%------------------------------------------------------------------------------
% File     : ITP294_4 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Example 00111_003949
% 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_00111_003949 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 12159 (4428 unt;1578 typ;   0 def)
%            Number of atoms       : 18600 (7321 equ)
%            Maximal formula atoms :   28 (   1 avg)
%            Number of connectives : 18438 (1445   ~; 276   |;2003   &)
%                                         (2025 <=>;12689  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   27 (   5 avg)
%            Maximal term depth    :   36 (   2 avg)
%            Number of FOOLs       :  743 ( 413 fml; 330 var)
%            Number of X terms     :  476 (   0  []; 433 ite;  43 let)
%            Number of types       :   20 (  19 usr)
%            Number of type conns  : 1382 (1194   >; 188   *;   0   +;   0  <<)
%            Number of predicates  :  217 ( 214 usr;   2 prp; 0-13 aty)
%            Number of functors    : 1360 (1360 usr;  50 con; 0-7 aty)
%            Number of variables   : 32077 (29186   !; 397   ?;32077   :)
%                                         (2494  !>;   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 23:52:31.968
%------------------------------------------------------------------------------
% Could-be-implicit typings (33)
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_Heap_Oheap_Oheap__ext,type,
    heap_ext: $tType > $tType ).

tff(ty_t_Product__Type_Ounit,type,
    product_unit: $tType ).

tff(ty_t_Product__Type_Oprod,type,
    product_prod: ( $tType * $tType ) > $tType ).

tff(ty_t_Numeral__Type_Onum1,type,
    numeral_num1: $tType ).

tff(ty_t_Numeral__Type_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_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_String_Ochar,type,
    char: $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_b,type,
    b: $tType ).

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

% Explicit typings (1545)
tff(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
    comple5582772986160207858norder: 
      !>[A: $tType] : $o ).

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

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

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

tff(sy_cl_Nat_Osize,type,
    size: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_cl_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__0,type,
    comm_semiring_0: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_cl_Lattices_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_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_Real__Vector__Spaces_Oreal__algebra,type,
    real_V6157519004096292374lgebra: 
      !>[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_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_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_Quickcheck__Narrowing_Opartial__term__of,type,
    quickc6926020345158392990erm_of: 
      !>[A: $tType] : $o ).

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

tff(sy_cl_Topological__Spaces_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_Euclidean__Division_Ounique__euclidean__semiring,type,
    euclid3128863361964157862miring: 
      !>[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_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] : fun(A,B) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__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] : ( 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] : fun(A,B) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__zf____,type,
    aTP_Lamp_zf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zg____,type,
    aTP_Lamp_zg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zh____,type,
    aTP_Lamp_zh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zi____,type,
    aTP_Lamp_zi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__zz____,type,
    aTP_Lamp_zz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_Archimedean__Field_Oceiling,type,
    archimedean_ceiling: 
      !>[A: $tType] : ( A > int ) ).

tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor,type,
    archim6421214686448440834_floor: 
      !>[A: $tType] : ( A > int ) ).

tff(sy_c_Archimedean__Field_Ofrac,type,
    archimedean_frac: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Archimedean__Field_Oround,type,
    archimedean_round: 
      !>[A: $tType] : ( A > int ) ).

tff(sy_c_Assertions_Oentails,type,
    entails: ( assn * assn ) > $o ).

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_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 ) > fun(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 > fun(A,A) ) ).

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

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

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

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

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

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

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

tff(sy_c_Bit__Operations_Osemiring__bits__class_Opossible__bit,type,
    bit_se6407376104438227557le_bit: 
      !>[A: $tType] : ( ( itself(A) * nat ) > $o ) ).

tff(sy_c_Bit__Shifts__Infix__Syntax_Osemiring__bit__operations__class_Oshiftl,type,
    bit_Sh4282982442137083160shiftl: 
      !>[A: $tType] : ( ( A * nat ) > A ) ).

tff(sy_c_Bit__Shifts__Infix__Syntax_Osemiring__bit__operations__class_Oshiftr,type,
    bit_Sh4282982442137083166shiftr: 
      !>[A: $tType] : ( ( A * nat ) > A ) ).

tff(sy_c_Bit__Shifts__Infix__Syntax_Osshiftr,type,
    bit_Sh8784991116023147202shiftr: 
      !>[A: $tType] : ( ( 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: fun(code_integer,code_integer) ).

tff(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff,type,
    boolea3799213064322606851m_diff: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,A)) * fun(A,A) * A * A * fun(A,fun(A,A)) ) > $o ) ).

tff(sy_c_Code__Numeral_Obit__cut__integer,type,
    code_bit_cut_integer: code_integer > product_prod(code_integer,$o) ).

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

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

tff(sy_c_Code__Numeral_Odup,type,
    code_dup: fun(code_integer,code_integer) ).

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

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

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

tff(sy_c_Code__Numeral_Ointeger__of__num,type,
    code_integer_of_num: num > code_integer ).

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

tff(sy_c_Code__Numeral_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__Int_Opositive,type,
    code_Target_positive: 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_Oquickcheck__narrowing__samples_Opartial__term__of__sample,type,
    code_T4081349890594273596sample: 
      !>[A: $tType] : ( ( fun(code_integer,product_prod(A,A)) * A * code_integer ) > A ) ).

tff(sy_c_Code__Target__Word__Base_Oset__bits__aux,type,
    code_T2661198915054445665ts_aux: 
      !>[A: $tType] : ( ( fun(nat,$o) * nat * word(A) ) > word(A) ) ).

tff(sy_c_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_Oadjust__mod,type,
    adjust_mod: ( int * int ) > int ).

tff(sy_c_Divides_Odivmod__nat,type,
    divmod_nat: ( nat * nat ) > product_prod(nat,nat) ).

tff(sy_c_Divides_Oeucl__rel__int,type,
    eucl_rel_int: ( int * int * product_prod(int,int) ) > $o ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type,
    unique5940410009612947441es_aux: 
      !>[A: $tType] : ( product_prod(A,A) > $o ) ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod,type,
    unique8689654367752047608divmod: 
      !>[A: $tType] : ( ( num * num ) > product_prod(A,A) ) ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step,type,
    unique1321980374590559556d_step: 
      !>[A: $tType] : ( ( num * product_prod(A,A) ) > product_prod(A,A) ) ).

tff(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer,type,
    comm_s3205402744901411588hammer: 
      !>[A: $tType] : ( ( A * nat ) > A ) ).

tff(sy_c_Factorial_Osemiring__char__0__class_Ofact,type,
    semiring_char_0_fact: 
      !>[A: $tType] : ( nat > A ) ).

tff(sy_c_Fields_Oinverse__class_Oinverse,type,
    inverse_inverse: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Filter_Oat__bot,type,
    at_bot: 
      !>[A: $tType] : filter(A) ).

tff(sy_c_Filter_Oat__top,type,
    at_top: 
      !>[A: $tType] : filter(A) ).

tff(sy_c_Filter_Oeventually,type,
    eventually: 
      !>[A: $tType] : ( ( fun(A,$o) * filter(A) ) > $o ) ).

tff(sy_c_Filter_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: $tType] : fun(set(A),$o) ).

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

tff(sy_c_Fun_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_Omap__fun,type,
    map_fun: 
      !>[C: $tType,A: $tType,B: $tType,D: $tType] : ( ( fun(C,A) * fun(B,D) * fun(A,B) ) > fun(C,D) ) ).

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_Obezw,type,
    bezw: ( nat * nat ) > product_prod(int,int) ).

tff(sy_c_GCD_Obezw__rel,type,
    bezw_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)) ).

tff(sy_c_GCD_Ogcd__class_Ogcd,type,
    gcd_gcd: 
      !>[A: $tType] : ( ( A * A ) > 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] : fun(A,fun(A,A)) ).

tff(sy_c_Groups_Osgn__class_Osgn,type,
    sgn_sgn: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Groups_Otimes__class_Otimes,type,
    times_times: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Groups_Ouminus__class_Ouminus,type,
    uminus_uminus: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Groups_Ozero__class_Ozero,type,
    zero_zero: 
      !>[A: $tType] : A ).

tff(sy_c_Groups__Big_Ocomm__monoid__add__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_Groups__List_Omonoid__add__class_Osum__list,type,
    groups8242544230860333062m_list: 
      !>[A: $tType] : ( list(A) > A ) ).

tff(sy_c_HOL_ONO__MATCH,type,
    nO_MATCH: 
      !>[A: $tType,B: $tType] : ( ( A * B ) > $o ) ).

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

tff(sy_c_HOL_Oundefined,type,
    undefined: 
      !>[A: $tType] : A ).

tff(sy_c_Hash__Instances_Ohash__code__list,type,
    hash_hash_code_list: 
      !>[A: $tType] : ( ( fun(A,uint32) * list(A) ) > uint32 ) ).

tff(sy_c_Hash__Instances_Ohash__code__option,type,
    hash_h1887023736457453652option: 
      !>[A: $tType] : ( ( fun(A,uint32) * option(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_Heap__Time__Monad_Obind,type,
    heap_Time_bind: 
      !>[A: $tType,B: $tType] : ( ( heap_Time_Heap(A) * fun(A,heap_Time_Heap(B)) ) > heap_Time_Heap(B) ) ).

tff(sy_c_Heap__Time__Monad_Olift,type,
    heap_Time_lift: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > fun(A,heap_Time_Heap(B)) ) ).

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

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

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

tff(sy_c_Int_Oint__ge__less__than,type,
    int_ge_less_than: int > set(product_prod(int,int)) ).

tff(sy_c_Int_Oint__ge__less__than2,type,
    int_ge_less_than2: int > set(product_prod(int,int)) ).

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

tff(sy_c_Int_Oring__1__class_OInts,type,
    ring_1_Ints: 
      !>[A: $tType] : set(A) ).

tff(sy_c_Int_Oring__1__class_Oof__int,type,
    ring_1_of_int: 
      !>[A: $tType] : fun(int,A) ).

tff(sy_c_Lattices_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_Oenumerate,type,
    enumerate: 
      !>[A: $tType] : ( ( nat * list(A) ) > list(product_prod(nat,A)) ) ).

tff(sy_c_List_Ofilter,type,
    filter2: 
      !>[A: $tType] : ( ( fun(A,$o) * list(A) ) > list(A) ) ).

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

tff(sy_c_List_Ofoldr,type,
    foldr: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * list(A) ) > fun(B,B) ) ).

tff(sy_c_List_Olinorder_Oinsort__insert__key,type,
    insort_insert_key: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(A,$o)) > fun(fun(B,A),fun(B,fun(list(B),list(B)))) ) ).

tff(sy_c_List_Olinorder_Oinsort__key,type,
    insort_key: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(A,$o)) > fun(fun(B,A),fun(B,fun(list(B),list(B)))) ) ).

tff(sy_c_List_Olinorder_Osorted__key__list__of__set,type,
    sorted8670434370408473282of_set: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(A,$o)) > fun(fun(B,A),fun(set(B),list(B))) ) ).

tff(sy_c_List_Olinorder_Osorted__list__of__set,type,
    sorted_list_of_set: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(set(A),list(A)) ) ).

tff(sy_c_List_Olinorder__class_Osorted__list__of__set,type,
    linord4507533701916653071of_set: 
      !>[A: $tType] : ( set(A) > list(A) ) ).

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

tff(sy_c_List_Olist_ONil,type,
    nil: 
      !>[A: $tType] : list(A) ).

tff(sy_c_List_Olist_Omap,type,
    map: 
      !>[A: $tType,Aa: $tType] : ( fun(A,Aa) > fun(list(A),list(Aa)) ) ).

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

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

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

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

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

tff(sy_c_List_Oproduct,type,
    product: 
      !>[A: $tType,B: $tType] : ( ( list(A) * list(B) ) > list(product_prod(A,B)) ) ).

tff(sy_c_List_Oproduct__lists,type,
    product_lists: 
      !>[A: $tType] : ( list(list(A)) > list(list(A)) ) ).

tff(sy_c_List_Oremove1,type,
    remove1: 
      !>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).

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

tff(sy_c_List_Orotate1,type,
    rotate1: 
      !>[A: $tType] : ( list(A) > list(A) ) ).

tff(sy_c_List_Oset__Cons,type,
    set_Cons: 
      !>[A: $tType] : ( ( set(A) * set(list(A)) ) > set(list(A)) ) ).

tff(sy_c_List_Osorted__wrt,type,
    sorted_wrt: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * list(A) ) > $o ) ).

tff(sy_c_List_Osubseqs,type,
    subseqs: 
      !>[A: $tType] : ( list(A) > list(list(A)) ) ).

tff(sy_c_List_Otranspose,type,
    transpose: 
      !>[A: $tType] : ( list(list(A)) > list(list(A)) ) ).

tff(sy_c_List_Ounion,type,
    union: 
      !>[A: $tType] : ( ( list(A) * list(A) ) > list(A) ) ).

tff(sy_c_List_Oupt,type,
    upt: ( nat * nat ) > list(nat) ).

tff(sy_c_List_Oupto,type,
    upto: ( int * int ) > list(int) ).

tff(sy_c_List_Oupto__aux,type,
    upto_aux: ( int * int * list(int) ) > list(int) ).

tff(sy_c_List_Oupto__rel,type,
    upto_rel: fun(product_prod(int,int),fun(product_prod(int,int),$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_Oinv__on,type,
    inv_on: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) ) > fun(B,A) ) ).

tff(sy_c_Misc_Omergesort__by__rel__split,type,
    merges295452479951948502_split: 
      !>[A: $tType] : ( ( product_prod(list(A),list(A)) * list(A) ) > product_prod(list(A),list(A)) ) ).

tff(sy_c_Misc_Omergesort__remdups,type,
    mergesort_remdups: 
      !>[A: $tType] : ( list(A) > list(A) ) ).

tff(sy_c_Misc_Orel__of,type,
    rel_of: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * fun(product_prod(A,B),$o) ) > set(product_prod(A,B)) ) ).

tff(sy_c_Misc_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_Multiset_Olinorder__class_Opart,type,
    linorder_part: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * A * list(B) ) > product_prod(list(B),product_prod(list(B),list(B))) ) ).

tff(sy_c_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] : fun(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_Num_Oring__1__class_Oiszero,type,
    ring_1_iszero: 
      !>[A: $tType] : ( A > $o ) ).

tff(sy_c_Numeral__Type_OAbs__bit0_H,type,
    numeral_Abs_bit0: 
      !>[A: $tType] : ( int > numeral_bit0(A) ) ).

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

tff(sy_c_Numeral__Type_Obit0_OAbs__bit0,type,
    numeral_Abs_bit02: 
      !>[A: $tType] : fun(int,numeral_bit0(A)) ).

tff(sy_c_Numeral__Type_Obit0_ORep__bit0,type,
    numeral_Rep_bit0: 
      !>[A: $tType] : fun(numeral_bit0(A),int) ).

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_Numeral__Type_Omod__ring,type,
    numeral_mod_ring: 
      !>[A: $tType] : ( ( int * fun(A,int) * fun(int,A) ) > $o ) ).

tff(sy_c_Numeral__Type_Omod__type,type,
    numeral_mod_type: 
      !>[A: $tType] : ( ( int * fun(A,int) * fun(int,A) ) > $o ) ).

tff(sy_c_Option_Ooption_OSome,type,
    some: 
      !>[A: $tType] : ( A > option(A) ) ).

tff(sy_c_Option_Ooption_Osize__option,type,
    size_option: 
      !>[A: $tType] : ( ( fun(A,nat) * option(A) ) > nat ) ).

tff(sy_c_Option_Ooption_Othe,type,
    the2: 
      !>[A: $tType] : ( option(A) > A ) ).

tff(sy_c_Orderings_Obot__class_Obot,type,
    bot_bot: 
      !>[A: $tType] : A ).

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

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

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

tff(sy_c_Orderings_Oord__class_Omax,type,
    ord_max: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Orderings_Oord__class_Omin,type,
    ord_min: 
      !>[A: $tType] : ( ( A * A ) > A ) ).

tff(sy_c_Orderings_Oorder_OGreatest,type,
    greatest: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(fun(A,$o),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_Product__Type_Oprod_Ofst,type,
    product_fst: 
      !>[A: $tType,B: $tType] : fun(product_prod(A,B),A) ).

tff(sy_c_Product__Type_Oprod_Osnd,type,
    product_snd: 
      !>[A: $tType,B: $tType] : fun(product_prod(A,B),B) ).

tff(sy_c_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_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_Simple__TBOUND__Cond_Ocond__TBOUND,type,
    simple8792101892335181110TBOUND: 
      !>[A: $tType] : ( ( assn * heap_Time_Heap(A) * nat ) > $o ) ).

tff(sy_c_String_Oascii__of,type,
    ascii_of: char > char ).

tff(sy_c_String_Ochar_OChar,type,
    char2: ( $o * $o * $o * $o * $o * $o * $o ) > fun($o,char) ).

tff(sy_c_String_Ochar__of__integer,type,
    char_of_integer: code_integer > char ).

tff(sy_c_String_Ocomm__semiring__1__class_Oof__char,type,
    comm_s6883823935334413003f_char: 
      !>[A: $tType] : fun(char,A) ).

tff(sy_c_String_Ointeger__of__char,type,
    integer_of_char: char > code_integer ).

tff(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of,type,
    unique5772411509450598832har_of: 
      !>[A: $tType] : fun(A,char) ).

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_Time__Reasoning_Othe__res,type,
    time_the_res: 
      !>[A: $tType] : ( ( heap_Time_Heap(A) * heap_ext(product_unit) ) > A ) ).

tff(sy_c_Topological__Spaces_Ocontinuous,type,
    topolo3448309680560233919inuous: 
      !>[A: $tType,B: $tType] : ( ( filter(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_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] : ( 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: fun(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))) ).

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_ORep__uint32,type,
    rep_uint322: fun(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__internal_Oreplicatei,type,
    vEBT_VEBT_replicatei: 
      !>[A: $tType] : ( ( nat * heap_Time_Heap(A) ) > heap_Time_Heap(list(A)) ) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei__rel,type,
    vEBT_V8535202610096940344ei_rel: 
      !>[A: $tType] : fun(product_prod(nat,heap_Time_Heap(A)),fun(product_prod(nat,heap_Time_Heap(A)),$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_Otest,type,
    vEBT_test: ( nat * list(nat) * list(nat) ) > heap_Time_Heap(list(nat)) ).

tff(sy_c_VEBT__Example__Setup_OmIf,type,
    vEBT_Example_mIf: 
      !>[A: $tType] : ( ( heap_Time_Heap($o) * heap_Time_Heap(A) * heap_Time_Heap(A) ) > heap_Time_Heap(A) ) ).

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) ) > fun(B,heap_Time_Heap(B)) ) ).

tff(sy_c_VEBT__Example__Setup_Ommap,type,
    vEBT_Example_mmap: 
      !>[A: $tType,B: $tType] : ( ( fun(A,heap_Time_Heap(B)) * list(A) ) > heap_Time_Heap(list(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__List__Assn_OlistI__assn,type,
    vEBT_List_listI_assn: 
      !>[A: $tType,B: $tType] : ( ( set(nat) * fun(A,fun(B,assn)) * list(A) * list(B) ) > assn ) ).

tff(sy_c_VEBT__List__Assn_Olist__assn,type,
    vEBT_List_list_assn: 
      !>[A: $tType,C: $tType] : ( ( fun(A,fun(C,assn)) * list(A) * list(C) ) > assn ) ).

tff(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
    vEBT_VEBT_bit_concat: ( nat * nat * nat ) > nat ).

tff(sy_c_VEBT__Pred_Ois__pred__in__set,type,
    vEBT_is_pred_in_set: ( set(nat) * nat * nat ) > $o ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d,type,
    vEBT_V8646137997579335489_i_l_d: nat > nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p,type,
    vEBT_V8346862874174094_d_u_p: nat > nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p__rel,type,
    vEBT_V1247956027447740395_p_rel: fun(nat,fun(nat,$o)) ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d__rel,type,
    vEBT_V5144397997797733112_d_rel: fun(nat,fun(nat,$o)) ).

tff(sy_c_VEBT__SuccPredImperative_Ovebt__predi,type,
    vEBT_vebt_predi: ( vEBT_VEBTi * nat ) > heap_Time_Heap(option(nat)) ).

tff(sy_c_VEBT__SuccPredImperative_Ovebt__succi,type,
    vEBT_vebt_succi: ( vEBT_VEBTi * nat ) > heap_Time_Heap(option(nat)) ).

tff(sy_c_VEBT__Succ_Ois__succ__in__set,type,
    vEBT_is_succ_in_set: ( set(nat) * nat * nat ) > $o ).

tff(sy_c_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_Oudvd,type,
    udvd: 
      !>[A: $tType] : ( ( word(A) * word(A) ) > $o ) ).

tff(sy_c_Word_Oword__cat,type,
    word_cat: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( word(A) * word(B) ) > word(C) ) ).

tff(sy_c_Word_Oword__int__case,type,
    word_int_case: 
      !>[B: $tType,A: $tType] : ( ( fun(int,B) * word(A) ) > B ) ).

tff(sy_c_Word_Oword__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: $tType] : fun(word(A),fun(word(A),$o)) ).

tff(sy_c_Word_Oword__sless,type,
    word_sless: 
      !>[A: $tType] : fun(word(A),fun(word(A),$o)) ).

tff(sy_c_Word_Oword__split,type,
    word_split: 
      !>[A: $tType,B: $tType,C: $tType] : ( word(A) > product_prod(word(B),word(C)) ) ).

tff(sy_c_Word_Oword__succ,type,
    word_succ: 
      !>[A: $tType] : ( word(A) > word(A) ) ).

tff(sy_c_Word__Type__Copies_Oword__type__copy__misc,type,
    word_T1870391057261780392y_misc: 
      !>[B: $tType,A: $tType] : ( ( fun(word(B),A) * fun(A,word(B)) * fun(nat,fun(A,A)) * fun(nat,A) * fun(A,nat) * fun(int,A) * fun(A,int) * fun(code_integer,A) * fun(A,code_integer) * nat * fun(fun(nat,$o),fun(nat,fun(A,A))) ) > $o ) ).

tff(sy_c_Word__Type__Copies_Oword__type__copy__misc__axioms,type,
    word_T8964210463127689547axioms: 
      !>[A: $tType,B: $tType] : ( ( fun(A,word(B)) * nat * fun(fun(nat,$o),fun(nat,fun(A,A))) ) > $o ) ).

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: $tType] : ( A > fun(A,$o) ) ).

tff(sy_c_member,type,
    member: 
      !>[A: $tType] : ( ( A * set(A) ) > $o ) ).

tff(sy_v_S,type,
    s: set(nat) ).

tff(sy_v_a____,type,
    a2: nat ).

tff(sy_v_b,type,
    b2: ( list(nat) * nat ) > nat ).

tff(sy_v_n,type,
    n: nat ).

tff(sy_v_ti,type,
    ti: vEBT_VEBTi ).

tff(sy_v_ysa____,type,
    ysa: list(nat) ).

% Relevant facts (9181)
tff(fact_0_ceiling__add__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(num,A,numeral_numeral(A),V))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V)) ) ).

% ceiling_add_numeral
tff(fact_1_ceiling__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(num,A,numeral_numeral(A),V)) ) ) ).

% ceiling_le_numeral
tff(fact_2_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_3_of__nat__add,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat,Na: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Na)) ) ).

% of_nat_add
tff(fact_4_of__nat__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Na))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ).

% of_nat_le_iff
tff(fact_5_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_6_distrib__left__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [V: num,Ba: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),Ba)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),C2)) ) ).

% distrib_left_numeral
tff(fact_7_distrib__right__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [Aa2: A,Ba: A,V: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)),aa(num,A,numeral_numeral(A),V)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),aa(num,A,numeral_numeral(A),V))) ) ).

% distrib_right_numeral
tff(fact_8_pred__time__pure,axiom,
    ! [Na: nat,Sa: set(nat),Tia: vEBT_VEBTi,Aa2: nat] : simple8792101892335181110TBOUND(option(nat),aa(vEBT_VEBTi,assn,vEBT_Intf_vebt_assn(Na,Sa),Tia),vEBT_vebt_predi(Tia,Aa2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit1(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit1(bit1(one2)))),aa(int,nat,nat2,archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na))))))) ).

% pred_time_pure
tff(fact_9_member__time__pure,axiom,
    ! [Na: nat,Sa: set(nat),Tia: vEBT_VEBTi,Aa2: nat] : simple8792101892335181110TBOUND($o,aa(vEBT_VEBTi,assn,vEBT_Intf_vebt_assn(Na,Sa),Tia),vEBT_vebt_memberi(Tia,Aa2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(one2)))),aa(int,nat,nat2,archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na))))))) ).

% member_time_pure
tff(fact_10_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_11_ceiling__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num] : archimedean_ceiling(A,aa(num,A,numeral_numeral(A),V)) = aa(num,int,numeral_numeral(int),V) ) ).

% ceiling_numeral
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_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_14_numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: num,Na: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Na))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),Na) ) ) ).

% numeral_le_iff
tff(fact_15_add__numeral__left,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [V: num,W: num,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),W)),Z)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),V),W))),Z) ) ).

% add_numeral_left
tff(fact_16_numeral__plus__numeral,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [M: num,Na: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Na)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),Na)) ) ).

% numeral_plus_numeral
tff(fact_17_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_18_mult__numeral__left__semiring__numeral,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [V: num,W: num,Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),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),V),W))),Z) ) ).

% mult_numeral_left_semiring_numeral
tff(fact_19_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_20_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% nat_add_left_cancel_le
tff(fact_21_add__One__commute,axiom,
    ! [Na: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),Na) = aa(num,num,aa(num,fun(num,num),plus_plus(num),Na),one2) ).

% add_One_commute
tff(fact_22_le__num__One__iff,axiom,
    ! [X: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),X),one2)
    <=> ( X = one2 ) ) ).

% le_num_One_iff
tff(fact_23_ceiling__add__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y))),aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,X)),archimedean_ceiling(A,Y))) ) ).

% ceiling_add_le
tff(fact_24_is__num__normalize_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Aa2: A,Ba: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),C2)) ) ).

% is_num_normalize(1)
tff(fact_25_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun(nat,$o),K: nat,Ba: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [Y2: nat] :
            ( aa(nat,$o,P,Y2)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y2),Ba) )
       => ? [X2: nat] :
            ( aa(nat,$o,P,X2)
            & ! [Y3: nat] :
                ( aa(nat,$o,P,Y3)
               => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y3),X2) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_26_nat__le__linear,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
      | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M) ) ).

% nat_le_linear
tff(fact_27_le__antisym,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
       => ( M = Na ) ) ) ).

% le_antisym
tff(fact_28_eq__imp__le,axiom,
    ! [M: nat,Na: nat] :
      ( ( M = Na )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% eq_imp_le
tff(fact_29_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),K) ) ) ).

% le_trans
tff(fact_30_le__refl,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),Na) ).

% le_refl
tff(fact_31_real__arch__simple,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% real_arch_simple
tff(fact_32_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_33_nat__le__iff__add,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
    <=> ? [K2: nat] : Na = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K2) ) ).

% nat_le_iff_add
tff(fact_34_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),J)) ) ).

% trans_le_add2
tff(fact_35_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),M)) ) ).

% trans_le_add1
tff(fact_36_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) ) ).

% add_le_mono1
tff(fact_37_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),L)) ) ) ).

% add_le_mono
tff(fact_38_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),L)
     => ? [N: nat] : L = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N) ) ).

% le_Suc_ex
tff(fact_39_add__leD2,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na) ) ).

% add_leD2
tff(fact_40_add__leD1,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% add_leD1
tff(fact_41_le__add2,axiom,
    ! [Na: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)) ).

% le_add2
tff(fact_42_le__add1,axiom,
    ! [Na: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M)) ).

% le_add1
tff(fact_43_add__leE,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),Na)
     => ~ ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na) ) ) ).

% add_leE
tff(fact_44_mem__Collect__eq,axiom,
    ! [A: $tType,Aa2: A,P: fun(A,$o)] :
      ( member(A,Aa2,collect(A,P))
    <=> aa(A,$o,P,Aa2) ) ).

% mem_Collect_eq
tff(fact_45_Collect__mem__eq,axiom,
    ! [A: $tType,A2: set(A)] : collect(A,aTP_Lamp_ab(set(A),fun(A,$o),A2)) = A2 ).

% Collect_mem_eq
tff(fact_46_Collect__cong,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ! [X2: A] :
          ( aa(A,$o,P,X2)
        <=> aa(A,$o,Q,X2) )
     => ( collect(A,P) = collect(A,Q) ) ) ).

% Collect_cong
tff(fact_47_ext,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),G: fun(A,B)] :
      ( ! [X2: A] : aa(A,B,F2,X2) = aa(A,B,G,X2)
     => ( F2 = G ) ) ).

% ext
tff(fact_48_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Na)) ).

% add_mult_distrib2
tff(fact_49_add__mult__distrib,axiom,
    ! [M: nat,Na: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),K)) ).

% add_mult_distrib
tff(fact_50_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),I)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),J)) ) ).

% mult_le_mono2
tff(fact_51_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),K)) ) ).

% mult_le_mono1
tff(fact_52_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),L)) ) ) ).

% mult_le_mono
tff(fact_53_le__square,axiom,
    ! [M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),M)) ).

% le_square
tff(fact_54_le__cube,axiom,
    ! [M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),M))) ).

% le_cube
tff(fact_55_ceiling__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Y)),archimedean_ceiling(A,X)) ) ) ).

% ceiling_mono
tff(fact_56_numeral__Bit0,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Na: num] : aa(num,A,numeral_numeral(A),bit0(Na)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Na)),aa(num,A,numeral_numeral(A),Na)) ) ).

% numeral_Bit0
tff(fact_57_mult__numeral__1__right,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(num,A,numeral_numeral(A),one2)) = Aa2 ) ).

% mult_numeral_1_right
tff(fact_58_mult__numeral__1,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),one2)),Aa2) = Aa2 ) ).

% mult_numeral_1
tff(fact_59_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one2 )
     => ( ! [X22: num] : Y != bit0(X22)
       => ~ ! [X3: num] : Y != bit1(X3) ) ) ).

% num.exhaust
tff(fact_60_of__nat__mono,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [I: nat,J: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),I)),aa(nat,A,semiring_1_of_nat(A),J)) ) ) ).

% of_nat_mono
tff(fact_61_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,aa(A,fun(A,A),plus_plus(A),m),m) ) ) ).

% numeral_code(2)
tff(fact_62_of__nat__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),R),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,archimedean_ceiling(A,R)))) ) ).

% of_nat_ceiling
tff(fact_63_left__add__twice,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Aa2)),Ba) ) ).

% left_add_twice
tff(fact_64_mult__2__right,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),Z),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),Z) ) ).

% mult_2_right
tff(fact_65_mult__2,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Z) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),Z) ) ).

% mult_2
tff(fact_66_nat__ceiling__le__eq,axiom,
    ! [X: real,Aa2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,archimedean_ceiling(real,X))),Aa2)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(nat,real,semiring_1_of_nat(real),Aa2)) ) ).

% nat_ceiling_le_eq
tff(fact_67_numeral__le__real__of__nat__iff,axiom,
    ! [Na: num,M: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(num,real,numeral_numeral(real),Na)),aa(nat,real,semiring_1_of_nat(real),M))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),Na)),M) ) ).

% numeral_le_real_of_nat_iff
tff(fact_68_nat__numeral,axiom,
    ! [K: num] : aa(int,nat,nat2,aa(num,int,numeral_numeral(int),K)) = aa(num,nat,numeral_numeral(nat),K) ).

% nat_numeral
tff(fact_69_int__eq__iff__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( aa(nat,int,semiring_1_of_nat(int),M) = aa(num,int,numeral_numeral(int),V) )
    <=> ( M = aa(num,nat,numeral_numeral(nat),V) ) ) ).

% int_eq_iff_numeral
tff(fact_70_semiring__norm_I3_J,axiom,
    ! [Na: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),bit0(Na)) = bit1(Na) ).

% semiring_norm(3)
tff(fact_71_semiring__norm_I4_J,axiom,
    ! [Na: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),bit1(Na)) = bit0(aa(num,num,aa(num,fun(num,num),plus_plus(num),Na),one2)) ).

% semiring_norm(4)
tff(fact_72_semiring__norm_I5_J,axiom,
    ! [M: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),bit0(M)),one2) = bit1(M) ).

% semiring_norm(5)
tff(fact_73_semiring__norm_I8_J,axiom,
    ! [M: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),bit1(M)),one2) = bit0(aa(num,num,aa(num,fun(num,num),plus_plus(num),M),one2)) ).

% semiring_norm(8)
tff(fact_74_semiring__norm_I10_J,axiom,
    ! [M: num,Na: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),bit1(M)),bit1(Na)) = bit0(aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),Na)),one2)) ).

% semiring_norm(10)
tff(fact_75_nat__int,axiom,
    ! [Na: nat] : aa(int,nat,nat2,aa(nat,int,semiring_1_of_nat(int),Na)) = Na ).

% nat_int
tff(fact_76_enat__ord__number_I1_J,axiom,
    ! [M: num,Na: num] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),aa(num,extended_enat,numeral_numeral(extended_enat),M)),aa(num,extended_enat,numeral_numeral(extended_enat),Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),Na)) ) ).

% enat_ord_number(1)
tff(fact_77_semiring__norm_I70_J,axiom,
    ! [M: num] : ~ aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),bit1(M)),one2) ).

% semiring_norm(70)
tff(fact_78_semiring__norm_I87_J,axiom,
    ! [M: num,Na: num] :
      ( ( bit0(M) = bit0(Na) )
    <=> ( M = Na ) ) ).

% semiring_norm(87)
tff(fact_79_semiring__norm_I90_J,axiom,
    ! [M: num,Na: num] :
      ( ( bit1(M) = bit1(Na) )
    <=> ( M = Na ) ) ).

% semiring_norm(90)
tff(fact_80_semiring__norm_I85_J,axiom,
    ! [M: num] : bit0(M) != one2 ).

% semiring_norm(85)
tff(fact_81_semiring__norm_I83_J,axiom,
    ! [Na: num] : one2 != bit0(Na) ).

% semiring_norm(83)
tff(fact_82_semiring__norm_I89_J,axiom,
    ! [M: num,Na: num] : bit1(M) != bit0(Na) ).

% semiring_norm(89)
tff(fact_83_semiring__norm_I88_J,axiom,
    ! [M: num,Na: num] : bit0(M) != bit1(Na) ).

% semiring_norm(88)
tff(fact_84_semiring__norm_I86_J,axiom,
    ! [M: num] : bit1(M) != one2 ).

% semiring_norm(86)
tff(fact_85_semiring__norm_I84_J,axiom,
    ! [Na: num] : one2 != bit1(Na) ).

% semiring_norm(84)
tff(fact_86_semiring__norm_I6_J,axiom,
    ! [M: num,Na: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),bit0(M)),bit0(Na)) = bit0(aa(num,num,aa(num,fun(num,num),plus_plus(num),M),Na)) ).

% semiring_norm(6)
tff(fact_87_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_88_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_89_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_90_semiring__norm_I71_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),bit0(M)),bit0(Na))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),Na) ) ).

% semiring_norm(71)
tff(fact_91_semiring__norm_I68_J,axiom,
    ! [Na: num] : aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),one2),Na) ).

% semiring_norm(68)
tff(fact_92_semiring__norm_I73_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),bit1(M)),bit1(Na))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),Na) ) ).

% semiring_norm(73)
tff(fact_93_semiring__norm_I2_J,axiom,
    aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),one2) = bit0(one2) ).

% semiring_norm(2)
tff(fact_94_semiring__norm_I9_J,axiom,
    ! [M: num,Na: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),bit1(M)),bit0(Na)) = bit1(aa(num,num,aa(num,fun(num,num),plus_plus(num),M),Na)) ).

% semiring_norm(9)
tff(fact_95_semiring__norm_I7_J,axiom,
    ! [M: num,Na: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),bit0(M)),bit1(Na)) = bit1(aa(num,num,aa(num,fun(num,num),plus_plus(num),M),Na)) ).

% semiring_norm(7)
tff(fact_96_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_97_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_98_semiring__norm_I69_J,axiom,
    ! [M: num] : ~ aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),bit0(M)),one2) ).

% semiring_norm(69)
tff(fact_99_semiring__norm_I72_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),bit0(M)),bit1(Na))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),Na) ) ).

% semiring_norm(72)
tff(fact_100_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,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(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_101_complete__real,axiom,
    ! [Sa: set(real)] :
      ( ? [X4: real] : member(real,X4,Sa)
     => ( ? [Z2: real] :
          ! [X2: real] :
            ( member(real,X2,Sa)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),Z2) )
       => ? [Y2: real] :
            ( ! [X4: real] :
                ( member(real,X4,Sa)
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),Y2) )
            & ! [Z2: real] :
                ( ! [X2: real] :
                    ( member(real,X2,Sa)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),Z2) )
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y2),Z2) ) ) ) ) ).

% complete_real
tff(fact_102_int__distrib_I1_J,axiom,
    ! [Z1: int,Z22: int,W: int] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z1),Z22)),W) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Z1),W)),aa(int,int,aa(int,fun(int,int),times_times(int),Z22),W)) ).

% int_distrib(1)
tff(fact_103_int__distrib_I2_J,axiom,
    ! [W: int,Z1: int,Z22: int] : aa(int,int,aa(int,fun(int,int),times_times(int),W),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z1),Z22)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z1)),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z22)) ).

% int_distrib(2)
tff(fact_104_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_105_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),K)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),U)),K) ).

% left_add_mult_distrib
tff(fact_106_zle__int,axiom,
    ! [M: nat,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% zle_int
tff(fact_107_zadd__int__left,axiom,
    ! [M: nat,Na: nat,Z: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),Na)),Z)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na))),Z) ).

% zadd_int_left
tff(fact_108_nat__mono,axiom,
    ! [X: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X),Y)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,X)),aa(int,nat,nat2,Y)) ) ).

% nat_mono
tff(fact_109_zle__iff__zadd,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W),Z)
    <=> ? [N2: nat] : Z = aa(int,int,aa(int,fun(int,int),plus_plus(int),W),aa(nat,int,semiring_1_of_nat(int),N2)) ) ).

% zle_iff_zadd
tff(fact_110_real__nat__ceiling__ge,axiom,
    ! [X: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(nat,real,semiring_1_of_nat(real),aa(int,nat,nat2,archimedean_ceiling(real,X)))) ).

% real_nat_ceiling_ge
tff(fact_111_nat__le__iff,axiom,
    ! [X: int,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,X)),Na)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X),aa(nat,int,semiring_1_of_nat(int),Na)) ) ).

% nat_le_iff
tff(fact_112_nat__int__add,axiom,
    ! [Aa2: nat,Ba: nat] : aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),Aa2)),aa(nat,int,semiring_1_of_nat(int),Ba))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Aa2),Ba) ).

% nat_int_add
tff(fact_113_add__le__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),Aa2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),Ba))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba) ) ) ).

% add_le_cancel_left
tff(fact_114_add__le__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Aa2: A,C2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),C2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba) ) ) ).

% add_le_cancel_right
tff(fact_115__092_060open_062_092_060section_062_Avebt__assn_An_AS_Ati_A_092_060section_062_ATBOUND_Areturn_A_091_093_Ab_A_091_093_An_092_060close_062,axiom,
    simple8792101892335181110TBOUND(list(nat),aa(vEBT_VEBTi,assn,vEBT_Intf_vebt_assn(n,s),ti),aa(list(nat),heap_Time_Heap(list(nat)),heap_Time_return(list(nat)),nil(nat)),b2(nil(nat),n)) ).

% \<open>\<section> vebt_assn n S ti \<section> TBOUND return [] b [] n\<close>
tff(fact_116_nat__plus__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X4),Xa) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),X4)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_plus_as_int
tff(fact_117_Heap__Time__Monad_Obind__bind,axiom,
    ! [C: $tType,A: $tType,B: $tType,F2: heap_Time_Heap(C),G: fun(C,heap_Time_Heap(B)),K: fun(B,heap_Time_Heap(A))] : heap_Time_bind(B,A,heap_Time_bind(C,B,F2,G),K) = heap_Time_bind(C,A,F2,aa(fun(B,heap_Time_Heap(A)),fun(C,heap_Time_Heap(A)),aTP_Lamp_ac(fun(C,heap_Time_Heap(B)),fun(fun(B,heap_Time_Heap(A)),fun(C,heap_Time_Heap(A))),G),K)) ).

% Heap_Time_Monad.bind_bind
tff(fact_118_nat__times__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),X4),Xa) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),X4)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_times_as_int
tff(fact_119_nat__leq__as__int,axiom,
    ! [X4: nat,Xa: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X4),Xa)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),X4)),aa(nat,int,semiring_1_of_nat(int),Xa)) ) ).

% nat_leq_as_int
tff(fact_120_b__def,axiom,
    ! [Ysb: list(nat),Na: nat] : b2(Ysb,Na) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(nat),nat,size_size(list(nat)),Ysb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(one2)))),aa(int,nat,nat2,archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na))))))),aa(num,nat,numeral_numeral(nat),bit1(bit0(bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit1(bit1(one2)))),aa(int,nat,nat2,archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na)))))))) ).

% b_def
tff(fact_121_int__plus,axiom,
    ! [Na: nat,M: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M)) = aa(int,int,aa(int,fun(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_122_int__ops_I5_J,axiom,
    ! [Aa2: nat,Ba: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Aa2),Ba)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),Aa2)),aa(nat,int,semiring_1_of_nat(int),Ba)) ).

% int_ops(5)
tff(fact_123_nat__int__comparison_I3_J,axiom,
    ! [Aa2: nat,Ba: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Aa2),Ba)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),Aa2)),aa(nat,int,semiring_1_of_nat(int),Ba)) ) ).

% nat_int_comparison(3)
tff(fact_124_nat__numeral__as__int,axiom,
    ! [X4: num] : aa(num,nat,numeral_numeral(nat),X4) = aa(int,nat,nat2,aa(num,int,numeral_numeral(int),X4)) ).

% nat_numeral_as_int
tff(fact_125_verit__eq__simplify_I8_J,axiom,
    ! [X23: num,Y22: num] :
      ( ( bit0(X23) = bit0(Y22) )
    <=> ( X23 = Y22 ) ) ).

