sup_213,axiom,
    semilattice_sup($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice_214,axiom,
    bounded_lattice($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder__top_215,axiom,
    order_top($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder__bot_216,axiom,
    order_bot($o) ).

tff(tcon_HOL_Obool___Orderings_Opreorder_217,axiom,
    preorder($o) ).

tff(tcon_HOL_Obool___Orderings_Olinorder_218,axiom,
    linorder($o) ).

tff(tcon_HOL_Obool___Finite__Set_Ofinite_219,axiom,
    finite_finite($o) ).

tff(tcon_HOL_Obool___Lattices_Olattice_220,axiom,
    lattice($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder_221,axiom,
    order($o) ).

tff(tcon_HOL_Obool___Orderings_Otop_222,axiom,
    top($o) ).

tff(tcon_HOL_Obool___Orderings_Oord_223,axiom,
    ord($o) ).

tff(tcon_HOL_Obool___Orderings_Obot_224,axiom,
    bot($o) ).

tff(tcon_HOL_Obool___Groups_Ouminus_225,axiom,
    uminus($o) ).

tff(tcon_HOL_Obool___Groups_Ominus_226,axiom,
    minus($o) ).

tff(tcon_List_Olist___Nat_Osize_227,axiom,
    ! [A11: $tType] : size(list(A11)) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_228,axiom,
    condit6923001295902523014norder(real) ).

tff(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_229,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_230,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_231,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_232,axiom,
    strict9044650504122735259up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_233,axiom,
    ordere580206878836729694up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_234,axiom,
    ordere2412721322843649153imp_le(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__comm__semiring__strict_235,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_236,axiom,
    strict7427464778891057005id_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add_237,axiom,
    ordere8940638589300402666id_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Otopological__space_238,axiom,
    topolo4958980785337419405_space(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_239,axiom,
    topolo1944317154257567458pology(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__field,axiom,
    real_V3459762299906320749_field(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__div__algebra,axiom,
    real_V5047593784448816457lgebra(real) ).

tff(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field_240,axiom,
    archim462609752435547400_field(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1__strict_241,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_242,axiom,
    unboun7993243217541854897norder(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__comm__monoid__add_243,axiom,
    topolo5987344860129210374id_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_244,axiom,
    topolo2564578578187576103pology(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_245,axiom,
    semiri2026040879449505780visors(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_246,axiom,
    linord181362715937106298miring(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__algebra__1,axiom,
    real_V2191834092415804123ebra_1(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Ocomplete__space,axiom,
    real_V8037385150606011577_space(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__semigroup__mult_247,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_248,axiom,
    linord8928482502909563296strict(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_249,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_250,axiom,
    ordere6658533253407199908up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_251,axiom,
    ordere166539214618696060dd_abs(real) ).

tff(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling_252,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_253,axiom,
    ordere6911136660526730532id_add(real) ).

tff(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_254,axiom,
    linord5086331880401160121up_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_255,axiom,
    cancel2418104881723323429up_add(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_256,axiom,
    ring_15535105094025558882visors(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__field,axiom,
    real_V7773925162809079976_field(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__monoid__add_257,axiom,
    topolo6943815403480290642id_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_258,axiom,
    cancel1802427076303600483id_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring__strict_259,axiom,
    linord4710134922213307826strict(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__1__cancel_260,axiom,
    comm_s4317794764714335236cancel(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__group__add,axiom,
    topolo1633459387980952147up_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ot2__space_261,axiom,
    topological_t2_space(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ot1__space_262,axiom,
    topological_t1_space(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__comm__semiring_263,axiom,
    ordere2520102378445227354miring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1_264,axiom,
    linord6961819062388156250ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add_265,axiom,
    ordered_ab_group_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_266,axiom,
    cancel_semigroup_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring_267,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_268,axiom,
    ordered_semiring_0(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semidom_269,axiom,
    linordered_semidom(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__linorder_270,axiom,
    dense_linorder(real) ).

tff(tcon_Real_Oreal___Lattices_Osemilattice__sup_271,axiom,
    semilattice_sup(real) ).

tff(tcon_Real_Oreal___Groups_Oab__semigroup__mult_272,axiom,
    ab_semigroup_mult(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__cancel_273,axiom,
    semiring_1_cancel(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_274,axiom,
    comm_monoid_mult(real) ).

tff(tcon_Real_Oreal___Groups_Oab__semigroup__add_275,axiom,
    ab_semigroup_add(real) ).

tff(tcon_Real_Oreal___Fields_Olinordered__field_276,axiom,
    linordered_field(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__semiring_277,axiom,
    ordered_semiring(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring__abs_278,axiom,
    ordered_ring_abs(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__add_279,axiom,
    comm_monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring_280,axiom,
    linordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__idom_281,axiom,
    linordered_idom(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__1_282,axiom,
    comm_semiring_1(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__order_283,axiom,
    dense_order(real) ).

tff(tcon_Real_Oreal___Groups_Osemigroup__mult_284,axiom,
    semigroup_mult(real) ).

tff(tcon_Real_Oreal___Rings_Osemidom__divide_285,axiom,
    semidom_divide(real) ).

tff(tcon_Real_Oreal___Num_Osemiring__numeral_286,axiom,
    semiring_numeral(real) ).

tff(tcon_Real_Oreal___Groups_Osemigroup__add_287,axiom,
    semigroup_add(real) ).

tff(tcon_Real_Oreal___Fields_Odivision__ring_288,axiom,
    division_ring(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__less__one_289,axiom,
    zero_less_one(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring_290,axiom,
    comm_semiring(real) ).

tff(tcon_Real_Oreal___Nat_Osemiring__char__0_291,axiom,
    semiring_char_0(real) ).

tff(tcon_Real_Oreal___Groups_Oab__group__add_292,axiom,
    ab_group_add(real) ).

tff(tcon_Real_Oreal___Fields_Ofield__char__0_293,axiom,
    field_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__neq__one_294,axiom,
    zero_neq_one(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring_295,axiom,
    ordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom__abs__sgn_296,axiom,
    idom_abs_sgn(real) ).

tff(tcon_Real_Oreal___Orderings_Opreorder_297,axiom,
    preorder(real) ).

tff(tcon_Real_Oreal___Orderings_Olinorder_298,axiom,
    linorder(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__mult_299,axiom,
    monoid_mult(real) ).

tff(tcon_Real_Oreal___Transcendental_Oln,axiom,
    ln(real) ).

tff(tcon_Real_Oreal___Rings_Oidom__divide_300,axiom,
    idom_divide(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring__1_301,axiom,
    comm_ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__add_302,axiom,
    monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1_303,axiom,
    semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__0_304,axiom,
    semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__top_305,axiom,
    no_top(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__bot_306,axiom,
    no_bot(real) ).

tff(tcon_Real_Oreal___Lattices_Olattice_307,axiom,
    lattice(real) ).

tff(tcon_Real_Oreal___Groups_Ogroup__add_308,axiom,
    group_add(real) ).

tff(tcon_Real_Oreal___Rings_Omult__zero_309,axiom,
    mult_zero(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring_310,axiom,
    comm_ring(real) ).

tff(tcon_Real_Oreal___Orderings_Oorder_311,axiom,
    order(real) ).

tff(tcon_Real_Oreal___Num_Oneg__numeral_312,axiom,
    neg_numeral(real) ).

tff(tcon_Real_Oreal___Nat_Oring__char__0_313,axiom,
    ring_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring_314,axiom,
    semiring(real) ).

tff(tcon_Real_Oreal___Fields_Oinverse_315,axiom,
    inverse(real) ).

tff(tcon_Real_Oreal___Rings_Osemidom_316,axiom,
    semidom(real) ).

tff(tcon_Real_Oreal___Orderings_Oord_317,axiom,
    ord(real) ).

tff(tcon_Real_Oreal___Groups_Ouminus_318,axiom,
    uminus(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1_319,axiom,
    ring_1(real) ).

tff(tcon_Real_Oreal___Rings_Oabs__if_320,axiom,
    abs_if(real) ).

tff(tcon_Real_Oreal___Groups_Ominus_321,axiom,
    minus(real) ).