% verit_eq_simplify(8)
tff(fact_126_add__right__cancel,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [Ba: A,Aa2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),Aa2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),Aa2) )
        <=> ( Ba = C2 ) ) ) ).

% add_right_cancel
tff(fact_127_add__left__cancel,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),C2) )
        <=> ( Ba = C2 ) ) ) ).

% add_left_cancel
tff(fact_128_verit__eq__simplify_I9_J,axiom,
    ! [X32: num,Y32: num] :
      ( ( bit1(X32) = bit1(Y32) )
    <=> ( X32 = Y32 ) ) ).

% verit_eq_simplify(9)
tff(fact_129_mult__1,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),one_one(A)),Aa2) = Aa2 ) ).

% mult_1
tff(fact_130_mult_Oright__neutral,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),one_one(A)) = Aa2 ) ).

% mult.right_neutral
tff(fact_131_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_132_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_133_ceiling__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_ceiling(A,one_one(A)) = one_one(int) ) ) ).

% ceiling_one
tff(fact_134_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_135_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_136_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_137_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_138_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_139_ceiling__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,X)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),one_one(A)) ) ) ).

% ceiling_le_one
tff(fact_140_ceiling__add__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),one_one(A))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,X)),one_one(int)) ) ).

% ceiling_add_one
tff(fact_141_one__add__one,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)) = aa(num,A,numeral_numeral(A),bit0(one2)) ) ) ).

% one_add_one
tff(fact_142_one__plus__numeral,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Na: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(num,A,numeral_numeral(A),Na)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),Na)) ) ).

% one_plus_numeral
tff(fact_143_numeral__plus__one,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Na: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Na)),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),Na),one2)) ) ).

% numeral_plus_one
tff(fact_144_numeral__le__one__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Na)),one_one(A))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Na),one2) ) ) ).

% numeral_le_one_iff
tff(fact_145_one__reorient,axiom,
    ! [A: $tType] :
      ( one(A)
     => ! [X: A] :
          ( ( one_one(A) = X )
        <=> ( X = one_one(A) ) ) ) ).

% one_reorient
tff(fact_146_size__neq__size__imp__neq,axiom,
    ! [A: $tType] :
      ( size(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,nat,size_size(A),X) != aa(A,nat,size_size(A),Y) )
         => ( X != Y ) ) ) ).

% size_neq_size_imp_neq
tff(fact_147_mult_Ocomm__neutral,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),one_one(A)) = Aa2 ) ).

% mult.comm_neutral
tff(fact_148_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),one_one(A)),Aa2) = Aa2 ) ).

% comm_monoid_mult_class.mult_1
tff(fact_149_le__numeral__extra_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),one_one(A)) ) ).

% le_numeral_extra(4)
tff(fact_150_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_151_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_152_one__le__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(num,A,numeral_numeral(A),Na)) ) ).

% one_le_numeral
tff(fact_153_numeral__One,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ( aa(num,A,numeral_numeral(A),one2) = one_one(A) ) ) ).

% numeral_One
tff(fact_154_one__plus__numeral__commute,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(num,A,numeral_numeral(A),X)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),X)),one_one(A)) ) ).

% one_plus_numeral_commute
tff(fact_155_numerals_I1_J,axiom,
    aa(num,nat,numeral_numeral(nat),one2) = one_one(nat) ).

% numerals(1)
tff(fact_156_verit__la__disequality,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: A,Ba: A] :
          ( ( Aa2 = Ba )
          | ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
          | ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2) ) ) ).

% verit_la_disequality
tff(fact_157_verit__comp__simplify1_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Aa2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Aa2) ) ).

% verit_comp_simplify1(2)
tff(fact_158_add__right__imp__eq,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [Ba: A,Aa2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),Aa2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),Aa2) )
         => ( Ba = C2 ) ) ) ).

% add_right_imp_eq
tff(fact_159_add__left__imp__eq,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),C2) )
         => ( Ba = C2 ) ) ) ).

% add_left_imp_eq
tff(fact_160_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [Ba: A,Aa2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),C2)) ) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_161_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),Aa2) ) ).

% ab_semigroup_add_class.add.commute
tff(fact_162_add_Oright__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [Ba: A,Aa2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),Aa2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),Aa2) )
        <=> ( Ba = C2 ) ) ) ).

% add.right_cancel
tff(fact_163_add_Oleft__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),C2) )
        <=> ( Ba = C2 ) ) ) ).

% add.left_cancel
tff(fact_164_add_Oassoc,axiom,
    ! [A: $tType] :
      ( semigroup_add(A)
     => ! [Aa2: A,Ba: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),C2)) ) ).

% add.assoc
tff(fact_165_group__cancel_Oadd2,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [B2: A,K: A,Ba: A,Aa2: A] :
          ( ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),Ba) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)) ) ) ) ).

% group_cancel.add2
tff(fact_166_group__cancel_Oadd1,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: A,K: A,Aa2: A,Ba: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),Aa2) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),Ba) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)) ) ) ) ).

% group_cancel.add1
tff(fact_167_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( ( I = J )
            & ( K = L ) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K) = aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L) ) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_168_numeral__Bit1,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Na: num] : aa(num,A,numeral_numeral(A),bit1(Na)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Na)),aa(num,A,numeral_numeral(A),Na))),one_one(A)) ) ).

% numeral_Bit1
tff(fact_169_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [Ba: A,Aa2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),Ba),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2)) ) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_170_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba) = aa(A,A,aa(A,fun(A,A),times_times(A),Ba),Aa2) ) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_171_mult_Oassoc,axiom,
    ! [A: $tType] :
      ( semigroup_mult(A)
     => ! [Aa2: A,Ba: 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),Aa2),Ba)),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2)) ) ).

% mult.assoc
tff(fact_172_fold__if__return,axiom,
    ! [A: $tType,Ba: $o,C2: A,D2: A] :
      $ite((Ba),aa(A,heap_Time_Heap(A),heap_Time_return(A),C2),aa(A,heap_Time_Heap(A),heap_Time_return(A),D2)) = aa(A,heap_Time_Heap(A),heap_Time_return(A),
        $ite((Ba),C2,D2)) ).

% fold_if_return
tff(fact_173_verit__la__generic,axiom,
    ! [Aa2: int,X: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Aa2),X)
      | ( Aa2 = X )
      | aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X),Aa2) ) ).

% verit_la_generic
tff(fact_174_nat__int__comparison_I1_J,axiom,
    ! [Aa2: nat,Ba: nat] :
      ( ( Aa2 = Ba )
    <=> ( aa(nat,int,semiring_1_of_nat(int),Aa2) = aa(nat,int,semiring_1_of_nat(int),Ba) ) ) ).

% nat_int_comparison(1)
tff(fact_175_int__if,axiom,
    ! [P: $o,Aa2: nat,Ba: nat] :
      aa(nat,int,semiring_1_of_nat(int),
        $ite((P),Aa2,Ba)) = $ite((P),aa(nat,int,semiring_1_of_nat(int),Aa2),aa(nat,int,semiring_1_of_nat(int),Ba)) ).

% int_if
tff(fact_176_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,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),m),m)),one_one(A)) ) ) ).

% numeral_code(3)
tff(fact_177_nat__1__add__1,axiom,
    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% nat_1_add_1
tff(fact_178_distrib__if__bind,axiom,
    ! [A: $tType,B: $tType,Ba: $o,C2: heap_Time_Heap(B),D2: heap_Time_Heap(B),F2: fun(B,heap_Time_Heap(A))] :
      heap_Time_bind(B,A,
        $ite((Ba),C2,D2),
        F2) = $ite((Ba),heap_Time_bind(B,A,C2,F2),heap_Time_bind(B,A,D2,F2)) ).

% distrib_if_bind
tff(fact_179_add__le__imp__le__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Aa2: A,C2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),C2))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba) ) ) ).

% add_le_imp_le_right
tff(fact_180_add__le__imp__le__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),Aa2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),Ba))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba) ) ) ).

% add_le_imp_le_left
tff(fact_181_le__iff__add,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
        <=> ? [C3: A] : Ba = aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),C3) ) ) ).

% le_iff_add
tff(fact_182_add__right__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),C2)) ) ) ).

% add_right_mono
tff(fact_183_less__eqE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => ~ ! [C4: A] : Ba != aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),C4) ) ) ).

% less_eqE
tff(fact_184_add__left__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),Aa2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),Ba)) ) ) ).

% add_left_mono
tff(fact_185_add__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [Aa2: A,Ba: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),D2)) ) ) ) ).

% add_mono
tff(fact_186_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_187_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( ( I = J )
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_188_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),J)
            & ( K = L ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_189_verit__eq__simplify_I10_J,axiom,
    ! [X23: num] : one2 != bit0(X23) ).

% verit_eq_simplify(10)
tff(fact_190_verit__eq__simplify_I14_J,axiom,
    ! [X23: num,X32: num] : bit0(X23) != bit1(X32) ).

% verit_eq_simplify(14)
tff(fact_191_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] : one2 != bit1(X32) ).

% verit_eq_simplify(12)
tff(fact_192_int__ops_I7_J,axiom,
    ! [Aa2: nat,Ba: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Aa2),Ba)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),Aa2)),aa(nat,int,semiring_1_of_nat(int),Ba)) ).

% int_ops(7)
tff(fact_193_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_194_list__decomp__2,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),L) = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
     => ? [A3: A,B3: A] : L = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B3),nil(A))) ) ).

% list_decomp_2
tff(fact_195_mmap_Oelims,axiom,
    ! [A: $tType,B: $tType,X: fun(B,heap_Time_Heap(A)),Xa2: list(B),Y: heap_Time_Heap(list(A))] :
      ( ( vEBT_Example_mmap(B,A,X,Xa2) = Y )
     => ( ( ( Xa2 = nil(B) )
         => ( Y != aa(list(A),heap_Time_Heap(list(A)),heap_Time_return(list(A)),nil(A)) ) )
       => ~ ! [X2: B,Xs: list(B)] :
              ( ( Xa2 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs) )
             => ( Y != heap_Time_bind(A,list(A),aa(B,heap_Time_Heap(A),X,X2),aa(list(B),fun(A,heap_Time_Heap(list(A))),aTP_Lamp_ae(fun(B,heap_Time_Heap(A)),fun(list(B),fun(A,heap_Time_Heap(list(A)))),X),Xs)) ) ) ) ) ).

% mmap.elims
tff(fact_196_mmap_Osimps_I2_J,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,heap_Time_Heap(A)),X: B,Xs2: list(B)] : vEBT_Example_mmap(B,A,F2,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs2)) = heap_Time_bind(A,list(A),aa(B,heap_Time_Heap(A),F2,X),aa(list(B),fun(A,heap_Time_Heap(list(A))),aTP_Lamp_ae(fun(B,heap_Time_Heap(A)),fun(list(B),fun(A,heap_Time_Heap(list(A)))),F2),Xs2)) ).

% mmap.simps(2)
tff(fact_197_list__decomp__1,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),L) = one_one(nat) )
     => ? [A3: A] : L = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) ) ).

% list_decomp_1
tff(fact_198_length__compl__induct,axiom,
    ! [A: $tType,P: fun(list(A),$o),L: list(A)] :
      ( aa(list(A),$o,P,nil(A))
     => ( ! [E: A,L2: list(A)] :
            ( ! [Ll: list(A)] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ll)),aa(list(A),nat,size_size(list(A)),L2))
               => aa(list(A),$o,P,Ll) )
           => aa(list(A),$o,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E),L2)) )
       => aa(list(A),$o,P,L) ) ) ).

% length_compl_induct
tff(fact_199_mmap_Osimps_I1_J,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,heap_Time_Heap(A))] : vEBT_Example_mmap(B,A,F2,nil(B)) = aa(list(A),heap_Time_Heap(list(A)),heap_Time_return(list(A)),nil(A)) ).

% mmap.simps(1)
tff(fact_200_cond__TBOUND__return,axiom,
    ! [A: $tType,P: assn,X: A] : simple8792101892335181110TBOUND(A,P,aa(A,heap_Time_Heap(A),heap_Time_return(A),X),one_one(nat)) ).

% cond_TBOUND_return
tff(fact_201_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_202_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_203_mlex__leI,axiom,
    ! [Aa2: nat,A4: nat,Ba: nat,B4: nat,N3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Aa2),A4)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ba),B4)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Aa2),N3)),Ba)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A4),N3)),B4)) ) ) ).

% mlex_leI
tff(fact_204_impossible__Cons,axiom,
    ! [A: $tType,Xs2: list(A),Ysb: list(A),X: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs2)),aa(list(A),nat,size_size(list(A)),Ysb))
     => ( Xs2 != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ysb) ) ) ).

% impossible_Cons
tff(fact_205_list__induct2,axiom,
    ! [A: $tType,B: $tType,Xs2: list(A),Ysb: list(B),P: fun(list(A),fun(list(B),$o))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(B),nat,size_size(list(B)),Ysb) )
     => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P,nil(A)),nil(B))
       => ( ! [X2: A,Xs: list(A),Y2: B,Ys: list(B)] :
              ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
             => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P,Xs),Ys)
               => aa(list(B),$o,aa(list(A),fun(list(B),$o),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys)) ) )
         => aa(list(B),$o,aa(list(A),fun(list(B),$o),P,Xs2),Ysb) ) ) ) ).

% list_induct2
tff(fact_206_list_Oinject,axiom,
    ! [A: $tType,X21: A,X222: list(A),Y21: A,Y222: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y21),Y222) )
    <=> ( ( X21 = Y21 )
        & ( X222 = Y222 ) ) ) ).

% list.inject
tff(fact_207_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_208_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_209_dbl__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [X: A] : neg_numeral_dbl(A,X) = aa(A,A,aa(A,fun(A,A),plus_plus(A),X),X) ) ).

% dbl_def
tff(fact_210_int__ops_I2_J,axiom,
    aa(nat,int,semiring_1_of_nat(int),one_one(nat)) = one_one(int) ).

% int_ops(2)
tff(fact_211_nat__one__as__int,axiom,
    one_one(nat) = aa(int,nat,nat2,one_one(int)) ).

% nat_one_as_int
tff(fact_212_int__ge__induct,axiom,
    ! [K: int,I: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I)
     => ( aa(int,$o,P,K)
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I2)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_ge_induct
tff(fact_213_ord__eq__le__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Aa2: A,Ba: A,C2: A,D2: A] :
          ( ( Aa2 = Ba )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),C2)
           => ( ( C2 = D2 )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),D2) ) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_214_not__Cons__self2,axiom,
    ! [A: $tType,X: A,Xs2: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs2) != Xs2 ).

% not_Cons_self2
tff(fact_215_list__tail__coinc,axiom,
    ! [A: $tType,N1: A,R1: list(A),N22: A,R2: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),N1),R1) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),N22),R2) )
     => ( ( N1 = N22 )
        & ( R1 = R2 ) ) ) ).

% list_tail_coinc
tff(fact_216_neq__if__length__neq,axiom,
    ! [A: $tType,Xs2: list(A),Ysb: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs2) != aa(list(A),nat,size_size(list(A)),Ysb) )
     => ( Xs2 != Ysb ) ) ).

% neq_if_length_neq
tff(fact_217_Ex__list__of__length,axiom,
    ! [A: $tType,Na: nat] :
    ? [Xs: list(A)] : aa(list(A),nat,size_size(list(A)),Xs) = Na ).

% Ex_list_of_length
tff(fact_218_dbl__inc__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [X: A] : neg_numeral_dbl_inc(A,X) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),X)),one_one(A)) ) ).

% dbl_inc_def
tff(fact_219_list_Odistinct_I1_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : nil(A) != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222) ).

% list.distinct(1)
tff(fact_220_list_OdiscI,axiom,
    ! [A: $tType,List: list(A),X21: A,X222: list(A)] :
      ( ( List = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222) )
     => ( List != nil(A) ) ) ).

% list.discI
tff(fact_221_list_Oexhaust,axiom,
    ! [A: $tType,Y: list(A)] :
      ( ( Y != nil(A) )
     => ~ ! [X212: A,X223: list(A)] : Y != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X212),X223) ) ).

% list.exhaust
tff(fact_222_min__list_Ocases,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: list(A)] :
          ( ! [X2: A,Xs: list(A)] : X != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)
         => ( X = nil(A) ) ) ) ).

% min_list.cases
tff(fact_223_remdups__adj_Ocases,axiom,
    ! [A: $tType,X: list(A)] :
      ( ( X != nil(A) )
     => ( ! [X2: A] : X != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A))
       => ~ ! [X2: A,Y2: A,Xs: list(A)] : X != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs)) ) ) ).

% remdups_adj.cases
tff(fact_224_neq__NilE,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( L != nil(A) )
     => ~ ! [X2: A,Xs: list(A)] : L != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs) ) ).

% neq_NilE
tff(fact_225_neq__Nil__conv,axiom,
    ! [A: $tType,Xs2: list(A)] :
      ( ( Xs2 != nil(A) )
    <=> ? [Y4: A,Ys2: list(A)] : Xs2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys2) ) ).

% neq_Nil_conv
tff(fact_226_list__induct2_H,axiom,
    ! [A: $tType,B: $tType,P: fun(list(A),fun(list(B),$o)),Xs2: list(A),Ysb: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P,nil(A)),nil(B))
     => ( ! [X2: A,Xs: list(A)] : aa(list(B),$o,aa(list(A),fun(list(B),$o),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)),nil(B))
       => ( ! [Y2: B,Ys: list(B)] : aa(list(B),$o,aa(list(A),fun(list(B),$o),P,nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys))
         => ( ! [X2: A,Xs: list(A),Y2: B,Ys: list(B)] :
                ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P,Xs),Ys)
               => aa(list(B),$o,aa(list(A),fun(list(B),$o),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys)) )
           => aa(list(B),$o,aa(list(A),fun(list(B),$o),P,Xs2),Ysb) ) ) ) ) ).

% list_induct2'
tff(fact_227_list__2pre__induct,axiom,
    ! [A: $tType,B: $tType,P: fun(list(A),fun(list(B),$o)),W1: list(A),W2: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P,nil(A)),nil(B))
     => ( ! [E: A,W12: list(A),W22: list(B)] :
            ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P,W12),W22)
           => aa(list(B),$o,aa(list(A),fun(list(B),$o),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E),W12)),W22) )
       => ( ! [E: B,W13: list(A),W23: list(B)] :
              ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P,W13),W23)
             => aa(list(B),$o,aa(list(A),fun(list(B),$o),P,W13),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),E),W23)) )
         => aa(list(B),$o,aa(list(A),fun(list(B),$o),P,W1),W2) ) ) ) ).

% list_2pre_induct
tff(fact_228_list__induct__first2,axiom,
    ! [A: $tType,P: fun(list(A),$o),Xs2: list(A)] :
      ( aa(list(A),$o,P,nil(A))
     => ( ! [X2: A] : aa(list(A),$o,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)))
       => ( ! [X1: A,X22: A,Xs: list(A)] :
              ( aa(list(A),$o,P,Xs)
             => aa(list(A),$o,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),Xs))) )
         => aa(list(A),$o,P,Xs2) ) ) ) ).

% list_induct_first2
tff(fact_229_list__nonempty__induct,axiom,
    ! [A: $tType,Xs2: list(A),P: fun(list(A),$o)] :
      ( ( Xs2 != nil(A) )
     => ( ! [X2: A] : aa(list(A),$o,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)))
       => ( ! [X2: A,Xs: list(A)] :
              ( ( Xs != nil(A) )
             => ( aa(list(A),$o,P,Xs)
               => aa(list(A),$o,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)) ) )
         => aa(list(A),$o,P,Xs2) ) ) ) ).

% list_nonempty_induct
tff(fact_230_mergesort__by__rel__merge__induct,axiom,
    ! [A: $tType,B: $tType,P: fun(list(A),fun(list(B),$o)),R3: fun(A,fun(B,$o)),Xs2: list(A),Ysb: list(B)] :
      ( ! [Xs: list(A)] : aa(list(B),$o,aa(list(A),fun(list(B),$o),P,Xs),nil(B))
     => ( ! [Ys: list(B)] : aa(list(B),$o,aa(list(A),fun(list(B),$o),P,nil(A)),Ys)
       => ( ! [X2: A,Xs: list(A),Y2: B,Ys: list(B)] :
              ( aa(B,$o,aa(A,fun(B,$o),R3,X2),Y2)
             => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P,Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys))
               => aa(list(B),$o,aa(list(A),fun(list(B),$o),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys)) ) )
         => ( ! [X2: A,Xs: list(A),Y2: B,Ys: list(B)] :
                ( ~ aa(B,$o,aa(A,fun(B,$o),R3,X2),Y2)
               => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)),Ys)
                 => aa(list(B),$o,aa(list(A),fun(list(B),$o),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys)) ) )
           => aa(list(B),$o,aa(list(A),fun(list(B),$o),P,Xs2),Ysb) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_231_transpose_Ocases,axiom,
    ! [A: $tType,X: list(list(A))] :
      ( ( X != nil(list(A)) )
     => ( ! [Xss: list(list(A))] : X != aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss)
       => ~ ! [X2: A,Xs: list(A),Xss: list(list(A))] : X != aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)),Xss) ) ) ).

% transpose.cases
tff(fact_232_mIf__def,axiom,
    ! [A: $tType,Ba: heap_Time_Heap($o),T: heap_Time_Heap(A),E2: heap_Time_Heap(A)] : vEBT_Example_mIf(A,Ba,T,E2) = heap_Time_bind($o,A,Ba,aa(heap_Time_Heap(A),fun($o,heap_Time_Heap(A)),aTP_Lamp_af(heap_Time_Heap(A),fun(heap_Time_Heap(A),fun($o,heap_Time_Heap(A))),T),E2)) ).

% mIf_def
tff(fact_233_cond__TBOUND__mono,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Ba: nat,B4: nat] :
      ( simple8792101892335181110TBOUND(A,P,C2,Ba)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ba),B4)
       => simple8792101892335181110TBOUND(A,P,C2,B4) ) ) ).

% cond_TBOUND_mono
tff(fact_234_list__induct4,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,Xs2: list(A),Ysb: list(B),Zs: list(C),Ws: list(D),P: fun(list(A),fun(list(B),fun(list(C),fun(list(D),$o))))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(B),nat,size_size(list(B)),Ysb) )
     => ( ( aa(list(B),nat,size_size(list(B)),Ysb) = aa(list(C),nat,size_size(list(C)),Zs) )
       => ( ( aa(list(C),nat,size_size(list(C)),Zs) = aa(list(D),nat,size_size(list(D)),Ws) )
         => ( aa(list(D),$o,aa(list(C),fun(list(D),$o),aa(list(B),fun(list(C),fun(list(D),$o)),aa(list(A),fun(list(B),fun(list(C),fun(list(D),$o))),P,nil(A)),nil(B)),nil(C)),nil(D))
           => ( ! [X2: A,Xs: list(A),Y2: B,Ys: list(B),Z3: C,Zs2: list(C),W3: D,Ws2: list(D)] :
                  ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
                 => ( ( aa(list(B),nat,size_size(list(B)),Ys) = aa(list(C),nat,size_size(list(C)),Zs2) )
                   => ( ( aa(list(C),nat,size_size(list(C)),Zs2) = aa(list(D),nat,size_size(list(D)),Ws2) )
                     => ( aa(list(D),$o,aa(list(C),fun(list(D),$o),aa(list(B),fun(list(C),fun(list(D),$o)),aa(list(A),fun(list(B),fun(list(C),fun(list(D),$o))),P,Xs),Ys),Zs2),Ws2)
                       => aa(list(D),$o,aa(list(C),fun(list(D),$o),aa(list(B),fun(list(C),fun(list(D),$o)),aa(list(A),fun(list(B),fun(list(C),fun(list(D),$o))),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),Z3),Zs2)),aa(list(D),list(D),aa(D,fun(list(D),list(D)),cons(D),W3),Ws2)) ) ) ) )
             => aa(list(D),$o,aa(list(C),fun(list(D),$o),aa(list(B),fun(list(C),fun(list(D),$o)),aa(list(A),fun(list(B),fun(list(C),fun(list(D),$o))),P,Xs2),Ysb),Zs),Ws) ) ) ) ) ) ).

% list_induct4
tff(fact_235_list__induct3,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs2: list(A),Ysb: list(B),Zs: list(C),P: fun(list(A),fun(list(B),fun(list(C),$o)))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(B),nat,size_size(list(B)),Ysb) )
     => ( ( aa(list(B),nat,size_size(list(B)),Ysb) = aa(list(C),nat,size_size(list(C)),Zs) )
       => ( aa(list(C),$o,aa(list(B),fun(list(C),$o),aa(list(A),fun(list(B),fun(list(C),$o)),P,nil(A)),nil(B)),nil(C))
         => ( ! [X2: A,Xs: list(A),Y2: B,Ys: list(B),Z3: C,Zs2: list(C)] :
                ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
               => ( ( aa(list(B),nat,size_size(list(B)),Ys) = aa(list(C),nat,size_size(list(C)),Zs2) )
                 => ( aa(list(C),$o,aa(list(B),fun(list(C),$o),aa(list(A),fun(list(B),fun(list(C),$o)),P,Xs),Ys),Zs2)
                   => aa(list(C),$o,aa(list(B),fun(list(C),$o),aa(list(A),fun(list(B),fun(list(C),$o)),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),Z3),Zs2)) ) ) )
           => aa(list(C),$o,aa(list(B),fun(list(C),$o),aa(list(A),fun(list(B),fun(list(C),$o)),P,Xs2),Ysb),Zs) ) ) ) ) ).

% list_induct3
tff(fact_236_Abs__fnat__hom__add,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Aa2: nat,Ba: nat] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Aa2)),aa(nat,A,semiring_1_of_nat(A),Ba)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Aa2),Ba)) ) ).

% Abs_fnat_hom_add
tff(fact_237_uint32_Osize__eq,axiom,
    ! [P2: uint32] : aa(uint32,nat,size_size(uint32),P2) = aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(one2)))))) ).

% uint32.size_eq
tff(fact_238_list__encode_Ocases,axiom,
    ! [X: list(nat)] :
      ( ( X != nil(nat) )
     => ~ ! [X2: nat,Xs: list(nat)] : X != aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X2),Xs) ) ).

% list_encode.cases
tff(fact_239_size__union,axiom,
    ! [A: $tType,M2: multiset(A),N3: multiset(A)] : aa(multiset(A),nat,size_size(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M2),N3)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(multiset(A),nat,size_size(multiset(A)),M2)),aa(multiset(A),nat,size_size(multiset(A)),N3)) ).

% size_union
tff(fact_240_order__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),X) ) ).

% order_refl
tff(fact_241_dual__order_Orefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Aa2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Aa2) ) ).

% dual_order.refl
tff(fact_242_lambda__one,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ( aTP_Lamp_ag(A,A) = aa(A,fun(A,A),times_times(A),one_one(A)) ) ) ).

% lambda_one
tff(fact_243_mfold_Oelims,axiom,
    ! [A: $tType,B: $tType,X: fun(B,fun(A,heap_Time_Heap(A))),Xa2: list(B),Xb: A,Y: heap_Time_Heap(A)] :
      ( ( aa(A,heap_Time_Heap(A),vEBT_Example_mfold(B,A,X,Xa2),Xb) = Y )
     => ( ( ( Xa2 = nil(B) )
         => ( Y != aa(A,heap_Time_Heap(A),heap_Time_return(A),Xb) ) )
       => ~ ! [X2: B,Xs: list(B)] :
              ( ( Xa2 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs) )
             => ( Y != heap_Time_bind(A,A,aa(A,heap_Time_Heap(A),aa(B,fun(A,heap_Time_Heap(A)),X,X2),Xb),vEBT_Example_mfold(B,A,X,Xs)) ) ) ) ) ).

% mfold.elims
tff(fact_244_bind__lift,axiom,
    ! [A: $tType,B: $tType,F2: heap_Time_Heap(B),G: fun(B,A)] : heap_Time_bind(B,A,F2,heap_Time_lift(B,A,G)) = heap_Time_bind(B,A,F2,aTP_Lamp_ah(fun(B,A),fun(B,heap_Time_Heap(A)),G)) ).

% bind_lift
tff(fact_245_lift__collapse,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),X: B] : aa(B,heap_Time_Heap(A),heap_Time_lift(B,A,F2),X) = aa(A,heap_Time_Heap(A),heap_Time_return(A),aa(B,A,F2,X)) ).

% lift_collapse
tff(fact_246_mset__distrib,axiom,
    ! [A: $tType,A2: multiset(A),B2: multiset(A),M2: multiset(A),N3: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A2),B2) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M2),N3) )
     => ~ ! [Am: multiset(A),An: multiset(A)] :
            ( ( A2 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Am),An) )
           => ! [Bm: multiset(A),Bn: multiset(A)] :
                ( ( B2 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Bm),Bn) )
               => ( ( M2 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Am),Bm) )
                 => ( N3 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),An),Bn) ) ) ) ) ) ).

% mset_distrib
tff(fact_247_union__assoc,axiom,
    ! [A: $tType,M2: multiset(A),N3: multiset(A),K3: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M2),N3)),K3) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),N3),K3)) ).

% union_assoc
tff(fact_248_union__lcomm,axiom,
    ! [A: $tType,M2: multiset(A),N3: multiset(A),K3: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),N3),K3)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),N3),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M2),K3)) ).

% union_lcomm
tff(fact_249_union__commute,axiom,
    ! [A: $tType,M2: multiset(A),N3: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M2),N3) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),N3),M2) ).

% union_commute
tff(fact_250_union__left__cancel,axiom,
    ! [A: $tType,K3: multiset(A),M2: multiset(A),N3: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),K3),M2) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),K3),N3) )
    <=> ( M2 = N3 ) ) ).

% union_left_cancel
tff(fact_251_union__right__cancel,axiom,
    ! [A: $tType,M2: multiset(A),K3: multiset(A),N3: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M2),K3) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),N3),K3) )
    <=> ( M2 = N3 ) ) ).

% union_right_cancel
tff(fact_252_multi__union__self__other__eq,axiom,
    ! [A: $tType,A2: multiset(A),X5: multiset(A),Y5: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A2),X5) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A2),Y5) )
     => ( X5 = Y5 ) ) ).

% multi_union_self_other_eq
tff(fact_253_order__antisym__conv,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
          <=> ( X = Y ) ) ) ) ).

% order_antisym_conv
tff(fact_254_linorder__le__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) ) ) ).

% linorder_le_cases
tff(fact_255_ord__le__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [Aa2: A,Ba: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => ( ( aa(A,B,F2,Ba) = C2 )
           => ( ! [X2: A,Y2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y2)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X2)),aa(A,B,F2,Y2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Aa2)),C2) ) ) ) ) ).

% ord_le_eq_subst
tff(fact_256_ord__eq__le__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [Aa2: A,F2: fun(B,A),Ba: B,C2: B] :
          ( ( Aa2 = aa(B,A,F2,Ba) )
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Ba),C2)
           => ( ! [X2: B,Y2: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),Y2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,X2)),aa(B,A,F2,Y2)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),aa(B,A,F2,C2)) ) ) ) ) ).

% ord_eq_le_subst
tff(fact_257_linorder__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
          | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) ) ) ).

% linorder_linear
tff(fact_258_order__eq__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( ( X = Y )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ).

% order_eq_refl
tff(fact_259_order__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [Aa2: A,Ba: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Ba)),C2)
           => ( ! [X2: A,Y2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y2)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X2)),aa(A,B,F2,Y2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Aa2)),C2) ) ) ) ) ).

% order_subst2
tff(fact_260_order__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [Aa2: A,F2: fun(B,A),Ba: B,C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),aa(B,A,F2,Ba))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Ba),C2)
           => ( ! [X2: B,Y2: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),Y2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,X2)),aa(B,A,F2,Y2)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),aa(B,A,F2,C2)) ) ) ) ) ).

% order_subst1
tff(fact_261_Orderings_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Aa2: A,Ba: A] :
          ( ( Aa2 = Ba )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2) ) ) ) ).

% Orderings.order_eq_iff
tff(fact_262_le__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F2),G)
        <=> ! [X6: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X6)),aa(A,B,G,X6)) ) ) ).

% le_fun_def
tff(fact_263_le__funI,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( ! [X2: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X2)),aa(A,B,G,X2))
         => aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F2),G) ) ) ).

% le_funI
tff(fact_264_le__funE,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B),X: A] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F2),G)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,G,X)) ) ) ).

% le_funE
tff(fact_265_le__funD,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B),X: A] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F2),G)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,G,X)) ) ) ).

% le_funD
tff(fact_266_antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2)
           => ( Aa2 = Ba ) ) ) ) ).

% antisym
tff(fact_267_dual__order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Ba: A,Aa2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),Ba)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),Aa2) ) ) ) ).

% dual_order.trans
tff(fact_268_dual__order_Oantisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
           => ( Aa2 = Ba ) ) ) ) ).

% dual_order.antisym
tff(fact_269_dual__order_Oeq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Aa2: A,Ba: A] :
          ( ( Aa2 = Ba )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba) ) ) ) ).

% dual_order.eq_iff
tff(fact_270_linorder__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,fun(A,$o)),Aa2: A,Ba: A] :
          ( ! [A3: A,B3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
             => aa(A,$o,aa(A,fun(A,$o),P,A3),B3) )
         => ( ! [A3: A,B3: A] :
                ( aa(A,$o,aa(A,fun(A,$o),P,B3),A3)
               => aa(A,$o,aa(A,fun(A,$o),P,A3),B3) )
           => aa(A,$o,aa(A,fun(A,$o),P,Aa2),Ba) ) ) ) ).

% linorder_wlog
tff(fact_271_order__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z) ) ) ) ).

% order_trans
tff(fact_272_order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),C2) ) ) ) ).

% order.trans
tff(fact_273_order__antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
           => ( X = Y ) ) ) ) ).

% order_antisym
tff(fact_274_ord__le__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => ( ( Ba = C2 )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),C2) ) ) ) ).

% ord_le_eq_trans
tff(fact_275_ord__eq__le__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( ( Aa2 = Ba )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),C2) ) ) ) ).

% ord_eq_le_trans
tff(fact_276_order__class_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( ( X = Y )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) ) ) ) ).

% order_class.order_eq_iff
tff(fact_277_le__cases3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Z: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
           => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) )
         => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z) )
           => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z)
               => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Y) )
             => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Y)
                 => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) )
               => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z)
                   => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),X) )
                 => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),X)
                     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ) ) ) ) ) ).

% le_cases3
tff(fact_278_nle__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: A,Ba: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2)
            & ( Ba != Aa2 ) ) ) ) ).

% nle_le
tff(fact_279_Abs__fnat__hom__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: nat,Ba: 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)),Aa2)),aa(nat,word(A),semiring_1_of_nat(word(A)),Ba)) = aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Aa2),Ba)) ) ).

% Abs_fnat_hom_mult
tff(fact_280_mfold_Osimps_I2_J,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,fun(A,heap_Time_Heap(A))),X: B,Xs2: list(B),S: A] : aa(A,heap_Time_Heap(A),vEBT_Example_mfold(B,A,F2,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs2)),S) = heap_Time_bind(A,A,aa(A,heap_Time_Heap(A),aa(B,fun(A,heap_Time_Heap(A)),F2,X),S),vEBT_Example_mfold(B,A,F2,Xs2)) ).

% mfold.simps(2)
tff(fact_281_mfold_Osimps_I1_J,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,fun(A,heap_Time_Heap(A))),S: A] : aa(A,heap_Time_Heap(A),vEBT_Example_mfold(B,A,F2,nil(B)),S) = aa(A,heap_Time_Heap(A),heap_Time_return(A),S) ).

% mfold.simps(1)
tff(fact_282_combine__common__factor,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [Aa2: A,E2: A,Ba: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),E2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),E2)),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)),E2)),C2) ) ).

% combine_common_factor
tff(fact_283_distrib__right,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [Aa2: A,Ba: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2)) ) ).

% distrib_right
tff(fact_284_distrib__left,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [Aa2: A,Ba: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)) ) ).

% distrib_left
tff(fact_285_comm__semiring__class_Odistrib,axiom,
    ! [A: $tType] :
      ( comm_semiring(A)
     => ! [Aa2: A,Ba: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2)) ) ).

% comm_semiring_class.distrib
tff(fact_286_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [Aa2: A,Ba: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)) ) ).

% ring_class.ring_distribs(1)
tff(fact_287_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [Aa2: A,Ba: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2)) ) ).

% ring_class.ring_distribs(2)
tff(fact_288_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,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% xor_numerals(4)
tff(fact_289_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,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% xor_numerals(6)
tff(fact_290_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,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% or_numerals(4)
tff(fact_291_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,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% or_numerals(6)
tff(fact_292_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,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% or_numerals(7)
tff(fact_293_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,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% and_numerals(7)
tff(fact_294_hash__code__list__simps_I1_J,axiom,
    ! [A: $tType,H_a: fun(A,uint32)] : hash_hash_code_list(A,H_a,nil(A)) = aa(num,uint32,numeral_numeral(uint32),bit0(bit0(bit1(bit1(bit1(bit1(bit1(bit0(bit0(bit0(bit1(bit0(bit1(bit1(bit0(bit0(bit1(bit1(bit1(bit0(bit0(bit0(bit1(bit0(bit1(bit1(bit1(bit1(bit0(bit0(one2))))))))))))))))))))))))))))))) ).

% hash_code_list_simps(1)
tff(fact_295_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),aa(A,fun(list(A),list(A)),cons(A),X),Xa2)) = aa(uint32,uint32,aa(uint32,fun(uint32,uint32),plus_plus(uint32),aa(uint32,uint32,aa(uint32,fun(uint32,uint32),times_times(uint32),aa(A,uint32,H_a,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,aa(uint32,fun(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_296_return__bind,axiom,
    ! [A: $tType,B: $tType,X: B,F2: fun(B,heap_Time_Heap(A))] : heap_Time_bind(B,A,aa(B,heap_Time_Heap(B),heap_Time_return(B),X),F2) = heap_Time_bind(product_unit,A,heap_Time_wait(one_one(nat)),aa(fun(B,heap_Time_Heap(A)),fun(product_unit,heap_Time_Heap(A)),aTP_Lamp_ai(B,fun(fun(B,heap_Time_Heap(A)),fun(product_unit,heap_Time_Heap(A))),X),F2)) ).

% return_bind
tff(fact_297_bind__return,axiom,
    ! [A: $tType,F2: heap_Time_Heap(A)] : heap_Time_bind(A,A,F2,heap_Time_return(A)) = heap_Time_bind(product_unit,A,heap_Time_wait(one_one(nat)),aTP_Lamp_aj(heap_Time_Heap(A),fun(product_unit,heap_Time_Heap(A)),F2)) ).

% bind_return
tff(fact_298_quicksort_Ocases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: list(A)] :
          ( ( X != nil(A) )
         => ~ ! [X2: A,Xs: list(A)] : X != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs) ) ) ).

% quicksort.cases
tff(fact_299_and_Oidem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Aa2),Aa2) = Aa2 ) ).

% and.idem
tff(fact_300_and_Oleft__idem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Aa2),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Aa2),Ba)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Aa2),Ba) ) ).

% and.left_idem
tff(fact_301_and_Oright__idem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A,Ba: 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),Aa2),Ba)),Ba) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Aa2),Ba) ) ).

% and.right_idem
tff(fact_302_or_Oidem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Aa2),Aa2) = Aa2 ) ).

% or.idem
tff(fact_303_or_Oleft__idem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Aa2),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Aa2),Ba)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Aa2),Ba) ) ).

% or.left_idem
tff(fact_304_or_Oright__idem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Aa2),Ba)),Ba) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Aa2),Ba) ) ).

% or.right_idem
tff(fact_305_bit_Oxor__left__self,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),Y)) = Y ) ).

% bit.xor_left_self
tff(fact_306_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_307_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_308_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_309_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_310_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_311_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_312_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_313_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_314_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_315_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_316_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_317_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_318_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_319_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_320_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_321_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_322_or_Oassoc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A,Ba: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Aa2),Ba)),C2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Aa2),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Ba),C2)) ) ).

% or.assoc
tff(fact_323_and_Oassoc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A,Ba: A,C2: 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),Aa2),Ba)),C2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Aa2),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Ba),C2)) ) ).

% and.assoc
tff(fact_324_xor_Oassoc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A,Ba: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Aa2),Ba)),C2) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Aa2),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Ba),C2)) ) ).

% xor.assoc
tff(fact_325_or_Ocommute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Aa2),Ba) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Ba),Aa2) ) ).

% or.commute
tff(fact_326_and_Ocommute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Aa2),Ba) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Ba),Aa2) ) ).

% and.commute
tff(fact_327_xor_Ocommute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Aa2),Ba) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Ba),Aa2) ) ).

% xor.commute
tff(fact_328_or_Oleft__commute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Ba: A,Aa2: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Ba),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Aa2),C2)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Aa2),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Ba),C2)) ) ).

% or.left_commute
tff(fact_329_bit_Oconj__xor__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),Y)),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),Z)) ) ).

% bit.conj_xor_distrib
tff(fact_330_and_Oleft__commute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Ba: A,Aa2: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Ba),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Aa2),C2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Aa2),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Ba),C2)) ) ).

% and.left_commute
tff(fact_331_xor_Oleft__commute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Ba: A,Aa2: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Ba),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Aa2),C2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Aa2),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Ba),C2)) ) ).

% xor.left_commute
tff(fact_332_bit_Oconj__disj__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),Y)),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),Z)) ) ).

% bit.conj_disj_distrib
tff(fact_333_bit_Oconj__xor__distrib2,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Y: A,Z: A,X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Y),Z)),X) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Y),X)),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Z),X)) ) ).

% bit.conj_xor_distrib2
tff(fact_334_bit_Odisj__conj__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),Y)),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),Z)) ) ).

% bit.disj_conj_distrib
tff(fact_335_bit_Oconj__disj__distrib2,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Y: A,Z: A,X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Y),Z)),X) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Y),X)),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Z),X)) ) ).

% bit.conj_disj_distrib2
tff(fact_336_bit_Odisj__conj__distrib2,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Y: A,Z: A,X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Y),Z)),X) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Y),X)),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Z),X)) ) ).

% bit.disj_conj_distrib2
tff(fact_337_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_338_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_339_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_340_plus__and__or,axiom,
    ! [X: int,Y: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),X),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),X),Y) ).