tff(tcon_Real_Oreal___Fields_Ofield_322,axiom,
    field(real) ).

tff(tcon_Real_Oreal___Power_Opower_323,axiom,
    power(real) ).

tff(tcon_Real_Oreal___Num_Onumeral_324,axiom,
    numeral(real) ).

tff(tcon_Real_Oreal___Groups_Ozero_325,axiom,
    zero(real) ).

tff(tcon_Real_Oreal___Groups_Oplus_326,axiom,
    plus(real) ).

tff(tcon_Real_Oreal___Rings_Oring_327,axiom,
    ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom_328,axiom,
    idom(real) ).

tff(tcon_Real_Oreal___Groups_Oone_329,axiom,
    one(real) ).

tff(tcon_Real_Oreal___Rings_Odvd_330,axiom,
    dvd(real) ).

tff(tcon_Word_Oword___Bit__Operations_Osemiring__bit__operations_331,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => bit_se359711467146920520ations(word(A11)) ) ).

tff(tcon_Word_Oword___Bit__Comprehension_Obit__comprehension_332,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => bit_bi6583157726757044596ension(word(A11)) ) ).

tff(tcon_Word_Oword___Bit__Operations_Oring__bit__operations_333,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => bit_ri3973907225187159222ations(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Ocancel__ab__semigroup__add_334,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => cancel2418104881723323429up_add(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Ocancel__comm__monoid__add_335,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => cancel1802427076303600483id_add(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__semiring__1__cancel_336,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => comm_s4317794764714335236cancel(word(A11)) ) ).

tff(tcon_Word_Oword___Bit__Operations_Osemiring__bits_337,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => bit_semiring_bits(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Ocancel__semigroup__add_338,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => cancel_semigroup_add(word(A11)) ) ).

tff(tcon_Word_Oword___Least__significant__bit_Olsb_339,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => least_6119777620449941438nt_lsb(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Oab__semigroup__mult_340,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => ab_semigroup_mult(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring__1__cancel_341,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => semiring_1_cancel(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Ocomm__monoid__mult_342,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => comm_monoid_mult(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Oab__semigroup__add_343,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => ab_semigroup_add(word(A11)) ) ).

tff(tcon_Word_Oword___Generic__set__bit_Oset__bit_344,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => generic_set_set_bit(word(A11)) ) ).

tff(tcon_Word_Oword___Parity_Osemiring__parity_345,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => semiring_parity(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Ocomm__monoid__add_346,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => comm_monoid_add(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring__modulo_347,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => semiring_modulo(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__semiring__1_348,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => comm_semiring_1(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Osemigroup__mult_349,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => semigroup_mult(word(A11)) ) ).

tff(tcon_Word_Oword___Num_Osemiring__numeral_350,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => semiring_numeral(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Osemigroup__add_351,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => semigroup_add(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__semiring_352,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => comm_semiring(word(A11)) ) ).

tff(tcon_Word_Oword___Orderings_Owellorder_353,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => wellorder(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Oab__group__add_354,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => ab_group_add(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Ozero__neq__one_355,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => zero_neq_one(word(A11)) ) ).

tff(tcon_Word_Oword___Parity_Oring__parity_356,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => ring_parity(word(A11)) ) ).

tff(tcon_Word_Oword___Orderings_Opreorder_357,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => preorder(word(A11)) ) ).

tff(tcon_Word_Oword___Orderings_Olinorder_358,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => linorder(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Omonoid__mult_359,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => monoid_mult(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__ring__1_360,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => comm_ring_1(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Omonoid__add_361,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => monoid_add(word(A11)) ) ).

tff(tcon_Word_Oword___Finite__Set_Ofinite_362,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => finite_finite(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring__1_363,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => semiring_1(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring__0_364,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => semiring_0(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Ogroup__add_365,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => group_add(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Omult__zero_366,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => mult_zero(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__ring_367,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => comm_ring(word(A11)) ) ).

tff(tcon_Word_Oword___Orderings_Oorder_368,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => order(word(A11)) ) ).

tff(tcon_Word_Oword___Num_Oneg__numeral_369,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => neg_numeral(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring_370,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => semiring(word(A11)) ) ).

tff(tcon_Word_Oword___Orderings_Oord_371,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => ord(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Ouminus_372,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => uminus(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Oring__1_373,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => ring_1(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Ominus_374,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => minus(word(A11)) ) ).

tff(tcon_Word_Oword___Power_Opower_375,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => power(word(A11)) ) ).

tff(tcon_Word_Oword___Num_Onumeral_376,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => numeral(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Ozero_377,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => zero(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Oplus_378,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => plus(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Oring_379,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => ring(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Oone_380,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => one(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Odvd_381,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => dvd(word(A11)) ) ).

tff(tcon_Word_Oword___Nat_Osize_382,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => size(word(A11)) ) ).

tff(tcon_Sum__Type_Osum___Finite__Set_Ofinite_383,axiom,
    ! [A11: $tType,A13: $tType] :
      ( ( finite_finite(A11)
        & finite_finite(A13) )
     => finite_finite(sum_sum(A11,A13)) ) ).

tff(tcon_Sum__Type_Osum___Nat_Osize_384,axiom,
    ! [A11: $tType,A13: $tType] : size(sum_sum(A11,A13)) ).

tff(tcon_Enum_Ofinite__1___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_385,axiom,
    condit6923001295902523014norder(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__ab__semigroup__monoid__add__imp__le_386,axiom,
    ordere1937475149494474687imp_le(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Osemiring__no__zero__divisors__cancel_387,axiom,
    semiri6575147826004484403cancel(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ostrict__ordered__ab__semigroup__add_388,axiom,
    strict9044650504122735259up_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__cancel__comm__monoid__diff_389,axiom,
    ordere1170586879665033532d_diff(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__cancel__ab__semigroup__add_390,axiom,
    ordere580206878836729694up_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__ab__semigroup__add__imp__le_391,axiom,
    ordere2412721322843649153imp_le(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Olinordered__comm__semiring__strict_392,axiom,
    linord2810124833399127020strict(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ostrict__ordered__comm__monoid__add_393,axiom,
    strict7427464778891057005id_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__cancel__comm__monoid__add_394,axiom,
    ordere8940638589300402666id_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ocanonically__ordered__monoid__add_395,axiom,
    canoni5634975068530333245id_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Lattices_Obounded__semilattice__sup__bot_396,axiom,
    bounde4967611905675639751up_bot(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Olinordered__semiring__strict_397,axiom,
    linord8928482502909563296strict(finite_1) ).

tff(tcon_Enum_Ofinite__1___Boolean__Algebras_Oboolean__algebra_398,axiom,
    boolea8198339166811842893lgebra(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Osemiring__no__zero__divisors_399,axiom,
    semiri3467727345109120633visors(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__ab__semigroup__add_400,axiom,
    ordere6658533253407199908up_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__ab__group__add__abs_401,axiom,
    ordere166539214618696060dd_abs(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__comm__monoid__add_402,axiom,
    ordere6911136660526730532id_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Olinordered__ab__group__add_403,axiom,
    linord5086331880401160121up_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ocancel__ab__semigroup__add_404,axiom,
    cancel2418104881723323429up_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ocancel__comm__monoid__add_405,axiom,
    cancel1802427076303600483id_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Olinordered__ring__strict_406,axiom,
    linord4710134922213307826strict(finite_1) ).

tff(tcon_Enum_Ofinite__1___Lattices_Obounded__lattice__top_407,axiom,
    bounded_lattice_top(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oordered__comm__semiring_408,axiom,
    ordere2520102378445227354miring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__ab__group__add_409,axiom,
    ordered_ab_group_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ocancel__semigroup__add_410,axiom,
    cancel_semigroup_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Olinordered__semiring_411,axiom,
    linordered_semiring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oordered__semiring__0_412,axiom,
    ordered_semiring_0(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Odense__linorder_413,axiom,
    dense_linorder(finite_1) ).