% plus_and_or
tff(fact_341_or__nat__def,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),Na) = aa(int,nat,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_342_xor__nat__def,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),Na) = aa(int,nat,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_343_and__nat__def,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),Na) = aa(int,nat,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_344_crossproduct__eq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [W: A,Y: A,X: A,Z: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),W),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),X),Y)) )
        <=> ( ( W = X )
            | ( Y = Z ) ) ) ) ).

% crossproduct_eq
tff(fact_345_crossproduct__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [Aa2: A,Ba: A,C2: A,D2: A] :
          ( ( ( Aa2 != Ba )
            & ( C2 != D2 ) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),D2)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),D2)),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2)) ) ) ) ).

% crossproduct_noteq
tff(fact_346_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,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),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_347_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,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),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_348_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,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),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_349_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_350_vebt__memberi__rule,axiom,
    ! [Na: nat,S: set(nat),Tia: vEBT_VEBTi,X: nat] : time_htt($o,aa(vEBT_VEBTi,assn,vEBT_Intf_vebt_assn(Na,S),Tia),vEBT_vebt_memberi(Tia,X),aa(nat,fun($o,assn),aa(vEBT_VEBTi,fun(nat,fun($o,assn)),aa(set(nat),fun(vEBT_VEBTi,fun(nat,fun($o,assn))),aTP_Lamp_ak(nat,fun(set(nat),fun(vEBT_VEBTi,fun(nat,fun($o,assn)))),Na),S),Tia),X),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(one2)))),aa(int,nat,nat2,archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na))))))) ).

% vebt_memberi_rule
tff(fact_351_product__lists_Osimps_I1_J,axiom,
    ! [A: $tType] : product_lists(A,nil(list(A))) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),nil(list(A))) ).

% product_lists.simps(1)
tff(fact_352_rotate1__length01,axiom,
    ! [A: $tType,Xs2: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs2)),one_one(nat))
     => ( rotate1(A,Xs2) = Xs2 ) ) ).

% rotate1_length01
tff(fact_353_le__log2__of__power,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),M)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))) ) ).

% le_log2_of_power
tff(fact_354_neg__equal__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [Aa2: A,Ba: A] :
          ( ( aa(A,A,uminus_uminus(A),Aa2) = aa(A,A,uminus_uminus(A),Ba) )
        <=> ( Aa2 = Ba ) ) ) ).

% neg_equal_iff_equal
tff(fact_355_add_Oinverse__inverse,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [Aa2: A] : aa(A,A,uminus_uminus(A),aa(A,A,uminus_uminus(A),Aa2)) = Aa2 ) ).

% add.inverse_inverse
tff(fact_356_verit__minus__simplify_I4_J,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [Ba: A] : aa(A,A,uminus_uminus(A),aa(A,A,uminus_uminus(A),Ba)) = Ba ) ).

% verit_minus_simplify(4)
tff(fact_357_bit_Ocompl__eq__compl__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,bit_ri4277139882892585799ns_not(A),X) = aa(A,A,bit_ri4277139882892585799ns_not(A),Y) )
        <=> ( X = Y ) ) ) ).

% bit.compl_eq_compl_iff
tff(fact_358_bit_Odouble__compl,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)) = X ) ).

% bit.double_compl
tff(fact_359_neg__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Ba)),aa(A,A,uminus_uminus(A),Aa2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba) ) ) ).

% neg_le_iff_le
tff(fact_360_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_361_minus__add__distrib,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),Aa2)),aa(A,A,uminus_uminus(A),Ba)) ) ).

% minus_add_distrib
tff(fact_362_minus__add__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),Aa2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)) = Ba ) ).

% minus_add_cancel
tff(fact_363_add__minus__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),Aa2)),Ba)) = Ba ) ).

% add_minus_cancel
tff(fact_364_mult__minus__right,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(A,A,uminus_uminus(A),Ba)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)) ) ).

% mult_minus_right
tff(fact_365_minus__mult__minus,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),Aa2)),aa(A,A,uminus_uminus(A),Ba)) = aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba) ) ).

% minus_mult_minus
tff(fact_366_mult__minus__left,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),Aa2)),Ba) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)) ) ).

% mult_minus_left
tff(fact_367_negative__zle,axiom,
    ! [Na: nat,M: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),Na))),aa(nat,int,semiring_1_of_nat(int),M)) ).

% negative_zle
tff(fact_368_bit_Oxor__compl__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),aa(A,A,bit_ri4277139882892585799ns_not(A),Y)) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),Y)) ) ).

% bit.xor_compl_right
tff(fact_369_bit_Oxor__compl__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),Y) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),Y)) ) ).

% bit.xor_compl_left
tff(fact_370_rotate1__is__Nil__conv,axiom,
    ! [A: $tType,Xs2: list(A)] :
      ( ( rotate1(A,Xs2) = nil(A) )
    <=> ( Xs2 = nil(A) ) ) ).

% rotate1_is_Nil_conv
tff(fact_371_length__rotate1,axiom,
    ! [A: $tType,Xs2: list(A)] : aa(list(A),nat,size_size(list(A)),rotate1(A,Xs2)) = aa(list(A),nat,size_size(list(A)),Xs2) ).

% length_rotate1
tff(fact_372_add__neg__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,Na: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Na))) ) ).

% add_neg_numeral_simps(3)
tff(fact_373_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_374_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_375_and_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,uminus_uminus(A),one_one(A))),Aa2) = Aa2 ) ).

% and.left_neutral
tff(fact_376_and_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Aa2),aa(A,A,uminus_uminus(A),one_one(A))) = Aa2 ) ).

% and.right_neutral
tff(fact_377_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_378_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_379_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_380_bit_Ode__Morgan__disj,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),Y)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),aa(A,A,bit_ri4277139882892585799ns_not(A),Y)) ) ).

% bit.de_Morgan_disj
tff(fact_381_bit_Ode__Morgan__conj,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),Y)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),aa(A,A,bit_ri4277139882892585799ns_not(A),Y)) ) ).

% bit.de_Morgan_conj
tff(fact_382_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_383_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_384_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_385_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_386_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_387_ceiling__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num] : archimedean_ceiling(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)) ) ).

% ceiling_neg_numeral
tff(fact_388_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_389_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_390_semiring__norm_I168_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [V: num,W: num,Y: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),V),W)))),Y) ) ).

% semiring_norm(168)
tff(fact_391_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_392_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_393_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_394_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_395_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_396_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_397_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_398_semiring__norm_I170_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [V: 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),V))),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),V),W)))),Y) ) ).

% semiring_norm(170)
tff(fact_399_semiring__norm_I171_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [V: num,W: num,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),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),V),W)))),Y) ) ).

% semiring_norm(171)
tff(fact_400_semiring__norm_I172_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [V: 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),V))),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),V),W))),Y) ) ).

% semiring_norm(172)
tff(fact_401_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_402_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_403_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_404_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_405_neg__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Na: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na)))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Na),M) ) ) ).

% neg_numeral_le_iff
tff(fact_406_not__neg__one__le__neg__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)))
        <=> ( M != one2 ) ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_407_add__neg__numeral__special_I9_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% add_neg_numeral_special(9)
tff(fact_408_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_409_ceiling__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) ) ) ).

% ceiling_le_neg_numeral
tff(fact_410_or__int__def,axiom,
    ! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,bit_ri4277139882892585799ns_not(int),K)),aa(int,int,bit_ri4277139882892585799ns_not(int),L))) ).

% or_int_def
tff(fact_411_xor__int__def,axiom,
    ! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),aa(int,int,bit_ri4277139882892585799ns_not(int),L))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,bit_ri4277139882892585799ns_not(int),K)),L)) ).

% xor_int_def
tff(fact_412_minus__equation__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [Aa2: A,Ba: A] :
          ( ( aa(A,A,uminus_uminus(A),Aa2) = Ba )
        <=> ( aa(A,A,uminus_uminus(A),Ba) = Aa2 ) ) ) ).

% minus_equation_iff
tff(fact_413_equation__minus__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [Aa2: A,Ba: A] :
          ( ( Aa2 = aa(A,A,uminus_uminus(A),Ba) )
        <=> ( Ba = aa(A,A,uminus_uminus(A),Aa2) ) ) ) ).

% equation_minus_iff
tff(fact_414_verit__negate__coefficient_I3_J,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Aa2: A,Ba: A] :
          ( ( Aa2 = Ba )
         => ( aa(A,A,uminus_uminus(A),Aa2) = aa(A,A,uminus_uminus(A),Ba) ) ) ) ).

% verit_negate_coefficient(3)
tff(fact_415_minus__eq__not__plus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Aa2: A] : aa(A,A,uminus_uminus(A),Aa2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Aa2)),one_one(A)) ) ).

% minus_eq_not_plus_1
tff(fact_416_le__imp__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Ba)),aa(A,A,uminus_uminus(A),Aa2)) ) ) ).

% le_imp_neg_le
tff(fact_417_minus__le__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Aa2)),Ba)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Ba)),Aa2) ) ) ).

% minus_le_iff
tff(fact_418_le__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),aa(A,A,uminus_uminus(A),Ba))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),aa(A,A,uminus_uminus(A),Aa2)) ) ) ).

% le_minus_iff
tff(fact_419_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_420_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_421_add_Oinverse__distrib__swap,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),Ba)),aa(A,A,uminus_uminus(A),Aa2)) ) ).

% add.inverse_distrib_swap
tff(fact_422_group__cancel_Oneg1,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,K: A,Aa2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),Aa2) )
         => ( aa(A,A,uminus_uminus(A),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),K)),aa(A,A,uminus_uminus(A),Aa2)) ) ) ) ).

% group_cancel.neg1
tff(fact_423_is__num__normalize_I8_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),Ba)),aa(A,A,uminus_uminus(A),Aa2)) ) ).

% is_num_normalize(8)
tff(fact_424_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_425_minus__mult__commute,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),Aa2)),Ba) = aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(A,A,uminus_uminus(A),Ba)) ) ).

% minus_mult_commute
tff(fact_426_square__eq__iff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Aa2: A,Ba: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Aa2) = aa(A,A,aa(A,fun(A,A),times_times(A),Ba),Ba) )
        <=> ( ( Aa2 = Ba )
            | ( Aa2 = aa(A,A,uminus_uminus(A),Ba) ) ) ) ) ).

% square_eq_iff
tff(fact_427_real__minus__mult__self__le,axiom,
    ! [U: real,X: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),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_428_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_429_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_430_rotate1_Osimps_I1_J,axiom,
    ! [A: $tType] : rotate1(A,nil(A)) = nil(A) ).

% rotate1.simps(1)
tff(fact_431_htt__elim,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),Ba: nat] :
      ( time_htt(A,P,C2,Q,Ba)
     => simple8792101892335181110TBOUND(A,P,C2,Ba) ) ).

% htt_elim
tff(fact_432_not__numeral__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Na: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))) ) ).

% not_numeral_le_neg_numeral
tff(fact_433_neg__numeral__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Na: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),Na)) ) ).

% neg_numeral_le_numeral
tff(fact_434_le__minus__one__simps_I4_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% le_minus_one_simps(4)
tff(fact_435_le__minus__one__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A)) ) ).

% le_minus_one_simps(2)
tff(fact_436_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_437_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_438_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_439_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_440_or__eq__not__not__and,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Aa2),Ba) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Aa2)),aa(A,A,bit_ri4277139882892585799ns_not(A),Ba))) ) ).

% or_eq_not_not_and
tff(fact_441_and__eq__not__not__or,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Aa2),Ba) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Aa2)),aa(A,A,bit_ri4277139882892585799ns_not(A),Ba))) ) ).

% and_eq_not_not_or
tff(fact_442_and__eq__minus__1__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Aa2: A,Ba: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Aa2),Ba) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> ( ( Aa2 = aa(A,A,uminus_uminus(A),one_one(A)) )
            & ( Ba = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).

% and_eq_minus_1_iff
tff(fact_443_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_444_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_445_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_446_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_447_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_448_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_449_not__one__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) ) ).

% not_one_le_neg_numeral
tff(fact_450_not__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% not_numeral_le_neg_one
tff(fact_451_neg__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% neg_numeral_le_neg_one
tff(fact_452_neg__one__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),M)) ) ).

% neg_one_le_numeral
tff(fact_453_neg__numeral__le__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A)) ) ).

% neg_numeral_le_one
tff(fact_454_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_455_mult__1s__ring__1_I1_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Ba: 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))),Ba) = aa(A,A,uminus_uminus(A),Ba) ) ).

% mult_1s_ring_1(1)
tff(fact_456_mult__1s__ring__1_I2_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Ba: A] : aa(A,A,aa(A,fun(A,A),times_times(A),Ba),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),one2))) = aa(A,A,uminus_uminus(A),Ba) ) ).

% mult_1s_ring_1(2)
tff(fact_457_bit_Oxor__def2,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),Y) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),Y)),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),aa(A,A,bit_ri4277139882892585799ns_not(A),Y))) ) ).

% bit.xor_def2
tff(fact_458_bit_Oxor__def,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),Y) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),aa(A,A,bit_ri4277139882892585799ns_not(A),Y))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),Y)) ) ).

% bit.xor_def
tff(fact_459_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_460_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_461_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_462_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_463_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_464_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_465_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_466_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,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),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_467_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,I: num,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),I)),Na))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),I)),Na)) ) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_468_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: num,Na: nat,X: nat] :
          ( aa(A,$o,aa(A,fun(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,aa(nat,fun(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_469_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_470_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Aa2: A,Na: nat] : aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),Aa2)),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,Aa2),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_471_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,Ba: nat,W: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),Ba)),W))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),aa(nat,nat,power_power(nat,Ba),W)) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_472_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Ba: nat,W: nat,X: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),Ba)),W)),aa(nat,A,semiring_1_of_nat(A),X))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,Ba),W)),X) ) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_473_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_474_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_475_power2__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Aa2: A] : aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),Aa2)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,power_power(A,Aa2),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% power2_minus
tff(fact_476_power__add__numeral2,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Aa2: A,M: num,Na: num,Ba: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,Aa2),aa(num,nat,numeral_numeral(nat),M))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,Aa2),aa(num,nat,numeral_numeral(nat),Na))),Ba)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,Aa2),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),Na)))),Ba) ) ).

% power_add_numeral2
tff(fact_477_power__add__numeral,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Aa2: A,M: num,Na: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,Aa2),aa(num,nat,numeral_numeral(nat),M))),aa(nat,A,power_power(A,Aa2),aa(num,nat,numeral_numeral(nat),Na))) = aa(nat,A,power_power(A,Aa2),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),Na))) ) ).

% power_add_numeral
tff(fact_478_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_479_power__one__right,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Aa2: A] : aa(nat,A,power_power(A,Aa2),one_one(nat)) = Aa2 ) ).

% power_one_right
tff(fact_480_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [X: nat,Ba: 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),Ba)),W) )
        <=> ( X = aa(nat,nat,power_power(nat,Ba),W) ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_481_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Ba: nat,W: nat,X: nat] :
          ( ( aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),Ba)),W) = aa(nat,A,semiring_1_of_nat(A),X) )
        <=> ( aa(nat,nat,power_power(nat,Ba),W) = X ) ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_482_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_483_power__mult__numeral,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Aa2: A,M: num,Na: num] : aa(nat,A,power_power(A,aa(nat,A,power_power(A,Aa2),aa(num,nat,numeral_numeral(nat),M))),aa(num,nat,numeral_numeral(nat),Na)) = aa(nat,A,power_power(A,Aa2),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),times_times(num),M),Na))) ) ).

% power_mult_numeral
tff(fact_484_left__minus__one__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Na: nat,Aa2: 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)),Aa2)) = Aa2 ) ).

% left_minus_one_mult_self
tff(fact_485_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_486_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,Na: nat] :
      ( ( aa(int,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_487_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) = aa(int,nat,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_488_nat__le__numeral__power__cancel__iff,axiom,
    ! [Aa2: int,X: num,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,Aa2)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),X)),Na))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Aa2),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),Na)) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_489_numeral__power__le__nat__cancel__iff,axiom,
    ! [X: num,Na: nat,Aa2: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),X)),Na)),aa(int,nat,nat2,Aa2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),Na)),Aa2) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_490_Bernoulli__inequality,axiom,
    ! [X: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),X))),aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X)),Na)) ) ).

% Bernoulli_inequality
tff(fact_491_realpow__square__minus__le,axiom,
    ! [U: real,X: real] : aa(real,$o,aa(real,fun(real,$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_492_two__realpow__ge__one,axiom,
    ! [Na: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),bit0(one2))),Na)) ).

% two_realpow_ge_one
tff(fact_493_power__commutes,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Aa2: A,Na: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,Aa2),Na)),Aa2) = aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(nat,A,power_power(A,Aa2),Na)) ) ).

% power_commutes
tff(fact_494_power__mult__distrib,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Aa2: A,Ba: A,Na: nat] : aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)),Na) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,Aa2),Na)),aa(nat,A,power_power(A,Ba),Na)) ) ).

% power_mult_distrib
tff(fact_495_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_496_power__mult,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Aa2: A,M: nat,Na: nat] : aa(nat,A,power_power(A,Aa2),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,Aa2),M)),Na) ) ).

% power_mult
tff(fact_497_one__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Aa2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Aa2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(nat,A,power_power(A,Aa2),Na)) ) ) ).

% one_le_power
tff(fact_498_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_499_power__add,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Aa2: A,M: nat,Na: nat] : aa(nat,A,power_power(A,Aa2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,Aa2),M)),aa(nat,A,power_power(A,Aa2),Na)) ) ).

% power_add
tff(fact_500_power__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Aa2: A,Na: nat] : aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),Aa2)),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,Aa2),Na)) ) ).

% power_minus
tff(fact_501_power__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat,N3: nat,Aa2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Aa2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,Aa2),Na)),aa(nat,A,power_power(A,Aa2),N3)) ) ) ) ).

% power_increasing
tff(fact_502_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_503_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_504_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_505_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_506_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_507_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_508_power2__eq__square,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Aa2: A] : aa(nat,A,power_power(A,Aa2),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Aa2) ) ).

% power2_eq_square
tff(fact_509_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_510_power3__eq__cube,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Aa2: A] : aa(nat,A,power_power(A,Aa2),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),Aa2),Aa2)),Aa2) ) ).

% power3_eq_cube
tff(fact_511_power__even__eq,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Aa2: A,Na: nat] : aa(nat,A,power_power(A,Aa2),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,Aa2),Na)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% power_even_eq
tff(fact_512_self__le__ge2__pow,axiom,
    ! [K: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,power_power(nat,K),M)) ) ).

% self_le_ge2_pow
tff(fact_513_power2__nat__le__eq__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,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,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% power2_nat_le_eq_le
tff(fact_514_power2__nat__le__imp__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% power2_nat_le_imp_le
tff(fact_515_power2__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [Aa2: A] :
          ( ( aa(nat,A,power_power(A,Aa2),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(A) )
        <=> ( ( Aa2 = one_one(A) )
            | ( Aa2 = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).

% power2_eq_1_iff
tff(fact_516_power2__sum,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [X: A,Y: A] : aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,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_517_square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),X)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),one_one(A))
           => aa(A,$o,aa(A,fun(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_518_minus__power__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Aa2: 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),Aa2)),Na)),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),Aa2)),Na)) = aa(nat,A,power_power(A,Aa2),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_519_sum__squares__bound,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),X)),Y)),aa(A,A,aa(A,fun(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_520_L2__set__mult__ineq__lemma,axiom,
    ! [Aa2: real,C2: real,Ba: real,D2: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,aa(real,fun(real,real),times_times(real),Aa2),C2))),aa(real,real,aa(real,fun(real,real),times_times(real),Ba),D2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,power_power(real,Aa2),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,Ba),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_521_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_522_VEBT__internal_Otwo__realpow__ge__two,axiom,
    ! [Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),Na))
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),bit0(one2))),Na)) ) ).

% VEBT_internal.two_realpow_ge_two
tff(fact_523_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_524_compl__le__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) ) ) ).

% compl_le_compl_iff
tff(fact_525_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_526_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_527_ceiling__log__nat__eq__if,axiom,
    ! [Ba: nat,Na: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,power_power(nat,Ba),Na)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(nat,nat,power_power(nat,Ba),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat))))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Ba)
         => ( archimedean_ceiling(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),Ba)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),Na)),one_one(int)) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_528_VEBT__internal_Obit__concat__def,axiom,
    ! [H: nat,L: nat,D2: nat] : vEBT_VEBT_bit_concat(H,L,D2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),D2))),L) ).

% VEBT_internal.bit_concat_def
tff(fact_529_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_530_numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: num,Na: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Na))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),Na) ) ) ).

% numeral_less_iff
tff(fact_531_add__less__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Aa2: A,C2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),C2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba) ) ) ).

% add_less_cancel_right
tff(fact_532_add__less__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),Aa2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),Ba))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba) ) ) ).

% add_less_cancel_left
tff(fact_533_neg__less__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),Ba)),aa(A,A,uminus_uminus(A),Aa2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba) ) ) ).

% neg_less_iff_less
tff(fact_534_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ).

% nat_add_left_cancel_less
tff(fact_535_word__order_Oextremum,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Aa2),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) ) ).

% word_order.extremum
tff(fact_536_word__order_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),Aa2)
        <=> ( Aa2 = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% word_order.extremum_unique
tff(fact_537_word__n1__ge,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) ) ).

% word_n1_ge
tff(fact_538_max__word__max,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Na),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) ) ).

% max_word_max
tff(fact_539_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_540_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_541_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_542_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_543_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_544_neg__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Na: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na)))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),Na),M) ) ) ).

% neg_numeral_less_iff
tff(fact_545_power__inject__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Aa2: A,M: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Aa2)
         => ( ( aa(nat,A,power_power(A,Aa2),M) = aa(nat,A,power_power(A,Aa2),Na) )
          <=> ( M = Na ) ) ) ) ).

% power_inject_exp
tff(fact_546_of__nat__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Na))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ).

% of_nat_less_iff
tff(fact_547_word__add__not,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),word(A),aa(word(A),fun(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_548_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_549_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_550_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_551_one__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(num,A,numeral_numeral(A),Na))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),one2),Na) ) ) ).

% one_less_numeral_iff
tff(fact_552_power__strict__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Ba: A,X: nat,Y: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,Ba),X)),aa(nat,A,power_power(A,Ba),Y))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Y) ) ) ) ).

% power_strict_increasing_iff
tff(fact_553_numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(num,int,numeral_numeral(int),V)),archimedean_ceiling(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),V)),X) ) ) ).

% numeral_less_ceiling
tff(fact_554_one__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),archimedean_ceiling(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),X) ) ) ).

% one_less_ceiling
tff(fact_555_neg__numeral__less__neg__one__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A)))
        <=> ( M != one2 ) ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_556_power__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Ba: A,X: nat,Y: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,Ba),X)),aa(nat,A,power_power(A,Ba),Y))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Y) ) ) ) ).

% power_increasing_iff
tff(fact_557_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_558_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_559_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Ba: nat,W: nat,X: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),Ba)),W)),aa(nat,A,semiring_1_of_nat(A),X))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,power_power(nat,Ba),W)),X) ) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_560_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,Ba: nat,W: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),Ba)),W))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),aa(nat,nat,power_power(nat,Ba),W)) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_561_neg__numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archimedean_ceiling(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),X) ) ) ).

% neg_numeral_less_ceiling
tff(fact_562_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,I: num,Na: nat] :
          ( aa(A,$o,aa(A,fun(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,aa(nat,fun(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_563_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: num,Na: nat,X: nat] :
          ( aa(A,$o,aa(A,fun(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,aa(nat,fun(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_564_word__order_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),Aa2)
         => ( Aa2 = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% word_order.extremum_uniqueI
tff(fact_565_plus__1__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),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_566_overflow__plus__one__self,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),P2)),P2)
        <=> ( P2 = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% overflow_plus_one_self
tff(fact_567_word__plus__and__or__coroll2,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)),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_568_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_569_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_570_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_571_infinite__descent__measure,axiom,
    ! [A: $tType,P: fun(A,$o),V2: fun(A,nat),X: A] :
      ( ! [X2: A] :
          ( ~ aa(A,$o,P,X2)
         => ? [Y3: A] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,V2,Y3)),aa(A,nat,V2,X2))
              & ~ aa(A,$o,P,Y3) ) )
     => aa(A,$o,P,X) ) ).

% infinite_descent_measure
tff(fact_572_measure__induct__rule,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F2: fun(A,B),P: fun(A,$o),Aa2: A] :
          ( ! [X2: A] :
              ( ! [Y3: A] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Y3)),aa(A,B,F2,X2))
                 => aa(A,$o,P,Y3) )
             => aa(A,$o,P,X2) )
         => aa(A,$o,P,Aa2) ) ) ).

% measure_induct_rule
tff(fact_573_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Y)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),X) ) ) ).

% linorder_neqE_nat
tff(fact_574_infinite__descent,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( ! [N: nat] :
          ( ~ aa(nat,$o,P,N)
         => ? [M3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),N)
              & ~ aa(nat,$o,P,M3) ) )
     => aa(nat,$o,P,Na) ) ).

% infinite_descent
tff(fact_575_nat__less__induct,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( ! [N: nat] :
          ( ! [M3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),N)
             => aa(nat,$o,P,M3) )
         => aa(nat,$o,P,N) )
     => aa(nat,$o,P,Na) ) ).

% nat_less_induct
tff(fact_576_less__irrefl__nat,axiom,
    ! [Na: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),Na) ).

% less_irrefl_nat
tff(fact_577_measure__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F2: fun(A,B),P: fun(A,$o),Aa2: A] :
          ( ! [X2: A] :
              ( ! [Y3: A] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Y3)),aa(A,B,F2,X2))
                 => aa(A,$o,P,Y3) )
             => aa(A,$o,P,X2) )
         => aa(A,$o,P,Aa2) ) ) ).

% measure_induct
tff(fact_578_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),S),T)
     => ( S != T ) ) ).

% less_not_refl3
tff(fact_579_less__not__refl2,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
     => ( M != Na ) ) ).

% less_not_refl2
tff(fact_580_less__not__refl,axiom,
    ! [Na: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),Na) ).

% less_not_refl
tff(fact_581_nat__neq__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( M != Na )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M) ) ) ).

% nat_neq_iff
tff(fact_582_verit__comp__simplify1_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Aa2: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Aa2) ) ).

% verit_comp_simplify1(1)
tff(fact_583_order__less__imp__not__less,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ).

% order_less_imp_not_less
tff(fact_584_order__less__imp__not__eq2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ( Y != X ) ) ) ).

% order_less_imp_not_eq2
tff(fact_585_order__less__imp__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ( X != Y ) ) ) ).

% order_less_imp_not_eq
tff(fact_586_linorder__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
          | ( X = Y )
          | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ).

% linorder_less_linear
tff(fact_587_order__less__imp__triv,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,P: $o] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X)
           => (P) ) ) ) ).

% order_less_imp_triv
tff(fact_588_order__less__not__sym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ).

% order_less_not_sym
tff(fact_589_order__less__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [Aa2: A,Ba: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Ba)),C2)
           => ( ! [X2: A,Y2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Y2)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X2)),aa(A,B,F2,Y2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Aa2)),C2) ) ) ) ) ).

% order_less_subst2
tff(fact_590_order__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [Aa2: A,F2: fun(B,A),Ba: B,C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),aa(B,A,F2,Ba))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Ba),C2)
           => ( ! [X2: B,Y2: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X2),Y2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,X2)),aa(B,A,F2,Y2)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),aa(B,A,F2,C2)) ) ) ) ) ).

% order_less_subst1
tff(fact_591_order__less__irrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),X) ) ).

% order_less_irrefl
tff(fact_592_ord__less__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [Aa2: A,Ba: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
         => ( ( aa(A,B,F2,Ba) = C2 )
           => ( ! [X2: A,Y2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Y2)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X2)),aa(A,B,F2,Y2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Aa2)),C2) ) ) ) ) ).

% ord_less_eq_subst
tff(fact_593_ord__eq__less__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [Aa2: A,F2: fun(B,A),Ba: B,C2: B] :
          ( ( Aa2 = aa(B,A,F2,Ba) )
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Ba),C2)
           => ( ! [X2: B,Y2: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X2),Y2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,X2)),aa(B,A,F2,Y2)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),aa(B,A,F2,C2)) ) ) ) ) ).

% ord_eq_less_subst
tff(fact_594_order__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z) ) ) ) ).

% order_less_trans
tff(fact_595_order__less__asym_H,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),Aa2) ) ) ).

% order_less_asym'
tff(fact_596_linorder__neq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ( X != Y )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ) ).

% linorder_neq_iff
tff(fact_597_order__less__asym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ).

% order_less_asym
tff(fact_598_linorder__neqE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ( X != Y )
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ) ).

% linorder_neqE
tff(fact_599_linorder__neqE__linordered__idom,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( ( X != Y )
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_600_dual__order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),Aa2)
         => ( Aa2 != Ba ) ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_601_order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
         => ( Aa2 != Ba ) ) ) ).

% order.strict_implies_not_eq
tff(fact_602_dual__order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Ba: A,Aa2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),Ba)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),Aa2) ) ) ) ).

% dual_order.strict_trans
tff(fact_603_not__less__iff__gr__or__eq,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X)
            | ( X = Y ) ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_604_order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),C2) ) ) ) ).

% order.strict_trans
tff(fact_605_linorder__less__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,fun(A,$o)),Aa2: A,Ba: A] :
          ( ! [A3: A,B3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
             => aa(A,$o,aa(A,fun(A,$o),P,A3),B3) )
         => ( ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),P,A3),A3)
           => ( ! [A3: A,B3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),P,B3),A3)
                 => aa(A,$o,aa(A,fun(A,$o),P,A3),B3) )
             => aa(A,$o,aa(A,fun(A,$o),P,Aa2),Ba) ) ) ) ) ).

% linorder_less_wlog
tff(fact_606_exists__least__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o)] :
          ( ? [X_1: A] : aa(A,$o,P,X_1)
        <=> ? [N2: A] :
              ( aa(A,$o,P,N2)
              & ! [M4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M4),N2)
                 => ~ aa(A,$o,P,M4) ) ) ) ) ).

% exists_least_iff
tff(fact_607_dual__order_Oirrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Aa2: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Aa2) ) ).

% dual_order.irrefl
tff(fact_608_dual__order_Oasym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),Aa2)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba) ) ) ).

% dual_order.asym
tff(fact_609_linorder__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ( ( X != Y )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ) ).

% linorder_cases
tff(fact_610_antisym__conv3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,X: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
          <=> ( X = Y ) ) ) ) ).

% antisym_conv3
tff(fact_611_less__induct,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),Aa2: A] :
          ( ! [X2: A] :
              ( ! [Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),X2)
                 => aa(A,$o,P,Y3) )
             => aa(A,$o,P,X2) )
         => aa(A,$o,P,Aa2) ) ) ).

% less_induct
tff(fact_612_ord__less__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
         => ( ( Ba = C2 )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),C2) ) ) ) ).

% ord_less_eq_trans
tff(fact_613_ord__eq__less__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( ( Aa2 = Ba )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),C2) ) ) ) ).

% ord_eq_less_trans
tff(fact_614_order_Oasym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),Aa2) ) ) ).

% order.asym
tff(fact_615_less__imp__neq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ( X != Y ) ) ) ).

% less_imp_neq
tff(fact_616_dense,axiom,
    ! [A: $tType] :
      ( dense_order(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ? [Z3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),Y) ) ) ) ).

% dense
tff(fact_617_gt__ex,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [X: A] :
        ? [X_12: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),X_12) ) ).

% gt_ex
tff(fact_618_lt__ex,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [X: A] :
        ? [Y2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y2),X) ) ).

% lt_ex
tff(fact_619_less__imp__of__nat__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Na)) ) ) ).

% less_imp_of_nat_less
tff(fact_620_of__nat__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Na))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ).

% of_nat_less_imp_less
tff(fact_621_ceiling__less__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,X)),archimedean_ceiling(A,Y))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y) ) ) ).

% ceiling_less_cancel
tff(fact_622_power__less__imp__less__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Aa2: A,M: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,Aa2),M)),aa(nat,A,power_power(A,Aa2),Na))
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ) ).

% power_less_imp_less_exp
tff(fact_623_power__strict__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat,N3: nat,Aa2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),N3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Aa2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,Aa2),Na)),aa(nat,A,power_power(A,Aa2),N3)) ) ) ) ).

% power_strict_increasing
tff(fact_624_order__le__imp__less__or__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
            | ( X = Y ) ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_625_linorder__le__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
          | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ).

% linorder_le_less_linear
tff(fact_626_order__less__le__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [Aa2: A,Ba: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Ba)),C2)
           => ( ! [X2: A,Y2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Y2)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X2)),aa(A,B,F2,Y2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Aa2)),C2) ) ) ) ) ).

% order_less_le_subst2
tff(fact_627_order__less__le__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [Aa2: A,F2: fun(B,A),Ba: B,C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),aa(B,A,F2,Ba))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Ba),C2)
           => ( ! [X2: B,Y2: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),Y2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,X2)),aa(B,A,F2,Y2)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),aa(B,A,F2,C2)) ) ) ) ) ).

% order_less_le_subst1
tff(fact_628_order__le__less__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [Aa2: A,Ba: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Ba)),C2)
           => ( ! [X2: A,Y2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y2)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X2)),aa(A,B,F2,Y2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Aa2)),C2) ) ) ) ) ).

% order_le_less_subst2
tff(fact_629_order__le__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [Aa2: A,F2: fun(B,A),Ba: B,C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),aa(B,A,F2,Ba))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Ba),C2)
           => ( ! [X2: B,Y2: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X2),Y2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,X2)),aa(B,A,F2,Y2)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),aa(B,A,F2,C2)) ) ) ) ) ).

% order_le_less_subst1
tff(fact_630_order__less__le__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z) ) ) ) ).

% order_less_le_trans
tff(fact_631_order__le__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z) ) ) ) ).

% order_le_less_trans
tff(fact_632_order__neq__le__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Aa2: A,Ba: A] :
          ( ( Aa2 != Ba )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba) ) ) ) ).

% order_neq_le_trans
tff(fact_633_order__le__neq__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => ( ( Aa2 != Ba )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba) ) ) ) ).

% order_le_neq_trans
tff(fact_634_order__less__imp__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ).

% order_less_imp_le
tff(fact_635_linorder__not__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) ) ) ).

% linorder_not_less
tff(fact_636_linorder__not__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ).

% linorder_not_le
tff(fact_637_order__less__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
            & ( X != Y ) ) ) ) ).

% order_less_le
tff(fact_638_order__le__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
            | ( X = Y ) ) ) ) ).

% order_le_less
tff(fact_639_dual__order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),Aa2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2) ) ) ).

% dual_order.strict_implies_order
tff(fact_640_order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba) ) ) ).

% order.strict_implies_order
tff(fact_641_dual__order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),Aa2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2)
            & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba) ) ) ) ).

% dual_order.strict_iff_not
tff(fact_642_dual__order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Ba: A,Aa2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),Ba)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),Aa2) ) ) ) ).

% dual_order.strict_trans2
tff(fact_643_dual__order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Ba: A,Aa2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),Ba)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),Aa2) ) ) ) ).

% dual_order.strict_trans1
tff(fact_644_dual__order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),Aa2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2)
            & ( Aa2 != Ba ) ) ) ) ).

% dual_order.strict_iff_order
tff(fact_645_dual__order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),Aa2)
            | ( Aa2 = Ba ) ) ) ) ).

% dual_order.order_iff_strict
tff(fact_646_dense__le__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [X: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ( ! [W3: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),W3)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),W3),Y)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),W3),Z) ) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ) ).

% dense_le_bounded
tff(fact_647_dense__ge__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z: A,X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),X)
         => ( ! [W3: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),W3)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),W3),X)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),W3) ) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ) ).

% dense_ge_bounded
tff(fact_648_order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
            & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2) ) ) ) ).

% order.strict_iff_not
tff(fact_649_order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),C2) ) ) ) ).

% order.strict_trans2
tff(fact_650_order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),C2) ) ) ) ).

% order.strict_trans1
tff(fact_651_order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
            & ( Aa2 != Ba ) ) ) ) ).

% order.strict_iff_order
tff(fact_652_order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
            | ( Aa2 = Ba ) ) ) ) ).

% order.order_iff_strict
tff(fact_653_not__le__imp__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,X: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y) ) ) ).

% not_le_imp_less
tff(fact_654_less__le__not__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
            & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) ) ) ) ).

% less_le_not_le
tff(fact_655_dense__le,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Y: A,Z: A] :
          ( ! [X2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Y)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Z) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ).

% dense_le
tff(fact_656_dense__ge,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z: A,Y: A] :
          ( ! [X2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),X2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X2) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ).

% dense_ge
tff(fact_657_antisym__conv2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
          <=> ( X = Y ) ) ) ) ).

% antisym_conv2
tff(fact_658_antisym__conv1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
          <=> ( X = Y ) ) ) ) ).

% antisym_conv1
tff(fact_659_nless__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Aa2: A,Ba: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
        <=> ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
            | ( Aa2 = Ba ) ) ) ) ).

% nless_le
tff(fact_660_leI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) ) ) ).

% leI
tff(fact_661_leD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y) ) ) ).

% leD
tff(fact_662_verit__comp__simplify1_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B4: A,A4: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B4),A4)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B4) ) ) ).

% verit_comp_simplify1(3)
tff(fact_663_add__less__imp__less__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Aa2: A,C2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),C2))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba) ) ) ).

% add_less_imp_less_right
tff(fact_664_add__less__imp__less__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),Aa2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),Ba))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba) ) ) ).

% add_less_imp_less_left
tff(fact_665_add__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),C2)) ) ) ).

% add_strict_right_mono
tff(fact_666_add__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),Aa2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),Ba)) ) ) ).

% add_strict_left_mono
tff(fact_667_add__strict__mono,axiom,
    ! [A: $tType] :
      ( strict9044650504122735259up_add(A)
     => ! [Aa2: A,Ba: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),D2)) ) ) ) ).

% add_strict_mono
tff(fact_668_add__mono__thms__linordered__field_I1_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),J)
            & ( K = L ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_669_add__mono__thms__linordered__field_I2_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( ( I = J )
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_670_add__mono__thms__linordered__field_I5_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_671_less__numeral__extra_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),one_one(A)) ) ).

% less_numeral_extra(4)
tff(fact_672_minus__less__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),Aa2)),Ba)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),Ba)),Aa2) ) ) ).

% minus_less_iff
tff(fact_673_less__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),aa(A,A,uminus_uminus(A),Ba))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),aa(A,A,uminus_uminus(A),Aa2)) ) ) ).

% less_minus_iff
tff(fact_674_verit__negate__coefficient_I2_J,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),Ba)),aa(A,A,uminus_uminus(A),Aa2)) ) ) ).

% verit_negate_coefficient(2)
tff(fact_675_reals__Archimedean2,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% reals_Archimedean2
tff(fact_676_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),L)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),L) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Na) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ).

% less_add_eq_less
tff(fact_677_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),J)) ) ).

% trans_less_add2
tff(fact_678_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),M)) ) ).

% trans_less_add1
tff(fact_679_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) ) ).

% add_less_mono1
tff(fact_680_not__add__less2,axiom,
    ! [J: nat,I: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I)),I) ).

% not_add_less2
tff(fact_681_not__add__less1,axiom,
    ! [I: nat,J: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),I) ).

% not_add_less1
tff(fact_682_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),L)) ) ) ).

% add_less_mono
tff(fact_683_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),K)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),K) ) ).

% add_lessD1
tff(fact_684_less__mono__imp__le__mono,axiom,
    ! [F2: fun(nat,nat),I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J2)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,F2,I2)),aa(nat,nat,F2,J2)) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,F2,I)),aa(nat,nat,F2,J)) ) ) ).

% less_mono_imp_le_mono
tff(fact_685_le__neq__implies__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( ( M != Na )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ).

% le_neq_implies_less
tff(fact_686_less__or__eq__imp__le,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
        | ( M = Na ) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% less_or_eq_imp_le
tff(fact_687_le__eq__less__or__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
        | ( M = Na ) ) ) ).

% le_eq_less_or_eq
tff(fact_688_less__imp__le__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% less_imp_le_nat
tff(fact_689_nat__less__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
        & ( M != Na ) ) ) ).

% nat_less_le
tff(fact_690_exists__leI,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( ( ! [N4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N4),Na)
           => ~ aa(nat,$o,P,N4) )
       => aa(nat,$o,P,Na) )
     => ? [N5: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N5),Na)
          & aa(nat,$o,P,N5) ) ) ).

% exists_leI
tff(fact_691_length__induct,axiom,
    ! [A: $tType,P: fun(list(A),$o),Xs2: list(A)] :
      ( ! [Xs: list(A)] :
          ( ! [Ys3: list(A)] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Ys3)),aa(list(A),nat,size_size(list(A)),Xs))
             => aa(list(A),$o,P,Ys3) )
         => aa(list(A),$o,P,Xs) )
     => aa(list(A),$o,P,Xs2) ) ).

% length_induct
tff(fact_692_add__less__le__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [Aa2: A,Ba: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),D2)) ) ) ) ).

% add_less_le_mono
tff(fact_693_add__le__less__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [Aa2: A,Ba: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),D2)) ) ) ) ).

% add_le_less_mono
tff(fact_694_add__mono__thms__linordered__field_I3_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_695_add__mono__thms__linordered__field_I4_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_696_not__numeral__less__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),Na)),one_one(A)) ) ).

% not_numeral_less_one
tff(fact_697_less__add__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Aa2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),one_one(A))) ) ).

% less_add_one
tff(fact_698_add__mono1,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),one_one(A))) ) ) ).

% add_mono1
tff(fact_699_less__1__mult,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: A,Na: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Na)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),M),Na)) ) ) ) ).

% less_1_mult
tff(fact_700_neg__numeral__less__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Na: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),Na)) ) ).

% neg_numeral_less_numeral
tff(fact_701_not__numeral__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Na: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))) ) ).

% not_numeral_less_neg_numeral
tff(fact_702_less__minus__one__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A)) ) ).

% less_minus_one_simps(2)
tff(fact_703_less__minus__one__simps_I4_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% less_minus_one_simps(4)
tff(fact_704_mono__nat__linear__lb,axiom,
    ! [F2: fun(nat,nat),M: nat,K: nat] :
      ( ! [M5: nat,N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M5),N)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,F2,M5)),aa(nat,nat,F2,N)) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,F2,M)),K)),aa(nat,nat,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K))) ) ).