tff(tcon_Enum_Ofinite__1___Lattices_Osemilattice__sup_414,axiom,
    semilattice_sup(finite_1) ).

tff(tcon_Enum_Ofinite__1___Lattices_Obounded__lattice_415,axiom,
    bounded_lattice(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oab__semigroup__mult_416,axiom,
    ab_semigroup_mult(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ocomm__monoid__mult_417,axiom,
    comm_monoid_mult(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ocomm__monoid__diff_418,axiom,
    comm_monoid_diff(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oab__semigroup__add_419,axiom,
    ab_semigroup_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oordered__semiring_420,axiom,
    ordered_semiring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oordered__ring__abs_421,axiom,
    ordered_ring_abs(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ocomm__monoid__add_422,axiom,
    comm_monoid_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Olinordered__ring_423,axiom,
    linordered_ring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Odense__order_424,axiom,
    dense_order(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Osemigroup__mult_425,axiom,
    semigroup_mult(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Osemigroup__add_426,axiom,
    semigroup_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Ocomm__semiring_427,axiom,
    comm_semiring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Owellorder_428,axiom,
    wellorder(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Oorder__top_429,axiom,
    order_top(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Oorder__bot_430,axiom,
    order_bot(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oab__group__add_431,axiom,
    ab_group_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oordered__ring_432,axiom,
    ordered_ring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Opreorder_433,axiom,
    preorder(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Olinorder_434,axiom,
    linorder(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Omonoid__mult_435,axiom,
    monoid_mult(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Omonoid__add_436,axiom,
    monoid_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Finite__Set_Ofinite_437,axiom,
    finite_finite(finite_1) ).

tff(tcon_Enum_Ofinite__1___Type__Length_Olen0,axiom,
    type_len0(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Osemiring__0_438,axiom,
    semiring_0(finite_1) ).

tff(tcon_Enum_Ofinite__1___Lattices_Olattice_439,axiom,
    lattice(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ogroup__add_440,axiom,
    group_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Type__Length_Olen,axiom,
    type_len(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Omult__zero_441,axiom,
    mult_zero(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Ocomm__ring_442,axiom,
    comm_ring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Oorder_443,axiom,
    order(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Osemiring_444,axiom,
    semiring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Fields_Oinverse_445,axiom,
    inverse(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Otop_446,axiom,
    top(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Oord_447,axiom,
    ord(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Obot_448,axiom,
    bot(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ouminus_449,axiom,
    uminus(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oabs__if_450,axiom,
    abs_if(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ominus_451,axiom,
    minus(finite_1) ).

tff(tcon_Enum_Ofinite__1___Power_Opower_452,axiom,
    power(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ozero_453,axiom,
    zero(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oplus_454,axiom,
    plus(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oring_455,axiom,
    ring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oone_456,axiom,
    one(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Odvd_457,axiom,
    dvd(finite_1) ).

tff(tcon_Enum_Ofinite__2___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_458,axiom,
    condit6923001295902523014norder(finite_2) ).

tff(tcon_Enum_Ofinite__2___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_459,axiom,
    semiri1453513574482234551roduct(finite_2) ).

tff(tcon_Enum_Ofinite__2___Euclidean__Division_Oeuclidean__semiring__cancel_460,axiom,
    euclid4440199948858584721cancel(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__no__zero__divisors__cancel_461,axiom,
    semiri6575147826004484403cancel(finite_2) ).

tff(tcon_Enum_Ofinite__2___Euclidean__Division_Oeuclidean__semiring_462,axiom,
    euclid3725896446679973847miring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Lattices_Obounded__semilattice__sup__bot_463,axiom,
    bounde4967611905675639751up_bot(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__1__no__zero__divisors_464,axiom,
    semiri2026040879449505780visors(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__no__zero__divisors_465,axiom,
    semiri3467727345109120633visors(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ocancel__ab__semigroup__add_466,axiom,
    cancel2418104881723323429up_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oring__1__no__zero__divisors_467,axiom,
    ring_15535105094025558882visors(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ocancel__comm__monoid__add_468,axiom,
    cancel1802427076303600483id_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring__1__cancel_469,axiom,
    comm_s4317794764714335236cancel(finite_2) ).

tff(tcon_Enum_Ofinite__2___Lattices_Obounded__lattice__top_470,axiom,
    bounded_lattice_top(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ocancel__semigroup__add_471,axiom,
    cancel_semigroup_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Lattices_Osemilattice__sup_472,axiom,
    semilattice_sup(finite_2) ).

tff(tcon_Enum_Ofinite__2___Lattices_Obounded__lattice_473,axiom,
    bounded_lattice(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Oab__semigroup__mult_474,axiom,
    ab_semigroup_mult(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__1__cancel_475,axiom,
    semiring_1_cancel(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oalgebraic__semidom_476,axiom,
    algebraic_semidom(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ocomm__monoid__mult_477,axiom,
    comm_monoid_mult(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Oab__semigroup__add_478,axiom,
    ab_semigroup_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ocomm__monoid__add_479,axiom,
    comm_monoid_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__modulo_480,axiom,
    semiring_modulo(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring__1_481,axiom,
    comm_semiring_1(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Osemigroup__mult_482,axiom,
    semigroup_mult(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemidom__modulo_483,axiom,
    semidom_modulo(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemidom__divide_484,axiom,
    semidom_divide(finite_2) ).

tff(tcon_Enum_Ofinite__2___Num_Osemiring__numeral_485,axiom,
    semiring_numeral(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Osemigroup__add_486,axiom,
    semigroup_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Fields_Odivision__ring_487,axiom,
    division_ring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring_488,axiom,
    comm_semiring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Owellorder_489,axiom,
    wellorder(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Oorder__top_490,axiom,
    order_top(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Oorder__bot_491,axiom,
    order_bot(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Oab__group__add_492,axiom,
    ab_group_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ozero__neq__one_493,axiom,
    zero_neq_one(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oidom__abs__sgn_494,axiom,
    idom_abs_sgn(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Opreorder_495,axiom,
    preorder(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Olinorder_496,axiom,
    linorder(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Omonoid__mult_497,axiom,
    monoid_mult(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oidom__modulo_498,axiom,
    idom_modulo(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oidom__divide_499,axiom,
    idom_divide(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ocomm__ring__1_500,axiom,
    comm_ring_1(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Omonoid__add_501,axiom,
    monoid_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Finite__Set_Ofinite_502,axiom,
    finite_finite(finite_2) ).

tff(tcon_Enum_Ofinite__2___Type__Length_Olen0_503,axiom,
    type_len0(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__1_504,axiom,
    semiring_1(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__0_505,axiom,
    semiring_0(finite_2) ).

tff(tcon_Enum_Ofinite__2___Lattices_Olattice_506,axiom,
    lattice(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ogroup__add_507,axiom,
    group_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Type__Length_Olen_508,axiom,
    type_len(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Omult__zero_509,axiom,
    mult_zero(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ocomm__ring_510,axiom,
    comm_ring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Oorder_511,axiom,
    order(finite_2) ).

tff(tcon_Enum_Ofinite__2___Num_Oneg__numeral_512,axiom,
    neg_numeral(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring_513,axiom,
    semiring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Fields_Oinverse_514,axiom,
    inverse(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemidom_515,axiom,
    semidom(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Otop_516,axiom,
    top(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Oord_517,axiom,
    ord(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Obot_518,axiom,
    bot(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ouminus_519,axiom,
    uminus(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oring__1_520,axiom,
    ring_1(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ominus_521,axiom,
    minus(finite_2) ).

tff(tcon_Enum_Ofinite__2___Fields_Ofield_522,axiom,
    field(finite_2) ).

tff(tcon_Enum_Ofinite__2___Power_Opower_523,axiom,
    power(finite_2) ).

tff(tcon_Enum_Ofinite__2___Num_Onumeral_524,axiom,
    numeral(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ozero_525,axiom,
    zero(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Oplus_526,axiom,
    plus(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oring_527,axiom,
    ring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oidom_528,axiom,
    idom(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Oone_529,axiom,
    one(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Odvd_530,axiom,
    dvd(finite_2) ).

tff(tcon_Enum_Ofinite__3___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_531,axiom,
    condit6923001295902523014norder(finite_3) ).

tff(tcon_Enum_Ofinite__3___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_532,axiom,
    semiri1453513574482234551roduct(finite_3) ).

tff(tcon_Enum_Ofinite__3___Euclidean__Division_Oeuclidean__semiring__cancel_533,axiom,
    euclid4440199948858584721cancel(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__no__zero__divisors__cancel_534,axiom,
    semiri6575147826004484403cancel(finite_3) ).

tff(tcon_Enum_Ofinite__3___Euclidean__Division_Oeuclidean__semiring_535,axiom,
    euclid3725896446679973847miring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Lattices_Obounded__semilattice__sup__bot_536,axiom,
    bounde4967611905675639751up_bot(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__1__no__zero__divisors_537,axiom,
    semiri2026040879449505780visors(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__no__zero__divisors_538,axiom,
    semiri3467727345109120633visors(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ocancel__ab__semigroup__add_539,axiom,
    cancel2418104881723323429up_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oring__1__no__zero__divisors_540,axiom,
    ring_15535105094025558882visors(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ocancel__comm__monoid__add_541,axiom,
    cancel1802427076303600483id_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring__1__cancel_542,axiom,
    comm_s4317794764714335236cancel(finite_3) ).

tff(tcon_Enum_Ofinite__3___Lattices_Obounded__lattice__top_543,axiom,
    bounded_lattice_top(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ocancel__semigroup__add_544,axiom,
    cancel_semigroup_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Lattices_Osemilattice__sup_545,axiom,
    semilattice_sup(finite_3) ).

tff(tcon_Enum_Ofinite__3___Lattices_Obounded__lattice_546,axiom,
    bounded_lattice(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Oab__semigroup__mult_547,axiom,
    ab_semigroup_mult(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__1__cancel_548,axiom,
    semiring_1_cancel(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oalgebraic__semidom_549,axiom,
    algebraic_semidom(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ocomm__monoid__mult_550,axiom,
    comm_monoid_mult(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Oab__semigroup__add_551,axiom,
    ab_semigroup_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ocomm__monoid__add_552,axiom,
    comm_monoid_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__modulo_553,axiom,
    semiring_modulo(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring__1_554,axiom,
    comm_semiring_1(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Osemigroup__mult_555,axiom,
    semigroup_mult(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemidom__modulo_556,axiom,
    semidom_modulo(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemidom__divide_557,axiom,
    semidom_divide(finite_3) ).

tff(tcon_Enum_Ofinite__3___Num_Osemiring__numeral_558,axiom,
    semiring_numeral(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Osemigroup__add_559,axiom,
    semigroup_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Fields_Odivision__ring_560,axiom,
    division_ring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring_561,axiom,
    comm_semiring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Owellorder_562,axiom,
    wellorder(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Oorder__top_563,axiom,
    order_top(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Oorder__bot_564,axiom,
    order_bot(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Oab__group__add_565,axiom,
    ab_group_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ozero__neq__one_566,axiom,
    zero_neq_one(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oidom__abs__sgn_567,axiom,
    idom_abs_sgn(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Opreorder_568,axiom,
    preorder(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Olinorder_569,axiom,
    linorder(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Omonoid__mult_570,axiom,
    monoid_mult(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oidom__modulo_571,axiom,
    idom_modulo(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oidom__divide_572,axiom,
    idom_divide(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ocomm__ring__1_573,axiom,
    comm_ring_1(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Omonoid__add_574,axiom,
    monoid_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Finite__Set_Ofinite_575,axiom,
    finite_finite(finite_3) ).

tff(tcon_Enum_Ofinite__3___Type__Length_Olen0_576,axiom,
    type_len0(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__1_577,axiom,
    semiring_1(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__0_578,axiom,
    semiring_0(finite_3) ).

tff(tcon_Enum_Ofinite__3___Lattices_Olattice_579,axiom,
    lattice(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ogroup__add_580,axiom,
    group_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Type__Length_Olen_581,axiom,
    type_len(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Omult__zero_582,axiom,
    mult_zero(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ocomm__ring_583,axiom,
    comm_ring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Oorder_584,axiom,
    order(finite_3) ).

tff(tcon_Enum_Ofinite__3___Num_Oneg__numeral_585,axiom,
    neg_numeral(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring_586,axiom,
    semiring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Fields_Oinverse_587,axiom,
    inverse(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemidom_588,axiom,
    semidom(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Otop_589,axiom,
    top(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Oord_590,axiom,
    ord(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Obot_591,axiom,
    bot(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ouminus_592,axiom,
    uminus(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oring__1_593,axiom,
    ring_1(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ominus_594,axiom,
    minus(finite_3) ).

tff(tcon_Enum_Ofinite__3___Fields_Ofield_595,axiom,
    field(finite_3) ).

tff(tcon_Enum_Ofinite__3___Power_Opower_596,axiom,
    power(finite_3) ).

tff(tcon_Enum_Ofinite__3___Num_Onumeral_597,axiom,
    numeral(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ozero_598,axiom,
    zero(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Oplus_599,axiom,
    plus(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oring_600,axiom,
    ring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oidom_601,axiom,
    idom(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Oone_602,axiom,
    one(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Odvd_603,axiom,
    dvd(finite_3) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_604,axiom,
    ! [A11: $tType] : bounde4967611905675639751up_bot(filter(A11)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice__top_605,axiom,
    ! [A11: $tType] : bounded_lattice_top(filter(A11)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_606,axiom,
    ! [A11: $tType] : semilattice_sup(filter(A11)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice_607,axiom,
    ! [A11: $tType] : bounded_lattice(filter(A11)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__top_608,axiom,
    ! [A11: $tType] : order_top(filter(A11)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__bot_609,axiom,
    ! [A11: $tType] : order_bot(filter(A11)) ).

tff(tcon_Filter_Ofilter___Orderings_Opreorder_610,axiom,
    ! [A11: $tType] : preorder(filter(A11)) ).

tff(tcon_Filter_Ofilter___Lattices_Olattice_611,axiom,
    ! [A11: $tType] : lattice(filter(A11)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder_612,axiom,
    ! [A11: $tType] : order(filter(A11)) ).

tff(tcon_Filter_Ofilter___Orderings_Otop_613,axiom,
    ! [A11: $tType] : top(filter(A11)) ).

tff(tcon_Filter_Ofilter___Orderings_Oord_614,axiom,
    ! [A11: $tType] : ord(filter(A11)) ).

tff(tcon_Filter_Ofilter___Orderings_Obot_615,axiom,
    ! [A11: $tType] : bot(filter(A11)) ).

tff(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_616,axiom,
    ! [A11: $tType] :
      ( comple5582772986160207858norder(A11)
     => condit6923001295902523014norder(option(A11)) ) ).

tff(tcon_Option_Ooption___Lattices_Obounded__semilattice__sup__bot_617,axiom,
    ! [A11: $tType] :
      ( lattice(A11)
     => bounde4967611905675639751up_bot(option(A11)) ) ).

tff(tcon_Option_Ooption___Lattices_Obounded__lattice__top_618,axiom,
    ! [A11: $tType] :
      ( bounded_lattice_top(A11)
     => bounded_lattice_top(option(A11)) ) ).

tff(tcon_Option_Ooption___Lattices_Osemilattice__sup_619,axiom,
    ! [A11: $tType] :
      ( semilattice_sup(A11)
     => semilattice_sup(option(A11)) ) ).

tff(tcon_Option_Ooption___Lattices_Obounded__lattice_620,axiom,
    ! [A11: $tType] :
      ( bounded_lattice_top(A11)
     => bounded_lattice(option(A11)) ) ).

tff(tcon_Option_Ooption___Orderings_Owellorder_621,axiom,
    ! [A11: $tType] :
      ( wellorder(A11)
     => wellorder(option(A11)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder__top_622,axiom,
    ! [A11: $tType] :
      ( order_top(A11)
     => order_top(option(A11)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder__bot_623,axiom,
    ! [A11: $tType] :
      ( order(A11)
     => order_bot(option(A11)) ) ).