% mono_nat_linear_lb
tff(fact_705_len__greater__imp__nonempty,axiom,
    ! [A: $tType,X: nat,L: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),aa(list(A),nat,size_size(list(A)),L))
     => ( L != nil(A) ) ) ).

% len_greater_imp_nonempty
tff(fact_706_mlex__snd__decrI,axiom,
    ! [Aa2: nat,A4: nat,Ba: nat,B4: nat,N3: nat] :
      ( ( Aa2 = A4 )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ba),B4)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Aa2),N3)),Ba)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A4),N3)),B4)) ) ) ).

% mlex_snd_decrI
tff(fact_707_mlex__fst__decrI,axiom,
    ! [Aa2: nat,A4: nat,Ba: nat,N3: nat,B4: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Aa2),A4)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ba),N3)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B4),N3)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Aa2),N3)),Ba)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A4),N3)),B4)) ) ) ) ).

% mlex_fst_decrI
tff(fact_708_mlex__bound,axiom,
    ! [Aa2: nat,A2: nat,Ba: nat,N3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Aa2),A2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ba),N3)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Aa2),N3)),Ba)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),N3)) ) ) ).

% mlex_bound
tff(fact_709_nat__less__real__le,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,semiring_1_of_nat(real),Na)),one_one(real))),aa(nat,real,semiring_1_of_nat(real),M)) ) ).

% nat_less_real_le
tff(fact_710_neg__numeral__less__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A)) ) ).

% neg_numeral_less_one
tff(fact_711_neg__one__less__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),M)) ) ).

% neg_one_less_numeral
tff(fact_712_not__numeral__less__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% not_numeral_less_neg_one
tff(fact_713_not__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) ) ).

% not_one_less_neg_numeral
tff(fact_714_not__neg__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) ) ).

% not_neg_one_less_neg_numeral
tff(fact_715_power__gt1__lemma,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Aa2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Aa2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(nat,A,power_power(A,Aa2),Na))) ) ) ).

% power_gt1_lemma
tff(fact_716_power__less__power__Suc,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Aa2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Aa2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,Aa2),Na)),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(nat,A,power_power(A,Aa2),Na))) ) ) ).

% power_less_power_Suc
tff(fact_717_word__le__add,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),Y)
         => ? [N: nat] : Y = aa(word(A),word(A),aa(word(A),fun(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_718_power__le__imp__le__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Aa2: A,M: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,Aa2),M)),aa(nat,A,power_power(A,Aa2),Na))
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ) ).

% power_le_imp_le_exp
tff(fact_719_of__nat__less__two__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Na)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) ) ).

% of_nat_less_two_power
tff(fact_720_compl__le__swap2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Y)),X)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),X)),Y) ) ) ).

% compl_le_swap2
tff(fact_721_compl__le__swap1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(A,A,uminus_uminus(A),X))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),Y)) ) ) ).

% compl_le_swap1
tff(fact_722_compl__mono,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Y)),aa(A,A,uminus_uminus(A),X)) ) ) ).

% compl_mono
tff(fact_723_ex__power__ivl2,axiom,
    ! [Ba: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Ba)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
       => ? [N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,power_power(nat,Ba),N)),K)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(nat,nat,power_power(nat,Ba),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))) ) ) ) ).

% ex_power_ivl2
tff(fact_724_ex__power__ivl1,axiom,
    ! [Ba: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Ba)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),K)
       => ? [N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,Ba),N)),K)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,power_power(nat,Ba),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))) ) ) ) ).

% ex_power_ivl1
tff(fact_725_power__2__mult__step__le,axiom,
    ! [N6: nat,Na: nat,K4: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N6)),K4)),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,$o,aa(nat,fun(nat,$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,aa(nat,fun(nat,nat),plus_plus(nat),K4),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_726_nat__add__offset__less,axiom,
    ! [Y: nat,Na: nat,X: nat,M: nat,Sz: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
       => ( ( Sz = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na) )
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),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_727_insert__time__pure,axiom,
    ! [Aa2: nat,Na: nat,Sa: set(nat),Tia: vEBT_VEBTi] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Aa2),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
     => simple8792101892335181110TBOUND(vEBT_VEBTi,aa(vEBT_VEBTi,assn,vEBT_Intf_vebt_assn(Na,Sa),Tia),vEBT_vebt_inserti(Tia,Aa2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(bit1(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(bit1(one2))))),aa(int,nat,nat2,archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na))))))) ) ).

% insert_time_pure
tff(fact_728_n__less__equal__power__2,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$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_729_ceiling__log__nat__eq__powr__iff,axiom,
    ! [Ba: nat,K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Ba)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => ( ( archimedean_ceiling(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),Ba)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),Na)),one_one(int)) )
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,power_power(nat,Ba),Na)),K)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(nat,nat,power_power(nat,Ba),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat)))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_730_discrete,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),one_one(A))),Ba) ) ) ).

% discrete
tff(fact_731_log2__of__power__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),Na)) ) ) ).

% log2_of_power_le
tff(fact_732_vebt__predi__rule,axiom,
    ! [Na: nat,S: set(nat),Tia: vEBT_VEBTi,X: nat,Y: nat] : time_htt(option(nat),aa(vEBT_VEBTi,assn,vEBT_Intf_vebt_assn(Na,S),Tia),vEBT_vebt_predi(Tia,X),aa(nat,fun(option(nat),assn),aa(nat,fun(nat,fun(option(nat),assn)),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(option(nat),assn))),aa(set(nat),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(option(nat),assn)))),aTP_Lamp_al(nat,fun(set(nat),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(option(nat),assn))))),Na),S),Tia),X),Y),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit1(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit1(bit1(one2)))),aa(int,nat,nat2,archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na))))))) ).

% vebt_predi_rule
tff(fact_733_ex__has__greatest__nat__lemma,axiom,
    ! [A: $tType,P: fun(A,$o),K: A,F2: fun(A,nat),Na: nat] :
      ( aa(A,$o,P,K)
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
           => ? [Y3: A] :
                ( aa(A,$o,P,Y3)
                & ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,Y3)),aa(A,nat,F2,X2)) ) )
       => ? [Y2: A] :
            ( aa(A,$o,P,Y2)
            & ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Y2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F2,K)),Na)) ) ) ) ).

% ex_has_greatest_nat_lemma
tff(fact_734_le__zero__eq,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Na: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Na),zero_zero(A))
        <=> ( Na = zero_zero(A) ) ) ) ).

% le_zero_eq
tff(fact_735_not__gr__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Na: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Na)
        <=> ( Na = zero_zero(A) ) ) ) ).

% not_gr_zero
tff(fact_736_double__eq__0__iff,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [Aa2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Aa2) = zero_zero(A) )
        <=> ( Aa2 = zero_zero(A) ) ) ) ).

% double_eq_0_iff
tff(fact_737_add_Oright__neutral,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),zero_zero(A)) = Aa2 ) ).

% add.right_neutral
tff(fact_738_double__zero__sym,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [Aa2: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Aa2) )
        <=> ( Aa2 = zero_zero(A) ) ) ) ).

% double_zero_sym
tff(fact_739_add__cancel__left__left,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [Ba: A,Aa2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),Aa2) = Aa2 )
        <=> ( Ba = zero_zero(A) ) ) ) ).

% add_cancel_left_left
tff(fact_740_add__cancel__left__right,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [Aa2: A,Ba: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba) = Aa2 )
        <=> ( Ba = zero_zero(A) ) ) ) ).

% add_cancel_left_right
tff(fact_741_add__cancel__right__left,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [Aa2: A,Ba: A] :
          ( ( Aa2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),Aa2) )
        <=> ( Ba = zero_zero(A) ) ) ) ).

% add_cancel_right_left
tff(fact_742_add__cancel__right__right,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [Aa2: A,Ba: A] :
          ( ( Aa2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba) )
        <=> ( Ba = zero_zero(A) ) ) ) ).

% add_cancel_right_right
tff(fact_743_add__eq__0__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y) = zero_zero(A) )
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_744_zero__eq__add__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [X: A,Y: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y) )
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_745_add__0,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),Aa2) = Aa2 ) ).

% add_0
tff(fact_746_mult__zero__left,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),zero_zero(A)),Aa2) = zero_zero(A) ) ).

% mult_zero_left
tff(fact_747_mult__zero__right,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),zero_zero(A)) = zero_zero(A) ) ).

% mult_zero_right
tff(fact_748_mult__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [Aa2: A,Ba: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba) = zero_zero(A) )
        <=> ( ( Aa2 = zero_zero(A) )
            | ( Ba = zero_zero(A) ) ) ) ) ).

% mult_eq_0_iff
tff(fact_749_mult__cancel__left,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C2: A,Aa2: A,Ba: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),Aa2) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),Ba) )
        <=> ( ( C2 = zero_zero(A) )
            | ( Aa2 = Ba ) ) ) ) ).

% mult_cancel_left
tff(fact_750_mult__cancel__right,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [Aa2: A,C2: A,Ba: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( Aa2 = Ba ) ) ) ) ).

% mult_cancel_right
tff(fact_751_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_752_neg__0__equal__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [Aa2: A] :
          ( ( zero_zero(A) = aa(A,A,uminus_uminus(A),Aa2) )
        <=> ( zero_zero(A) = Aa2 ) ) ) ).

% neg_0_equal_iff_equal
tff(fact_753_neg__equal__0__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [Aa2: A] :
          ( ( aa(A,A,uminus_uminus(A),Aa2) = zero_zero(A) )
        <=> ( Aa2 = zero_zero(A) ) ) ) ).

% neg_equal_0_iff_equal
tff(fact_754_equal__neg__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [Aa2: A] :
          ( ( Aa2 = aa(A,A,uminus_uminus(A),Aa2) )
        <=> ( Aa2 = zero_zero(A) ) ) ) ).

% equal_neg_zero
tff(fact_755_neg__equal__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [Aa2: A] :
          ( ( aa(A,A,uminus_uminus(A),Aa2) = Aa2 )
        <=> ( Aa2 = zero_zero(A) ) ) ) ).

% neg_equal_zero
tff(fact_756_word__size__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(word(A),nat,size_size(word(A)),W)) ) ).

% word_size_gt_0
tff(fact_757_bot__nat__0_Onot__eq__extremum,axiom,
    ! [Aa2: nat] :
      ( ( Aa2 != zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Aa2) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_758_neq0__conv,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ).

% neq0_conv
tff(fact_759_less__nat__zero__code,axiom,
    ! [Na: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),zero_zero(nat)) ).

% less_nat_zero_code
tff(fact_760_add__is__0,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na) = zero_zero(nat) )
    <=> ( ( M = zero_zero(nat) )
        & ( Na = zero_zero(nat) ) ) ) ).

% add_is_0
tff(fact_761_Nat_Oadd__0__right,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),zero_zero(nat)) = M ).

% Nat.add_0_right
tff(fact_762_bot__nat__0_Oextremum,axiom,
    ! [Aa2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),Aa2) ).

% bot_nat_0.extremum
tff(fact_763_le0,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),Na) ).

% le0
tff(fact_764_and__zero__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Aa2),zero_zero(A)) = zero_zero(A) ) ).

% and_zero_eq
tff(fact_765_zero__and__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),zero_zero(A)),Aa2) = zero_zero(A) ) ).

% zero_and_eq
tff(fact_766_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_767_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_768_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_769_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_770_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_771_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_772_or_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),zero_zero(A)),Aa2) = Aa2 ) ).

% or.left_neutral
tff(fact_773_or_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Aa2),zero_zero(A)) = Aa2 ) ).

% or.right_neutral
tff(fact_774_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_775_xor__self__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Aa2),Aa2) = zero_zero(A) ) ).

% xor_self_eq
tff(fact_776_xor_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),zero_zero(A)),Aa2) = Aa2 ) ).

% xor.left_neutral
tff(fact_777_xor_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Aa2),zero_zero(A)) = Aa2 ) ).

% xor.right_neutral
tff(fact_778_ceiling__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_ceiling(A,zero_zero(A)) = zero_zero(int) ) ) ).

% ceiling_zero
tff(fact_779_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_780_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_781_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_782_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_783_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_784_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_785_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_786_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_787_semiring__norm_I78_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less(num),bit0(M)),bit0(Na))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),Na) ) ).

% semiring_norm(78)
tff(fact_788_semiring__norm_I75_J,axiom,
    ! [M: num] : ~ aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),one2) ).

% semiring_norm(75)
tff(fact_789_semiring__norm_I80_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less(num),bit1(M)),bit1(Na))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),Na) ) ).

% semiring_norm(80)
tff(fact_790_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_791_add__le__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),Aa2)),Ba)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),zero_zero(A)) ) ) ).

% add_le_same_cancel1
tff(fact_792_add__le__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)),Ba)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),zero_zero(A)) ) ) ).

% add_le_same_cancel2
tff(fact_793_le__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Ba) ) ) ).

% le_add_same_cancel1
tff(fact_794_le__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),Aa2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Ba) ) ) ).

% le_add_same_cancel2
tff(fact_795_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Aa2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),zero_zero(A)) ) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_796_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Aa2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2) ) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_797_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Aa2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2) ) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_798_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Aa2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),zero_zero(A)) ) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_799_less__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),Aa2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Ba) ) ) ).

% less_add_same_cancel2
tff(fact_800_less__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Ba) ) ) ).

% less_add_same_cancel1
tff(fact_801_add__less__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)),Ba)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),zero_zero(A)) ) ) ).

% add_less_same_cancel2
tff(fact_802_add__less__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),Aa2)),Ba)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),zero_zero(A)) ) ) ).

% add_less_same_cancel1
tff(fact_803_sum__squares__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)) = zero_zero(A) )
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_804_mult__cancel__right2,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [Aa2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2) = C2 )
        <=> ( ( C2 = zero_zero(A) )
            | ( Aa2 = one_one(A) ) ) ) ) ).

% mult_cancel_right2
tff(fact_805_mult__cancel__right1,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C2: A,Ba: A] :
          ( ( C2 = aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( Ba = one_one(A) ) ) ) ) ).

% mult_cancel_right1
tff(fact_806_mult__cancel__left2,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C2: A,Aa2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),Aa2) = C2 )
        <=> ( ( C2 = zero_zero(A) )
            | ( Aa2 = one_one(A) ) ) ) ) ).

% mult_cancel_left2
tff(fact_807_mult__cancel__left1,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C2: A,Ba: A] :
          ( ( C2 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),Ba) )
        <=> ( ( C2 = zero_zero(A) )
            | ( Ba = one_one(A) ) ) ) ) ).

% mult_cancel_left1
tff(fact_808_neg__less__eq__nonneg,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Aa2)),Aa2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2) ) ) ).

% neg_less_eq_nonneg
tff(fact_809_less__eq__neg__nonpos,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),aa(A,A,uminus_uminus(A),Aa2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),zero_zero(A)) ) ) ).

% less_eq_neg_nonpos
tff(fact_810_neg__le__0__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Aa2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2) ) ) ).

% neg_le_0_iff_le
tff(fact_811_neg__0__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),Aa2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),zero_zero(A)) ) ) ).

% neg_0_le_iff_le
tff(fact_812_less__neg__neg,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),aa(A,A,uminus_uminus(A),Aa2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),zero_zero(A)) ) ) ).

% less_neg_neg
tff(fact_813_neg__less__pos,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),Aa2)),Aa2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2) ) ) ).

% neg_less_pos
tff(fact_814_neg__0__less__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),Aa2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),zero_zero(A)) ) ) ).

% neg_0_less_iff_less
tff(fact_815_neg__less__0__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),Aa2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2) ) ) ).

% neg_less_0_iff_less
tff(fact_816_ab__left__minus,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),Aa2)),Aa2) = zero_zero(A) ) ).

% ab_left_minus
tff(fact_817_add_Oright__inverse,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),aa(A,A,uminus_uminus(A),Aa2)) = zero_zero(A) ) ).

% add.right_inverse
tff(fact_818_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_819_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_820_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_821_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_822_add__gr__0,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ) ).

% add_gr_0
tff(fact_823_less__one,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),one_one(nat))
    <=> ( Na = zero_zero(nat) ) ) ).

% less_one
tff(fact_824_length__0__conv,axiom,
    ! [A: $tType,Xs2: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs2) = zero_zero(nat) )
    <=> ( Xs2 = nil(A) ) ) ).

% length_0_conv
tff(fact_825_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Na))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_826_nat__0__less__mult__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ) ).

% nat_0_less_mult_iff
tff(fact_827_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),K))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ).

% mult_less_cancel2
tff(fact_828_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_829_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_830_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_831_semiring__norm_I76_J,axiom,
    ! [Na: num] : aa(num,$o,aa(num,fun(num,$o),ord_less(num),one2),bit0(Na)) ).

% semiring_norm(76)
tff(fact_832_semiring__norm_I81_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less(num),bit1(M)),bit0(Na))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),Na) ) ).

% semiring_norm(81)
tff(fact_833_semiring__norm_I77_J,axiom,
    ! [Na: num] : aa(num,$o,aa(num,fun(num,$o),ord_less(num),one2),bit1(Na)) ).

% semiring_norm(77)
tff(fact_834_word__plus__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)),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),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y) ) ).

% word_plus_and_or
tff(fact_835_enat__ord__number_I2_J,axiom,
    ! [M: num,Na: num] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),aa(num,extended_enat,numeral_numeral(extended_enat),M)),aa(num,extended_enat,numeral_numeral(extended_enat),Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),Na)) ) ).

% enat_ord_number(2)
tff(fact_836_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_837_add__neg__numeral__special_I7_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) = zero_zero(A) ) ) ).

% add_neg_numeral_special(7)
tff(fact_838_add__neg__numeral__special_I8_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A)) = zero_zero(A) ) ) ).

% add_neg_numeral_special(8)
tff(fact_839_of__nat__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),M)),zero_zero(A))
        <=> ( M = zero_zero(nat) ) ) ) ).

% of_nat_le_0_iff
tff(fact_840_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_841_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_842_length__greater__0__conv,axiom,
    ! [A: $tType,Xs2: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(list(A),nat,size_size(list(A)),Xs2))
    <=> ( Xs2 != nil(A) ) ) ).

% length_greater_0_conv
tff(fact_843_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Na))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_844_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),K))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ).

% mult_le_cancel2
tff(fact_845_ceiling__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,X)),zero_zero(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),zero_zero(A)) ) ) ).

% ceiling_le_zero
tff(fact_846_zero__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),archimedean_ceiling(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),X) ) ) ).

% zero_less_ceiling
tff(fact_847_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(num,real,numeral_numeral(real),W)),aa(nat,real,semiring_1_of_nat(real),Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,numeral_numeral(nat),W)),Na) ) ).

% numeral_less_real_of_nat_iff
tff(fact_848_real__of__nat__less__numeral__iff,axiom,
    ! [Na: nat,W: num] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(num,real,numeral_numeral(real),W))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(num,nat,numeral_numeral(nat),W)) ) ).

% real_of_nat_less_numeral_iff
tff(fact_849_nat__neg__numeral,axiom,
    ! [K: num] : aa(int,nat,nat2,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))) = zero_zero(nat) ).

% nat_neg_numeral
tff(fact_850_nat__zminus__int,axiom,
    ! [Na: nat] : aa(int,nat,nat2,aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),Na))) = zero_zero(nat) ).

% nat_zminus_int
tff(fact_851_zle__add1__eq__le,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),one_one(int)))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W),Z) ) ).

% zle_add1_eq_le
tff(fact_852_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_853_semiring__norm_I79_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less(num),bit0(M)),bit1(Na))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),Na) ) ).

% semiring_norm(79)
tff(fact_854_semiring__norm_I74_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),bit1(M)),bit0(Na))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),Na) ) ).

% semiring_norm(74)
tff(fact_855_zero__eq__power2,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [Aa2: A] :
          ( ( aa(nat,A,power_power(A,Aa2),aa(num,nat,numeral_numeral(nat),bit0(one2))) = zero_zero(A) )
        <=> ( Aa2 = zero_zero(A) ) ) ) ).

% zero_eq_power2
tff(fact_856_power__strict__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Ba: A,M: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),one_one(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,Ba),M)),aa(nat,A,power_power(A,Ba),Na))
            <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M) ) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_857_power__mono__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Aa2: A,Ba: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Ba)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,Aa2),Na)),aa(nat,A,power_power(A,Ba),Na))
              <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba) ) ) ) ) ) ).

% power_mono_iff
tff(fact_858_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_859_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_860_of__nat__0__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),Na))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ) ).

% of_nat_0_less_iff
tff(fact_861_ceiling__less__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,X)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),zero_zero(A)) ) ) ).

% ceiling_less_one
tff(fact_862_one__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),archimedean_ceiling(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),X) ) ) ).

% one_le_ceiling
tff(fact_863_power2__less__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,Aa2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),zero_zero(A))
        <=> ( Aa2 = zero_zero(A) ) ) ) ).

% power2_less_eq_zero_iff
tff(fact_864_power2__eq__iff__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
         => ( aa(A,$o,aa(A,fun(A,$o),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_865_zero__less__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,power_power(A,Aa2),aa(num,nat,numeral_numeral(nat),bit0(one2))))
        <=> ( Aa2 != zero_zero(A) ) ) ) ).

% zero_less_power2
tff(fact_866_sum__power2__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(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_867_power__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Ba: A,M: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),one_one(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,Ba),M)),aa(nat,A,power_power(A,Ba),Na))
            <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M) ) ) ) ) ).

% power_decreasing_iff
tff(fact_868_of__nat__zero__less__power__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(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,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),X)
            | ( Na = zero_zero(nat) ) ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_869_numeral__power__less__nat__cancel__iff,axiom,
    ! [X: num,Na: nat,Aa2: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),X)),Na)),aa(int,nat,nat2,Aa2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),Na)),Aa2) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_870_nat__less__numeral__power__cancel__iff,axiom,
    ! [Aa2: int,X: num,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(int,nat,nat2,Aa2)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),X)),Na))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Aa2),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),Na)) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_871_olen__add__eqv,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),X))
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),X)) ) ) ).

% olen_add_eqv
tff(fact_872_le__no__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Ba: word(A),Aa2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),Ba)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Aa2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Aa2),Ba))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Aa2),Ba)) ) ) ) ).

% le_no_overflow
tff(fact_873_word__le__less__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),Y)
        <=> ( ( X = Y )
            | aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),X),Y) ) ) ) ).

% word_le_less_eq
tff(fact_874_word__random,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),X7: word(A),X: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),P2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P2),X7))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),X7)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),P2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P2),X)) ) ) ) ).

% word_random
tff(fact_875_word__le__plus,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A),C2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Aa2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Aa2),Ba))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),C2),Ba)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Aa2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Aa2),C2)) ) ) ) ).

% word_le_plus
tff(fact_876_word__sub__mono2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A),C2: word(A),D2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Aa2),Ba)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),C2),D2))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C2),Aa2)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Ba),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Aa2),Ba))
             => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),D2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),C2),D2))
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Ba),D2) ) ) ) ) ) ).

% word_sub_mono2
tff(fact_877_word__le__plus__either,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A),Z: word(A)] :
          ( ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),Y)
            | aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),Z) )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),Z))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),Z)) ) ) ) ).

% word_le_plus_either
tff(fact_878_word__plus__mcs__3,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: word(A),W: word(A),X: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),V),W)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),X)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)) ) ) ) ).

% word_plus_mcs_3
tff(fact_879_word__plus__mcs__4,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: word(A),X: word(A),W: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),X)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),X))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),V),W) ) ) ) ).

% word_plus_mcs_4
tff(fact_880_word__plus__mono__right,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Z: word(A),X: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Z))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Z)) ) ) ) ).

% word_plus_mono_right
tff(fact_881_word__add__le__iff2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),J),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K))
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K))
            <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),J) ) ) ) ) ).

% word_add_le_iff2
tff(fact_882_word__plus__mcs__4_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),V: word(A),W: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),V)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),W))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),V))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),V),W) ) ) ) ).

% word_plus_mcs_4'
tff(fact_883_word__plus__mono__right2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A),C2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Aa2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Aa2),Ba))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C2),Ba)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Aa2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Aa2),C2)) ) ) ) ).

% word_plus_mono_right2
tff(fact_884_word__add__increasing,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),W: word(A),X: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P2),W)),X)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),P2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P2),W))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),P2),X) ) ) ) ).

% word_add_increasing
tff(fact_885_word__plus__mono__left,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Z: word(A),X: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Z))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),X)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Z),X)) ) ) ) ).

% word_plus_mono_left
tff(fact_886_plus__le__left__cancel__wrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y6: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y6)),X)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y)),X)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y6)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y))
            <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y6),Y) ) ) ) ) ).

% plus_le_left_cancel_wrap
tff(fact_887_plus__le__left__cancel__nowrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y6: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y6))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y))
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y6)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y))
            <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y6),Y) ) ) ) ) ).

% plus_le_left_cancel_nowrap
tff(fact_888_word__plus__strict__mono__right,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Z: word(A),X: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Z))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Z)) ) ) ) ).

% word_plus_strict_mono_right
tff(fact_889_le__word__or1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [C2: word(A),Y: word(A)] : aa(word(A),$o,aa(word(A),fun(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_890_le__word__or2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] : aa(word(A),$o,aa(word(A),fun(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_891_word__and__le1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Aa2: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Y),Aa2)),Aa2) ) ).

% word_and_le1
tff(fact_892_word__and__le2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Y: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Aa2),Y)),Aa2) ) ).

% word_and_le2
tff(fact_893_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_894_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_895_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_896_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_897_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_898_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_899_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_900_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_901_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_902_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_903_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_904_leao,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W4: word(A),X7: word(A),Y6: word(A)] :
          ( ( W4 = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),X7),Y6) )
         => ( X7 = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),X7),W4) ) ) ) ).

% leao
tff(fact_905_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_906_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_907_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_908_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_909_word__ao__equiv,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),W4: word(A)] :
          ( ( W = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),W),W4) )
        <=> ( W4 = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),W4) ) ) ) ).

% word_ao_equiv
tff(fact_910_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_911_Heap_Osize__neq,axiom,
    ! [A: $tType,X: heap_Time_Heap(A)] : aa(heap_Time_Heap(A),nat,size_size(heap_Time_Heap(A)),X) != zero_zero(nat) ).

% Heap.size_neq
tff(fact_912_size__0__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),V: word(A)] :
          ( ( aa(word(A),nat,size_size(word(A)),W) = zero_zero(nat) )
         => ( V = W ) ) ) ).

% size_0_eq
tff(fact_913_lens__not__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(word(A),nat,size_size(word(A)),W) != zero_zero(nat) ) ).

% lens_not_0
tff(fact_914_size__0__same_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),V: word(A)] :
          ( ( aa(word(A),nat,size_size(word(A)),W) = zero_zero(nat) )
         => ( W = V ) ) ) ).

% size_0_same'
tff(fact_915_zero__reorient,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [X: A] :
          ( ( zero_zero(A) = X )
        <=> ( X = zero_zero(A) ) ) ) ).

% zero_reorient
tff(fact_916_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_917_le__some__optE,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M: A,X: option(A)] :
          ( aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less_eq(option(A)),some(A,M)),X)
         => ~ ! [M6: A] :
                ( ( X = some(A,M6) )
               => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),M6) ) ) ) ).

% le_some_optE
tff(fact_918_of__nat__gt__0,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [K: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),K) != zero_zero(A) )
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K) ) ) ).

% of_nat_gt_0
tff(fact_919_less__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less(fun(A,B)),F2),G)
        <=> ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F2),G)
            & ~ aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),G),F2) ) ) ) ).

% less_fun_def
tff(fact_920_less__eq__real__def,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),Y)
        | ( X = Y ) ) ) ).

% less_eq_real_def
tff(fact_921_le__numeral__extra_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),zero_zero(A)) ) ).

% le_numeral_extra(3)
tff(fact_922_zero__le,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X) ) ).

% zero_le
tff(fact_923_gr__zeroI,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Na: A] :
          ( ( Na != zero_zero(A) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Na) ) ) ).

% gr_zeroI
tff(fact_924_not__less__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Na: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Na),zero_zero(A)) ) ).

% not_less_zero
tff(fact_925_gr__implies__not__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [M: A,Na: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M),Na)
         => ( Na != zero_zero(A) ) ) ) ).

% gr_implies_not_zero
tff(fact_926_zero__less__iff__neq__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Na: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Na)
        <=> ( Na != zero_zero(A) ) ) ) ).

% zero_less_iff_neq_zero
tff(fact_927_less__numeral__extra_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),zero_zero(A)) ) ).

% less_numeral_extra(3)
tff(fact_928_field__lbound__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D1: A,D22: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D1)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D22)
           => ? [E: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),E),D1)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),E),D22) ) ) ) ) ).

% field_lbound_gt_zero
tff(fact_929_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_930_union__less__mono,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: multiset(A),C5: multiset(A),B2: multiset(A),D3: multiset(A)] :
          ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),A2),C5)
         => ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),B2),D3)
           => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A2),B2)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C5),D3)) ) ) ) ).

% union_less_mono
tff(fact_931_union__le__mono2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: multiset(A),D3: multiset(A),C5: multiset(A)] :
          ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),B2),D3)
         => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C5),B2)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C5),D3)) ) ) ).

% union_le_mono2
tff(fact_932_union__le__mono1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: multiset(A),D3: multiset(A),C5: multiset(A)] :
          ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),B2),D3)
         => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B2),C5)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),D3),C5)) ) ) ).

% union_le_mono1
tff(fact_933_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),Aa2) = Aa2 ) ).

% comm_monoid_add_class.add_0
tff(fact_934_add_Ocomm__neutral,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),zero_zero(A)) = Aa2 ) ).

% add.comm_neutral
tff(fact_935_add_Ogroup__left__neutral,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),Aa2) = Aa2 ) ).

% add.group_left_neutral
tff(fact_936_verit__sum__simplify,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),zero_zero(A)) = Aa2 ) ).

% verit_sum_simplify
tff(fact_937_add__0__iff,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [Ba: A,Aa2: A] :
          ( ( Ba = aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),Aa2) )
        <=> ( Aa2 = zero_zero(A) ) ) ) ).

% add_0_iff
tff(fact_938_zero__neq__one,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( zero_zero(A) != one_one(A) ) ) ).

% zero_neq_one
tff(fact_939_mult__not__zero,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [Aa2: A,Ba: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba) != zero_zero(A) )
         => ( ( Aa2 != zero_zero(A) )
            & ( Ba != zero_zero(A) ) ) ) ) ).

% mult_not_zero
tff(fact_940_divisors__zero,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [Aa2: A,Ba: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba) = zero_zero(A) )
         => ( ( Aa2 = zero_zero(A) )
            | ( Ba = zero_zero(A) ) ) ) ) ).

% divisors_zero
tff(fact_941_no__zero__divisors,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [Aa2: A,Ba: A] :
          ( ( Aa2 != zero_zero(A) )
         => ( ( Ba != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba) != zero_zero(A) ) ) ) ) ).

% no_zero_divisors
tff(fact_942_mult__left__cancel,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C2: A,Aa2: A,Ba: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),Aa2) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),Ba) )
          <=> ( Aa2 = Ba ) ) ) ) ).

% mult_left_cancel
tff(fact_943_mult__right__cancel,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C2: A,Aa2: A,Ba: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2) )
          <=> ( Aa2 = Ba ) ) ) ) ).

% mult_right_cancel
tff(fact_944_plus__one__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Na: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Na),one_one(word(A))))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),Na) ) ) ).

% plus_one_helper
tff(fact_945_word__le__plus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Na: word(A),Aa2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),Na))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Aa2),Na)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),Aa2)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),Aa2)),one_one(word(A)))) ) ) ) ).

% word_le_plus_1
tff(fact_946_inc__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),M: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),M)
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),one_one(word(A)))),M) ) ) ).

% inc_le
tff(fact_947_inc__i,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),M: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),one_one(word(A))),I)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),M)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),one_one(word(A))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),one_one(word(A))))
              & aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),one_one(word(A)))),M) ) ) ) ) ).

% inc_i
tff(fact_948_num_Osize_I4_J,axiom,
    aa(num,nat,size_size(num),one2) = zero_zero(nat) ).

% num.size(4)
tff(fact_949_no__plus__overflow__neg,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),X),aa(word(A),word(A),uminus_uminus(word(A)),Y))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y)) ) ) ).

% no_plus_overflow_neg
tff(fact_950_bot__nat__0_Oextremum__strict,axiom,
    ! [Aa2: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Aa2),zero_zero(nat)) ).

% bot_nat_0.extremum_strict
tff(fact_951_gr0I,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero(nat) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ).

% gr0I
tff(fact_952_not__gr0,axiom,
    ! [Na: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
    <=> ( Na = zero_zero(nat) ) ) ).

% not_gr0
tff(fact_953_not__less0,axiom,
    ! [Na: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),zero_zero(nat)) ).

% not_less0
tff(fact_954_less__zeroE,axiom,
    ! [Na: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),zero_zero(nat)) ).

% less_zeroE
tff(fact_955_gr__implies__not0,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => ( Na != zero_zero(nat) ) ) ).

% gr_implies_not0
tff(fact_956_infinite__descent0,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ! [N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
           => ( ~ aa(nat,$o,P,N)
             => ? [M3: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),N)
                  & ~ aa(nat,$o,P,M3) ) ) )
       => aa(nat,$o,P,Na) ) ) ).

% infinite_descent0
tff(fact_957_infinite__descent0__measure,axiom,
    ! [A: $tType,V2: fun(A,nat),P: fun(A,$o),X: A] :
      ( ! [X2: A] :
          ( ( aa(A,nat,V2,X2) = zero_zero(nat) )
         => aa(A,$o,P,X2) )
     => ( ! [X2: A] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,V2,X2))
           => ( ~ aa(A,$o,P,X2)
             => ? [Y3: A] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,V2,Y3)),aa(A,nat,V2,X2))
                  & ~ aa(A,$o,P,Y3) ) ) )
       => aa(A,$o,P,X) ) ) ).

% infinite_descent0_measure
tff(fact_958_plus__nat_Oadd__0,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),zero_zero(nat)),Na) = Na ).

% plus_nat.add_0
tff(fact_959_add__eq__self__zero,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na) = M )
     => ( Na = zero_zero(nat) ) ) ).

% add_eq_self_zero
tff(fact_960_less__eq__nat_Osimps_I1_J,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),Na) ).

% less_eq_nat.simps(1)
tff(fact_961_bot__nat__0_Oextremum__unique,axiom,
    ! [Aa2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Aa2),zero_zero(nat))
    <=> ( Aa2 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_unique
tff(fact_962_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [Aa2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Aa2),zero_zero(nat))
     => ( Aa2 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_963_le__0__eq,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),zero_zero(nat))
    <=> ( Na = zero_zero(nat) ) ) ).

% le_0_eq
tff(fact_964_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_965_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_966_or__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A,Ba: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Aa2),Ba) = zero_zero(A) )
        <=> ( ( Aa2 = zero_zero(A) )
            & ( Ba = zero_zero(A) ) ) ) ) ).

% or_eq_0_iff
tff(fact_967_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_968_power__eq__imp__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Aa2: A,Na: nat,Ba: A] :
          ( ( aa(nat,A,power_power(A,Aa2),Na) = aa(nat,A,power_power(A,Ba),Na) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Ba)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
               => ( Aa2 = Ba ) ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_969_power__eq__iff__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat,Aa2: A,Ba: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Ba)
             => ( ( aa(nat,A,power_power(A,Aa2),Na) = aa(nat,A,power_power(A,Ba),Na) )
              <=> ( Aa2 = Ba ) ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_970_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),K)),I)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),K)),J)) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_971_lambda__zero,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ( aTP_Lamp_am(A,A) = aa(A,fun(A,A),times_times(A),zero_zero(A)) ) ) ).

% lambda_zero
tff(fact_972_power__strict__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Aa2: A,Ba: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,Aa2),Na)),aa(nat,A,power_power(A,Ba),Na)) ) ) ) ) ).

% power_strict_mono
tff(fact_973_nat__int__comparison_I2_J,axiom,
    ! [Aa2: nat,Ba: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Aa2),Ba)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),Aa2)),aa(nat,int,semiring_1_of_nat(int),Ba)) ) ).

% nat_int_comparison(2)
tff(fact_974_zero__le__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(num,A,numeral_numeral(A),Na)) ) ).

% zero_le_numeral
tff(fact_975_not__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Na)),zero_zero(A)) ) ).

% not_numeral_le_zero
tff(fact_976_not__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),Na)),zero_zero(A)) ) ).

% not_numeral_less_zero
tff(fact_977_zero__less__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(num,A,numeral_numeral(A),Na)) ) ).

% zero_less_numeral
tff(fact_978_add__decreasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Aa2: A,C2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),Ba)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),C2)),Ba) ) ) ) ).

% add_decreasing
tff(fact_979_add__increasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),C2)) ) ) ) ).

% add_increasing
tff(fact_980_add__decreasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C2: A,Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),C2)),Ba) ) ) ) ).

% add_decreasing2
tff(fact_981_add__increasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C2: A,Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),C2)) ) ) ) ).

% add_increasing2
tff(fact_982_add__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Ba)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)) ) ) ) ).

% add_nonneg_nonneg
tff(fact_983_add__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)),zero_zero(A)) ) ) ) ).

% add_nonpos_nonpos
tff(fact_984_add__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y) = zero_zero(A) )
            <=> ( ( X = zero_zero(A) )
                & ( Y = zero_zero(A) ) ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_985_add__nonpos__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),zero_zero(A))
           => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y) = zero_zero(A) )
            <=> ( ( X = zero_zero(A) )
                & ( Y = zero_zero(A) ) ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_986_not__one__le__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),zero_zero(A)) ) ).

% not_one_le_zero
tff(fact_987_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),one_one(A)) ) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_988_zero__less__one__class_Ozero__le__one,axiom,
    ! [A: $tType] :
      ( zero_less_one(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),one_one(A)) ) ).

% zero_less_one_class.zero_le_one
tff(fact_989_mult__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [Aa2: A,Ba: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Ba)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),D2)) ) ) ) ) ) ).

% mult_mono
tff(fact_990_mult__mono_H,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [Aa2: A,Ba: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),D2)) ) ) ) ) ) ).

% mult_mono'
tff(fact_991_zero__le__square,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [Aa2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Aa2)) ) ).

% zero_le_square
tff(fact_992_split__mult__pos__le,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [Aa2: A,Ba: A] :
          ( ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Ba) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),zero_zero(A)) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)) ) ) ).

% split_mult_pos_le
tff(fact_993_mult__left__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [Ba: A,Aa2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Aa2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Ba)) ) ) ) ).

% mult_left_mono_neg
tff(fact_994_mult__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)) ) ) ) ).

% mult_nonpos_nonpos
tff(fact_995_mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Aa2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Ba)) ) ) ) ).

% mult_left_mono
tff(fact_996_mult__right__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [Ba: A,Aa2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2)) ) ) ) ).

% mult_right_mono_neg
tff(fact_997_mult__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2)) ) ) ) ).

% mult_right_mono
tff(fact_998_mult__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)),zero_zero(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Ba) ) ) ) ) ).

% mult_le_0_iff
tff(fact_999_split__mult__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [Aa2: A,Ba: A] :
          ( ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Ba) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)),zero_zero(A)) ) ) ).

% split_mult_neg_le
tff(fact_1000_mult__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Ba)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)) ) ) ) ).

% mult_nonneg_nonneg
tff(fact_1001_mult__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)),zero_zero(A)) ) ) ) ).

% mult_nonneg_nonpos
tff(fact_1002_mult__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Ba)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)),zero_zero(A)) ) ) ) ).

% mult_nonpos_nonneg
tff(fact_1003_mult__nonneg__nonpos2,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),Aa2)),zero_zero(A)) ) ) ) ).

% mult_nonneg_nonpos2
tff(fact_1004_zero__le__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Ba) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),zero_zero(A)) ) ) ) ) ).

% zero_le_mult_iff
tff(fact_1005_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordere2520102378445227354miring(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Aa2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Ba)) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_1006_pos__add__strict,axiom,
    ! [A: $tType] :
      ( strict7427464778891057005id_add(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),C2)) ) ) ) ).

% pos_add_strict
tff(fact_1007_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
         => ~ ! [C4: A] :
                ( ( Ba = aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),C4) )
               => ( C4 = zero_zero(A) ) ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_1008_add__pos__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Ba)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)) ) ) ) ).

% add_pos_pos
tff(fact_1009_add__neg__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)),zero_zero(A)) ) ) ) ).

% add_neg_neg
tff(fact_1010_add__less__zeroD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),zero_zero(A))
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),zero_zero(A)) ) ) ) ).

% add_less_zeroD
tff(fact_1011_less__numeral__extra_I1_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),one_one(A)) ) ).

% less_numeral_extra(1)
tff(fact_1012_not__one__less__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),zero_zero(A)) ) ).

% not_one_less_zero
tff(fact_1013_zero__less__one,axiom,
    ! [A: $tType] :
      ( zero_less_one(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),one_one(A)) ) ).

% zero_less_one
tff(fact_1014_mult__neg__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)) ) ) ) ).

% mult_neg_neg
tff(fact_1015_not__square__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [Aa2: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Aa2)),zero_zero(A)) ) ).

% not_square_less_zero
tff(fact_1016_mult__less__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)),zero_zero(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Ba) ) ) ) ) ).

% mult_less_0_iff
tff(fact_1017_mult__neg__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Ba)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)),zero_zero(A)) ) ) ) ).

% mult_neg_pos
tff(fact_1018_mult__pos__neg,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)),zero_zero(A)) ) ) ) ).

% mult_pos_neg
tff(fact_1019_mult__pos__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Ba)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)) ) ) ) ).

% mult_pos_pos
tff(fact_1020_mult__pos__neg2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),Aa2)),zero_zero(A)) ) ) ) ).

% mult_pos_neg2
tff(fact_1021_zero__less__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Ba) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),zero_zero(A)) ) ) ) ) ).

% zero_less_mult_iff
tff(fact_1022_zero__less__mult__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Ba) ) ) ) ).

% zero_less_mult_pos
tff(fact_1023_zero__less__mult__pos2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),Aa2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Ba) ) ) ) ).

% zero_less_mult_pos2
tff(fact_1024_mult__less__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Aa2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Ba))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),Aa2) ) ) ) ).

% mult_less_cancel_left_neg
tff(fact_1025_mult__less__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Aa2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Ba))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba) ) ) ) ).