tff(tcon_Option_Ooption___Orderings_Opreorder_624,axiom,
    ! [A11: $tType] :
      ( preorder(A11)
     => preorder(option(A11)) ) ).

tff(tcon_Option_Ooption___Orderings_Olinorder_625,axiom,
    ! [A11: $tType] :
      ( linorder(A11)
     => linorder(option(A11)) ) ).

tff(tcon_Option_Ooption___Finite__Set_Ofinite_626,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => finite_finite(option(A11)) ) ).

tff(tcon_Option_Ooption___Lattices_Olattice_627,axiom,
    ! [A11: $tType] :
      ( lattice(A11)
     => lattice(option(A11)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder_628,axiom,
    ! [A11: $tType] :
      ( order(A11)
     => order(option(A11)) ) ).

tff(tcon_Option_Ooption___Orderings_Otop_629,axiom,
    ! [A11: $tType] :
      ( order_top(A11)
     => top(option(A11)) ) ).

tff(tcon_Option_Ooption___Orderings_Oord_630,axiom,
    ! [A11: $tType] :
      ( preorder(A11)
     => ord(option(A11)) ) ).

tff(tcon_Option_Ooption___Orderings_Obot_631,axiom,
    ! [A11: $tType] :
      ( order(A11)
     => bot(option(A11)) ) ).

tff(tcon_Option_Ooption___Nat_Osize_632,axiom,
    ! [A11: $tType] : size(option(A11)) ).

tff(tcon_Uint32_Ouint32___Bit__Operations_Osemiring__bit__operations_633,axiom,
    bit_se359711467146920520ations(uint32) ).

tff(tcon_Uint32_Ouint32___Bit__Comprehension_Obit__comprehension_634,axiom,
    bit_bi6583157726757044596ension(uint32) ).

tff(tcon_Uint32_Ouint32___Bit__Operations_Oring__bit__operations_635,axiom,
    bit_ri3973907225187159222ations(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocancel__ab__semigroup__add_636,axiom,
    cancel2418104881723323429up_add(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocancel__comm__monoid__add_637,axiom,
    cancel1802427076303600483id_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__1__cancel_638,axiom,
    comm_s4317794764714335236cancel(uint32) ).

tff(tcon_Uint32_Ouint32___Bit__Operations_Osemiring__bits_639,axiom,
    bit_semiring_bits(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocancel__semigroup__add_640,axiom,
    cancel_semigroup_add(uint32) ).

tff(tcon_Uint32_Ouint32___Least__significant__bit_Olsb_641,axiom,
    least_6119777620449941438nt_lsb(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oab__semigroup__mult_642,axiom,
    ab_semigroup_mult(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring__1__cancel_643,axiom,
    semiring_1_cancel(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocomm__monoid__mult_644,axiom,
    comm_monoid_mult(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oab__semigroup__add_645,axiom,
    ab_semigroup_add(uint32) ).

tff(tcon_Uint32_Ouint32___Generic__set__bit_Oset__bit_646,axiom,
    generic_set_set_bit(uint32) ).

tff(tcon_Uint32_Ouint32___Parity_Osemiring__parity_647,axiom,
    semiring_parity(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocomm__monoid__add_648,axiom,
    comm_monoid_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring__modulo_649,axiom,
    semiring_modulo(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__1_650,axiom,
    comm_semiring_1(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Osemigroup__mult_651,axiom,
    semigroup_mult(uint32) ).

tff(tcon_Uint32_Ouint32___Num_Osemiring__numeral_652,axiom,
    semiring_numeral(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Osemigroup__add_653,axiom,
    semigroup_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__semiring_654,axiom,
    comm_semiring(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oab__group__add_655,axiom,
    ab_group_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ozero__neq__one_656,axiom,
    zero_neq_one(uint32) ).

tff(tcon_Uint32_Ouint32___Parity_Oring__parity_657,axiom,
    ring_parity(uint32) ).

tff(tcon_Uint32_Ouint32___Orderings_Opreorder_658,axiom,
    preorder(uint32) ).

tff(tcon_Uint32_Ouint32___Orderings_Olinorder_659,axiom,
    linorder(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Omonoid__mult_660,axiom,
    monoid_mult(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__ring__1_661,axiom,
    comm_ring_1(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Omonoid__add_662,axiom,
    monoid_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring__1_663,axiom,
    semiring_1(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring__0_664,axiom,
    semiring_0(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ogroup__add_665,axiom,
    group_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Omult__zero_666,axiom,
    mult_zero(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__ring_667,axiom,
    comm_ring(uint32) ).

tff(tcon_Uint32_Ouint32___Orderings_Oorder_668,axiom,
    order(uint32) ).

tff(tcon_Uint32_Ouint32___Num_Oneg__numeral_669,axiom,
    neg_numeral(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring_670,axiom,
    semiring(uint32) ).

tff(tcon_Uint32_Ouint32___Orderings_Oord_671,axiom,
    ord(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ouminus_672,axiom,
    uminus(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Oring__1_673,axiom,
    ring_1(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ominus_674,axiom,
    minus(uint32) ).

tff(tcon_Uint32_Ouint32___Power_Opower_675,axiom,
    power(uint32) ).

tff(tcon_Uint32_Ouint32___Num_Onumeral_676,axiom,
    numeral(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ozero_677,axiom,
    zero(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oplus_678,axiom,
    plus(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Oring_679,axiom,
    ring(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oone_680,axiom,
    one(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Odvd_681,axiom,
    dvd(uint32) ).

tff(tcon_Uint32_Ouint32___Nat_Osize_682,axiom,
    size(uint32) ).

tff(tcon_String_Oliteral___Groups_Osemigroup__add_683,axiom,
    semigroup_add(literal) ).

tff(tcon_String_Oliteral___Orderings_Opreorder_684,axiom,
    preorder(literal) ).

tff(tcon_String_Oliteral___Orderings_Olinorder_685,axiom,
    linorder(literal) ).

tff(tcon_String_Oliteral___Groups_Omonoid__add_686,axiom,
    monoid_add(literal) ).

tff(tcon_String_Oliteral___Orderings_Oorder_687,axiom,
    order(literal) ).

tff(tcon_String_Oliteral___Orderings_Oord_688,axiom,
    ord(literal) ).

tff(tcon_String_Oliteral___Groups_Ozero_689,axiom,
    zero(literal) ).

tff(tcon_String_Oliteral___Groups_Oplus_690,axiom,
    plus(literal) ).

tff(tcon_String_Oliteral___Nat_Osize_691,axiom,
    size(literal) ).

tff(tcon_Assertions_Oassn___Lattices_Obounded__semilattice__sup__bot_692,axiom,
    bounde4967611905675639751up_bot(assn) ).

tff(tcon_Assertions_Oassn___Boolean__Algebras_Oboolean__algebra_693,axiom,
    boolea8198339166811842893lgebra(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Obounded__lattice__top_694,axiom,
    bounded_lattice_top(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Osemilattice__sup_695,axiom,
    semilattice_sup(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Obounded__lattice_696,axiom,
    bounded_lattice(assn) ).

tff(tcon_Assertions_Oassn___Groups_Oab__semigroup__mult_697,axiom,
    ab_semigroup_mult(assn) ).

tff(tcon_Assertions_Oassn___Groups_Ocomm__monoid__mult_698,axiom,
    comm_monoid_mult(assn) ).

tff(tcon_Assertions_Oassn___Groups_Osemigroup__mult_699,axiom,
    semigroup_mult(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oorder__top_700,axiom,
    order_top(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oorder__bot_701,axiom,
    order_bot(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Opreorder_702,axiom,
    preorder(assn) ).

tff(tcon_Assertions_Oassn___Groups_Omonoid__mult_703,axiom,
    monoid_mult(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Olattice_704,axiom,
    lattice(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oorder_705,axiom,
    order(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Otop_706,axiom,
    top(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oord_707,axiom,
    ord(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Obot_708,axiom,
    bot(assn) ).

tff(tcon_Assertions_Oassn___Groups_Ouminus_709,axiom,
    uminus(assn) ).