% mult_less_cancel_left_pos
tff(fact_1026_mult__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Ba: A,Aa2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Aa2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Ba)) ) ) ) ).

% mult_strict_left_mono_neg
tff(fact_1027_mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Aa2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Ba)) ) ) ) ).

% mult_strict_left_mono
tff(fact_1028_mult__less__cancel__left__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Aa2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Ba))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),Aa2) ) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_1029_mult__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Ba: A,Aa2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2)) ) ) ) ).

% mult_strict_right_mono_neg
tff(fact_1030_mult__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2)) ) ) ) ).

% mult_strict_right_mono
tff(fact_1031_mult__less__cancel__right__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Aa2: A,C2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),Aa2) ) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_1032_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord2810124833399127020strict(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Aa2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Ba)) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1033_add__scale__eq__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [R: A,Aa2: A,Ba: A,C2: A,D2: A] :
          ( ( R != zero_zero(A) )
         => ( ( ( Aa2 = Ba )
              & ( C2 != D2 ) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),R),C2)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),aa(A,A,aa(A,fun(A,A),times_times(A),R),D2)) ) ) ) ) ).

% add_scale_eq_noteq
tff(fact_1034_power__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Aa2: A,Ba: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,Aa2),Na)),aa(nat,A,power_power(A,Ba),Na)) ) ) ) ).

% power_mono
tff(fact_1035_zero__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Aa2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,Aa2),Na)) ) ) ).

% zero_le_power
tff(fact_1036_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_1037_int__gr__induct,axiom,
    ! [K: int,I: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),I)
     => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int)))
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),I2)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_gr_induct
tff(fact_1038_zless__add1__eq,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),one_one(int)))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),Z)
        | ( W = Z ) ) ) ).

% zless_add1_eq
tff(fact_1039_neg__eq__iff__add__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [Aa2: A,Ba: A] :
          ( ( aa(A,A,uminus_uminus(A),Aa2) = Ba )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba) = zero_zero(A) ) ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_1040_eq__neg__iff__add__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [Aa2: A,Ba: A] :
          ( ( Aa2 = aa(A,A,uminus_uminus(A),Ba) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba) = zero_zero(A) ) ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_1041_add_Oinverse__unique,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [Aa2: A,Ba: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba) = zero_zero(A) )
         => ( aa(A,A,uminus_uminus(A),Aa2) = Ba ) ) ) ).

% add.inverse_unique
tff(fact_1042_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),Aa2)),Aa2) = zero_zero(A) ) ).

% ab_group_add_class.ab_left_minus
tff(fact_1043_add__eq__0__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [Aa2: A,Ba: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba) = zero_zero(A) )
        <=> ( Ba = aa(A,A,uminus_uminus(A),Aa2) ) ) ) ).

% add_eq_0_iff
tff(fact_1044_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_1045_of__nat__0__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),Na)) ) ).

% of_nat_0_le_iff
tff(fact_1046_not__int__zless__negative,axiom,
    ! [Na: nat,M: nat] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),Na)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),M))) ).

% not_int_zless_negative
tff(fact_1047_of__nat__less__0__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M)),zero_zero(A)) ) ).

% of_nat_less_0_iff
tff(fact_1048_real__arch__pow,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),X)
     => ? [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(nat,real,power_power(real,X),N)) ) ).

% real_arch_pow
tff(fact_1049_word__not__simps_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A)] : ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),Y) ) ).

% word_not_simps(3)
tff(fact_1050_word__order_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A)] : ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),Aa2) ) ).

% word_order.extremum_strict
tff(fact_1051_word__order_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A)] :
          ( ( Aa2 != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Aa2),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) ) ) ).

% word_order.not_eq_extremum
tff(fact_1052_word__add__no__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(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,aa(word(A),fun(word(A),$o),ord_less(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),one_one(word(A)))) ) ) ).

% word_add_no_overflow
tff(fact_1053_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,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),Y)
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),one_one(word(A)))) ) ) ) ).

% word_le_make_less
tff(fact_1054_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,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),one_one(word(A))))
          <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),X)
              | ( Y = X ) ) ) ) ) ).

% less_x_plus_1
tff(fact_1055_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,aa(word(A),fun(word(A),$o),ord_less(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),K),one_one(word(A))))
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),K) ) ) ) ).

% word_Suc_leq
tff(fact_1056_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,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),one_one(word(A)))),K)
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),X),K) ) ) ) ).

% word_Suc_le
tff(fact_1057_power__0,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [Aa2: A] : aa(nat,A,power_power(A,Aa2),zero_zero(nat)) = one_one(A) ) ).

% power_0
tff(fact_1058_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ? [K5: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K5)
          & ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K5) = J ) ) ) ).

% less_imp_add_positive
tff(fact_1059_ex__least__nat__le,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( aa(nat,$o,P,Na)
     => ( ~ aa(nat,$o,P,zero_zero(nat))
       => ? [K5: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K5),Na)
            & ! [I3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),K5)
               => ~ aa(nat,$o,P,I3) )
            & aa(nat,$o,P,K5) ) ) ) ).

% ex_least_nat_le
tff(fact_1060_word__1__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A),X: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Aa2),one_one(word(A)))),Ba)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Aa2),aa(nat,word(A),semiring_1_of_nat(word(A)),X))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Aa2),Ba) ) ) ) ).

% word_1_0
tff(fact_1061_nat__geq__1__eq__neqz,axiom,
    ! [X: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),X)
    <=> ( X != zero_zero(nat) ) ) ).

% nat_geq_1_eq_neqz
tff(fact_1062_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Na) )
      <=> ( M = Na ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_1063_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Na))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ).

% nat_mult_less_cancel1
tff(fact_1064_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),I)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),J)) ) ) ).

% mult_less_mono2
tff(fact_1065_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),K)) ) ) ).

% mult_less_mono1
tff(fact_1066_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_1067_nat__less__as__int,axiom,
    ! [X4: nat,Xa: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X4),Xa)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),X4)),aa(nat,int,semiring_1_of_nat(int),Xa)) ) ).

% nat_less_as_int
tff(fact_1068_log__of__power__less,axiom,
    ! [M: nat,Ba: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),M)),aa(nat,real,power_power(real,Ba),Na))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Ba)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(Ba),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),Na)) ) ) ) ).

% log_of_power_less
tff(fact_1069_log__of__power__le,axiom,
    ! [M: nat,Ba: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),M)),aa(nat,real,power_power(real,Ba),Na))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Ba)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(Ba),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),Na)) ) ) ) ).

% log_of_power_le
tff(fact_1070_add__neg__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)),zero_zero(A)) ) ) ) ).

% add_neg_nonpos
tff(fact_1071_add__nonneg__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Ba)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)) ) ) ) ).

% add_nonneg_pos
tff(fact_1072_add__nonpos__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)),zero_zero(A)) ) ) ) ).

% add_nonpos_neg
tff(fact_1073_add__pos__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Ba)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)) ) ) ) ).

% add_pos_nonneg
tff(fact_1074_add__strict__increasing,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),C2)) ) ) ) ).

% add_strict_increasing
tff(fact_1075_add__strict__increasing2,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),C2)) ) ) ) ).

% add_strict_increasing2
tff(fact_1076_mult__le__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Aa2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Ba))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2) ) ) ) ) ).

% mult_le_cancel_left
tff(fact_1077_mult__le__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Aa2: A,C2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2) ) ) ) ) ).

% mult_le_cancel_right
tff(fact_1078_mult__left__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [C2: A,Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Aa2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Ba))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba) ) ) ) ).

% mult_left_less_imp_less
tff(fact_1079_mult__strict__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [Aa2: A,Ba: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Ba)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),D2)) ) ) ) ) ) ).

% mult_strict_mono
tff(fact_1080_mult__less__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Aa2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Ba))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),Aa2) ) ) ) ) ).

% mult_less_cancel_left
tff(fact_1081_mult__right__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [Aa2: A,C2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba) ) ) ) ).

% mult_right_less_imp_less
tff(fact_1082_mult__strict__mono_H,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [Aa2: A,Ba: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),D2)) ) ) ) ) ) ).

% mult_strict_mono'
tff(fact_1083_mult__less__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Aa2: A,C2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),Aa2) ) ) ) ) ).

% mult_less_cancel_right
tff(fact_1084_mult__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Aa2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Ba))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2) ) ) ) ).

% mult_le_cancel_left_neg
tff(fact_1085_mult__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Aa2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Ba))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba) ) ) ) ).

% mult_le_cancel_left_pos
tff(fact_1086_mult__left__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [C2: A,Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Aa2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Ba))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba) ) ) ) ).

% mult_left_le_imp_le
tff(fact_1087_mult__right__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [Aa2: A,C2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba) ) ) ) ).

% mult_right_le_imp_le
tff(fact_1088_mult__le__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [Aa2: A,Ba: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),D2)) ) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_1089_mult__less__le__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [Aa2: A,Ba: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),D2)) ) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_1090_sum__squares__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),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_1091_sum__squares__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [X: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))) ) ).

% sum_squares_ge_zero
tff(fact_1092_mult__left__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Y),X)),X) ) ) ) ) ).

% mult_left_le_one_le
tff(fact_1093_mult__right__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Y)),X) ) ) ) ) ).

% mult_right_le_one_le
tff(fact_1094_mult__le__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Ba)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)),one_one(A)) ) ) ) ) ).

% mult_le_one
tff(fact_1095_mult__left__le,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [C2: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),Aa2) ) ) ) ).

% mult_left_le
tff(fact_1096_zero__less__two,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A))) ) ).

% zero_less_two
tff(fact_1097_sum__squares__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),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_1098_not__sum__squares__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [X: A,Y: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),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_1099_power__less__imp__less__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Aa2: A,Na: nat,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,Aa2),Na)),aa(nat,A,power_power(A,Ba),Na))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Ba)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba) ) ) ) ).

% power_less_imp_less_base
tff(fact_1100_neg__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))),zero_zero(A)) ) ).

% neg_numeral_le_zero
tff(fact_1101_not__zero__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))) ) ).

% not_zero_le_neg_numeral
tff(fact_1102_neg__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))),zero_zero(A)) ) ).

% neg_numeral_less_zero
tff(fact_1103_not__zero__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))) ) ).

% not_zero_less_neg_numeral
tff(fact_1104_power__le__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Aa2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,Aa2),Na)),one_one(A)) ) ) ) ).

% power_le_one
tff(fact_1105_le__minus__one__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),zero_zero(A)) ) ).

% le_minus_one_simps(1)
tff(fact_1106_le__minus__one__simps_I3_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% le_minus_one_simps(3)
tff(fact_1107_less__minus__one__simps_I3_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% less_minus_one_simps(3)
tff(fact_1108_less__minus__one__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),zero_zero(A)) ) ).

% less_minus_one_simps(1)
tff(fact_1109_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(int,nat,nat2,Z))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),M)),Z) ) ).

% zless_nat_eq_int_zless
tff(fact_1110_zless__imp__add1__zle,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),Z)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),W),one_one(int))),Z) ) ).

% zless_imp_add1_zle
tff(fact_1111_add1__zle__eq,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),W),one_one(int))),Z)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),Z) ) ).

% add1_zle_eq
tff(fact_1112_ex__less__of__nat__mult,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),X)
         => ? [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),X)) ) ) ).

% ex_less_of_nat_mult
tff(fact_1113_bounded__Max__nat,axiom,
    ! [P: fun(nat,$o),X: nat,M2: nat] :
      ( aa(nat,$o,P,X)
     => ( ! [X2: nat] :
            ( aa(nat,$o,P,X2)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X2),M2) )
       => ~ ! [M5: nat] :
              ( aa(nat,$o,P,M5)
             => ~ ! [X4: nat] :
                    ( aa(nat,$o,P,X4)
                   => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X4),M5) ) ) ) ) ).

% bounded_Max_nat
tff(fact_1114_ex__has__least__nat,axiom,
    ! [A: $tType,P: fun(A,$o),K: A,M: fun(A,nat)] :
      ( aa(A,$o,P,K)
     => ? [X2: A] :
          ( aa(A,$o,P,X2)
          & ! [Y3: A] :
              ( aa(A,$o,P,Y3)
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,M,X2)),aa(A,nat,M,Y3)) ) ) ) ).

% ex_has_least_nat
tff(fact_1115_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Na))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ).

% nat_mult_le_cancel1
tff(fact_1116_int__cases4,axiom,
    ! [M: int] :
      ( ! [N: nat] : M != aa(nat,int,semiring_1_of_nat(int),N)
     => ~ ! [N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
           => ( M != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N)) ) ) ) ).

% int_cases4
tff(fact_1117_int__zle__neg,axiom,
    ! [Na: nat,M: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),Na)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),M)))
    <=> ( ( Na = zero_zero(nat) )
        & ( M = zero_zero(nat) ) ) ) ).

% int_zle_neg
tff(fact_1118_log2__of__power__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),Na)) ) ) ).

% log2_of_power_less
tff(fact_1119_mult__le__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Ba))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Ba) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),one_one(A)) ) ) ) ) ).

% mult_le_cancel_left1
tff(fact_1120_mult__le__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Aa2)),C2)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Aa2) ) ) ) ) ).

% mult_le_cancel_left2
tff(fact_1121_mult__le__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Ba) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),one_one(A)) ) ) ) ) ).

% mult_le_cancel_right1
tff(fact_1122_mult__le__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Aa2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),C2)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Aa2) ) ) ) ) ).

% mult_le_cancel_right2
tff(fact_1123_mult__less__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Ba))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Ba) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),one_one(A)) ) ) ) ) ).

% mult_less_cancel_left1
tff(fact_1124_mult__less__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Aa2)),C2)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Aa2) ) ) ) ) ).

% mult_less_cancel_left2
tff(fact_1125_mult__less__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Ba) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),one_one(A)) ) ) ) ) ).

% mult_less_cancel_right1
tff(fact_1126_mult__less__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Aa2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),C2)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Aa2) ) ) ) ) ).

% mult_less_cancel_right2
tff(fact_1127_convex__bound__le,axiom,
    ! [A: $tType] :
      ( linord6961819062388156250ring_1(A)
     => ! [X: A,Aa2: A,Y: A,U: A,V: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Aa2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),U)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),V)
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V) = one_one(A) )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),U),X)),aa(A,A,aa(A,fun(A,A),times_times(A),V),Y))),Aa2) ) ) ) ) ) ) ).

% convex_bound_le
tff(fact_1128_power__Suc__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Aa2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(nat,A,power_power(A,Aa2),Na))),aa(nat,A,power_power(A,Aa2),Na)) ) ) ) ).

% power_Suc_less
tff(fact_1129_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_1130_power__strict__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat,N3: nat,Aa2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),N3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,Aa2),N3)),aa(nat,A,power_power(A,Aa2),Na)) ) ) ) ) ).

% power_strict_decreasing
tff(fact_1131_power__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat,N3: nat,Aa2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,Aa2),N3)),aa(nat,A,power_power(A,Aa2),Na)) ) ) ) ) ).

% power_decreasing
tff(fact_1132_nat__le__real__less,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,semiring_1_of_nat(real),M)),one_one(real))) ) ).

% nat_le_real_less
tff(fact_1133_self__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Aa2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Aa2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),aa(nat,A,power_power(A,Aa2),Na)) ) ) ) ).

% self_le_power
tff(fact_1134_pos2,axiom,
    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% pos2
tff(fact_1135_one__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Aa2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Aa2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(nat,A,power_power(A,Aa2),Na)) ) ) ) ).

% one_less_power
tff(fact_1136_bit_Ocomplement__unique,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Aa2: A,X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Aa2),X) = zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Aa2),X) = aa(A,A,uminus_uminus(A),one_one(A)) )
           => ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Aa2),Y) = zero_zero(A) )
             => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Aa2),Y) = aa(A,A,uminus_uminus(A),one_one(A)) )
               => ( X = Y ) ) ) ) ) ) ).

% bit.complement_unique
tff(fact_1137_log__of__power__eq,axiom,
    ! [M: nat,Ba: real,Na: nat] :
      ( ( aa(nat,real,semiring_1_of_nat(real),M) = aa(nat,real,power_power(real,Ba),Na) )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Ba)
       => ( aa(nat,real,semiring_1_of_nat(real),Na) = aa(real,real,log(Ba),aa(nat,real,semiring_1_of_nat(real),M)) ) ) ) ).

% log_of_power_eq
tff(fact_1138_less__log__of__power,axiom,
    ! [Ba: real,Na: nat,M: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,power_power(real,Ba),Na)),M)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Ba)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(real,real,log(Ba),M)) ) ) ).

% less_log_of_power
tff(fact_1139_convex__bound__lt,axiom,
    ! [A: $tType] :
      ( linord715952674999750819strict(A)
     => ! [X: A,Aa2: A,Y: A,U: A,V: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Aa2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),U)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),V)
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V) = one_one(A) )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),U),X)),aa(A,A,aa(A,fun(A,A),times_times(A),V),Y))),Aa2) ) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_1140_zero__le__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Aa2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,Aa2),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% zero_le_power2
tff(fact_1141_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,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
             => ( X = Y ) ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_1142_power2__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(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,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ) ).

% power2_le_imp_le
tff(fact_1143_power2__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Aa2: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,Aa2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),zero_zero(A)) ) ).

% power2_less_0
tff(fact_1144_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,aa(nat,fun(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_1145_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,aa(nat,fun(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_1146_le__log__of__power,axiom,
    ! [Ba: real,Na: nat,M: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,power_power(real,Ba),Na)),M)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Ba)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(real,real,log(Ba),M)) ) ) ).

% le_log_of_power
tff(fact_1147_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_1148_mult__ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Ba)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba))),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(A,Aa2)),archimedean_ceiling(A,Ba))) ) ) ) ).

% mult_ceiling_le
tff(fact_1149_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_1150_power2__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(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,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y) ) ) ) ).

% power2_less_imp_less
tff(fact_1151_sum__power2__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,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_1152_sum__power2__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,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_1153_not__sum__power2__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,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_1154_sum__power2__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,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_1155_zero__le__even__power_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Aa2: A,Na: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,Aa2),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_1156_Lattices__Big_Oex__has__greatest__nat,axiom,
    ! [A: $tType,P: fun(A,$o),K: A,F2: fun(A,nat),Ba: nat] :
      ( aa(A,$o,P,K)
     => ( ! [Y2: A] :
            ( aa(A,$o,P,Y2)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Y2)),Ba) )
       => ? [X2: A] :
            ( aa(A,$o,P,X2)
            & ! [Y3: A] :
                ( aa(A,$o,P,Y3)
               => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,Y3)),aa(A,nat,F2,X2)) ) ) ) ) ).

% Lattices_Big.ex_has_greatest_nat
tff(fact_1157_less__log2__of__power,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),M)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))) ) ).

% less_log2_of_power
tff(fact_1158_hash__code__option__simps_I2_J,axiom,
    ! [A: $tType,H_a: fun(A,uint32),X: A] : hash_h1887023736457453652option(A,H_a,some(A,X)) = aa(uint32,uint32,aa(uint32,fun(uint32,uint32),plus_plus(uint32),aa(uint32,uint32,aa(uint32,fun(uint32,uint32),times_times(uint32),aa(A,uint32,H_a,X)),aa(num,uint32,numeral_numeral(uint32),bit1(bit1(bit1(bit0(bit1(bit0(bit0(bit1(bit1(bit1(bit1(bit0(bit0(bit1(bit1(bit0(bit0(bit0(bit1(bit0(bit1(bit1(bit1(bit1(bit0(bit0(bit0(bit1(bit0(one2)))))))))))))))))))))))))))))))),aa(num,uint32,numeral_numeral(uint32),bit0(bit0(bit0(bit1(bit0(bit0(bit0(bit0(bit0(bit1(bit1(bit0(bit1(bit1(bit1(bit1(bit0(bit1(bit1(bit0(bit1(bit0(bit1(bit1(bit0(bit1(bit0(bit0(bit0(one2))))))))))))))))))))))))))))))) ).

% hash_code_option_simps(2)
tff(fact_1159_arcosh__1,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,arcosh(A),one_one(A)) = zero_zero(A) ) ) ).

% arcosh_1
tff(fact_1160_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,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
         => aa(nat,$o,P,Na) ) ) ) ).

% nat_induct2
tff(fact_1161_TBOUND__mfold,axiom,
    ! [Xs2: list(nat),Na: nat,Sa: set(nat),Tia: vEBT_VEBTi] :
      ( ! [X2: nat] :
          ( member(nat,X2,aa(list(nat),set(nat),set2(nat),Xs2))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X2),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) )
     => simple8792101892335181110TBOUND(vEBT_VEBTi,aa(vEBT_VEBTi,assn,vEBT_Intf_vebt_assn(Na,Sa),Tia),aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),vEBT_Example_mfold(nat,vEBT_VEBTi,aTP_Lamp_an(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),Xs2),Tia),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(nat),nat,size_size(list(nat)),Xs2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(bit1(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(bit1(one2))))),aa(int,nat,nat2,archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na)))))))),one_one(nat))) ) ).

% TBOUND_mfold
tff(fact_1162_less__eq__option__Some,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less_eq(option(A)),some(A,X)),some(A,Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ).

% less_eq_option_Some
tff(fact_1163_field__le__mult__one__interval,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( ! [Z3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z3)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),one_one(A))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z3),X)),Y) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ).

% field_le_mult_one_interval
tff(fact_1164_vebt__succi__rule,axiom,
    ! [Na: nat,S: set(nat),Tia: vEBT_VEBTi,X: nat,Y: nat] : time_htt(option(nat),aa(vEBT_VEBTi,assn,vEBT_Intf_vebt_assn(Na,S),Tia),vEBT_vebt_succi(Tia,X),aa(nat,fun(option(nat),assn),aa(nat,fun(nat,fun(option(nat),assn)),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(option(nat),assn))),aa(set(nat),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(option(nat),assn)))),aTP_Lamp_ao(nat,fun(set(nat),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(option(nat),assn))))),Na),S),Tia),X),Y),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit1(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit1(bit1(one2)))),aa(int,nat,nat2,archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na))))))) ).

% vebt_succi_rule
tff(fact_1165_test__def,axiom,
    ! [Na: nat,Xs2: list(nat),Ysb: list(nat)] : vEBT_test(Na,Xs2,Ysb) = heap_Time_bind(vEBT_VEBTi,list(nat),vEBT_vebt_buildupi(Na),aa(list(nat),fun(vEBT_VEBTi,heap_Time_Heap(list(nat))),aTP_Lamp_ar(list(nat),fun(list(nat),fun(vEBT_VEBTi,heap_Time_Heap(list(nat)))),Xs2),Ysb)) ).

% test_def
tff(fact_1166_mult__le__cancel__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ) ).

% mult_le_cancel_iff1
tff(fact_1167_mult__le__cancel__iff2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ) ).

% mult_le_cancel_iff2
tff(fact_1168_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
    ! [A: $tType,X: multiset(A),Y: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(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_1169_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),aa(multiset(A),fun(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_1170_empty__eq__union,axiom,
    ! [A: $tType,M2: multiset(A),N3: multiset(A)] :
      ( ( zero_zero(multiset(A)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M2),N3) )
    <=> ( ( M2 = zero_zero(multiset(A)) )
        & ( N3 = zero_zero(multiset(A)) ) ) ) ).

% empty_eq_union
tff(fact_1171_union__eq__empty,axiom,
    ! [A: $tType,M2: multiset(A),N3: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M2),N3) = zero_zero(multiset(A)) )
    <=> ( ( M2 = zero_zero(multiset(A)) )
        & ( N3 = zero_zero(multiset(A)) ) ) ) ).

% union_eq_empty
tff(fact_1172_word__coorder_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Aa2),zero_zero(word(A)))
        <=> ( Aa2 = zero_zero(word(A)) ) ) ) ).

% word_coorder.extremum_unique
tff(fact_1173_word__le__0__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),zero_zero(word(A)))
        <=> ( X = zero_zero(word(A)) ) ) ) ).

% word_le_0_iff
tff(fact_1174_i0__less,axiom,
    ! [Na: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),zero_zero(extended_enat)),Na)
    <=> ( Na != zero_zero(extended_enat) ) ) ).

% i0_less
tff(fact_1175_word__gt__0__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),aa(num,word(A),numeral_numeral(word(A)),Y))
        <=> ( zero_zero(word(A)) != aa(num,word(A),numeral_numeral(word(A)),Y) ) ) ) ).

% word_gt_0_no
tff(fact_1176_word__less__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),X),one_one(word(A)))
        <=> ( X = zero_zero(word(A)) ) ) ) ).

% word_less_1
tff(fact_1177_real__add__minus__iff,axiom,
    ! [X: real,Aa2: real] :
      ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,uminus_uminus(real),Aa2)) = zero_zero(real) )
    <=> ( X = Aa2 ) ) ).

% real_add_minus_iff
tff(fact_1178_log__one,axiom,
    ! [Aa2: real] : aa(real,real,log(Aa2),one_one(real)) = zero_zero(real) ).

% log_one
tff(fact_1179_size__empty,axiom,
    ! [A: $tType] : aa(multiset(A),nat,size_size(multiset(A)),zero_zero(multiset(A))) = zero_zero(nat) ).

% size_empty
tff(fact_1180_size__eq__0__iff__empty,axiom,
    ! [A: $tType,M2: multiset(A)] :
      ( ( aa(multiset(A),nat,size_size(multiset(A)),M2) = zero_zero(nat) )
    <=> ( M2 = zero_zero(multiset(A)) ) ) ).

% size_eq_0_iff_empty
tff(fact_1181_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_1182_set__rotate1,axiom,
    ! [A: $tType,Xs2: list(A)] : aa(list(A),set(A),set2(A),rotate1(A,Xs2)) = aa(list(A),set(A),set2(A),Xs2) ).

% set_rotate1
tff(fact_1183_log__eq__one,axiom,
    ! [Aa2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Aa2)
     => ( ( Aa2 != one_one(real) )
       => ( aa(real,real,log(Aa2),Aa2) = one_one(real) ) ) ) ).

% log_eq_one
tff(fact_1184_log__less__cancel__iff,axiom,
    ! [Aa2: real,X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Aa2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(Aa2),X)),aa(real,real,log(Aa2),Y))
          <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),Y) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_1185_log__less__one__cancel__iff,axiom,
    ! [Aa2: real,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Aa2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(Aa2),X)),one_one(real))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),Aa2) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_1186_one__less__log__cancel__iff,axiom,
    ! [Aa2: real,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Aa2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),aa(real,real,log(Aa2),X))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Aa2),X) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_1187_log__less__zero__cancel__iff,axiom,
    ! [Aa2: real,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Aa2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(Aa2),X)),zero_zero(real))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),one_one(real)) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_1188_zero__less__log__cancel__iff,axiom,
    ! [Aa2: real,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Aa2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,log(Aa2),X))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),X) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_1189_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
        | aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) ) ) ).

% and_nonnegative_int_iff
tff(fact_1190_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)),zero_zero(int))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ) ).

% and_negative_int_iff
tff(fact_1191_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
        & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) ) ) ).

% or_nonnegative_int_iff
tff(fact_1192_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)),zero_zero(int))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
        | aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ) ).

% or_negative_int_iff
tff(fact_1193_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) ) ) ).

% xor_nonnegative_int_iff
tff(fact_1194_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L)),zero_zero(int))
    <=> ~ ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ) ).

% xor_negative_int_iff
tff(fact_1195_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_1196_nat__le__0,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),zero_zero(int))
     => ( aa(int,nat,nat2,Z) = zero_zero(nat) ) ) ).

% nat_le_0
tff(fact_1197_nat__0__iff,axiom,
    ! [I: int] :
      ( ( aa(int,nat,nat2,I) = zero_zero(nat) )
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),zero_zero(int)) ) ).

% nat_0_iff
tff(fact_1198_zless__nat__conj,axiom,
    ! [W: int,Z: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(int,nat,nat2,W)),aa(int,nat,nat2,Z))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),Z) ) ) ).

% zless_nat_conj
tff(fact_1199_zero__le__log__cancel__iff,axiom,
    ! [Aa2: real,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Aa2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,log(Aa2),X))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),X) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_1200_log__le__zero__cancel__iff,axiom,
    ! [Aa2: real,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Aa2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(Aa2),X)),zero_zero(real))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),one_one(real)) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_1201_one__le__log__cancel__iff,axiom,
    ! [Aa2: real,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Aa2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(real,real,log(Aa2),X))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),X) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_1202_log__le__one__cancel__iff,axiom,
    ! [Aa2: real,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Aa2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(Aa2),X)),one_one(real))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Aa2) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_1203_log__le__cancel__iff,axiom,
    ! [Aa2: real,X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Aa2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(Aa2),X)),aa(real,real,log(Aa2),Y))
          <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_1204_int__nat__eq,axiom,
    ! [Z: int] :
      aa(nat,int,semiring_1_of_nat(int),aa(int,nat,nat2,Z)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z),Z,zero_zero(int)) ).

% int_nat_eq
tff(fact_1205_not__negative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4277139882892585799ns_not(int),K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% not_negative_int_iff
tff(fact_1206_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% not_nonnegative_int_iff
tff(fact_1207_zero__less__nat__eq,axiom,
    ! [Z: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(int,nat,nat2,Z))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z) ) ).

% zero_less_nat_eq
tff(fact_1208_log__pow__cancel,axiom,
    ! [Aa2: real,Ba: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Aa2)
     => ( ( Aa2 != one_one(real) )
       => ( aa(real,real,log(Aa2),aa(nat,real,power_power(real,Aa2),Ba)) = aa(nat,real,semiring_1_of_nat(real),Ba) ) ) ) ).

% log_pow_cancel
tff(fact_1209_ceiling__less__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,X)),zero_zero(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),one_one(A))) ) ) ).

% ceiling_less_zero
tff(fact_1210_zero__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),archimedean_ceiling(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),X) ) ) ).

% zero_le_ceiling
tff(fact_1211_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_1212_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_1213_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,Na: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),zero_zero(extended_enat)),aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),M),Na))
    <=> ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),zero_zero(extended_enat)),M)
        & aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),zero_zero(extended_enat)),Na) ) ) ).

% enat_0_less_mult_iff
tff(fact_1214_less__eq__multiset__def,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M2: multiset(A),N3: multiset(A)] :
          ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less_eq(multiset(A)),M2),N3)
        <=> ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),M2),N3)
            | ( M2 = N3 ) ) ) ) ).

% less_eq_multiset_def
tff(fact_1215_enat__less__induct,axiom,
    ! [P: fun(extended_enat,$o),Na: extended_enat] :
      ( ! [N: extended_enat] :
          ( ! [M3: extended_enat] :
              ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),M3),N)
             => aa(extended_enat,$o,P,M3) )
         => aa(extended_enat,$o,P,N) )
     => aa(extended_enat,$o,P,Na) ) ).

% enat_less_induct
tff(fact_1216_not__iless0,axiom,
    ! [Na: extended_enat] : ~ aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),Na),zero_zero(extended_enat)) ).

% not_iless0
tff(fact_1217_word__greater__zero__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),Aa2)
        <=> ( Aa2 != zero_zero(word(A)) ) ) ) ).

% word_greater_zero_iff
tff(fact_1218_mset__le__not__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M2: multiset(A)] : ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),M2),M2) ) ).

% mset_le_not_refl
tff(fact_1219_mset__le__not__sym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M2: multiset(A),N3: multiset(A)] :
          ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),M2),N3)
         => ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),N3),M2) ) ) ).

% mset_le_not_sym
tff(fact_1220_mset__le__irrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M2: multiset(A)] : ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),M2),M2) ) ).

% mset_le_irrefl
tff(fact_1221_mset__le__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K3: multiset(A),M2: multiset(A),N3: multiset(A)] :
          ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),K3),M2)
         => ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),M2),N3)
           => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),K3),N3) ) ) ) ).

% mset_le_trans
tff(fact_1222_mset__le__asym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M2: multiset(A),N3: multiset(A)] :
          ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),M2),N3)
         => ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),N3),M2) ) ) ).

% mset_le_asym
tff(fact_1223_word__neq__0__conv,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( ( W != zero_zero(word(A)) )
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),W) ) ) ).

% word_neq_0_conv
tff(fact_1224_word__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),Y)
        <=> ( zero_zero(word(A)) != Y ) ) ) ).

% word_gt_0
tff(fact_1225_word__coorder_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A)] : ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Aa2),zero_zero(word(A))) ) ).

% word_coorder.extremum_strict
tff(fact_1226_word__not__simps_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),X),zero_zero(word(A))) ) ).

% word_not_simps(1)
tff(fact_1227_subset__code_I1_J,axiom,
    ! [A: $tType,Xs2: list(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs2)),B2)
    <=> ! [X6: A] :
          ( member(A,X6,aa(list(A),set(A),set2(A),Xs2))
         => member(A,X6,B2) ) ) ).

% subset_code(1)
tff(fact_1228_set__subset__Cons,axiom,
    ! [A: $tType,Xs2: list(A),X: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs2)),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs2))) ).

% set_subset_Cons
tff(fact_1229_set__ConsD,axiom,
    ! [A: $tType,Y: A,X: A,Xs2: list(A)] :
      ( member(A,Y,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs2)))
     => ( ( Y = X )
        | member(A,Y,aa(list(A),set(A),set2(A),Xs2)) ) ) ).

% set_ConsD
tff(fact_1230_list_Oset__cases,axiom,
    ! [A: $tType,E2: A,Aa2: list(A)] :
      ( member(A,E2,aa(list(A),set(A),set2(A),Aa2))
     => ( ! [Z23: list(A)] : Aa2 != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),Z23)
       => ~ ! [Z12: A,Z23: list(A)] :
              ( ( Aa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z12),Z23) )
             => ~ member(A,E2,aa(list(A),set(A),set2(A),Z23)) ) ) ) ).

% list.set_cases
tff(fact_1231_list_Oset__intros_I1_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : member(A,X21,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222))) ).

% list.set_intros(1)
tff(fact_1232_list_Oset__intros_I2_J,axiom,
    ! [A: $tType,Y: A,X222: list(A),X21: A] :
      ( member(A,Y,aa(list(A),set(A),set2(A),X222))
     => member(A,Y,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222))) ) ).

% list.set_intros(2)
tff(fact_1233_less__eq__int__code_I1_J,axiom,
    aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),zero_zero(int)) ).

% less_eq_int_code(1)
tff(fact_1234_less__int__code_I1_J,axiom,
    ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),zero_zero(int)) ).

% less_int_code(1)
tff(fact_1235_uminus__int__code_I1_J,axiom,
    aa(int,int,uminus_uminus(int),zero_zero(int)) = zero_zero(int) ).

% uminus_int_code(1)
tff(fact_1236_plus__int__code_I2_J,axiom,
    ! [L: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),zero_zero(int)),L) = L ).

% plus_int_code(2)
tff(fact_1237_plus__int__code_I1_J,axiom,
    ! [K: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),K),zero_zero(int)) = K ).

% plus_int_code(1)
tff(fact_1238_word__coorder_Oextremum,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),zero_zero(word(A))),Aa2) ) ).

% word_coorder.extremum
tff(fact_1239_word__coorder_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Aa2),zero_zero(word(A)))
         => ( Aa2 = zero_zero(word(A)) ) ) ) ).

% word_coorder.extremum_uniqueI
tff(fact_1240_word__zero__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),zero_zero(word(A))),Y) ) ).

% word_zero_le
tff(fact_1241_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_1242_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_1243_empty__neutral_I2_J,axiom,
    ! [A: $tType,X: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),X),zero_zero(multiset(A))) = X ).

% empty_neutral(2)
tff(fact_1244_empty__neutral_I1_J,axiom,
    ! [A: $tType,X: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),zero_zero(multiset(A))),X) = X ).

% empty_neutral(1)
tff(fact_1245_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_1246_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_1247_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_1248_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_1249_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_1250_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_1251_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_1252_ile0__eq,axiom,
    ! [Na: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),Na),zero_zero(extended_enat))
    <=> ( Na = zero_zero(extended_enat) ) ) ).

% ile0_eq
tff(fact_1253_i0__lb,axiom,
    ! [Na: extended_enat] : aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),zero_zero(extended_enat)),Na) ).

% i0_lb
tff(fact_1254_int__ops_I1_J,axiom,
    aa(nat,int,semiring_1_of_nat(int),zero_zero(nat)) = zero_zero(int) ).

% int_ops(1)
tff(fact_1255_nat__zero__as__int,axiom,
    zero_zero(nat) = aa(int,nat,nat2,zero_zero(int)) ).

% nat_zero_as_int
tff(fact_1256_odd__nonzero,axiom,
    ! [Z: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z)),Z) != zero_zero(int) ).

% odd_nonzero
tff(fact_1257_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I)),aa(int,int,aa(int,fun(int,int),times_times(int),K),J)) ) ) ).

% zmult_zless_mono2
tff(fact_1258_nonneg__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ~ ! [N: nat] : K != aa(nat,int,semiring_1_of_nat(int),N) ) ).

% nonneg_int_cases
tff(fact_1259_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ? [N: nat] : K = aa(nat,int,semiring_1_of_nat(int),N) ) ).

% zero_le_imp_eq_int
tff(fact_1260_real__arch__pow__inv,axiom,
    ! [Y: real,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),one_one(real))
       => ? [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,power_power(real,X),N)),Y) ) ) ).

% real_arch_pow_inv
tff(fact_1261_ex__nat,axiom,
    ! [P: fun(nat,$o)] :
      ( ? [X_1: nat] : aa(nat,$o,P,X_1)
    <=> ? [X6: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X6)
          & aa(nat,$o,P,aa(int,nat,nat2,X6)) ) ) ).

% ex_nat
tff(fact_1262_all__nat,axiom,
    ! [P: fun(nat,$o)] :
      ( ! [X_1: nat] : aa(nat,$o,P,X_1)
    <=> ! [X6: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X6)
         => aa(nat,$o,P,aa(int,nat,nat2,X6)) ) ) ).

% all_nat
tff(fact_1263_eq__nat__nat__iff,axiom,
    ! [Z: int,Z4: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z4)
       => ( ( aa(int,nat,nat2,Z) = aa(int,nat,nat2,Z4) )
        <=> ( Z = Z4 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_1264_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),$o,P,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),N)) )
           => aa(word(A),$o,P,M) ) ) ) ).

% word_induct
tff(fact_1265_word__induct2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(word(A),$o),Na: word(A)] :
          ( aa(word(A),$o,P,zero_zero(word(A)))
         => ( ! [N: word(A)] :
                ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),N) != zero_zero(word(A)) )
               => ( aa(word(A),$o,P,N)
                 => aa(word(A),$o,P,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),N)) ) )
           => aa(word(A),$o,P,Na) ) ) ) ).

% word_induct2
tff(fact_1266_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,aa(word(A),fun(word(A),$o),ord_less(word(A)),N),M)
               => ( aa(word(A),$o,P,N)
                 => aa(word(A),$o,P,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),N)) ) )
           => aa(word(A),$o,P,M) ) ) ) ).

% word_induct_less
tff(fact_1267_word__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),one_one(word(A))))
          | ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),one_one(word(A))) = zero_zero(word(A)) ) ) ) ).

% word_overflow
tff(fact_1268_word__gr0__conv__Suc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),M)
         => ? [N: word(A)] : M = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),N),one_one(word(A))) ) ) ).

% word_gr0_conv_Suc
tff(fact_1269_less__is__non__zero__p1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),K: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Aa2),K)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Aa2),one_one(word(A))) != zero_zero(word(A)) ) ) ) ).

% less_is_non_zero_p1
tff(fact_1270_neq__0__no__wrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y))
         => ( ( X != zero_zero(word(A)) )
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y) != zero_zero(word(A)) ) ) ) ) ).

% neq_0_no_wrap
tff(fact_1271_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_1272_reals__Archimedean3,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ! [Y3: real] :
        ? [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y3),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_1273_real__add__less__0__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,uminus_uminus(real),X)) ) ).

% real_add_less_0_iff
tff(fact_1274_real__0__less__add__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),X)),Y) ) ).

% real_0_less_add_iff
tff(fact_1275_AND__lower,axiom,
    ! [X: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y)) ) ).

% AND_lower
tff(fact_1276_AND__upper1,axiom,
    ! [X: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y)),X) ) ).

% AND_upper1
tff(fact_1277_AND__upper2,axiom,
    ! [Y: int,X: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y)),Y) ) ).

% AND_upper2
tff(fact_1278_AND__upper1_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Y),Z)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Y),Ya)),Z) ) ) ).

% AND_upper1'
tff(fact_1279_AND__upper2_H,axiom,
    ! [Y: int,Z: int,X: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Y),Z)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y)),Z) ) ) ).

% AND_upper2'
tff(fact_1280_OR__lower,axiom,
    ! [X: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),X),Y)) ) ) ).

% OR_lower
tff(fact_1281_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)) ) ).

% or_greater_eq
tff(fact_1282_XOR__lower,axiom,
    ! [X: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),X),Y)) ) ) ).

% XOR_lower
tff(fact_1283_length__pos__if__in__set,axiom,
    ! [A: $tType,X: A,Xs2: list(A)] :
      ( member(A,X,aa(list(A),set(A),set2(A),Xs2))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(list(A),nat,size_size(list(A)),Xs2)) ) ).

% length_pos_if_in_set
tff(fact_1284_nat__mono__iff,axiom,
    ! [Z: int,W: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(int,nat,nat2,W)),aa(int,nat,nat2,Z))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),Z) ) ) ).

% nat_mono_iff
tff(fact_1285_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),Z)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z) ) ).

% int_one_le_iff_zero_less
tff(fact_1286_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z)),Z)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),zero_zero(int)) ) ).

% odd_less_0_iff
tff(fact_1287_pos__zmult__eq__1__iff,axiom,
    ! [M: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),M)
     => ( ( aa(int,int,aa(int,fun(int,int),times_times(int),M),Na) = one_one(int) )
      <=> ( ( M = one_one(int) )
          & ( Na = one_one(int) ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_1288_nonempty__has__size,axiom,
    ! [A: $tType,Sa: multiset(A)] :
      ( ( Sa != zero_zero(multiset(A)) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(multiset(A),nat,size_size(multiset(A)),Sa)) ) ).

% nonempty_has_size
tff(fact_1289_negative__zle__0,axiom,
    ! [Na: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),Na))),zero_zero(int)) ).

% negative_zle_0
tff(fact_1290_nonpos__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),zero_zero(int))
     => ~ ! [N: nat] : K != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N)) ) ).

% nonpos_int_cases
tff(fact_1291_real__add__le__0__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,uminus_uminus(real),X)) ) ).

% real_add_le_0_iff
tff(fact_1292_real__0__le__add__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),X)),Y) ) ).

% real_0_le_add_iff
tff(fact_1293_word__plus__one__nonzero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y))
         => ( ( Y != zero_zero(word(A)) )
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),one_one(word(A))) != zero_zero(word(A)) ) ) ) ) ).

% word_plus_one_nonzero
tff(fact_1294_plus__one__helper2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Na: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),Na)
         => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Na),one_one(word(A))) != zero_zero(word(A)) )
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Na),one_one(word(A)))) ) ) ) ).

% plus_one_helper2
tff(fact_1295_nat__0__le,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => ( aa(nat,int,semiring_1_of_nat(int),aa(int,nat,nat2,Z)) = Z ) ) ).

% nat_0_le
tff(fact_1296_int__eq__iff,axiom,
    ! [M: nat,Z: int] :
      ( ( aa(nat,int,semiring_1_of_nat(int),M) = Z )
    <=> ( ( M = aa(int,nat,nat2,Z) )
        & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z) ) ) ).

% int_eq_iff
tff(fact_1297_max__word__wrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(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_1298_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,X: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Y),Z)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y)),Z) ) ) ).

% AND_upper2''
tff(fact_1299_AND__upper1_H_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Y),Z)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Y),Ya)),Z) ) ) ).

% AND_upper1''
tff(fact_1300_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)),K) ) ).

% and_less_eq
tff(fact_1301_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),aa(word(A),fun(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_1302_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_1303_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_1304_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ? [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
          & ( K = aa(nat,int,semiring_1_of_nat(int),N) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_1305_pos__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ~ ! [N: nat] :
            ( ( K = aa(nat,int,semiring_1_of_nat(int),N) )
           => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ) ).

% pos_int_cases
tff(fact_1306_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero(int) )
     => ( ! [N: nat] :
            ( ( K = aa(nat,int,semiring_1_of_nat(int),N) )
           => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) )
       => ~ ! [N: nat] :
              ( ( K = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N)) )
             => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ) ) ).

% int_cases3
tff(fact_1307_nat__less__eq__zless,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),W)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(int,nat,nat2,W)),aa(int,nat,nat2,Z))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),Z) ) ) ).

% nat_less_eq_zless
tff(fact_1308_nat__eq__iff,axiom,
    ! [W: int,M: nat] :
      ( ( aa(int,nat,nat2,W) = M )
    <=> $ite(aa(int,$o,aa(int,fun(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_1309_nat__eq__iff2,axiom,
    ! [M: nat,W: int] :
      ( ( M = aa(int,nat,nat2,W) )
    <=> $ite(aa(int,$o,aa(int,fun(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_1310_split__nat,axiom,
    ! [P: fun(nat,$o),I: int] :
      ( aa(nat,$o,P,aa(int,nat,nat2,I))
    <=> ( ! [N2: nat] :
            ( ( I = aa(nat,int,semiring_1_of_nat(int),N2) )
           => aa(nat,$o,P,N2) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),zero_zero(int))
         => aa(nat,$o,P,zero_zero(nat)) ) ) ) ).

% split_nat
tff(fact_1311_nat__le__eq__zle,axiom,
    ! [W: int,Z: int] :
      ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),W)
        | aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,W)),aa(int,nat,nat2,Z))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W),Z) ) ) ).

% nat_le_eq_zle
tff(fact_1312_nat__add__distrib,axiom,
    ! [Z: int,Z4: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z4)
       => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),Z4)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z4)) ) ) ) ).

% nat_add_distrib
tff(fact_1313_le__imp__0__less,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z)) ) ).

% le_imp_0_less
tff(fact_1314_pos__mult__pos__ge,axiom,
    ! [X: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),X)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Na)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),Na),one_one(int))),aa(int,int,aa(int,fun(int,int),times_times(int),Na),X)) ) ) ).

% pos_mult_pos_ge
tff(fact_1315_le__nat__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(int,nat,nat2,K))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),Na)),K) ) ) ).

% le_nat_iff
tff(fact_1316_q__pos__lemma,axiom,
    ! [B4: int,Q2: int,R4: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B4),Q2)),R4))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R4),B4)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B4)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Q2) ) ) ) ).

% q_pos_lemma
tff(fact_1317_zdiv__mono2__lemma,axiom,
    ! [Ba: int,Q3: int,R: int,B4: int,Q2: int,R4: int] :
      ( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Ba),Q3)),R) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B4),Q2)),R4) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B4),Q2)),R4))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R4),B4)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B4)
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B4),Ba)
               => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q3),Q2) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_1318_zdiv__mono2__neg__lemma,axiom,
    ! [Ba: int,Q3: int,R: int,B4: int,Q2: int,R4: int] :
      ( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Ba),Q3)),R) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B4),Q2)),R4) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B4),Q2)),R4)),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R),Ba)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R4)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B4)
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B4),Ba)
               => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q2),Q3) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_1319_unique__quotient__lemma,axiom,
    ! [Ba: int,Q2: int,R4: int,Q3: int,R: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Ba),Q2)),R4)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Ba),Q3)),R))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R4)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R4),Ba)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R),Ba)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q2),Q3) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_1320_unique__quotient__lemma__neg,axiom,
    ! [Ba: int,Q2: int,R4: int,Q3: int,R: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Ba),Q2)),R4)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Ba),Q3)),R))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ba),R)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ba),R4)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q3),Q2) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_1321_nat__mult__distrib,axiom,
    ! [Z: int,Z4: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),times_times(int),Z),Z4)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z4)) ) ) ).

% nat_mult_distrib
tff(fact_1322_nat__power__eq,axiom,
    ! [Z: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => ( aa(int,nat,nat2,aa(nat,int,power_power(int,Z),Na)) = aa(nat,nat,power_power(nat,aa(int,nat,nat2,Z)),Na) ) ) ).

% nat_power_eq
tff(fact_1323_log__mult,axiom,
    ! [Aa2: real,X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Aa2)
     => ( ( Aa2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
           => ( aa(real,real,log(Aa2),aa(real,real,aa(real,fun(real,real),times_times(real),X),Y)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,log(Aa2),X)),aa(real,real,log(Aa2),Y)) ) ) ) ) ) ).

% log_mult
tff(fact_1324_log__nat__power,axiom,
    ! [X: real,Ba: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(real,real,log(Ba),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(Ba),X)) ) ) ).

% log_nat_power
tff(fact_1325_not__exp__less__eq__0__int,axiom,
    ! [Na: nat] : ~ aa(int,$o,aa(int,fun(int,$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_1326_neg__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
     => ~ ! [N: nat] :
            ( ( K = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N)) )
           => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ) ).

% neg_int_cases
tff(fact_1327_real__archimedian__rdiv__eq__0,axiom,
    ! [X: real,C2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),C2)
       => ( ! [M5: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M5)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M5)),X)),C2) )
         => ( X = zero_zero(real) ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_1328_nat__less__iff,axiom,
    ! [W: int,M: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),W)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(int,nat,nat2,W)),M)
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),aa(nat,int,semiring_1_of_nat(int),M)) ) ) ).

% nat_less_iff
tff(fact_1329_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z4: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),zero_zero(int))
     => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),times_times(int),Z),Z4)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),Z))),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),Z4))) ) ) ).

% nat_mult_distrib_neg
tff(fact_1330_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_1331_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_1332_int__bit__induct,axiom,
    ! [P: fun(int,$o),K: int] :
      ( aa(int,$o,P,zero_zero(int))
     => ( aa(int,$o,P,aa(int,int,uminus_uminus(int),one_one(int)))
       => ( ! [K5: int] :
              ( aa(int,$o,P,K5)
             => ( ( K5 != zero_zero(int) )
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),times_times(int),K5),aa(num,int,numeral_numeral(int),bit0(one2)))) ) )
         => ( ! [K5: int] :
                ( aa(int,$o,P,K5)
               => ( ( K5 != aa(int,int,uminus_uminus(int),one_one(int)) )
                 => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),K5),aa(num,int,numeral_numeral(int),bit0(one2))))) ) )
           => aa(int,$o,P,K) ) ) ) ) ).

% int_bit_induct
tff(fact_1333_OR__upper,axiom,
    ! [X: int,Na: nat,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Y),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))
         => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),X),Y)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ) ) ).

% OR_upper
tff(fact_1334_XOR__upper,axiom,
    ! [X: int,Na: nat,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Y),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))
         => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),X),Y)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ) ) ).

% XOR_upper
tff(fact_1335_mult__less__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y) ) ) ) ).

% mult_less_iff1
tff(fact_1336_field__le__epsilon,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( ! [E: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),E)) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ).

% field_le_epsilon
tff(fact_1337_linear__plus__1__le__power,axiom,
    ! [X: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),X)),one_one(real))),aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),one_one(real))),Na)) ) ).

% linear_plus_1_le_power
tff(fact_1338_incr__mult__lemma,axiom,
    ! [D2: int,P: fun(int,$o),K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( ! [X2: int] :
            ( aa(int,$o,P,X2)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D2)) )
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
         => ! [X4: int] :
              ( aa(int,$o,P,X4)
             => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),aa(int,int,aa(int,fun(int,int),times_times(int),K),D2))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_1339_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_1340_vebt__buildupi__rule,axiom,
    ! [Na: nat] : time_htt(vEBT_VEBTi,pure_assn(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)),vEBT_vebt_buildupi(Na),vEBT_Intf_vebt_assn(Na,bot_bot(set(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit1(bit0(one2))))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ).

% vebt_buildupi_rule
tff(fact_1341_n__lists__Nil,axiom,
    ! [A: $tType,Na: nat] :
      n_lists(A,Na,nil(A)) = $ite(Na = zero_zero(nat),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),nil(list(A))),nil(list(A))) ).

% n_lists_Nil
tff(fact_1342_is__succ__in__set__def,axiom,
    ! [Xs2: set(nat),X: nat,Y: nat] :
      ( vEBT_is_succ_in_set(Xs2,X,Y)
    <=> ( member(nat,Y,Xs2)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Y)
        & ! [X6: nat] :
            ( member(nat,X6,Xs2)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),X6)
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),X6) ) ) ) ) ).

% is_succ_in_set_def
tff(fact_1343_is__pred__in__set__def,axiom,
    ! [Xs2: set(nat),X: nat,Y: nat] :
      ( vEBT_is_pred_in_set(Xs2,X,Y)
    <=> ( member(nat,Y,Xs2)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),X)
        & ! [X6: nat] :
            ( member(nat,X6,Xs2)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X6),X)
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X6),Y) ) ) ) ) ).

% is_pred_in_set_def
tff(fact_1344_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_as(A,A),Na,zero_zero(A)) ) ).

% of_nat_code
tff(fact_1345_floor__log__nat__eq__powr__iff,axiom,
    ! [Ba: nat,K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Ba)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => ( ( archim6421214686448440834_floor(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),Ba)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(nat,int,semiring_1_of_nat(int),Na) )
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,Ba),Na)),K)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,power_power(nat,Ba),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat)))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_1346_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_1347_set__empty,axiom,
    ! [A: $tType,Xs2: list(A)] :
      ( ( aa(list(A),set(A),set2(A),Xs2) = bot_bot(set(A)) )
    <=> ( Xs2 = nil(A) ) ) ).

% set_empty
tff(fact_1348_set__empty2,axiom,
    ! [A: $tType,Xs2: list(A)] :
      ( ( bot_bot(set(A)) = aa(list(A),set(A),set2(A),Xs2) )
    <=> ( Xs2 = nil(A) ) ) ).

% set_empty2
tff(fact_1349_floor__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archim6421214686448440834_floor(A,zero_zero(A)) = zero_zero(int) ) ) ).

% floor_zero
tff(fact_1350_floor__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num] : archim6421214686448440834_floor(A,aa(num,A,numeral_numeral(A),V)) = aa(num,int,numeral_numeral(int),V) ) ).

% floor_numeral
tff(fact_1351_floor__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archim6421214686448440834_floor(A,one_one(A)) = one_one(int) ) ) ).

% floor_one
tff(fact_1352_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_1353_zero__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),archim6421214686448440834_floor(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X) ) ) ).

% zero_le_floor
tff(fact_1354_numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),V)),archim6421214686448440834_floor(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),V)),X) ) ) ).

% numeral_le_floor
tff(fact_1355_floor__less__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,X)),zero_zero(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),zero_zero(A)) ) ) ).

% floor_less_zero
tff(fact_1356_floor__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,X)),aa(num,int,numeral_numeral(int),V))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(num,A,numeral_numeral(A),V)) ) ) ).

% floor_less_numeral
tff(fact_1357_zero__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),archim6421214686448440834_floor(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),X) ) ) ).

% zero_less_floor
tff(fact_1358_floor__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,X)),zero_zero(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),one_one(A)) ) ) ).

% floor_le_zero
tff(fact_1359_one__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),archim6421214686448440834_floor(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),X) ) ) ).

% one_le_floor
tff(fact_1360_floor__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num] : archim6421214686448440834_floor(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)) ) ).

% floor_neg_numeral
tff(fact_1361_floor__less__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,X)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),one_one(A)) ) ) ).

% floor_less_one
tff(fact_1362_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_1363_numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(num,int,numeral_numeral(int),V)),archim6421214686448440834_floor(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V)),one_one(A))),X) ) ) ).

% numeral_less_floor
tff(fact_1364_floor__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,X)),aa(num,int,numeral_numeral(int),V))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V)),one_one(A))) ) ) ).

% floor_le_numeral
tff(fact_1365_one__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),archim6421214686448440834_floor(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),bit0(one2))),X) ) ) ).

% one_less_floor
tff(fact_1366_floor__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,X)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% floor_le_one
tff(fact_1367_neg__numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archim6421214686448440834_floor(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),X) ) ) ).

% neg_numeral_le_floor
tff(fact_1368_floor__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) ) ) ).

% floor_less_neg_numeral
tff(fact_1369_neg__numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archim6421214686448440834_floor(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),one_one(A))),X) ) ) ).

% neg_numeral_less_floor
tff(fact_1370_floor__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),one_one(A))) ) ) ).

% floor_le_neg_numeral
tff(fact_1371_iadd__is__0,axiom,
    ! [M: extended_enat,Na: extended_enat] :
      ( ( aa(extended_enat,extended_enat,aa(extended_enat,fun(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_1372_zero__one__enat__neq_I1_J,axiom,
    zero_zero(extended_enat) != one_one(extended_enat) ).

% zero_one_enat_neq(1)
tff(fact_1373_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_1374_bot__fun__def,axiom,
    ! [A: $tType,B: $tType] :
      ( bot(B)
     => ! [X4: A] : aa(A,B,bot_bot(fun(A,B)),X4) = bot_bot(B) ) ).

% bot_fun_def
tff(fact_1375_set__notEmptyE,axiom,
    ! [A: $tType,Sa: set(A)] :
      ( ( Sa != bot_bot(set(A)) )
     => ~ ! [X2: A] : ~ member(A,X2,Sa) ) ).

% set_notEmptyE
tff(fact_1376_memb__imp__not__empty,axiom,
    ! [A: $tType,X: A,Sa: set(A)] :
      ( member(A,X,Sa)
     => ( Sa != bot_bot(set(A)) ) ) ).

% memb_imp_not_empty
tff(fact_1377_length__n__lists__elem,axiom,
    ! [A: $tType,Ysb: list(A),Na: nat,Xs2: list(A)] :
      ( member(list(A),Ysb,aa(list(list(A)),set(list(A)),set2(list(A)),n_lists(A,Na,Xs2)))
     => ( aa(list(A),nat,size_size(list(A)),Ysb) = Na ) ) ).

% length_n_lists_elem
tff(fact_1378_bot_Oextremum,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Aa2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),bot_bot(A)),Aa2) ) ).

% bot.extremum
tff(fact_1379_bot_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),bot_bot(A))
        <=> ( Aa2 = bot_bot(A) ) ) ) ).

% bot.extremum_unique
tff(fact_1380_bot_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),bot_bot(A))
         => ( Aa2 = bot_bot(A) ) ) ) ).

% bot.extremum_uniqueI
tff(fact_1381_bot_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Aa2: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),bot_bot(A)) ) ).

% bot.extremum_strict
tff(fact_1382_bot_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Aa2: A] :
          ( ( Aa2 != bot_bot(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),bot_bot(A)),Aa2) ) ) ).

% bot.not_eq_extremum
tff(fact_1383_floor__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y)) ) ) ).

% floor_mono
tff(fact_1384_floor__less__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y) ) ) ).

% floor_less_cancel
tff(fact_1385_floor__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,X)),archimedean_ceiling(A,X)) ) ).

% floor_le_ceiling
tff(fact_1386_empty__set,axiom,
    ! [A: $tType] : bot_bot(set(A)) = aa(list(A),set(A),set2(A),nil(A)) ).

% empty_set
tff(fact_1387_in__set__product__lists__length,axiom,
    ! [A: $tType,Xs2: list(A),Xss2: list(list(A))] :
      ( member(list(A),Xs2,aa(list(list(A)),set(list(A)),set2(list(A)),product_lists(A,Xss2)))
     => ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(list(A)),nat,size_size(list(list(A))),Xss2) ) ) ).

% in_set_product_lists_length
tff(fact_1388_le__floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y))),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y))) ) ).

% le_floor_add
tff(fact_1389_nat__floor__neg,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),zero_zero(real))
     => ( aa(int,nat,nat2,archim6421214686448440834_floor(real,X)) = zero_zero(nat) ) ) ).

% nat_floor_neg
tff(fact_1390_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_1391_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_1392_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_1393_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_1394_of__nat__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),R)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,archim6421214686448440834_floor(A,R)))),R) ) ) ).

% of_nat_floor
tff(fact_1395_one__add__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),one_one(int)) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),one_one(A))) ) ).

% one_add_floor
tff(fact_1396_le__mult__nat__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Aa2: A,Ba: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(int,nat,nat2,archim6421214686448440834_floor(A,Aa2))),aa(int,nat,nat2,archim6421214686448440834_floor(A,Ba)))),aa(int,nat,nat2,archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)))) ) ).

% le_mult_nat_floor
tff(fact_1397_le__nat__floor,axiom,
    ! [X: nat,Aa2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),X)),Aa2)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),aa(int,nat,nat2,archim6421214686448440834_floor(real,Aa2))) ) ).

% le_nat_floor
tff(fact_1398_le__mult__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Ba)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(A,Aa2)),archim6421214686448440834_floor(A,Ba))),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba))) ) ) ) ).

% le_mult_floor
tff(fact_1399_set__n__lists,axiom,
    ! [A: $tType,Na: nat,Xs2: list(A)] : aa(list(list(A)),set(list(A)),set2(list(A)),n_lists(A,Na,Xs2)) = collect(list(A),aa(list(A),fun(list(A),$o),aTP_Lamp_at(nat,fun(list(A),fun(list(A),$o)),Na),Xs2)) ).

% set_n_lists
tff(fact_1400_minf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T: A] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z3)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),T),X4) ) ) ).

% minf(8)
tff(fact_1401_minf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T: A] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),T) ) ) ).

% minf(6)
tff(fact_1402_pinf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T: A] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X4)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),T),X4) ) ) ).

% pinf(8)
tff(fact_1403_pinf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T: A] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X4)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),T) ) ) ).

% pinf(6)
tff(fact_1404_conj__le__cong,axiom,
    ! [X: int,X7: int,P: $o,P3: $o] :
      ( ( X = X7 )
     => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X7)
         => ( (P)
          <=> (P3) ) )
       => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
            & (P) )
        <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X7)
            & (P3) ) ) ) ) ).

% conj_le_cong
tff(fact_1405_imp__le__cong,axiom,
    ! [X: int,X7: int,P: $o,P3: $o] :
      ( ( X = X7 )
     => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X7)
         => ( (P)
          <=> (P3) ) )
       => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
           => (P) )
        <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X7)
           => (P3) ) ) ) ) ).

% imp_le_cong
tff(fact_1406_n__lists_Osimps_I1_J,axiom,
    ! [A: $tType,Xs2: list(A)] : n_lists(A,zero_zero(nat),Xs2) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),nil(list(A))) ).

% n_lists.simps(1)
tff(fact_1407_floor__log__nat__eq__if,axiom,
    ! [Ba: nat,Na: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,Ba),Na)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,power_power(nat,Ba),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat))))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Ba)
         => ( archim6421214686448440834_floor(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),Ba)),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_1408_even__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A)] :
          ( aa(word(A),$o,even_word(A),Aa2)
        <=> ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Aa2),one_one(word(A))) = zero_zero(word(A)) ) ) ) ).

% even_word_iff
tff(fact_1409_length__n__lists,axiom,
    ! [A: $tType,Na: nat,Xs2: list(A)] : aa(list(list(A)),nat,size_size(list(list(A))),n_lists(A,Na,Xs2)) = aa(nat,nat,power_power(nat,aa(list(A),nat,size_size(list(A)),Xs2)),Na) ).

% length_n_lists
tff(fact_1410_ceiling__log__eq__powr__iff,axiom,
    ! [X: real,Ba: real,K: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Ba)
       => ( ( archimedean_ceiling(real,aa(real,real,log(Ba),X)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),K)),one_one(int)) )
        <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,Ba,aa(nat,real,semiring_1_of_nat(real),K))),X)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),powr(real,Ba,aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),one_one(nat))))) ) ) ) ) ).

% ceiling_log_eq_powr_iff
tff(fact_1411_floor__log2__div2,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => ( archim6421214686448440834_floor(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),one_one(int)) ) ) ).

% floor_log2_div2
tff(fact_1412_in__set__simps_I2_J,axiom,
    ! [A: $tType,X: A,Y: A] :
      ( member(A,X,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A))))
    <=> ( X = Y ) ) ).

% in_set_simps(2)
tff(fact_1413_length__nth__simps_I1_J,axiom,
    ! [A: $tType] : aa(list(A),nat,size_size(list(A)),nil(A)) = zero_zero(nat) ).

% length_nth_simps(1)
tff(fact_1414_bits__div__by__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),zero_zero(A)) = zero_zero(A) ) ).

% bits_div_by_0
tff(fact_1415_bits__div__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),zero_zero(A)),Aa2) = zero_zero(A) ) ).

% bits_div_0
tff(fact_1416_div__by__0,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),zero_zero(A)) = zero_zero(A) ) ).

% div_by_0
tff(fact_1417_div__0,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),zero_zero(A)),Aa2) = zero_zero(A) ) ).

% div_0
tff(fact_1418_div__by__1,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),one_one(A)) = Aa2 ) ).

% div_by_1
tff(fact_1419_bits__div__by__1,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),one_one(A)) = Aa2 ) ).

% bits_div_by_1
tff(fact_1420_times__divide__eq__right,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Aa2: A,Ba: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)),C2) ) ).

% times_divide_eq_right
tff(fact_1421_divide__divide__eq__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Aa2: A,Ba: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),Ba) ) ).

% divide_divide_eq_right
tff(fact_1422_divide__divide__eq__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Aa2: A,Ba: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba)),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2)) ) ).

% divide_divide_eq_left
tff(fact_1423_times__divide__eq__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Ba: A,C2: A,Aa2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2)),Aa2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),Aa2)),C2) ) ).

% times_divide_eq_left
tff(fact_1424_powr__one__eq__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Aa2: A] : powr(A,one_one(A),Aa2) = one_one(A) ) ).

% powr_one_eq_one
tff(fact_1425_zero__eq__1__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),Aa2) )
        <=> ( Aa2 = zero_zero(A) ) ) ) ).

% zero_eq_1_divide_iff
tff(fact_1426_one__divide__eq__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),Aa2) = zero_zero(A) )
        <=> ( Aa2 = zero_zero(A) ) ) ) ).

% one_divide_eq_0_iff
tff(fact_1427_eq__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Ba: A,Aa2: A] :
          ( ( one_one(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),Aa2) )
        <=> ( ( Aa2 != zero_zero(A) )
            & ( Aa2 = Ba ) ) ) ) ).

% eq_divide_eq_1
tff(fact_1428_divide__eq__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Ba: A,Aa2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),Aa2) = one_one(A) )
        <=> ( ( Aa2 != zero_zero(A) )
            & ( Aa2 = Ba ) ) ) ) ).

% divide_eq_eq_1
tff(fact_1429_divide__self__if,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Aa2: A] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Aa2) = $ite(Aa2 = zero_zero(A),zero_zero(A),one_one(A)) ) ).

% divide_self_if
tff(fact_1430_divide__self,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Aa2: A] :
          ( ( Aa2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Aa2) = one_one(A) ) ) ) ).

% divide_self
tff(fact_1431_one__eq__divide__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Aa2: A,Ba: A] :
          ( ( one_one(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba) )
        <=> ( ( Ba != zero_zero(A) )
            & ( Aa2 = Ba ) ) ) ) ).

% one_eq_divide_iff
tff(fact_1432_divide__eq__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Aa2: A,Ba: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba) = one_one(A) )
        <=> ( ( Ba != zero_zero(A) )
            & ( Aa2 = Ba ) ) ) ) ).

% divide_eq_1_iff
tff(fact_1433_div__self,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [Aa2: A] :
          ( ( Aa2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Aa2) = one_one(A) ) ) ) ).

% div_self
tff(fact_1434_mult__divide__mult__cancel__left__if,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,Aa2: A,Ba: A] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Aa2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Ba)) = $ite(C2 = zero_zero(A),zero_zero(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba)) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_1435_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,Aa2: A,Ba: 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),Aa2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Ba)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_1436_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,Aa2: A,Ba: 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),Aa2)),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_1437_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,Aa2: A,Ba: 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),Aa2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_1438_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,Aa2: A,Ba: 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),Aa2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Ba)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_1439_nonzero__mult__div__cancel__right,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [Ba: A,Aa2: A] :
          ( ( Ba != 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),Aa2),Ba)),Ba) = Aa2 ) ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1440_nonzero__mult__div__cancel__left,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [Aa2: A,Ba: A] :
          ( ( Aa2 != 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),Aa2),Ba)),Aa2) = Ba ) ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1441_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_1442_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_1443_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_1444_powr__nonneg__iff,axiom,
    ! [Aa2: real,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Aa2,X)),zero_zero(real))
    <=> ( Aa2 = zero_zero(real) ) ) ).

% powr_nonneg_iff
tff(fact_1445_divide__le__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),Aa2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),zero_zero(A)) ) ) ).

% divide_le_0_1_iff
tff(fact_1446_zero__le__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),Aa2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2) ) ) ).

% zero_le_divide_1_iff
tff(fact_1447_divide__less__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),Aa2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),zero_zero(A)) ) ) ).

% divide_less_0_1_iff
tff(fact_1448_divide__less__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),Aa2)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba) ) ) ) ).

% divide_less_eq_1_neg
tff(fact_1449_divide__less__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),Aa2)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),Aa2) ) ) ) ).

% divide_less_eq_1_pos
tff(fact_1450_less__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),Aa2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),Aa2) ) ) ) ).

% less_divide_eq_1_neg
tff(fact_1451_less__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),Aa2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba) ) ) ) ).

% less_divide_eq_1_pos
tff(fact_1452_zero__less__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),Aa2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2) ) ) ).

% zero_less_divide_1_iff
tff(fact_1453_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Aa2: A,Ba: A,W: num] :
          ( ( Aa2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),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),Aa2),aa(num,A,numeral_numeral(A),W)) = Ba,Aa2 = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_1454_divide__eq__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Ba: A,W: num,Aa2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),aa(num,A,numeral_numeral(A),W)) = Aa2 )
        <=> $ite(aa(num,A,numeral_numeral(A),W) != zero_zero(A),Ba = aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(num,A,numeral_numeral(A),W)),Aa2 = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_1455_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A,Ba: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),aa(num,A,numeral_numeral(A),W)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(num,A,numeral_numeral(A),W))),Ba) ) ) ).

% le_divide_eq_numeral1(1)
tff(fact_1456_divide__le__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Ba: A,W: num,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),aa(num,A,numeral_numeral(A),W))),Aa2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(num,A,numeral_numeral(A),W))) ) ) ).

% divide_le_eq_numeral1(1)
tff(fact_1457_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A,Ba: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),aa(num,A,numeral_numeral(A),W)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(num,A,numeral_numeral(A),W))),Ba) ) ) ).

% less_divide_eq_numeral1(1)
tff(fact_1458_divide__less__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Ba: A,W: num,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),aa(num,A,numeral_numeral(A),W))),Aa2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(num,A,numeral_numeral(A),W))) ) ) ).

% divide_less_eq_numeral1(1)
tff(fact_1459_nonzero__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Aa2: A,Ba: A] :
          ( ( Aa2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),Ba) ) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_1460_nonzero__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Ba: A,Aa2: A] :
          ( ( Ba != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),Aa2) ) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_1461_powr__one__gt__zero__iff,axiom,
    ! [X: real] :
      ( ( powr(real,X,one_one(real)) = X )
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X) ) ).

% powr_one_gt_zero_iff
tff(fact_1462_powr__one,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( powr(real,X,one_one(real)) = X ) ) ).

% powr_one
tff(fact_1463_powr__le__cancel__iff,axiom,
    ! [X: real,Aa2: real,Ba: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,X,Aa2)),powr(real,X,Ba))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),Ba) ) ) ).

% powr_le_cancel_iff
tff(fact_1464_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_1465_le__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),Aa2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba) ) ) ) ).

% le_divide_eq_1_pos
tff(fact_1466_le__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),Aa2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2) ) ) ) ).

% le_divide_eq_1_neg
tff(fact_1467_divide__le__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),Aa2)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2) ) ) ) ).

% divide_le_eq_1_pos
tff(fact_1468_divide__le__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),Aa2)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba) ) ) ) ).

% divide_le_eq_1_neg
tff(fact_1469_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Aa2: A,Ba: A,W: num] :
          ( ( Aa2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),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),Aa2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) = Ba,Aa2 = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_1470_divide__eq__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Ba: A,W: num,Aa2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) = Aa2 )
        <=> $ite(aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) != zero_zero(A),Ba = aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),Aa2 = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_1471_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A,Ba: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))) ) ) ).

% le_divide_eq_numeral1(2)
tff(fact_1472_divide__le__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Ba: A,W: num,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),Aa2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),Ba) ) ) ).

% divide_le_eq_numeral1(2)
tff(fact_1473_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A,Ba: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))) ) ) ).

% less_divide_eq_numeral1(2)
tff(fact_1474_divide__less__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Ba: A,W: num,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),Aa2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),Ba) ) ) ).

% divide_less_eq_numeral1(2)
tff(fact_1475_powr__log__cancel,axiom,
    ! [Aa2: real,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Aa2)
     => ( ( Aa2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
         => ( powr(real,Aa2,aa(real,real,log(Aa2),X)) = X ) ) ) ) ).

% powr_log_cancel
tff(fact_1476_log__powr__cancel,axiom,
    ! [Aa2: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Aa2)
     => ( ( Aa2 != one_one(real) )
       => ( aa(real,real,log(Aa2),powr(real,Aa2,Y)) = Y ) ) ) ).

% log_powr_cancel
tff(fact_1477_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_1478_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_1479_powr__numeral,axiom,
    ! [X: real,Na: num] :
      ( aa(real,$o,aa(real,fun(real,$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_1480_bot__nat__def,axiom,
    bot_bot(nat) = zero_zero(nat) ).

% bot_nat_def
tff(fact_1481_add__divide__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Aa2: A,Ba: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2)) ) ).

% add_divide_distrib
tff(fact_1482_divide__divide__eq__left_H,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Aa2: A,Ba: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba)),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Ba)) ) ).

% divide_divide_eq_left'
tff(fact_1483_divide__divide__times__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,Y: A,Z: A,W: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Z),W)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),W)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)) ) ).

% divide_divide_times_eq
tff(fact_1484_times__divide__times__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,Y: A,Z: A,W: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Z),W)) = aa(A,A,aa(A,fun(A,A),divide_divide(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),W)) ) ).

% times_divide_times_eq
tff(fact_1485_powr__minus__divide,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [X: A,Aa2: A] : powr(A,X,aa(A,A,uminus_uminus(A),Aa2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),powr(A,X,Aa2)) ) ).

% powr_minus_divide
tff(fact_1486_bot__enat__def,axiom,
    bot_bot(extended_enat) = zero_zero(extended_enat) ).

% bot_enat_def
tff(fact_1487_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_1488_powr__mono2,axiom,
    ! [Aa2: real,X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Aa2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,X,Aa2)),powr(real,Y,Aa2)) ) ) ) ).

% powr_mono2
tff(fact_1489_powr__ge__pzero,axiom,
    ! [X: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),powr(real,X,Y)) ).

% powr_ge_pzero
tff(fact_1490_powr__mono,axiom,
    ! [Aa2: real,Ba: real,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),Ba)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),X)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,X,Aa2)),powr(real,X,Ba)) ) ) ).

% powr_mono
tff(fact_1491_divide__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),C2)) ) ) ) ).

% divide_right_mono_neg
tff(fact_1492_divide__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)) ) ) ) ).

% divide_nonpos_nonpos
tff(fact_1493_divide__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),zero_zero(A)) ) ) ) ).

% divide_nonpos_nonneg
tff(fact_1494_divide__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),zero_zero(A)) ) ) ) ).

% divide_nonneg_nonpos
tff(fact_1495_divide__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)) ) ) ) ).

% divide_nonneg_nonneg
tff(fact_1496_zero__le__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Ba) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),zero_zero(A)) ) ) ) ) ).

% zero_le_divide_iff
tff(fact_1497_divide__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2)) ) ) ) ).

% divide_right_mono
tff(fact_1498_divide__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba)),zero_zero(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Ba) ) ) ) ) ).

% divide_le_0_iff
tff(fact_1499_right__inverse__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Ba: A,Aa2: A] :
          ( ( Ba != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba) = one_one(A) )
          <=> ( Aa2 = Ba ) ) ) ) ).

% right_inverse_eq
tff(fact_1500_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_1501_divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Ba: A,C2: A,Aa2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2) = Aa2 )
        <=> $ite(C2 != zero_zero(A),Ba = aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2),Aa2 = zero_zero(A)) ) ) ).

% divide_eq_eq
tff(fact_1502_eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( ( Aa2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2) )
        <=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2) = Ba,Aa2 = zero_zero(A)) ) ) ).

% eq_divide_eq
tff(fact_1503_divide__eq__imp,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,Ba: A,Aa2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( Ba = aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2) = Aa2 ) ) ) ) ).

% divide_eq_imp
tff(fact_1504_eq__divide__imp,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,Aa2: A,Ba: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2) = Ba )
           => ( Aa2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2) ) ) ) ) ).

% eq_divide_imp
tff(fact_1505_nonzero__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,Ba: A,Aa2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2) = Aa2 )
          <=> ( Ba = aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2) ) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_1506_nonzero__eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,Aa2: A,Ba: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( Aa2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2) = Ba ) ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_1507_divide__numeral__1,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(num,A,numeral_numeral(A),one2)) = Aa2 ) ).

% divide_numeral_1
tff(fact_1508_power__one__over,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Aa2: A,Na: nat] : aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),Aa2)),Na) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,power_power(A,Aa2),Na)) ) ).

% power_one_over
tff(fact_1509_powr__mono2_H,axiom,
    ! [Aa2: real,X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),zero_zero(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Y,Aa2)),powr(real,X,Aa2)) ) ) ) ).

% powr_mono2'
tff(fact_1510_powr__less__mono2,axiom,
    ! [Aa2: real,X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Aa2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),Y)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,X,Aa2)),powr(real,Y,Aa2)) ) ) ) ).

% powr_less_mono2
tff(fact_1511_ge__one__powr__ge__zero,axiom,
    ! [X: real,Aa2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Aa2)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),powr(real,X,Aa2)) ) ) ).

% ge_one_powr_ge_zero
tff(fact_1512_powr__mono__both,axiom,
    ! [Aa2: real,Ba: real,X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Aa2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),Ba)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),X)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,X,Aa2)),powr(real,Y,Ba)) ) ) ) ) ).

% powr_mono_both
tff(fact_1513_powr__le1,axiom,
    ! [Aa2: real,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Aa2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),one_one(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,X,Aa2)),one_one(real)) ) ) ) ).

% powr_le1
tff(fact_1514_powr__mult,axiom,
    ! [X: real,Y: real,Aa2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( powr(real,aa(real,real,aa(real,fun(real,real),times_times(real),X),Y),Aa2) = aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,X,Aa2)),powr(real,Y,Aa2)) ) ) ) ).

% powr_mult
tff(fact_1515_log__powr,axiom,
    ! [X: real,Ba: real,Y: real] :
      ( ( X != zero_zero(real) )
     => ( aa(real,real,log(Ba),powr(real,X,Y)) = aa(real,real,aa(real,fun(real,real),times_times(real),Y),aa(real,real,log(Ba),X)) ) ) ).

% log_powr
tff(fact_1516_divide__nonpos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),zero_zero(A)) ) ) ) ).

% divide_nonpos_pos
tff(fact_1517_divide__nonpos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)) ) ) ) ).

% divide_nonpos_neg
tff(fact_1518_divide__nonneg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)) ) ) ) ).

% divide_nonneg_pos
tff(fact_1519_divide__nonneg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),zero_zero(A)) ) ) ) ).

% divide_nonneg_neg
tff(fact_1520_divide__le__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A,C2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2) ) ) ) ) ).

% divide_le_cancel
tff(fact_1521_frac__less2,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A,W: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),X)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),W)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),W),Z)
               => aa(A,$o,aa(A,fun(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_1522_frac__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A,W: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),W)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),W),Z)
               => aa(A,$o,aa(A,fun(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_1523_frac__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,X: A,W: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),W)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),W),Z)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),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_1524_div__positive,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba)) ) ) ) ).

% div_positive
tff(fact_1525_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba) = zero_zero(A) ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_1526_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C2: A,Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba)),C2) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_1527_divide__less__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),Aa2)),one_one(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),Aa2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba) )
            | ( Aa2 = zero_zero(A) ) ) ) ) ).

% divide_less_eq_1
tff(fact_1528_less__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),Aa2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),Aa2) ) ) ) ) ).

% less_divide_eq_1
tff(fact_1529_divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Ba: A,C2: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2)),Aa2)
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),Ba),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2)) ) ) ) ).

% divide_less_eq
tff(fact_1530_less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),Ba),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),zero_zero(A))) ) ) ) ).

% less_divide_eq
tff(fact_1531_neg__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2)),Aa2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),Ba) ) ) ) ).

% neg_divide_less_eq
tff(fact_1532_neg__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)) ) ) ) ).

% neg_less_divide_eq
tff(fact_1533_pos__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2)),Aa2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)) ) ) ) ).

% pos_divide_less_eq
tff(fact_1534_pos__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),Ba) ) ) ) ).

% pos_less_divide_eq
tff(fact_1535_mult__imp__div__pos__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,X: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),Z) ) ) ) ).

% mult_imp_div_pos_less
tff(fact_1536_mult__imp__less__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)),X)
           => aa(A,$o,aa(A,fun(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_1537_divide__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Ba: A,Aa2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),Aa2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),Ba)) ) ) ) ) ).

% divide_strict_left_mono
tff(fact_1538_divide__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),Aa2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),Ba)) ) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_1539_eq__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W: num,Ba: A,C2: A] :
          ( ( aa(num,A,numeral_numeral(A),W) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),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) = Ba,aa(num,A,numeral_numeral(A),W) = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_1540_divide__eq__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Ba: A,C2: A,W: num] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2) = aa(num,A,numeral_numeral(A),W) )
        <=> $ite(C2 != zero_zero(A),Ba = 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_1541_divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z)),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),Z) ) ) ) ).

% divide_add_eq_iff
tff(fact_1542_add__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)),Y)),Z) ) ) ) ).

% add_divide_eq_iff
tff(fact_1543_add__num__frac,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,X: A] :
          ( ( Y != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))),Y) ) ) ) ).

% add_num_frac
tff(fact_1544_add__frac__num,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,X: A,Z: A] :
          ( ( Y != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),Z) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))),Y) ) ) ) ).

% add_frac_num
tff(fact_1545_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,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)) ) ) ) ) ).

% add_frac_eq
tff(fact_1546_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Aa2: A,Ba: A,Z: A] :
          aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),Z)) = $ite(Z = zero_zero(A),Aa2,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Z)),Ba)),Z)) ) ).

% add_divide_eq_if_simps(1)
tff(fact_1547_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Aa2: A,Z: A,Ba: A] :
          aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Z)),Ba) = $ite(Z = zero_zero(A),Ba,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),Z))),Z)) ) ).

% add_divide_eq_if_simps(2)
tff(fact_1548_gt__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)))),Ba) ) ) ).

% gt_half_sum
tff(fact_1549_less__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)))) ) ) ).

% less_half_sum
tff(fact_1550_divide__eq__minus__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Aa2: A,Ba: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> ( ( Ba != zero_zero(A) )
            & ( Aa2 = aa(A,A,uminus_uminus(A),Ba) ) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_1551_eq__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( ( Aa2 = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2)) )
        <=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2) = aa(A,A,uminus_uminus(A),Ba),Aa2 = zero_zero(A)) ) ) ).

% eq_minus_divide_eq
tff(fact_1552_minus__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Ba: A,C2: A,Aa2: A] :
          ( ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2)) = Aa2 )
        <=> $ite(C2 != zero_zero(A),aa(A,A,uminus_uminus(A),Ba) = aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2),Aa2 = zero_zero(A)) ) ) ).

% minus_divide_eq_eq
tff(fact_1553_nonzero__neg__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Ba: A,Aa2: A,C2: A] :
          ( ( Ba != zero_zero(A) )
         => ( ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba)) = C2 )
          <=> ( aa(A,A,uminus_uminus(A),Aa2) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),Ba) ) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_1554_nonzero__neg__divide__eq__eq2,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Ba: A,C2: A,Aa2: A] :
          ( ( Ba != zero_zero(A) )
         => ( ( C2 = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba)) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),Ba) = aa(A,A,uminus_uminus(A),Aa2) ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_1555_powr__add,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [X: A,Aa2: A,Ba: A] : powr(A,X,aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)) = aa(A,A,aa(A,fun(A,A),times_times(A),powr(A,X,Aa2)),powr(A,X,Ba)) ) ).

% powr_add
tff(fact_1556_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Na)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na) ) ) ).