tff(tcon_Assertions_Oassn___Groups_Ominus_710,axiom,
    minus(assn) ).

tff(tcon_Assertions_Oassn___Power_Opower_711,axiom,
    power(assn) ).

tff(tcon_Assertions_Oassn___Groups_Oone_712,axiom,
    one(assn) ).

tff(tcon_Assertions_Oassn___Rings_Odvd_713,axiom,
    dvd(assn) ).

tff(tcon_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_714,axiom,
    semiri1453513574482234551roduct(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ofirst__countable__topology_715,axiom,
    topolo3112930676232923870pology(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__div__algebra_716,axiom,
    real_V8999393235501362500lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra__1_717,axiom,
    real_V2822296259951069270ebra_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors__cancel_718,axiom,
    semiri6575147826004484403cancel(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra_719,axiom,
    real_V4412858255891104859lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_720,axiom,
    real_V822414075346904944vector(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_721,axiom,
    topolo4958980785337419405_space(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_722,axiom,
    real_V3459762299906320749_field(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__div__algebra_723,axiom,
    real_V5047593784448816457lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ouniformity__dist_724,axiom,
    real_V768167426530841204y_dist(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__comm__monoid__add_725,axiom,
    topolo5987344860129210374id_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1__no__zero__divisors_726,axiom,
    semiri2026040879449505780visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra__1_727,axiom,
    real_V2191834092415804123ebra_1(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ocomplete__space_728,axiom,
    real_V8037385150606011577_space(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__semigroup__mult_729,axiom,
    topolo4211221413907600880p_mult(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ouniform__space_730,axiom,
    topolo7287701948861334536_space(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Operfect__space_731,axiom,
    topolo8386298272705272623_space(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors_732,axiom,
    semiri3467727345109120633visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ometric__space_733,axiom,
    real_V7819770556892013058_space(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__ab__group__add_734,axiom,
    topolo1287966508704411220up_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_735,axiom,
    real_V4867850818363320053vector(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add_736,axiom,
    cancel2418104881723323429up_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors_737,axiom,
    ring_15535105094025558882visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__field_738,axiom,
    real_V7773925162809079976_field(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__monoid__add_739,axiom,
    topolo6943815403480290642id_add(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__comm__monoid__add_740,axiom,
    cancel1802427076303600483id_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1__cancel_741,axiom,
    comm_s4317794764714335236cancel(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__group__add_742,axiom,
    topolo1633459387980952147up_add(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ot2__space_743,axiom,
    topological_t2_space(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ot1__space_744,axiom,
    topological_t1_space(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__semigroup__add_745,axiom,
    cancel_semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Obanach_746,axiom,
    real_Vector_banach(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__semigroup__mult_747,axiom,
    ab_semigroup_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1__cancel_748,axiom,
    semiring_1_cancel(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__mult_749,axiom,
    comm_monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__semigroup__add_750,axiom,
    ab_semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__add_751,axiom,
    comm_monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1_752,axiom,
    comm_semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Osemigroup__mult_753,axiom,
    semigroup_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemidom__divide_754,axiom,
    semidom_divide(complex) ).

tff(tcon_Complex_Ocomplex___Num_Osemiring__numeral_755,axiom,
    semiring_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Osemigroup__add_756,axiom,
    semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Odivision__ring_757,axiom,
    division_ring(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring_758,axiom,
    comm_semiring(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Osemiring__char__0_759,axiom,
    semiring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__group__add_760,axiom,
    ab_group_add(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield__char__0_761,axiom,
    field_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ozero__neq__one_762,axiom,
    zero_neq_one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom__abs__sgn_763,axiom,
    idom_abs_sgn(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__mult_764,axiom,
    monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom__divide_765,axiom,
    idom_divide(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring__1_766,axiom,
    comm_ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__add_767,axiom,
    monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1_768,axiom,
    semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__0_769,axiom,
    semiring_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ogroup__add_770,axiom,
    group_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Omult__zero_771,axiom,
    mult_zero(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring_772,axiom,
    comm_ring(complex) ).

tff(tcon_Complex_Ocomplex___Num_Oneg__numeral_773,axiom,
    neg_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Oring__char__0_774,axiom,
    ring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring_775,axiom,
    semiring(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Oinverse_776,axiom,
    inverse(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemidom_777,axiom,
    semidom(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ouminus_778,axiom,
    uminus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1_779,axiom,
    ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ominus_780,axiom,
    minus(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield_781,axiom,
    field(complex) ).

tff(tcon_Complex_Ocomplex___Power_Opower_782,axiom,
    power(complex) ).

tff(tcon_Complex_Ocomplex___Num_Onumeral_783,axiom,
    numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ozero_784,axiom,
    zero(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oplus_785,axiom,
    plus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring_786,axiom,
    ring(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom_787,axiom,
    idom(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oone_788,axiom,
    one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Odvd_789,axiom,
    dvd(complex) ).

tff(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_790,axiom,
    condit6923001295902523014norder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_791,axiom,
    strict9044650504122735259up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_792,axiom,
    strict7427464778891057005id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_793,axiom,
    canoni5634975068530333245id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__sup__bot_794,axiom,
    bounde4967611905675639751up_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_795,axiom,
    linord181362715937106298miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_796,axiom,
    semiri3467727345109120633visors(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_797,axiom,
    ordere6658533253407199908up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_798,axiom,
    ordere6911136660526730532id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice__top_799,axiom,
    bounded_lattice_top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__comm__semiring_800,axiom,
    ordere2520102378445227354miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__sup_801,axiom,
    semilattice_sup(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice_802,axiom,
    bounded_lattice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_803,axiom,
    ab_semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__mult_804,axiom,
    comm_monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_805,axiom,
    ab_semigroup_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_806,axiom,
    ordered_semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_807,axiom,
    comm_monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_808,axiom,
    comm_semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_809,axiom,
    semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Osemiring__numeral_810,axiom,
    semiring_numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_811,axiom,
    semigroup_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__less__one_812,axiom,
    zero_less_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring_813,axiom,
    comm_semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Owellorder_814,axiom,
    wellorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_815,axiom,
    order_top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__bot_816,axiom,
    order_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_817,axiom,
    semiring_char_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__neq__one_818,axiom,
    zero_neq_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Opreorder_819,axiom,
    preorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Olinorder_820,axiom,
    linorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__mult_821,axiom,
    monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_822,axiom,
    monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_823,axiom,
    semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__0_824,axiom,
    semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Olattice_825,axiom,
    lattice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Omult__zero_826,axiom,
    mult_zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder_827,axiom,
    order(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring_828,axiom,
    semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Otop_829,axiom,
    top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oord_830,axiom,
    ord(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Obot_831,axiom,
    bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ominus_832,axiom,
    minus(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Power_Opower_833,axiom,
    power(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Onumeral_834,axiom,
    numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ozero_835,axiom,
    zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oplus_836,axiom,
    plus(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oone_837,axiom,
    one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Odvd_838,axiom,
    dvd(extended_enat) ).

tff(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add_839,axiom,
    ! [A11: $tType] :
      ( preorder(A11)
     => ordere6658533253407199908up_add(multiset(A11)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Ocancel__ab__semigroup__add_840,axiom,
    ! [A11: $tType] : cancel2418104881723323429up_add(multiset(A11)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocancel__comm__monoid__add_841,axiom,
    ! [A11: $tType] : cancel1802427076303600483id_add(multiset(A11)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocancel__semigroup__add_842,axiom,
    ! [A11: $tType] : cancel_semigroup_add(multiset(A11)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__diff_843,axiom,
    ! [A11: $tType] : comm_monoid_diff(multiset(A11)) ).

tff(tcon_Multiset_Omultiset___Groups_Oab__semigroup__add_844,axiom,
    ! [A11: $tType] : ab_semigroup_add(multiset(A11)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_845,axiom,
    ! [A11: $tType] : comm_monoid_add(multiset(A11)) ).