% nat_mult_div_cancel1
tff(fact_1557_powr__realpow,axiom,
    ! [X: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$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_1558_powr__less__iff,axiom,
    ! [Ba: real,X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Ba)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,Ba,Y)),X)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,log(Ba),X)) ) ) ) ).

% powr_less_iff
tff(fact_1559_less__powr__iff,axiom,
    ! [Ba: real,X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Ba)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),powr(real,Ba,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(Ba),X)),Y) ) ) ) ).

% less_powr_iff
tff(fact_1560_log__less__iff,axiom,
    ! [Ba: real,X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Ba)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(Ba),X)),Y)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),powr(real,Ba,Y)) ) ) ) ).

% log_less_iff
tff(fact_1561_less__log__iff,axiom,
    ! [Ba: real,X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Ba)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,log(Ba),X))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,Ba,Y)),X) ) ) ) ).

% less_log_iff
tff(fact_1562_divide__le__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),Aa2)),one_one(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba) )
            | ( Aa2 = zero_zero(A) ) ) ) ) ).

% divide_le_eq_1
tff(fact_1563_le__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),Aa2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2) ) ) ) ) ).

% le_divide_eq_1
tff(fact_1564_divide__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),Aa2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),Ba)) ) ) ) ) ).

% divide_left_mono_neg
tff(fact_1565_mult__imp__le__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)),X)
           => aa(A,$o,aa(A,fun(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_1566_mult__imp__div__pos__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,X: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),Z) ) ) ) ).

% mult_imp_div_pos_le
tff(fact_1567_pos__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),Ba) ) ) ) ).

% pos_le_divide_eq
tff(fact_1568_pos__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2)),Aa2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)) ) ) ) ).

% pos_divide_le_eq
tff(fact_1569_neg__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)) ) ) ) ).

% neg_le_divide_eq
tff(fact_1570_neg__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2)),Aa2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),Ba) ) ) ) ).

% neg_divide_le_eq
tff(fact_1571_divide__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Ba: A,Aa2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),Aa2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),Ba)) ) ) ) ) ).

% divide_left_mono
tff(fact_1572_le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),Ba),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),zero_zero(A))) ) ) ) ).

% le_divide_eq
tff(fact_1573_divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Ba: A,C2: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2)),Aa2)
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),Ba),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)) ) ) ) ).

% divide_le_eq
tff(fact_1574_less__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),W)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),Ba),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),W)),zero_zero(A))) ) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_1575_divide__less__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Ba: A,C2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2)),aa(num,A,numeral_numeral(A),W))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),Ba),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(num,A,numeral_numeral(A),W))) ) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_1576_field__sum__of__halves,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] : aa(A,A,aa(A,fun(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_1577_less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2)))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),aa(A,A,uminus_uminus(A),Ba)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),Ba)),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),zero_zero(A))) ) ) ) ).

% less_minus_divide_eq
tff(fact_1578_minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Ba: A,C2: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2))),Aa2)
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),Ba)),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),aa(A,A,uminus_uminus(A),Ba)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2)) ) ) ) ).

% minus_divide_less_eq
tff(fact_1579_neg__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),Ba)),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)) ) ) ) ).

% neg_less_minus_divide_eq
tff(fact_1580_neg__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2))),Aa2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),aa(A,A,uminus_uminus(A),Ba)) ) ) ) ).

% neg_minus_divide_less_eq
tff(fact_1581_pos__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),aa(A,A,uminus_uminus(A),Ba)) ) ) ) ).

% pos_less_minus_divide_eq
tff(fact_1582_pos__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2))),Aa2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),Ba)),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)) ) ) ) ).

% pos_minus_divide_less_eq
tff(fact_1583_eq__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W: num,Ba: 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),Ba),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) = Ba,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_1584_divide__eq__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Ba: A,C2: A,W: num] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) )
        <=> $ite(C2 != zero_zero(A),Ba = 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_1585_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Aa2: A,Z: A,Ba: A] :
          aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Z))),Ba) = $ite(Z = zero_zero(A),Ba,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),Aa2)),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),Z))),Z)) ) ).

% add_divide_eq_if_simps(3)
tff(fact_1586_minus__divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z))),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),Z) ) ) ) ).

% minus_divide_add_eq_iff
tff(fact_1587_msrevs_I1_J,axiom,
    ! [Na: nat,K: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Na)),M)),Na) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)),K) ) ) ).

% msrevs(1)
tff(fact_1588_powr__mult__base,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$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,aa(real,fun(real,real),plus_plus(real),one_one(real)),Y)) ) ) ).

% powr_mult_base
tff(fact_1589_powr__le__iff,axiom,
    ! [Ba: real,X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Ba)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Ba,Y)),X)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,log(Ba),X)) ) ) ) ).

% powr_le_iff
tff(fact_1590_le__powr__iff,axiom,
    ! [Ba: real,X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Ba)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),powr(real,Ba,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(Ba),X)),Y) ) ) ) ).

% le_powr_iff
tff(fact_1591_log__le__iff,axiom,
    ! [Ba: real,X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Ba)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(Ba),X)),Y)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),powr(real,Ba,Y)) ) ) ) ).

% log_le_iff
tff(fact_1592_le__log__iff,axiom,
    ! [Ba: real,X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Ba)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,log(Ba),X))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Ba,Y)),X) ) ) ) ).

% le_log_iff
tff(fact_1593_le__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),W)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),Ba),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),W)),zero_zero(A))) ) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_1594_divide__le__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Ba: A,C2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2)),aa(num,A,numeral_numeral(A),W))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),Ba),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(num,A,numeral_numeral(A),W))) ) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_1595_half__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2))))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2) ) ) ).

% half_gt_zero_iff
tff(fact_1596_half__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ).

% half_gt_zero
tff(fact_1597_field__less__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ).

% field_less_half_sum
tff(fact_1598_le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2)))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),aa(A,A,uminus_uminus(A),Ba)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Ba)),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),zero_zero(A))) ) ) ) ).

% le_minus_divide_eq
tff(fact_1599_minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Ba: A,C2: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2))),Aa2)
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Ba)),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),aa(A,A,uminus_uminus(A),Ba)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)) ) ) ) ).

% minus_divide_le_eq
tff(fact_1600_neg__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Ba)),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)) ) ) ) ).

% neg_le_minus_divide_eq
tff(fact_1601_neg__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2))),Aa2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),aa(A,A,uminus_uminus(A),Ba)) ) ) ) ).

% neg_minus_divide_le_eq
tff(fact_1602_pos__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),aa(A,A,uminus_uminus(A),Ba)) ) ) ) ).

% pos_le_minus_divide_eq
tff(fact_1603_pos__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2))),Aa2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Ba)),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)) ) ) ) ).

% pos_minus_divide_le_eq
tff(fact_1604_divide__less__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Ba: A,C2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),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,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),Ba),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))) ) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_1605_less__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(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),Ba),C2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),Ba),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),aa(A,$o,aa(A,fun(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_1606_div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Aa2: 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),Aa2),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),Aa2),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na))) ) ).

% div_exp_eq
tff(fact_1607_inverse__of__nat__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Na: nat,M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
         => ( ( Na != zero_zero(nat) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),M))),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),Na))) ) ) ) ).

% inverse_of_nat_le
tff(fact_1608_log__add__eq__powr,axiom,
    ! [Ba: real,X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Ba)
     => ( ( Ba != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
         => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,log(Ba),X)),Y) = aa(real,real,log(Ba),aa(real,real,aa(real,fun(real,real),times_times(real),X),powr(real,Ba,Y))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_1609_add__log__eq__powr,axiom,
    ! [Ba: real,X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Ba)
     => ( ( Ba != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
         => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),Y),aa(real,real,log(Ba),X)) = aa(real,real,log(Ba),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,Ba,Y)),X)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_1610_le__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),Ba),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),aa(A,$o,aa(A,fun(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_1611_divide__le__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Ba: A,C2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),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,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),Ba),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))) ) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_1612_in__set__simps_I1_J,axiom,
    ! [A: $tType,X: A,Y: A,Z: A,Ysb: list(A)] :
      ( member(A,X,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z),Ysb))))
    <=> ( ( X = Y )
        | member(A,X,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z),Ysb))) ) ) ).

% in_set_simps(1)
tff(fact_1613_in__set__simps_I3_J,axiom,
    ! [A: $tType,X: A] : ~ member(A,X,aa(list(A),set(A),set2(A),nil(A))) ).

% in_set_simps(3)
tff(fact_1614_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,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ) ) ).

% arith_geo_mean
tff(fact_1615_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_1616_add__self__div__2,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),M)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = M ).

% add_self_div_2
tff(fact_1617_div__mult__self__is__m,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na)),Na) = M ) ) ).

% div_mult_self_is_m
tff(fact_1618_div__mult__self1__is__m,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),M)),Na) = M ) ) ).

% div_mult_self1_is_m
tff(fact_1619_div__eq__dividend__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na) = M )
      <=> ( Na = one_one(nat) ) ) ) ).

% div_eq_dividend_iff
tff(fact_1620_div__mult__self4,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Ba: A,C2: A,Aa2: A] :
          ( ( Ba != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2)),Aa2)),Ba) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba)) ) ) ) ).

% div_mult_self4
tff(fact_1621_div__mult__self3,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Ba: A,C2: A,Aa2: A] :
          ( ( Ba != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Ba)),Aa2)),Ba) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba)) ) ) ) ).

% div_mult_self3
tff(fact_1622_div__mult__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Ba: A,Aa2: A,C2: A] :
          ( ( Ba != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2))),Ba) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba)) ) ) ) ).

% div_mult_self2
tff(fact_1623_zdiv__numeral__Bit0,axiom,
    ! [V: num,W: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),bit0(V))),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),V)),aa(num,int,numeral_numeral(int),W)) ).

% zdiv_numeral_Bit0
tff(fact_1624_div__mult__mult1__if,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,Aa2: A,Ba: A] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Aa2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Ba)) = $ite(C2 = zero_zero(A),zero_zero(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba)) ) ).

% div_mult_mult1_if
tff(fact_1625_div__mult__mult2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,Aa2: A,Ba: 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),Aa2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba) ) ) ) ).

% div_mult_mult2
tff(fact_1626_div__mult__mult1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,Aa2: A,Ba: 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),Aa2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Ba)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba) ) ) ) ).

% div_mult_mult1
tff(fact_1627_div__minus1__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),Aa2) ) ).

% div_minus1_right
tff(fact_1628_zdiv__numeral__Bit1,axiom,
    ! [V: num,W: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),bit1(V))),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),V)),aa(num,int,numeral_numeral(int),W)) ).

% zdiv_numeral_Bit1
tff(fact_1629_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),L)
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = zero_zero(int) ) ) ) ).

% div_pos_pos_trivial
tff(fact_1630_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),K)
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = zero_zero(int) ) ) ) ).

% div_neg_neg_trivial
tff(fact_1631_floor__divide__eq__div__numeral,axiom,
    ! [Aa2: num,Ba: num] : archim6421214686448440834_floor(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),Aa2)),aa(num,real,numeral_numeral(real),Ba))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),Aa2)),aa(num,int,numeral_numeral(int),Ba)) ).

% floor_divide_eq_div_numeral
tff(fact_1632_div__mult__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Ba: A,Aa2: A,C2: A] :
          ( ( Ba != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Ba))),Ba) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba)) ) ) ) ).

% div_mult_self1
tff(fact_1633_ceiling__divide__eq__div__numeral,axiom,
    ! [Aa2: num,Ba: num] : archimedean_ceiling(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),Aa2)),aa(num,real,numeral_numeral(real),Ba))) = 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),Aa2))),aa(num,int,numeral_numeral(int),Ba))) ).

% ceiling_divide_eq_div_numeral
tff(fact_1634_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2))))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% half_nonnegative_int_iff
tff(fact_1635_half__negative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% half_negative_int_iff
tff(fact_1636_floor__one__divide__eq__div__numeral,axiom,
    ! [Ba: num] : archim6421214686448440834_floor(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),Ba))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),one_one(int)),aa(num,int,numeral_numeral(int),Ba)) ).

% floor_one_divide_eq_div_numeral
tff(fact_1637_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [Aa2: num,Ba: 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),Aa2)),aa(num,real,numeral_numeral(real),Ba)))) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),Aa2)),aa(num,int,numeral_numeral(int),Ba))) ).

% ceiling_minus_divide_eq_div_numeral
tff(fact_1638_floor__minus__divide__eq__div__numeral,axiom,
    ! [Aa2: num,Ba: 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),Aa2)),aa(num,real,numeral_numeral(real),Ba)))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Aa2))),aa(num,int,numeral_numeral(int),Ba)) ).

% floor_minus_divide_eq_div_numeral
tff(fact_1639_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [Ba: 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),Ba)))) = 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),Ba)) ).

% floor_minus_one_divide_eq_div_numeral
tff(fact_1640_zdiv__zmult2__eq,axiom,
    ! [C2: int,Aa2: int,Ba: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),C2)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),Aa2),aa(int,int,aa(int,fun(int,int),times_times(int),Ba),C2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),Aa2),Ba)),C2) ) ) ).

% zdiv_zmult2_eq
tff(fact_1641_zdiv__int,axiom,
    ! [Aa2: nat,Ba: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Aa2),Ba)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,semiring_1_of_nat(int),Aa2)),aa(nat,int,semiring_1_of_nat(int),Ba)) ).

% zdiv_int
tff(fact_1642_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_1643_word__div__mult__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Aa2),Ba)),Ba)),Aa2) ) ).

% word_div_mult_le
tff(fact_1644_nat__div__distrib_H,axiom,
    ! [Y: int,X: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
     => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),divide_divide(int),X),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,X)),aa(int,nat,nat2,Y)) ) ) ).

% nat_div_distrib'
tff(fact_1645_nat__div__distrib,axiom,
    ! [X: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
     => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),divide_divide(int),X),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,X)),aa(int,nat,nat2,Y)) ) ) ).

% nat_div_distrib
tff(fact_1646_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [Aa2: int,Ba: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Aa2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),Aa2),Ba))
      <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ba),Aa2)
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Ba) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_1647_pos__imp__zdiv__nonneg__iff,axiom,
    ! [Ba: int,Aa2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Ba)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),Aa2),Ba))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Aa2) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_1648_neg__imp__zdiv__nonneg__iff,axiom,
    ! [Ba: int,Aa2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ba),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),Aa2),Ba))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Aa2),zero_zero(int)) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_1649_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),I),K))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_1650_div__nonpos__pos__le0,axiom,
    ! [Aa2: int,Ba: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Aa2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Ba)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),Aa2),Ba)),zero_zero(int)) ) ) ).

% div_nonpos_pos_le0
tff(fact_1651_div__nonneg__neg__le0,axiom,
    ! [Aa2: int,Ba: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Aa2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ba),zero_zero(int))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),Aa2),Ba)),zero_zero(int)) ) ) ).

% div_nonneg_neg_le0
tff(fact_1652_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),L),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L)) ) ) ).

% div_positive_int
tff(fact_1653_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L))
    <=> ( ( K = zero_zero(int) )
        | ( L = zero_zero(int) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
          & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ) ) ).

% div_int_pos_iff
tff(fact_1654_zdiv__mono2__neg,axiom,
    ! [Aa2: int,B4: int,Ba: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Aa2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B4)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B4),Ba)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),Aa2),B4)),aa(int,int,aa(int,fun(int,int),divide_divide(int),Aa2),Ba)) ) ) ) ).

% zdiv_mono2_neg
tff(fact_1655_zdiv__mono1__neg,axiom,
    ! [Aa2: int,A4: int,Ba: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Aa2),A4)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ba),zero_zero(int))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A4),Ba)),aa(int,int,aa(int,fun(int,int),divide_divide(int),Aa2),Ba)) ) ) ).

% zdiv_mono1_neg
tff(fact_1656_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa(int,int,aa(int,fun(int,int),divide_divide(int),I),K) = zero_zero(int) )
    <=> ( ( K = zero_zero(int) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I)
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),K) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),zero_zero(int))
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),I) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_1657_zdiv__mono2,axiom,
    ! [Aa2: int,B4: int,Ba: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Aa2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B4)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B4),Ba)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),Aa2),Ba)),aa(int,int,aa(int,fun(int,int),divide_divide(int),Aa2),B4)) ) ) ) ).

% zdiv_mono2
tff(fact_1658_zdiv__mono1,axiom,
    ! [Aa2: int,A4: int,Ba: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Aa2),A4)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Ba)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),Aa2),Ba)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A4),Ba)) ) ) ).

% zdiv_mono1
tff(fact_1659_int__div__less__self,axiom,
    ! [X: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),X)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),K)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),X),K)),X) ) ) ).

% int_div_less_self
tff(fact_1660_real__of__nat__div4,axiom,
    ! [Na: nat,X: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),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_1661_powr__divide,axiom,
    ! [X: real,Y: real,Aa2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( powr(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),X),Y),Aa2) = aa(real,real,aa(real,fun(real,real),divide_divide(real),powr(real,X,Aa2)),powr(real,Y,Aa2)) ) ) ) ).

% powr_divide
tff(fact_1662_div__lt__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),X: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),K),X))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),X)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),X)),K) ) ) ) ).

% div_lt_mult
tff(fact_1663_log__base__powr,axiom,
    ! [Aa2: real,Ba: real,X: real] :
      ( ( Aa2 != zero_zero(real) )
     => ( aa(real,real,log(powr(real,Aa2,Ba)),X) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,log(Aa2),X)),Ba) ) ) ).

% log_base_powr
tff(fact_1664_nat__div__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X4),Xa) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,semiring_1_of_nat(int),X4)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_div_as_int
tff(fact_1665_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_1666_verit__less__mono__div__int2,axiom,
    ! [A2: int,B2: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),B2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,uminus_uminus(int),Na))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),B2),Na)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),Na)) ) ) ).

% verit_less_mono_div_int2
tff(fact_1667_div__eq__minus1,axiom,
    ! [Ba: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Ba)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),one_one(int))),Ba) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ).

% div_eq_minus1
tff(fact_1668_log__base__change,axiom,
    ! [Aa2: real,Ba: real,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Aa2)
     => ( ( Aa2 != one_one(real) )
       => ( aa(real,real,log(Ba),X) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,log(Aa2),X)),aa(real,real,log(Aa2),Ba)) ) ) ) ).

% log_base_change
tff(fact_1669_div__le__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),X: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),K),X))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),X)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),X)),K) ) ) ) ).

% div_le_mult
tff(fact_1670_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)),zero_zero(int))
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% div_pos_neg_trivial
tff(fact_1671_int__div__pos__eq,axiom,
    ! [Aa2: int,Ba: int,Q3: int,R: int] :
      ( ( Aa2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Ba),Q3)),R) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R),Ba)
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),Aa2),Ba) = Q3 ) ) ) ) ).

% int_div_pos_eq
tff(fact_1672_int__div__neg__eq,axiom,
    ! [Aa2: int,Ba: int,Q3: int,R: int] :
      ( ( Aa2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Ba),Q3)),R) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ba),R)
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),Aa2),Ba) = Q3 ) ) ) ) ).

% int_div_neg_eq
tff(fact_1673_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,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
         => ! [I4: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less(int),J3),K)
                & ( Na = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
             => aa(int,$o,P,I4) ) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
         => ! [I4: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J3),zero_zero(int))
                & ( Na = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
             => aa(int,$o,P,I4) ) ) ) ) ).

% split_zdiv
tff(fact_1674_log__base__pow,axiom,
    ! [Aa2: real,Na: nat,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Aa2)
     => ( aa(real,real,log(aa(nat,real,power_power(real,Aa2),Na)),X) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,log(Aa2),X)),aa(nat,real,semiring_1_of_nat(real),Na)) ) ) ).

% log_base_pow
tff(fact_1675_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_1676_div__le__mono,axiom,
    ! [M: nat,Na: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),K)) ) ).

% div_le_mono
tff(fact_1677_div__le__dividend,axiom,
    ! [M: nat,Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)),M) ).

% div_le_dividend
tff(fact_1678_div__mult2__eq,axiom,
    ! [M: nat,Na: nat,Q3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),Q3)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)),Q3) ).

% div_mult2_eq
tff(fact_1679_pos__zdiv__mult__2,axiom,
    ! [Aa2: int,Ba: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Aa2)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Ba))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Aa2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),Ba),Aa2) ) ) ).

% pos_zdiv_mult_2
tff(fact_1680_neg__zdiv__mult__2,axiom,
    ! [Aa2: int,Ba: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Aa2),zero_zero(int))
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Ba))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Aa2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Ba),one_one(int))),Aa2) ) ) ).

% neg_zdiv_mult_2
tff(fact_1681_powr__neg__numeral,axiom,
    ! [X: real,Na: num] :
      ( aa(real,$o,aa(real,fun(real,$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_1682_div__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Aa2: A,M: nat,Na: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),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),Aa2),aa(nat,A,semiring_1_of_nat(A),M))),aa(nat,A,semiring_1_of_nat(A),Na)) ) ).

% div_mult2_eq'
tff(fact_1683_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_1684_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),Na))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)),I) ) ).

% less_mult_imp_div_less
tff(fact_1685_div__times__less__eq__dividend,axiom,
    ! [M: nat,Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)),Na)),M) ).

% div_times_less_eq_dividend
tff(fact_1686_times__div__less__eq__dividend,axiom,
    ! [Na: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na))),M) ).

% times_div_less_eq_dividend
tff(fact_1687_div__mult2__numeral__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Aa2: 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),Aa2),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),Aa2),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_1688_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_1689_div__add__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Ba: A,Aa2: A] :
          ( ( Ba != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)),Ba) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba)),one_one(A)) ) ) ) ).

% div_add_self2
tff(fact_1690_div__add__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Ba: A,Aa2: A] :
          ( ( Ba != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),Aa2)),Ba) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba)),one_one(A)) ) ) ) ).

% div_add_self1
tff(fact_1691_div__le__mono2,axiom,
    ! [M: nat,Na: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),K),Na)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),K),M)) ) ) ).

% div_le_mono2
tff(fact_1692_div__greater__zero__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ) ).

% div_greater_zero_iff
tff(fact_1693_div__less__dividend,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)),M) ) ) ).

% div_less_dividend
tff(fact_1694_div__less__iff__less__mult,axiom,
    ! [Q3: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Q3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Q3)),Na)
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),Q3)) ) ) ).

% div_less_iff_less_mult
tff(fact_1695_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_1696_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,aa(nat,fun(nat,$o),ord_less(nat),J3),Na)
             => ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),I4)),J3) )
               => aa(nat,$o,P,I4) ) ) ) ) ) ).

% split_div
tff(fact_1697_dividend__less__div__times,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)),Na))) ) ).

% dividend_less_div_times
tff(fact_1698_dividend__less__times__div,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)))) ) ).

% dividend_less_times_div
tff(fact_1699_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q3: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Q3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),Q3))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q3)),Na) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_1700_int__div__minus__is__minus1,axiom,
    ! [Aa2: int,Ba: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Aa2),zero_zero(int))
     => ( ( aa(int,int,aa(int,fun(int,int),divide_divide(int),Aa2),Ba) = aa(int,int,uminus_uminus(int),Aa2) )
      <=> ( Ba = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% int_div_minus_is_minus1
tff(fact_1701_int__div__same__is__1,axiom,
    ! [Aa2: int,Ba: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Aa2)
     => ( ( aa(int,int,aa(int,fun(int,int),divide_divide(int),Aa2),Ba) = Aa2 )
      <=> ( Ba = one_one(int) ) ) ) ).

% int_div_same_is_1
tff(fact_1702_two__pow__div__gt__le,axiom,
    ! [V: nat,Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),V),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,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% two_pow_div_gt_le
tff(fact_1703_z1pdiv2,axiom,
    ! [Ba: int] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Ba)),one_one(int))),aa(num,int,numeral_numeral(int),bit0(one2))) = Ba ).

% z1pdiv2
tff(fact_1704_axxdiv2,axiom,
    ! [X: int] :
      ( ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),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,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),zero_zero(int)),X)),X)),aa(num,int,numeral_numeral(int),bit0(one2))) = X ) ) ).

% axxdiv2
tff(fact_1705_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_1706_VEBT__internal_Opow__sum,axiom,
    ! [Aa2: nat,Ba: 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,aa(nat,fun(nat,nat),plus_plus(nat),Aa2),Ba))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Aa2)) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Ba) ).

% VEBT_internal.pow_sum
tff(fact_1707_td__gal,axiom,
    ! [C2: nat,Ba: nat,Aa2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),C2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ba),C2)),Aa2)
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ba),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Aa2),C2)) ) ) ).

% td_gal
tff(fact_1708_div__mult__le,axiom,
    ! [Aa2: nat,Ba: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Aa2),Ba)),Ba)),Aa2) ).

% div_mult_le
tff(fact_1709_zdiv__le__dividend,axiom,
    ! [Aa2: int,Ba: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Aa2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Ba)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),Aa2),Ba)),Aa2) ) ) ).

% zdiv_le_dividend
tff(fact_1710_zdiv__mult__self,axiom,
    ! [M: int,Aa2: int,Na: int] :
      ( ( M != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Aa2),aa(int,int,aa(int,fun(int,int),times_times(int),M),Na))),M) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),Aa2),M)),Na) ) ) ).

% zdiv_mult_self
tff(fact_1711_td__gal__lt,axiom,
    ! [C2: nat,Aa2: nat,Ba: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),C2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Aa2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ba),C2))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Aa2),C2)),Ba) ) ) ).

% td_gal_lt
tff(fact_1712_set__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A] : aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),zero_zero(nat)),Aa2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% set_bit_0
tff(fact_1713_ceiling__log2__div2,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => ( archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,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_1714_unset__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),zero_zero(nat)),Aa2) = 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),Aa2),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% unset_bit_0
tff(fact_1715_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,aa(A,fun(A,A),plus_plus(A),Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2)))),Na)) ) ).

% pochhammer_double
tff(fact_1716_central__binomial__lower__bound,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,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_1717_even__succ__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Aa2: A,Na: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),Aa2)),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),Aa2),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) ) ) ) ) ).

% even_succ_div_exp
tff(fact_1718_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,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),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_1719_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_1720_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_1721_diff__self,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [Aa2: A] : aa(A,A,minus_minus(A,Aa2),Aa2) = zero_zero(A) ) ).

% diff_self
tff(fact_1722_diff__0__right,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [Aa2: A] : aa(A,A,minus_minus(A,Aa2),zero_zero(A)) = Aa2 ) ).

% diff_0_right
tff(fact_1723_zero__diff,axiom,
    ! [A: $tType] :
      ( comm_monoid_diff(A)
     => ! [Aa2: A] : aa(A,A,minus_minus(A,zero_zero(A)),Aa2) = zero_zero(A) ) ).

% zero_diff
tff(fact_1724_diff__zero,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [Aa2: A] : aa(A,A,minus_minus(A,Aa2),zero_zero(A)) = Aa2 ) ).

% diff_zero
tff(fact_1725_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [Aa2: A] : aa(A,A,minus_minus(A,Aa2),Aa2) = zero_zero(A) ) ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1726_add__diff__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)),Ba) = Aa2 ) ).

% add_diff_cancel
tff(fact_1727_diff__add__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,Aa2),Ba)),Ba) = Aa2 ) ).

% diff_add_cancel
tff(fact_1728_add__diff__cancel__left,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [C2: A,Aa2: A,Ba: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),Aa2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),Ba)) = aa(A,A,minus_minus(A,Aa2),Ba) ) ).

% add_diff_cancel_left
tff(fact_1729_add__diff__cancel__left_H,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)),Aa2) = Ba ) ).

% add_diff_cancel_left'
tff(fact_1730_add__diff__cancel__right,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [Aa2: A,C2: A,Ba: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),C2)) = aa(A,A,minus_minus(A,Aa2),Ba) ) ).

% add_diff_cancel_right
tff(fact_1731_add__diff__cancel__right_H,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)),Ba) = Aa2 ) ).

% add_diff_cancel_right'
tff(fact_1732_dvd__0__right,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Aa2: A] : aa(A,$o,dvd_dvd(A,Aa2),zero_zero(A)) ) ).

% dvd_0_right
tff(fact_1733_dvd__0__left__iff,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Aa2: A] :
          ( aa(A,$o,dvd_dvd(A,zero_zero(A)),Aa2)
        <=> ( Aa2 = zero_zero(A) ) ) ) ).

% dvd_0_left_iff
tff(fact_1734_minus__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,uminus_uminus(A),aa(A,A,minus_minus(A,Aa2),Ba)) = aa(A,A,minus_minus(A,Ba),Aa2) ) ).

% minus_diff_eq
tff(fact_1735_mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),zero_zero(A)),Aa2) = zero_zero(A) ) ).

% mod_0
tff(fact_1736_mod__by__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),zero_zero(A)) = Aa2 ) ).

% mod_by_0
tff(fact_1737_mod__self,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),Aa2) = zero_zero(A) ) ).

% mod_self
tff(fact_1738_bits__mod__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),zero_zero(A)),Aa2) = zero_zero(A) ) ).

% bits_mod_0
tff(fact_1739_dvd__add__triv__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,dvd_dvd(A,Aa2),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba))
        <=> aa(A,$o,dvd_dvd(A,Aa2),Ba) ) ) ).

% dvd_add_triv_left_iff
tff(fact_1740_dvd__add__triv__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,dvd_dvd(A,Aa2),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),Aa2))
        <=> aa(A,$o,dvd_dvd(A,Aa2),Ba) ) ) ).

% dvd_add_triv_right_iff
tff(fact_1741_mod__add__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Ba: A,Aa2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),Aa2)),Ba) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),Ba) ) ).

% mod_add_self1
tff(fact_1742_mod__add__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)),Ba) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),Ba) ) ).

% mod_add_self2
tff(fact_1743_div__dvd__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,Aa2),Ba)
         => ( aa(A,$o,dvd_dvd(A,Aa2),C2)
           => ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),Aa2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),Aa2))
            <=> aa(A,$o,dvd_dvd(A,Ba),C2) ) ) ) ) ).

% div_dvd_div
tff(fact_1744_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_1745_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_1746_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_1747_diff__self__eq__0,axiom,
    ! [M: nat] : aa(nat,nat,minus_minus(nat,M),M) = zero_zero(nat) ).

% diff_self_eq_0
tff(fact_1748_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_1749_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,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) ).

% diff_diff_left
tff(fact_1750_diff__diff__cancel,axiom,
    ! [I: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),Na)
     => ( aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,minus_minus(nat,Na),I)) = I ) ) ).

% diff_diff_cancel
tff(fact_1751_unset__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),Na),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_1752_set__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),Na),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% set_bit_nonnegative_int_iff
tff(fact_1753_unset__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),Na),K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% unset_bit_negative_int_iff
tff(fact_1754_set__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),Na),K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% set_bit_negative_int_iff
tff(fact_1755_pochhammer__1,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Aa2: A] : comm_s3205402744901411588hammer(A,Aa2,one_one(nat)) = Aa2 ) ).

% pochhammer_1
tff(fact_1756_zero__comp__diff__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,minus_minus(A,Aa2),Ba))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2) ) ) ).

% zero_comp_diff_simps(1)
tff(fact_1757_diff__ge__0__iff__ge,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,minus_minus(A,Aa2),Ba))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2) ) ) ).

% diff_ge_0_iff_ge
tff(fact_1758_diff__gt__0__iff__gt,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,minus_minus(A,Aa2),Ba))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),Aa2) ) ) ).

% diff_gt_0_iff_gt
tff(fact_1759_zero__comp__diff__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,minus_minus(A,Aa2),Ba))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),Aa2) ) ) ).

% zero_comp_diff_simps(2)
tff(fact_1760_diff__add__zero,axiom,
    ! [A: $tType] :
      ( comm_monoid_diff(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,minus_minus(A,Aa2),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)) = zero_zero(A) ) ).

% diff_add_zero
tff(fact_1761_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_1762_le__add__diff__inverse,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),aa(A,A,minus_minus(A,Aa2),Ba)) = Aa2 ) ) ) ).

% le_add_diff_inverse
tff(fact_1763_le__add__diff__inverse2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,Aa2),Ba)),Ba) = Aa2 ) ) ) ).

% le_add_diff_inverse2
tff(fact_1764_left__diff__distrib__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ring(A) )
     => ! [Aa2: A,Ba: A,V: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Aa2),Ba)),aa(num,A,numeral_numeral(A),V)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),aa(num,A,numeral_numeral(A),V))) ) ).

% left_diff_distrib_numeral
tff(fact_1765_right__diff__distrib__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ring(A) )
     => ! [V: num,Ba: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,minus_minus(A,Ba),C2)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),Ba)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),C2)) ) ).

% right_diff_distrib_numeral
tff(fact_1766_verit__minus__simplify_I3_J,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [Ba: A] : aa(A,A,minus_minus(A,zero_zero(A)),Ba) = aa(A,A,uminus_uminus(A),Ba) ) ).

% verit_minus_simplify(3)
tff(fact_1767_diff__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [Aa2: A] : aa(A,A,minus_minus(A,zero_zero(A)),Aa2) = aa(A,A,uminus_uminus(A),Aa2) ) ).

% diff_0
tff(fact_1768_dvd__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: A,Aa2: A,Ba: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),Aa2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Ba))
        <=> ( ( C2 = zero_zero(A) )
            | aa(A,$o,dvd_dvd(A,Aa2),Ba) ) ) ) ).

% dvd_mult_cancel_left
tff(fact_1769_dvd__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Aa2: A,C2: A,Ba: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2))
        <=> ( ( C2 = zero_zero(A) )
            | aa(A,$o,dvd_dvd(A,Aa2),Ba) ) ) ) ).

% dvd_mult_cancel_right
tff(fact_1770_dvd__times__left__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( ( Aa2 != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2))
          <=> aa(A,$o,dvd_dvd(A,Ba),C2) ) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_1771_dvd__times__right__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( ( Aa2 != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),Ba),Aa2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Aa2))
          <=> aa(A,$o,dvd_dvd(A,Ba),C2) ) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_1772_uminus__add__conv__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),Aa2)),Ba) = aa(A,A,minus_minus(A,Ba),Aa2) ) ).

% uminus_add_conv_diff
tff(fact_1773_diff__minus__eq__add,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,minus_minus(A,Aa2),aa(A,A,uminus_uminus(A),Ba)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba) ) ).

% diff_minus_eq_add
tff(fact_1774_bits__mod__by__1,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),one_one(A)) = zero_zero(A) ) ).

% bits_mod_by_1
tff(fact_1775_mod__by__1,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),one_one(A)) = zero_zero(A) ) ).

% mod_by_1
tff(fact_1776_mod__mult__self2__is__0,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)),Ba) = zero_zero(A) ) ).

% mod_mult_self2_is_0
tff(fact_1777_mod__mult__self1__is__0,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Ba: A,Aa2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),Aa2)),Ba) = zero_zero(A) ) ).

% mod_mult_self1_is_0
tff(fact_1778_dvd__add__times__triv__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [Aa2: A,C2: A,Ba: A] :
          ( aa(A,$o,dvd_dvd(A,Aa2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Aa2)),Ba))
        <=> aa(A,$o,dvd_dvd(A,Aa2),Ba) ) ) ).

% dvd_add_times_triv_left_iff
tff(fact_1779_dvd__add__times__triv__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,Aa2),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Aa2)))
        <=> aa(A,$o,dvd_dvd(A,Aa2),Ba) ) ) ).

% dvd_add_times_triv_right_iff
tff(fact_1780_unit__prod,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,dvd_dvd(A,Aa2),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,Ba),one_one(A))
           => aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)),one_one(A)) ) ) ) ).

% unit_prod
tff(fact_1781_bits__mod__div__trivial,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),Ba)),Ba) = zero_zero(A) ) ).

% bits_mod_div_trivial
tff(fact_1782_mod__mult__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Aa2: A,C2: A,Ba: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Ba))),Ba) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),Ba) ) ).

% mod_mult_self1
tff(fact_1783_mod__mult__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Aa2: A,Ba: A,C2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2))),Ba) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),Ba) ) ).

% mod_mult_self2
tff(fact_1784_mod__mult__self3,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,Ba: A,Aa2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Ba)),Aa2)),Ba) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),Ba) ) ).

% mod_mult_self3
tff(fact_1785_mod__mult__self4,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Ba: A,C2: A,Aa2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2)),Aa2)),Ba) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),Ba) ) ).

% mod_mult_self4
tff(fact_1786_div__add,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,Aa2: A,Ba: A] :
          ( aa(A,$o,dvd_dvd(A,C2),Aa2)
         => ( aa(A,$o,dvd_dvd(A,C2),Ba)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2)) ) ) ) ) ).

% div_add
tff(fact_1787_unit__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,dvd_dvd(A,Aa2),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,Ba),one_one(A))
           => aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba)),one_one(A)) ) ) ) ).

% unit_div
tff(fact_1788_unit__div__1__unit,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Aa2: A] :
          ( aa(A,$o,dvd_dvd(A,Aa2),one_one(A))
         => aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),Aa2)),one_one(A)) ) ) ).

% unit_div_1_unit
tff(fact_1789_unit__div__1__div__1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Aa2: A] :
          ( aa(A,$o,dvd_dvd(A,Aa2),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)),Aa2)) = Aa2 ) ) ) ).

% unit_div_1_div_1
tff(fact_1790_dvd__div__mult__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,dvd_dvd(A,Aa2),Ba)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),Aa2)),Aa2) = Ba ) ) ) ).

% dvd_div_mult_self
tff(fact_1791_dvd__mult__div__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,dvd_dvd(A,Aa2),Ba)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),Aa2)) = Ba ) ) ) ).

% dvd_mult_div_cancel
tff(fact_1792_div__diff,axiom,
    ! [A: $tType] :
      ( idom_modulo(A)
     => ! [C2: A,Aa2: A,Ba: A] :
          ( aa(A,$o,dvd_dvd(A,C2),Aa2)
         => ( aa(A,$o,dvd_dvd(A,C2),Ba)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,Aa2),Ba)),C2) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2)) ) ) ) ) ).

% div_diff
tff(fact_1793_dvd__imp__mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,dvd_dvd(A,Aa2),Ba)
         => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Ba),Aa2) = zero_zero(A) ) ) ) ).

% dvd_imp_mod_0
tff(fact_1794_zero__less__diff,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,minus_minus(nat,Na),M))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ).

% zero_less_diff
tff(fact_1795_diff__is__0__eq_H,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( aa(nat,nat,minus_minus(nat,M),Na) = zero_zero(nat) ) ) ).

% diff_is_0_eq'
tff(fact_1796_diff__is__0__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,minus_minus(nat,M),Na) = zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% diff_is_0_eq
tff(fact_1797_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(nat,nat,minus_minus(nat,J),K)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),K) ) ) ).

% Nat.add_diff_assoc
tff(fact_1798_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,J),K)),I) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I)),K) ) ) ).

% Nat.add_diff_assoc2
tff(fact_1799_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,minus_minus(nat,I),aa(nat,nat,minus_minus(nat,J),K)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_1800_pochhammer__0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Aa2: A] : comm_s3205402744901411588hammer(A,Aa2,zero_zero(nat)) = one_one(A) ) ).

% pochhammer_0
tff(fact_1801_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_1802_mod__minus1__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),aa(A,A,uminus_uminus(A),one_one(A))) = zero_zero(A) ) ).

% mod_minus1_right
tff(fact_1803_unit__div__mult__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,dvd_dvd(A,Aa2),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),Ba),Aa2)),Aa2) = Ba ) ) ) ).

% unit_div_mult_self
tff(fact_1804_unit__mult__div__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,dvd_dvd(A,Aa2),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),Ba),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),Aa2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),Aa2) ) ) ) ).

% unit_mult_div_div
tff(fact_1805_floor__diff__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] : archim6421214686448440834_floor(A,aa(A,A,minus_minus(A,X),aa(num,A,numeral_numeral(A),V))) = aa(int,int,minus_minus(int,archim6421214686448440834_floor(A,X)),aa(num,int,numeral_numeral(int),V)) ) ).

% floor_diff_numeral
tff(fact_1806_ceiling__diff__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] : archimedean_ceiling(A,aa(A,A,minus_minus(A,X),aa(num,A,numeral_numeral(A),V))) = aa(int,int,minus_minus(int,archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V)) ) ).

% ceiling_diff_numeral
tff(fact_1807_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_1808_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,aa(num,fun(num,num),plus_plus(num),M),Na)) ) ).

% diff_numeral_simps(2)
tff(fact_1809_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,aa(num,fun(num,num),plus_plus(num),M),Na))) ) ).

% diff_numeral_simps(3)
tff(fact_1810_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_1811_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),L)
       => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L) = K ) ) ) ).

% mod_pos_pos_trivial
tff(fact_1812_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),K)
       => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L) = K ) ) ) ).

% mod_neg_neg_trivial
tff(fact_1813_odd__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Aa2: A,Ba: A] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba))
        <=> ~ ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2)
            <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Ba) ) ) ) ).

% odd_add
tff(fact_1814_even__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2)
          <=> aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Ba) ) ) ) ).

% even_add
tff(fact_1815_even__mult__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2)
            | aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Ba) ) ) ) ).

% even_mult_iff
tff(fact_1816_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_1817_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_1818_power__minus__odd,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Na: nat,Aa2: A] :
          ( ~ aa(nat,$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),Aa2)),Na) = aa(A,A,uminus_uminus(A),aa(nat,A,power_power(A,Aa2),Na)) ) ) ) ).

% power_minus_odd
tff(fact_1819_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Na: nat,Aa2: A] :
          ( aa(nat,$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),Aa2)),Na) = aa(nat,A,power_power(A,Aa2),Na) ) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_1820_even__mod__2__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Aa2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2))))
        <=> aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2) ) ) ).

% even_mod_2_iff
tff(fact_1821_even__not__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Aa2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,bit_ri4277139882892585799ns_not(A),Aa2))
        <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2) ) ) ).

% even_not_iff
tff(fact_1822_zmod__numeral__Bit0,axiom,
    ! [V: num,W: num] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(num,int,numeral_numeral(int),bit0(V))),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),V)),aa(num,int,numeral_numeral(int),W))) ).

% zmod_numeral_Bit0
tff(fact_1823_zero__le__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Aa2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,Aa2),aa(num,nat,numeral_numeral(nat),W)))
        <=> ( aa(nat,$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,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2) ) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_1824_power__less__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Aa2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,Aa2),Na)),zero_zero(A))
        <=> ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),zero_zero(A)) ) ) ) ).