tff(tcon_Multiset_Omultiset___Groups_Osemigroup__add_846,axiom,
    ! [A11: $tType] : semigroup_add(multiset(A11)) ).

tff(tcon_Multiset_Omultiset___Orderings_Opreorder_847,axiom,
    ! [A11: $tType] :
      ( preorder(A11)
     => preorder(multiset(A11)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Omonoid__add_848,axiom,
    ! [A11: $tType] : monoid_add(multiset(A11)) ).

tff(tcon_Multiset_Omultiset___Orderings_Oorder_849,axiom,
    ! [A11: $tType] :
      ( preorder(A11)
     => order(multiset(A11)) ) ).

tff(tcon_Multiset_Omultiset___Orderings_Oord_850,axiom,
    ! [A11: $tType] :
      ( preorder(A11)
     => ord(multiset(A11)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Ominus_851,axiom,
    ! [A11: $tType] : minus(multiset(A11)) ).

tff(tcon_Multiset_Omultiset___Groups_Ozero_852,axiom,
    ! [A11: $tType] : zero(multiset(A11)) ).

tff(tcon_Multiset_Omultiset___Groups_Oplus_853,axiom,
    ! [A11: $tType] : plus(multiset(A11)) ).

tff(tcon_Multiset_Omultiset___Nat_Osize_854,axiom,
    ! [A11: $tType] : size(multiset(A11)) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocancel__ab__semigroup__add_855,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => cancel2418104881723323429up_add(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocancel__comm__monoid__add_856,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => cancel1802427076303600483id_add(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__1__cancel_857,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_s4317794764714335236cancel(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocancel__semigroup__add_858,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => cancel_semigroup_add(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oab__semigroup__mult_859,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => ab_semigroup_mult(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Osemiring__1__cancel_860,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semiring_1_cancel(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocomm__monoid__mult_861,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_monoid_mult(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oab__semigroup__add_862,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => ab_semigroup_add(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocomm__monoid__add_863,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_monoid_add(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__1_864,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_semiring_1(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Osemigroup__mult_865,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semigroup_mult(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Num_Osemiring__numeral_866,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semiring_numeral(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Osemigroup__add_867,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semigroup_add(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring_868,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_semiring(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Owellorder_869,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => wellorder(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oab__group__add_870,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => ab_group_add(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ozero__neq__one_871,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => zero_neq_one(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Opreorder_872,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => preorder(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Olinorder_873,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => linorder(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Omonoid__mult_874,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => monoid_mult(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__ring__1_875,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_ring_1(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Omonoid__add_876,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => monoid_add(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Finite__Set_Ofinite_877,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => finite_finite(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Cardinality_Ocard2,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => card2(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Type__Length_Olen0_878,axiom,
    ! [A11: $tType] :
      ( type_len0(A11)
     => type_len0(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Osemiring__1_879,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semiring_1(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Osemiring__0_880,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semiring_0(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ogroup__add_881,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => group_add(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Type__Length_Olen_882,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => type_len(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Omult__zero_883,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => mult_zero(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__ring_884,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_ring(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Oorder_885,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => order(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Num_Oneg__numeral_886,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => neg_numeral(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Osemiring_887,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semiring(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Oord_888,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => ord(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ouminus_889,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => uminus(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Oring__1_890,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => ring_1(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ominus_891,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => minus(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Power_Opower_892,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => power(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Num_Onumeral_893,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => numeral(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ozero_894,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => zero(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oplus_895,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => plus(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Oring_896,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => ring(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oone_897,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => one(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Odvd_898,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => dvd(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocancel__ab__semigroup__add_899,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => cancel2418104881723323429up_add(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocancel__comm__monoid__add_900,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => cancel1802427076303600483id_add(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__1__cancel_901,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_s4317794764714335236cancel(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocancel__semigroup__add_902,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => cancel_semigroup_add(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oab__semigroup__mult_903,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => ab_semigroup_mult(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Osemiring__1__cancel_904,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semiring_1_cancel(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocomm__monoid__mult_905,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_monoid_mult(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oab__semigroup__add_906,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => ab_semigroup_add(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocomm__monoid__add_907,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_monoid_add(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__1_908,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_semiring_1(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Osemigroup__mult_909,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semigroup_mult(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Num_Osemiring__numeral_910,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semiring_numeral(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Osemigroup__add_911,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semigroup_add(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring_912,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_semiring(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Owellorder_913,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => wellorder(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oab__group__add_914,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => ab_group_add(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ozero__neq__one_915,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => zero_neq_one(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Opreorder_916,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => preorder(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Olinorder_917,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => linorder(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Omonoid__mult_918,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => monoid_mult(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__ring__1_919,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_ring_1(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Omonoid__add_920,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => monoid_add(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Finite__Set_Ofinite_921,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => finite_finite(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Cardinality_Ocard2_922,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => card2(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Type__Length_Olen0_923,axiom,
    ! [A11: $tType] :
      ( type_len0(A11)
     => type_len0(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Osemiring__1_924,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semiring_1(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Osemiring__0_925,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semiring_0(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ogroup__add_926,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => group_add(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Type__Length_Olen_927,axiom,
    ! [A11: $tType] :
      ( type_len0(A11)
     => type_len(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Omult__zero_928,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => mult_zero(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__ring_929,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_ring(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Oorder_930,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => order(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Num_Oneg__numeral_931,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => neg_numeral(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Osemiring_932,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semiring(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Oord_933,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => ord(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ouminus_934,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => uminus(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Oring__1_935,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => ring_1(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ominus_936,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => minus(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Power_Opower_937,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => power(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Num_Onumeral_938,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => numeral(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ozero_939,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => zero(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oplus_940,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => plus(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Oring_941,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => ring(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oone_942,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => one(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Odvd_943,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => dvd(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Onum0___Type__Length_Olen0_944,axiom,
    type_len0(numeral_num0) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ocancel__ab__semigroup__add_945,axiom,
    cancel2418104881723323429up_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ocancel__comm__monoid__add_946,axiom,
    cancel1802427076303600483id_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ocancel__semigroup__add_947,axiom,
    cancel_semigroup_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Oab__semigroup__mult_948,axiom,
    ab_semigroup_mult(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ocomm__monoid__mult_949,axiom,
    comm_monoid_mult(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Oab__semigroup__add_950,axiom,
    ab_semigroup_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ocomm__monoid__add_951,axiom,
    comm_monoid_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Osemigroup__mult_952,axiom,
    semigroup_mult(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Osemigroup__add_953,axiom,
    semigroup_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Ocomm__semiring_954,axiom,
    comm_semiring(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Orderings_Owellorder_955,axiom,
    wellorder(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Oab__group__add_956,axiom,
    ab_group_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Orderings_Opreorder_957,axiom,
    preorder(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Orderings_Olinorder_958,axiom,
    linorder(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Omonoid__mult_959,axiom,
    monoid_mult(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Cardinality_OCARD__1,axiom,
    cARD_1(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Omonoid__add_960,axiom,
    monoid_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Finite__Set_Ofinite_961,axiom,
    finite_finite(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Type__Length_Olen0_962,axiom,
    type_len0(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Osemiring__0_963,axiom,
    semiring_0(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ogroup__add_964,axiom,
    group_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Type__Length_Olen_965,axiom,
    type_len(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Omult__zero_966,axiom,
    mult_zero(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Ocomm__ring_967,axiom,
    comm_ring(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Orderings_Oorder_968,axiom,
    order(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Osemiring_969,axiom,
    semiring(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Orderings_Oord_970,axiom,
    ord(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ouminus_971,axiom,
    uminus(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ominus_972,axiom,
    minus(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Power_Opower_973,axiom,
    power(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Num_Onumeral_974,axiom,
    numeral(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ozero_975,axiom,
    zero(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Oplus_976,axiom,
    plus(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Oring_977,axiom,
    ring(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Oone_978,axiom,
    one(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Odvd_979,axiom,
    dvd(numeral_num1) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_980,axiom,
    ! [A11: $tType,A13: $tType] :
      ( ( topolo4958980785337419405_space(A11)
        & topolo4958980785337419405_space(A13) )
     => topolo4958980785337419405_space(product_prod(A11,A13)) ) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Ot2__space_981,axiom,
    ! [A11: $tType,A13: $tType] :
      ( ( topological_t2_space(A11)
        & topological_t2_space(A13) )
     => topological_t2_space(product_prod(A11,A13)) ) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Ot1__space_982,axiom,
    ! [A11: $tType,A13: $tType] :
      ( ( topological_t1_space(A11)
        & topological_t1_space(A13) )
     => topological_t1_space(product_prod(A11,A13)) ) ).

tff(tcon_Product__Type_Oprod___Finite__Set_Ofinite_983,axiom,
    ! [A11: $tType,A13: $tType] :
      ( ( finite_finite(A11)
        & finite_finite(A13) )
     => finite_finite(product_prod(A11,A13)) ) ).

tff(tcon_Product__Type_Oprod___Nat_Osize_984,axiom,
    ! [A11: $tType,A13: $tType] : size(product_prod(A11,A13)) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_985,axiom,
    condit6923001295902523014norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_986,axiom,
    bounde4967611905675639751up_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_987,axiom,
    boolea8198339166811842893lgebra(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top_988,axiom,
    bounded_lattice_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_989,axiom,
    semilattice_sup(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_990,axiom,
    bounded_lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Owellorder_991,axiom,
    wellorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__top_992,axiom,
    order_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__bot_993,axiom,
    order_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Opreorder_994,axiom,
    preorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Olinorder_995,axiom,
    linorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Finite__Set_Ofinite_996,axiom,
    finite_finite(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Olattice_997,axiom,
    lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder_998,axiom,
    order(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Otop_999,axiom,
    top(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oord_1000,axiom,
    ord(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Obot_1001,axiom,
    bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Groups_Ouminus_1002,axiom,
    uminus(product_unit) ).

tff(tcon_Product__Type_Ounit___Groups_Ominus_1003,axiom,
    minus(product_unit) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_1004,axiom,
    bit_un5681908812861735899ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_1005,axiom,
    semiri1453513574482234551roduct(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_1006,axiom,
    euclid5411537665997757685th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_1007,axiom,
    euclid8789492081693882211th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_1008,axiom,
    ordere1937475149494474687imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_1009,axiom,
    euclid4440199948858584721cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_1010,axiom,
    unique1627219031080169319umeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_1011,axiom,
    euclid8851590272496341667cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_1012,axiom,
    semiri6575147826004484403cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_1013,axiom,
    strict9044650504122735259up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_1014,axiom,
    ordere580206878836729694up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_1015,axiom,
    ordere2412721322843649153imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_1016,axiom,
    bit_se359711467146920520ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_1017,axiom,
    linord2810124833399127020strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_1018,axiom,
    strict7427464778891057005id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_1019,axiom,
    ordere8940638589300402666id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_1020,axiom,
    euclid3725896446679973847miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_1021,axiom,
    linord715952674999750819strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_1022,axiom,
    bit_ri3973907225187159222ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_1023,axiom,
    semiri2026040879449505780visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_1024,axiom,
    linord181362715937106298miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_1025,axiom,
    linord8928482502909563296strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_1026,axiom,
    semiri3467727345109120633visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_1027,axiom,
    ordere6658533253407199908up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_1028,axiom,
    ordere166539214618696060dd_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_1029,axiom,
    ordere6911136660526730532id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_1030,axiom,
    linord5086331880401160121up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_1031,axiom,
    cancel2418104881723323429up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_1032,axiom,
    ring_15535105094025558882visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_1033,axiom,
    cancel1802427076303600483id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_1034,axiom,
    linord4710134922213307826strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_1035,axiom,
    comm_s4317794764714335236cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_1036,axiom,
    bit_semiring_bits(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_1037,axiom,
    ordere2520102378445227354miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_1038,axiom,
    linord6961819062388156250ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_1039,axiom,
    ordered_ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_1040,axiom,
    cancel_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_1041,axiom,
    linordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Least__significant__bit_Olsb_1042,axiom,
    least_6119777620449941438nt_lsb(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_1043,axiom,
    ordered_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_1044,axiom,
    linordered_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_1045,axiom,
    ab_semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_1046,axiom,
    semiring_1_cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_1047,axiom,
    algebraic_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_1048,axiom,
    comm_monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_1049,axiom,
    ab_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Generic__set__bit_Oset__bit_1050,axiom,
    generic_set_set_bit(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_1051,axiom,
    ordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_1052,axiom,
    ordered_ring_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_1053,axiom,
    semiring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_1054,axiom,
    comm_monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_1055,axiom,
    semiring_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_1056,axiom,
    linordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_1057,axiom,
    linordered_idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_1058,axiom,
    comm_semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_1059,axiom,
    semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_1060,axiom,
    semidom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_1061,axiom,
    semidom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_1062,axiom,
    semiring_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_1063,axiom,
    semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_1064,axiom,
    zero_less_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_1065,axiom,
    comm_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_1066,axiom,
    semiring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_1067,axiom,
    ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_1068,axiom,
    zero_neq_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_1069,axiom,
    ordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_1070,axiom,
    idom_abs_sgn(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_1071,axiom,
    ring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_1072,axiom,
    preorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_1073,axiom,
    linorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_1074,axiom,
    monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__modulo_1075,axiom,
    idom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_1076,axiom,
    idom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_1077,axiom,
    comm_ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_1078,axiom,
    monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_1079,axiom,
    semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_1080,axiom,
    semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_1081,axiom,
    group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_1082,axiom,
    mult_zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_1083,axiom,
    comm_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oorder_1084,axiom,
    order(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_1085,axiom,
    neg_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_1086,axiom,
    ring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring_1087,axiom,
    semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom_1088,axiom,
    semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oord_1089,axiom,
    ord(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ouminus_1090,axiom,
    uminus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1_1091,axiom,
    ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_1092,axiom,
    abs_if(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ominus_1093,axiom,
    minus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Power_Opower_1094,axiom,
    power(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Onumeral_1095,axiom,
    numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ozero_1096,axiom,
    zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oplus_1097,axiom,
    plus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring_1098,axiom,
    ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom_1099,axiom,
    idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oone_1100,axiom,
    one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Odvd_1101,axiom,
    dvd(code_integer) ).

tff(tcon_Heap__Time__Monad_OHeap___Nat_Osize_1102,axiom,
    ! [A11: $tType] : size(heap_Time_Heap(A11)) ).

tff(tcon_VEBT__BuildupMemImp_OVEBTi___Nat_Osize_1103,axiom,
    size(vEBT_VEBTi) ).

% Helper facts (3)
tff(help_fequal_2_1_T,axiom,
    ! [A: $tType,X: A,Y: A] :
      ( ( X != Y )
      | aa(A,$o,aa(A,fun(A,$o),fequal(A),X),Y) ) ).

tff(help_fequal_1_1_T,axiom,
    ! [A: $tType,X: A,Y: A] :
      ( ~ aa(A,$o,aa(A,fun(A,$o),fequal(A),X),Y)
      | ( X = Y ) ) ).

tff(help_fChoice_1_1_T,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( aa(A,$o,P,fChoice(A,P))
      = ( ? [X9: A] : aa(A,$o,P,X9) ) ) ).

% Free types (1)
tff(tfree_0,hypothesis,
    semiring_1(a) ).

% Conjectures (2)
tff(conj_0,hypothesis,
    ! [X2: nat] :
      ( member(nat,X2,aa(list(nat),set(nat),set2(nat),xs))
     => aa(nat,$o,ord_less(nat,X2),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),n)) ) ).

tff(conj_1,conjecture,
    hoare_hoare_triple(vEBT_VEBTi,aa(vEBT_VEBTi,assn,vEBT_Intf_vebt_assn(n,s),t),vEBT_Example_mfold(nat,vEBT_VEBTi,aTP_Lamp_a(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),xs,t),vEBT_Intf_vebt_assn(n,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),s),aa(list(nat),set(nat),set2(nat),xs)))) ).