% power_less_zero_eq
tff(fact_1825_power__less__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Aa2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,Aa2),aa(num,nat,numeral_numeral(nat),W))),zero_zero(A))
        <=> ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W))
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),zero_zero(A)) ) ) ) ).

% power_less_zero_eq_numeral
tff(fact_1826_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_1827_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_1828_not__mod__2__eq__0__eq__1,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Aa2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2))) != zero_zero(A) )
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_1829_not__mod__2__eq__1__eq__0,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Aa2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2))) != one_one(A) )
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_1830_even__plus__one__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Aa2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),one_one(A)))
        <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2) ) ) ).

% even_plus_one_iff
tff(fact_1831_even__diff,axiom,
    ! [A: $tType] :
      ( ring_parity(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,minus_minus(A,Aa2),Ba))
        <=> aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)) ) ) ).

% even_diff
tff(fact_1832_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_1833_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_1834_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_1835_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_1836_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_1837_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,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
        | aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)) ) ) ).

% even_diff_nat
tff(fact_1838_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,aa(num,fun(num,num),plus_plus(num),one2),Na)) ) ).

% diff_numeral_special(3)
tff(fact_1839_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,aa(num,fun(num,num),plus_plus(num),M),one2))) ) ).

% diff_numeral_special(4)
tff(fact_1840_zero__less__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Aa2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,power_power(A,Aa2),aa(num,nat,numeral_numeral(nat),W)))
        <=> ( ( aa(num,nat,numeral_numeral(nat),W) = zero_zero(nat) )
            | ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W))
              & ( Aa2 != 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,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2) ) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_1841_even__succ__div__2,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Aa2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),Aa2)),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ) ).

% even_succ_div_2
tff(fact_1842_odd__succ__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Aa2: A] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),one_one(A))),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2)))),one_one(A)) ) ) ) ).

% odd_succ_div_two
tff(fact_1843_even__succ__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Aa2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),one_one(A))),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ) ).

% even_succ_div_two
tff(fact_1844_even__power,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Aa2: A,Na: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,power_power(A,Aa2),Na))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ) ) ).

% even_power
tff(fact_1845_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_1846_ceiling__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,minus_minus(A,aa(num,A,numeral_numeral(A),V)),one_one(A))) ) ) ).

% ceiling_less_numeral
tff(fact_1847_numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),V)),archimedean_ceiling(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,aa(num,A,numeral_numeral(A),V)),one_one(A))),X) ) ) ).

% numeral_le_ceiling
tff(fact_1848_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_1849_power__le__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Aa2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,Aa2),aa(num,nat,numeral_numeral(nat),W))),zero_zero(A))
        <=> ( aa(nat,$o,aa(nat,fun(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,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),zero_zero(A)) )
              | ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W))
                & ( Aa2 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_1850_odd__two__times__div__two__succ,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Aa2: A] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2))))),one_one(A)) = Aa2 ) ) ) ).

% odd_two_times_div_two_succ
tff(fact_1851_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_1852_zmod__numeral__Bit1,axiom,
    ! [V: num,W: num] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(num,int,numeral_numeral(int),bit1(V))),aa(num,int,numeral_numeral(int),bit0(W))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(num,int,numeral_numeral(int),V)),aa(num,int,numeral_numeral(int),W)))),one_one(int)) ).

% zmod_numeral_Bit1
tff(fact_1853_ceiling__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)))
        <=> aa(A,$o,aa(A,fun(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),V))),one_one(A))) ) ) ).

% ceiling_less_neg_numeral
tff(fact_1854_neg__numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archimedean_ceiling(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),one_one(A))),X) ) ) ).

% neg_numeral_le_ceiling
tff(fact_1855_even__succ__mod__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Aa2: A,Na: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
           => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),Aa2)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na))) ) ) ) ) ).

% even_succ_mod_exp
tff(fact_1856_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,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
        | aa(nat,$o,dvd_dvd(nat,M),Na) ) ) ).

% dvd_minus_self
tff(fact_1857_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_1858_strict__subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),collect(A,aTP_Lamp_au(A,fun(A,$o),Aa2))),collect(A,aTP_Lamp_au(A,fun(A,$o),Ba)))
        <=> ( aa(A,$o,dvd_dvd(A,Aa2),Ba)
            & ~ aa(A,$o,dvd_dvd(A,Ba),Aa2) ) ) ) ).

% strict_subset_divisors_dvd
tff(fact_1859_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_1860_dvd__refl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Aa2: A] : aa(A,$o,dvd_dvd(A,Aa2),Aa2) ) ).

% dvd_refl
tff(fact_1861_dvd__trans,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,Aa2),Ba)
         => ( aa(A,$o,dvd_dvd(A,Ba),C2)
           => aa(A,$o,dvd_dvd(A,Aa2),C2) ) ) ) ).

% dvd_trans
tff(fact_1862_dvd__mod__iff,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [C2: A,Ba: A,Aa2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),Ba)
         => ( aa(A,$o,dvd_dvd(A,C2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),Ba))
          <=> aa(A,$o,dvd_dvd(A,C2),Aa2) ) ) ) ).

% dvd_mod_iff
tff(fact_1863_dvd__minus__mod,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [Ba: A,Aa2: A] : aa(A,$o,dvd_dvd(A,Ba),aa(A,A,minus_minus(A,Aa2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),Ba))) ) ).

% dvd_minus_mod
tff(fact_1864_dvd__mod__imp__dvd,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [C2: A,Aa2: A,Ba: A] :
          ( aa(A,$o,dvd_dvd(A,C2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),Ba))
         => ( aa(A,$o,dvd_dvd(A,C2),Ba)
           => aa(A,$o,dvd_dvd(A,C2),Aa2) ) ) ) ).

% dvd_mod_imp_dvd
tff(fact_1865_diff__eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [Aa2: A,Ba: A,C2: A,D2: A] :
          ( ( aa(A,A,minus_minus(A,Aa2),Ba) = aa(A,A,minus_minus(A,C2),D2) )
         => ( ( Aa2 = Ba )
          <=> ( C2 = D2 ) ) ) ) ).

% diff_eq_diff_eq
tff(fact_1866_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [Aa2: A,C2: A,Ba: A] : aa(A,A,minus_minus(A,aa(A,A,minus_minus(A,Aa2),C2)),Ba) = aa(A,A,minus_minus(A,aa(A,A,minus_minus(A,Aa2),Ba)),C2) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_1867_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_1868_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_1869_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,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
         => aa(nat,$o,dvd_dvd(nat,K),M) ) ) ) ).

% dvd_diffD
tff(fact_1870_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,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
         => aa(nat,$o,dvd_dvd(nat,K),Na) ) ) ) ).

% dvd_diffD1
tff(fact_1871_less__eq__dvd__minus,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( aa(nat,$o,dvd_dvd(nat,M),Na)
      <=> aa(nat,$o,dvd_dvd(nat,M),aa(nat,nat,minus_minus(nat,Na),M)) ) ) ).

% less_eq_dvd_minus
tff(fact_1872_mod__eq__0__iff__dvd,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [Aa2: A,Ba: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),Ba) = zero_zero(A) )
        <=> aa(A,$o,dvd_dvd(A,Ba),Aa2) ) ) ).

% mod_eq_0_iff_dvd
tff(fact_1873_dvd__eq__mod__eq__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,dvd_dvd(A,Aa2),Ba)
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Ba),Aa2) = zero_zero(A) ) ) ) ).

% dvd_eq_mod_eq_0
tff(fact_1874_mod__0__imp__dvd,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [Aa2: A,Ba: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),Ba) = zero_zero(A) )
         => aa(A,$o,dvd_dvd(A,Ba),Aa2) ) ) ).

% mod_0_imp_dvd
tff(fact_1875_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_1876_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_1877_unit__imp__mod__eq__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,dvd_dvd(A,Ba),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),Ba) = zero_zero(A) ) ) ) ).

% unit_imp_mod_eq_0
tff(fact_1878_minus__mult__div__eq__mod,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,minus_minus(A,Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba))) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),Ba) ) ).

% minus_mult_div_eq_mod
tff(fact_1879_minus__mod__eq__mult__div,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,minus_minus(A,Aa2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),Ba)) = aa(A,A,aa(A,fun(A,A),times_times(A),Ba),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba)) ) ).

% minus_mod_eq_mult_div
tff(fact_1880_minus__mod__eq__div__mult,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,minus_minus(A,Aa2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),Ba)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba)),Ba) ) ).

% minus_mod_eq_div_mult
tff(fact_1881_minus__div__mult__eq__mod,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,minus_minus(A,Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba)),Ba)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),Ba) ) ).

% minus_div_mult_eq_mod
tff(fact_1882_zmde,axiom,
    ! [A: $tType] :
      ( ( group_add(A)
        & semiring_modulo(A) )
     => ! [Ba: A,Aa2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),Ba),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba)) = aa(A,A,minus_minus(A,Aa2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),Ba)) ) ).

% zmde
tff(fact_1883_inf__period_I4_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D2: A,D3: A,T: A] :
          ( aa(A,$o,dvd_dvd(A,D2),D3)
         => ! [X4: A,K6: A] :
              ( ~ aa(A,$o,dvd_dvd(A,D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),T))
            <=> ~ aa(A,$o,dvd_dvd(A,D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,X4),aa(A,A,aa(A,fun(A,A),times_times(A),K6),D3))),T)) ) ) ) ).

% inf_period(4)
tff(fact_1884_inf__period_I3_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D2: A,D3: A,T: A] :
          ( aa(A,$o,dvd_dvd(A,D2),D3)
         => ! [X4: A,K6: A] :
              ( aa(A,$o,dvd_dvd(A,D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),T))
            <=> aa(A,$o,dvd_dvd(A,D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,X4),aa(A,A,aa(A,fun(A,A),times_times(A),K6),D3))),T)) ) ) ) ).

% inf_period(3)
tff(fact_1885_of__nat__diff,axiom,
    ! [A: $tType] :
      ( semiring_1_cancel(A)
     => ! [Na: nat,M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,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_1886_mod__add__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Aa2: A,C2: A,Ba: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Ba),C2))),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)),C2) ) ).

% mod_add_eq
tff(fact_1887_mod__add__cong,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Aa2: A,C2: A,A4: A,Ba: A,B4: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A4),C2) )
         => ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Ba),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B4),C2) )
           => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B4)),C2) ) ) ) ) ).

% mod_add_cong
tff(fact_1888_mod__add__left__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Aa2: A,C2: A,Ba: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),C2)),Ba)),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)),C2) ) ).

% mod_add_left_eq
tff(fact_1889_mod__add__right__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Aa2: A,Ba: A,C2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Ba),C2))),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)),C2) ) ).

% mod_add_right_eq
tff(fact_1890_mod__mult__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Aa2: A,C2: A,Ba: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Ba),C2))),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)),C2) ) ).

% mod_mult_eq
tff(fact_1891_mod__mult__cong,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Aa2: A,C2: A,A4: A,Ba: A,B4: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A4),C2) )
         => ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Ba),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B4),C2) )
           => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B4)),C2) ) ) ) ) ).

% mod_mult_cong
tff(fact_1892_mod__mult__mult2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Aa2: A,C2: A,Ba: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),Ba)),C2) ) ).

% mod_mult_mult2
tff(fact_1893_mult__mod__right,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,Aa2: A,Ba: A] : aa(A,A,aa(A,fun(A,A),times_times(A),C2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),Ba)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Aa2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Ba)) ) ).

% mult_mod_right
tff(fact_1894_mod__mult__left__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Aa2: A,C2: A,Ba: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),C2)),Ba)),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)),C2) ) ).

% mod_mult_left_eq
tff(fact_1895_mod__mult__right__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Aa2: A,Ba: A,C2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Ba),C2))),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)),C2) ) ).

% mod_mult_right_eq
tff(fact_1896_dvd__0__left,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Aa2: A] :
          ( aa(A,$o,dvd_dvd(A,zero_zero(A)),Aa2)
         => ( Aa2 = zero_zero(A) ) ) ) ).

% dvd_0_left
tff(fact_1897_eq__iff__diff__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [Aa2: A,Ba: A] :
          ( ( Aa2 = Ba )
        <=> ( aa(A,A,minus_minus(A,Aa2),Ba) = zero_zero(A) ) ) ) ).

% eq_iff_diff_eq_0
tff(fact_1898_diff__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Aa2: A,Ba: A,D2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,Aa2),C2)),aa(A,A,minus_minus(A,Ba),D2)) ) ) ) ).

% diff_mono
tff(fact_1899_diff__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Ba: A,Aa2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,C2),Aa2)),aa(A,A,minus_minus(A,C2),Ba)) ) ) ).

% diff_left_mono
tff(fact_1900_diff__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,Aa2),C2)),aa(A,A,minus_minus(A,Ba),C2)) ) ) ).

% diff_right_mono
tff(fact_1901_diff__eq__diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Aa2: A,Ba: A,C2: A,D2: A] :
          ( ( aa(A,A,minus_minus(A,Aa2),Ba) = aa(A,A,minus_minus(A,C2),D2) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2) ) ) ) ).

% diff_eq_diff_less_eq
tff(fact_1902_dvd__add,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,Aa2),Ba)
         => ( aa(A,$o,dvd_dvd(A,Aa2),C2)
           => aa(A,$o,dvd_dvd(A,Aa2),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),C2)) ) ) ) ).

% dvd_add
tff(fact_1903_dvd__add__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [Aa2: A,C2: A,Ba: A] :
          ( aa(A,$o,dvd_dvd(A,Aa2),C2)
         => ( aa(A,$o,dvd_dvd(A,Aa2),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),C2))
          <=> aa(A,$o,dvd_dvd(A,Aa2),Ba) ) ) ) ).

% dvd_add_left_iff
tff(fact_1904_dvd__add__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,Aa2),Ba)
         => ( aa(A,$o,dvd_dvd(A,Aa2),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),C2))
          <=> aa(A,$o,dvd_dvd(A,Aa2),C2) ) ) ) ).

% dvd_add_right_iff
tff(fact_1905_dvd__unit__imp__unit,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,dvd_dvd(A,Aa2),Ba)
         => ( aa(A,$o,dvd_dvd(A,Ba),one_one(A))
           => aa(A,$o,dvd_dvd(A,Aa2),one_one(A)) ) ) ) ).

% dvd_unit_imp_unit
tff(fact_1906_unit__imp__dvd,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,dvd_dvd(A,Ba),one_one(A))
         => aa(A,$o,dvd_dvd(A,Ba),Aa2) ) ) ).

% unit_imp_dvd
tff(fact_1907_one__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Aa2: A] : aa(A,$o,dvd_dvd(A,one_one(A)),Aa2) ) ).

% one_dvd
tff(fact_1908_dvdE,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,dvd_dvd(A,Ba),Aa2)
         => ~ ! [K5: A] : Aa2 != aa(A,A,aa(A,fun(A,A),times_times(A),Ba),K5) ) ) ).

% dvdE
tff(fact_1909_dvdI,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [Aa2: A,Ba: A,K: A] :
          ( ( Aa2 = aa(A,A,aa(A,fun(A,A),times_times(A),Ba),K) )
         => aa(A,$o,dvd_dvd(A,Ba),Aa2) ) ) ).

% dvdI
tff(fact_1910_dvd__def,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,dvd_dvd(A,Ba),Aa2)
        <=> ? [K2: A] : Aa2 = aa(A,A,aa(A,fun(A,A),times_times(A),Ba),K2) ) ) ).

% dvd_def
tff(fact_1911_dvd__mult,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Aa2: A,C2: A,Ba: A] :
          ( aa(A,$o,dvd_dvd(A,Aa2),C2)
         => aa(A,$o,dvd_dvd(A,Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2)) ) ) ).

% dvd_mult
tff(fact_1912_dvd__mult2,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,Aa2),Ba)
         => aa(A,$o,dvd_dvd(A,Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2)) ) ) ).

% dvd_mult2
tff(fact_1913_dvd__mult__left,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)),C2)
         => aa(A,$o,dvd_dvd(A,Aa2),C2) ) ) ).

% dvd_mult_left
tff(fact_1914_dvd__triv__left,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Aa2: A,Ba: A] : aa(A,$o,dvd_dvd(A,Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)) ) ).

% dvd_triv_left
tff(fact_1915_mult__dvd__mono,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Aa2: A,Ba: A,C2: A,D2: A] :
          ( aa(A,$o,dvd_dvd(A,Aa2),Ba)
         => ( 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),Aa2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),D2)) ) ) ) ).

% mult_dvd_mono
tff(fact_1916_dvd__mult__right,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)),C2)
         => aa(A,$o,dvd_dvd(A,Ba),C2) ) ) ).

% dvd_mult_right
tff(fact_1917_dvd__triv__right,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Aa2: A,Ba: A] : aa(A,$o,dvd_dvd(A,Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),Aa2)) ) ).

% dvd_triv_right
tff(fact_1918_diff__strict__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Aa2: A,Ba: A,D2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),D2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,Aa2),C2)),aa(A,A,minus_minus(A,Ba),D2)) ) ) ) ).

% diff_strict_mono
tff(fact_1919_diff__eq__diff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Aa2: A,Ba: A,C2: A,D2: A] :
          ( ( aa(A,A,minus_minus(A,Aa2),Ba) = aa(A,A,minus_minus(A,C2),D2) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2) ) ) ) ).

% diff_eq_diff_less
tff(fact_1920_diff__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Ba: A,Aa2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),Aa2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,C2),Aa2)),aa(A,A,minus_minus(A,C2),Ba)) ) ) ).

% diff_strict_left_mono
tff(fact_1921_diff__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,Aa2),C2)),aa(A,A,minus_minus(A,Ba),C2)) ) ) ).

% diff_strict_right_mono
tff(fact_1922_add__diff__add,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Aa2: A,C2: A,Ba: A,D2: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),D2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,Aa2),Ba)),aa(A,A,minus_minus(A,C2),D2)) ) ).

% add_diff_add
tff(fact_1923_group__cancel_Osub1,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,K: A,Aa2: A,Ba: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),Aa2) )
         => ( aa(A,A,minus_minus(A,A2),Ba) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,minus_minus(A,Aa2),Ba)) ) ) ) ).

% group_cancel.sub1
tff(fact_1924_diff__eq__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( ( aa(A,A,minus_minus(A,Aa2),Ba) = C2 )
        <=> ( Aa2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),Ba) ) ) ) ).

% diff_eq_eq
tff(fact_1925_eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [Aa2: A,C2: A,Ba: A] :
          ( ( Aa2 = aa(A,A,minus_minus(A,C2),Ba) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba) = C2 ) ) ) ).

% eq_diff_eq
tff(fact_1926_add__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [Aa2: A,Ba: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),aa(A,A,minus_minus(A,Ba),C2)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)),C2) ) ).

% add_diff_eq
tff(fact_1927_diff__diff__eq2,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [Aa2: A,Ba: A,C2: A] : aa(A,A,minus_minus(A,Aa2),aa(A,A,minus_minus(A,Ba),C2)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),C2)),Ba) ) ).

% diff_diff_eq2
tff(fact_1928_diff__add__eq,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Aa2: A,Ba: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,Aa2),Ba)),C2) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),C2)),Ba) ) ).

% diff_add_eq
tff(fact_1929_diff__add__eq__diff__diff__swap,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [Aa2: A,Ba: A,C2: A] : aa(A,A,minus_minus(A,Aa2),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),C2)) = aa(A,A,minus_minus(A,aa(A,A,minus_minus(A,Aa2),C2)),Ba) ) ).

% diff_add_eq_diff_diff_swap
tff(fact_1930_add__implies__diff,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [C2: A,Ba: A,Aa2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),Ba) = Aa2 )
         => ( C2 = aa(A,A,minus_minus(A,Aa2),Ba) ) ) ) ).

% add_implies_diff
tff(fact_1931_diff__diff__eq,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [Aa2: A,Ba: A,C2: A] : aa(A,A,minus_minus(A,aa(A,A,minus_minus(A,Aa2),Ba)),C2) = aa(A,A,minus_minus(A,Aa2),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),C2)) ) ).

% diff_diff_eq
tff(fact_1932_inf__period_I1_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [P: fun(A,$o),D3: A,Q: fun(A,$o)] :
          ( ! [X2: A,K5: A] :
              ( aa(A,$o,P,X2)
            <=> aa(A,$o,P,aa(A,A,minus_minus(A,X2),aa(A,A,aa(A,fun(A,A),times_times(A),K5),D3))) )
         => ( ! [X2: A,K5: A] :
                ( aa(A,$o,Q,X2)
              <=> aa(A,$o,Q,aa(A,A,minus_minus(A,X2),aa(A,A,aa(A,fun(A,A),times_times(A),K5),D3))) )
           => ! [X4: A,K6: A] :
                ( ( aa(A,$o,P,X4)
                  & aa(A,$o,Q,X4) )
              <=> ( aa(A,$o,P,aa(A,A,minus_minus(A,X4),aa(A,A,aa(A,fun(A,A),times_times(A),K6),D3)))
                  & aa(A,$o,Q,aa(A,A,minus_minus(A,X4),aa(A,A,aa(A,fun(A,A),times_times(A),K6),D3))) ) ) ) ) ) ).

% inf_period(1)
tff(fact_1933_inf__period_I2_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [P: fun(A,$o),D3: A,Q: fun(A,$o)] :
          ( ! [X2: A,K5: A] :
              ( aa(A,$o,P,X2)
            <=> aa(A,$o,P,aa(A,A,minus_minus(A,X2),aa(A,A,aa(A,fun(A,A),times_times(A),K5),D3))) )
         => ( ! [X2: A,K5: A] :
                ( aa(A,$o,Q,X2)
              <=> aa(A,$o,Q,aa(A,A,minus_minus(A,X2),aa(A,A,aa(A,fun(A,A),times_times(A),K5),D3))) )
           => ! [X4: A,K6: A] :
                ( ( aa(A,$o,P,X4)
                  | aa(A,$o,Q,X4) )
              <=> ( aa(A,$o,P,aa(A,A,minus_minus(A,X4),aa(A,A,aa(A,fun(A,A),times_times(A),K6),D3)))
                  | aa(A,$o,Q,aa(A,A,minus_minus(A,X4),aa(A,A,aa(A,fun(A,A),times_times(A),K6),D3))) ) ) ) ) ) ).

% inf_period(2)
tff(fact_1934_left__diff__distrib,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [Aa2: A,Ba: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Aa2),Ba)),C2) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2)) ) ).

% left_diff_distrib
tff(fact_1935_right__diff__distrib,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [Aa2: A,Ba: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(A,A,minus_minus(A,Ba),C2)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)) ) ).

% right_diff_distrib
tff(fact_1936_left__diff__distrib_H,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [Ba: A,C2: A,Aa2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Ba),C2)),Aa2) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),Ba),Aa2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Aa2)) ) ).

% left_diff_distrib'
tff(fact_1937_right__diff__distrib_H,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [Aa2: A,Ba: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(A,A,minus_minus(A,Ba),C2)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)) ) ).

% right_diff_distrib'
tff(fact_1938_dvd__div__eq__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [C2: A,Aa2: A,Ba: A] :
          ( aa(A,$o,dvd_dvd(A,C2),Aa2)
         => ( aa(A,$o,dvd_dvd(A,C2),Ba)
           => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2) )
            <=> ( Aa2 = Ba ) ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_1939_dvd__div__eq__cancel,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [Aa2: A,C2: A,Ba: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2) )
         => ( aa(A,$o,dvd_dvd(A,C2),Aa2)
           => ( aa(A,$o,dvd_dvd(A,C2),Ba)
             => ( Aa2 = Ba ) ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_1940_div__div__div__same,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [D2: A,Ba: A,Aa2: A] :
          ( aa(A,$o,dvd_dvd(A,D2),Ba)
         => ( aa(A,$o,dvd_dvd(A,Ba),Aa2)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),D2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),D2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba) ) ) ) ) ).

% div_div_div_same
tff(fact_1941_minus__diff__commute,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Ba: A,Aa2: A] : aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),Ba)),Aa2) = aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),Aa2)),Ba) ) ).

% minus_diff_commute
tff(fact_1942_minus__diff__minus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),Aa2)),aa(A,A,uminus_uminus(A),Ba)) = aa(A,A,uminus_uminus(A),aa(A,A,minus_minus(A,Aa2),Ba)) ) ).

% minus_diff_minus
tff(fact_1943_minus__nat_Odiff__0,axiom,
    ! [M: nat] : aa(nat,nat,minus_minus(nat,M),zero_zero(nat)) = M ).

% minus_nat.diff_0
tff(fact_1944_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_1945_mod__plus__cong,axiom,
    ! [Ba: int,B4: int,X: int,X7: int,Y: int,Y6: int,Z4: int] :
      ( ( Ba = B4 )
     => ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),X),B4) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),X7),B4) )
       => ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Y),B4) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Y6),B4) )
         => ( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X7),Y6) = Z4 )
           => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),Y)),Ba) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Z4),B4) ) ) ) ) ) ).

% mod_plus_cong
tff(fact_1946_zmod__helper,axiom,
    ! [Na: int,M: int,K: int,Aa2: int] :
      ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Na),M) = K )
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Na),Aa2)),M) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),Aa2)),M) ) ) ).

% zmod_helper
tff(fact_1947_diff__less__mono2,axiom,
    ! [M: nat,Na: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,minus_minus(nat,L),Na)),aa(nat,nat,minus_minus(nat,L),M)) ) ) ).

% diff_less_mono2
tff(fact_1948_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),K)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,minus_minus(nat,J),Na)),K) ) ).

% less_imp_diff_less
tff(fact_1949_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,Na: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Na)) = aa(nat,nat,minus_minus(nat,M),Na) ).

% Nat.diff_cancel
tff(fact_1950_diff__cancel2,axiom,
    ! [M: nat,K: nat,Na: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),K)) = aa(nat,nat,minus_minus(nat,M),Na) ).

% diff_cancel2
tff(fact_1951_diff__add__inverse,axiom,
    ! [Na: nat,M: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M)),Na) = M ).

% diff_add_inverse
tff(fact_1952_diff__add__inverse2,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)),Na) = M ).

% diff_add_inverse2
tff(fact_1953_eq__diff__iff,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
       => ( ( aa(nat,nat,minus_minus(nat,M),K) = aa(nat,nat,minus_minus(nat,Na),K) )
        <=> ( M = Na ) ) ) ) ).

% eq_diff_iff
tff(fact_1954_le__diff__iff,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,M),K)),aa(nat,nat,minus_minus(nat,Na),K))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ) ).

% le_diff_iff
tff(fact_1955_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
       => ( aa(nat,nat,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_1956_diff__le__mono,axiom,
    ! [M: nat,Na: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,M),L)),aa(nat,nat,minus_minus(nat,Na),L)) ) ).

% diff_le_mono
tff(fact_1957_diff__le__self,axiom,
    ! [M: nat,Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,M),Na)),M) ).

% diff_le_self
tff(fact_1958_le__diff__iff_H,axiom,
    ! [Aa2: nat,C2: nat,Ba: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Aa2),C2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ba),C2)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,C2),Aa2)),aa(nat,nat,minus_minus(nat,C2),Ba))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ba),Aa2) ) ) ) ).

% le_diff_iff'
tff(fact_1959_diff__le__mono2,axiom,
    ! [M: nat,Na: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,L),Na)),aa(nat,nat,minus_minus(nat,L),M)) ) ).

% diff_le_mono2
tff(fact_1960_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_1961_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_1962_unset__bit__nat__def,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se2638667681897837118et_bit(nat),M),Na) = aa(int,nat,nat2,aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),M),aa(nat,int,semiring_1_of_nat(int),Na))) ).

% unset_bit_nat_def
tff(fact_1963_unset__bit__less__eq,axiom,
    ! [Na: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),Na),K)),K) ).

% unset_bit_less_eq
tff(fact_1964_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L))
    <=> ( aa(int,$o,dvd_dvd(int,L),K)
        | ( ( L = zero_zero(int) )
          & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) )
        | aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L) ) ) ).

% mod_int_pos_iff
tff(fact_1965_set__bit__greater__eq,axiom,
    ! [K: int,Na: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),Na),K)) ).

% set_bit_greater_eq
tff(fact_1966_dvd__minus__add,axiom,
    ! [Q3: nat,Na: nat,R: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Q3),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Q3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R),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,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R),M)),Q3))) ) ) ) ).

% dvd_minus_add
tff(fact_1967_subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),collect(A,aTP_Lamp_au(A,fun(A,$o),Aa2))),collect(A,aTP_Lamp_au(A,fun(A,$o),Ba)))
        <=> aa(A,$o,dvd_dvd(A,Aa2),Ba) ) ) ).

% subset_divisors_dvd
tff(fact_1968_even__iff__mod__2__eq__zero,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Aa2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2)
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_1969_odd__iff__mod__2__eq__one,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Aa2: A] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2)
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_1970_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),Ba)),Aa2) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_1971_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_1972_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_1973_pochhammer__product,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [M: nat,Na: nat,Z: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( comm_s3205402744901411588hammer(A,Z,Na) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z,M)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,nat,minus_minus(nat,Na),M))) ) ) ) ).

% pochhammer_product
tff(fact_1974_mod__eqE,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [Aa2: A,C2: A,Ba: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Ba),C2) )
         => ~ ! [D4: A] : Ba != aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D4)) ) ) ).

% mod_eqE
tff(fact_1975_div__add1__eq,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [Aa2: A,Ba: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Ba),C2))),C2)) ) ).

% div_add1_eq
tff(fact_1976_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_1977_pinf_I9_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D2: A,S: A] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X4)
         => ( aa(A,$o,dvd_dvd(A,D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),S))
          <=> aa(A,$o,dvd_dvd(A,D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),S)) ) ) ) ).

% pinf(9)
tff(fact_1978_pinf_I10_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D2: A,S: A] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X4)
         => ( ~ aa(A,$o,dvd_dvd(A,D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),S))
          <=> ~ aa(A,$o,dvd_dvd(A,D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),S)) ) ) ) ).

% pinf(10)
tff(fact_1979_minf_I9_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D2: A,S: A] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z3)
         => ( aa(A,$o,dvd_dvd(A,D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),S))
          <=> aa(A,$o,dvd_dvd(A,D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),S)) ) ) ) ).

% minf(9)
tff(fact_1980_minf_I10_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D2: A,S: A] :
        ? [Z3: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z3)
         => ( ~ aa(A,$o,dvd_dvd(A,D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),S))
          <=> ~ aa(A,$o,dvd_dvd(A,D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),S)) ) ) ) ).

% minf(10)
tff(fact_1981_le__iff__diff__le__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,Aa2),Ba)),zero_zero(A)) ) ) ).

% le_iff_diff_le_0
tff(fact_1982_less__iff__diff__less__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,Aa2),Ba)),zero_zero(A)) ) ) ).

% less_iff_diff_less_0
tff(fact_1983_dvd__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,dvd_dvd(A,Ba),Aa2)
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba) = zero_zero(A) )
          <=> ( Aa2 = zero_zero(A) ) ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_1984_unit__mult__right__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,Aa2),one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Ba),Aa2) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),Aa2) )
          <=> ( Ba = C2 ) ) ) ) ).

% unit_mult_right_cancel
tff(fact_1985_unit__mult__left__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,Aa2),one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba) = aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2) )
          <=> ( Ba = C2 ) ) ) ) ).

% unit_mult_left_cancel
tff(fact_1986_mult__unit__dvd__iff_H,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,Aa2),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)),C2)
          <=> aa(A,$o,dvd_dvd(A,Ba),C2) ) ) ) ).

% mult_unit_dvd_iff'
tff(fact_1987_dvd__mult__unit__iff_H,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Ba: A,Aa2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,Ba),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2))
          <=> aa(A,$o,dvd_dvd(A,Aa2),C2) ) ) ) ).

% dvd_mult_unit_iff'
tff(fact_1988_mult__unit__dvd__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Ba: A,Aa2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,Ba),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)),C2)
          <=> aa(A,$o,dvd_dvd(A,Aa2),C2) ) ) ) ).

% mult_unit_dvd_iff
tff(fact_1989_dvd__mult__unit__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Ba: A,Aa2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,Ba),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Ba))
          <=> aa(A,$o,dvd_dvd(A,Aa2),C2) ) ) ) ).

% dvd_mult_unit_iff
tff(fact_1990_is__unit__mult__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)),one_one(A))
        <=> ( aa(A,$o,dvd_dvd(A,Aa2),one_one(A))
            & aa(A,$o,dvd_dvd(A,Ba),one_one(A)) ) ) ) ).

% is_unit_mult_iff
tff(fact_1991_diff__le__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,Aa2),Ba)),C2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),Ba)) ) ) ).

% diff_le_eq
tff(fact_1992_le__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Aa2: A,C2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),aa(A,A,minus_minus(A,C2),Ba))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)),C2) ) ) ).

% le_diff_eq
tff(fact_1993_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,Ba),Aa2)),Aa2) = Ba ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add
tff(fact_1994_le__add__diff,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),C2)),Aa2)) ) ) ).

% le_add_diff
tff(fact_1995_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,minus_minus(A,Ba),Aa2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),Aa2)),Ba) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_1996_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,minus_minus(A,Ba),Aa2)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),Ba)),Aa2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_1997_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => ( aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),Ba)),Aa2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,minus_minus(A,Ba),Aa2)) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_1998_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,Ba),Aa2)),C2) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),C2)),Aa2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_1999_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => ( aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),C2)),Aa2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,Ba),Aa2)),C2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_2000_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => ( aa(A,A,minus_minus(A,C2),aa(A,A,minus_minus(A,Ba),Aa2)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),Aa2)),Ba) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_2001_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),aa(A,A,minus_minus(A,Ba),Aa2)) = Ba ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_2002_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
           => ( ( aa(A,A,minus_minus(A,Ba),Aa2) = C2 )
            <=> ( Ba = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),Aa2) ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_2003_add__le__imp__le__diff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: A,K: A,Na: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),Na)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),aa(A,A,minus_minus(A,Na),K)) ) ) ).

% add_le_imp_le_diff
tff(fact_2004_add__le__add__imp__diff__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: A,K: A,Na: A,J: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),Na)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Na),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),Na)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Na),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,Na),K)),J) ) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_2005_less__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Aa2: A,C2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),aa(A,A,minus_minus(A,C2),Ba))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)),C2) ) ) ).

% less_diff_eq
tff(fact_2006_diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,Aa2),Ba)),C2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),Ba)) ) ) ).

% diff_less_eq
tff(fact_2007_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Aa2: A,Ba: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),aa(A,A,minus_minus(A,Aa2),Ba)) = Aa2 ) ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_2008_div__plus__div__distrib__dvd__right,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,Ba: A,Aa2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),Ba)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2)) ) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_2009_div__plus__div__distrib__dvd__left,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,Aa2: A,Ba: A] :
          ( aa(A,$o,dvd_dvd(A,C2),Aa2)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2)) ) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_2010_dvd__div__unit__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Ba: A,Aa2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,Ba),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,Aa2),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),Ba))
          <=> aa(A,$o,dvd_dvd(A,Aa2),C2) ) ) ) ).

% dvd_div_unit_iff
tff(fact_2011_div__unit__dvd__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Ba: A,Aa2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,Ba),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba)),C2)
          <=> aa(A,$o,dvd_dvd(A,Aa2),C2) ) ) ) ).

% div_unit_dvd_iff
tff(fact_2012_unit__div__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,Aa2),one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),Aa2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),Aa2) )
          <=> ( Ba = C2 ) ) ) ) ).

% unit_div_cancel
tff(fact_2013_dvd__div__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,Ba: A,Aa2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),Ba)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2)),Aa2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),Aa2)),C2) ) ) ) ).

% dvd_div_mult
tff(fact_2014_div__mult__swap,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,Ba: A,Aa2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),Ba)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)),C2) ) ) ) ).

% div_mult_swap
tff(fact_2015_div__div__eq__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,Ba: A,Aa2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),Ba)
         => ( aa(A,$o,dvd_dvd(A,Ba),Aa2)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba)),C2) ) ) ) ) ).

% div_div_eq_right
tff(fact_2016_dvd__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Ba: A,C2: A,Aa2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2)),Aa2)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba)),C2) ) ) ) ).

% dvd_div_mult2_eq
tff(fact_2017_dvd__mult__imp__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Aa2: A,C2: A,Ba: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),Ba)
         => aa(A,$o,dvd_dvd(A,Aa2),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2)) ) ) ).

% dvd_mult_imp_div
tff(fact_2018_div__mult__div__if__dvd,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Ba: A,Aa2: A,D2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,Ba),Aa2)
         => ( 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),Aa2),Ba)),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),Aa2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),D2)) ) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_2019_mult__diff__mult,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [X: A,Y: A,Aa2: A,Ba: 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),Aa2),Ba)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(A,A,minus_minus(A,Y),Ba))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,X),Aa2)),Ba)) ) ).

% mult_diff_mult
tff(fact_2020_eq__add__iff1,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [Aa2: A,E2: A,C2: A,Ba: A,D2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),E2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),E2)),D2) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Aa2),Ba)),E2)),C2) = D2 ) ) ) ).

% eq_add_iff1
tff(fact_2021_eq__add__iff2,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [Aa2: A,E2: A,C2: A,Ba: A,D2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),E2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),E2)),D2) )
        <=> ( C2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Ba),Aa2)),E2)),D2) ) ) ) ).

% eq_add_iff2
tff(fact_2022_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,aa(A,fun(A,A),plus_plus(A),X),Y)),aa(A,A,minus_minus(A,X),Y)) ) ).

% square_diff_square_factored
tff(fact_2023_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,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ).

% diff_shunt_var
tff(fact_2024_dvd__div__neg,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,dvd_dvd(A,Ba),Aa2)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(A,A,uminus_uminus(A),Ba)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba)) ) ) ) ).

% dvd_div_neg
tff(fact_2025_dvd__neg__div,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,dvd_dvd(A,Ba),Aa2)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),Aa2)),Ba) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba)) ) ) ) ).

% dvd_neg_div
tff(fact_2026_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,minus_minus(A,Aa2),Ba) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),aa(A,A,uminus_uminus(A),Ba)) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_2027_diff__conv__add__uminus,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,minus_minus(A,Aa2),Ba) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),aa(A,A,uminus_uminus(A),Ba)) ) ).

% diff_conv_add_uminus
tff(fact_2028_group__cancel_Osub2,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [B2: A,K: A,Ba: A,Aa2: A] :
          ( ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),Ba) )
         => ( aa(A,A,minus_minus(A,Aa2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),K)),aa(A,A,minus_minus(A,Aa2),Ba)) ) ) ) ).

% group_cancel.sub2
tff(fact_2029_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,$o,aa(nat,fun(nat,$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_2030_power__le__dvd,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Aa2: A,Na: nat,Ba: A,M: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(nat,A,power_power(A,Aa2),Na)),Ba)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
           => aa(A,$o,dvd_dvd(A,aa(nat,A,power_power(A,Aa2),M)),Ba) ) ) ) ).

% power_le_dvd
tff(fact_2031_le__imp__power__dvd,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [M: nat,Na: nat,Aa2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => aa(A,$o,dvd_dvd(A,aa(nat,A,power_power(A,Aa2),M)),aa(nat,A,power_power(A,Aa2),Na)) ) ) ).

% le_imp_power_dvd
tff(fact_2032_nat__dvd__not__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
       => ~ aa(nat,$o,dvd_dvd(nat,Na),M) ) ) ).

% nat_dvd_not_less
tff(fact_2033_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),M)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),M),K)),M) ) ).

% zmod_le_nonneg_dividend
tff(fact_2034_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) )
    <=> ? [Q4: int] : M = aa(int,int,aa(int,fun(int,int),times_times(int),D2),Q4) ) ).

% zmod_eq_0_iff
tff(fact_2035_zmod__eq__0D,axiom,
    ! [M: int,D2: int] :
      ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),M),D2) = zero_zero(int) )
     => ? [Q5: int] : M = aa(int,int,aa(int,fun(int,int),times_times(int),D2),Q5) ) ).

% zmod_eq_0D
tff(fact_2036_not__diff__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,minus_minus(A,Aa2),Ba)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Aa2)),Ba) ) ).

% not_diff_distrib
tff(fact_2037_not__add__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)) = aa(A,A,minus_minus(A,aa(A,A,bit_ri4277139882892585799ns_not(A),Aa2)),Ba) ) ).

% not_add_distrib
tff(fact_2038_diff__less,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,minus_minus(nat,M),Na)),M) ) ) ).

% diff_less
tff(fact_2039_pochhammer__eq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Aa2: A,Na: nat,M: nat] :
          ( ( comm_s3205402744901411588hammer(A,Aa2,Na) = zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
           => ( comm_s3205402744901411588hammer(A,Aa2,M) = zero_zero(A) ) ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_2040_pochhammer__neq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Aa2: A,M: nat,Na: nat] :
          ( ( comm_s3205402744901411588hammer(A,Aa2,M) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
           => ( comm_s3205402744901411588hammer(A,Aa2,Na) != zero_zero(A) ) ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_2041_zdvd__antisym__nonneg,axiom,
    ! [M: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),M)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Na)
       => ( aa(int,$o,dvd_dvd(int,M),Na)
         => ( aa(int,$o,dvd_dvd(int,Na),M)
           => ( M = Na ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_2042_diff__add__0,axiom,
    ! [Na: nat,M: nat] : aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M)) = zero_zero(nat) ).

% diff_add_0
tff(fact_2043_zdvd__not__zless,axiom,
    ! [M: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),M)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),M),Na)
       => ~ aa(int,$o,dvd_dvd(int,Na),M) ) ) ).

% zdvd_not_zless
tff(fact_2044_add__diff__inverse__nat,axiom,
    ! [M: nat,Na: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(nat,nat,minus_minus(nat,M),Na)) = M ) ) ).

% add_diff_inverse_nat
tff(fact_2045_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,minus_minus(nat,J),K))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J) ) ).

% less_diff_conv
tff(fact_2046_diff__less__mono,axiom,
    ! [Aa2: nat,Ba: nat,C2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Aa2),Ba)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),C2),Aa2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,minus_minus(nat,Aa2),C2)),aa(nat,nat,minus_minus(nat,Ba),C2)) ) ) ).

% diff_less_mono
tff(fact_2047_less__diff__iff,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,minus_minus(nat,M),K)),aa(nat,nat,minus_minus(nat,Na),K))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ) ).

% less_diff_iff
tff(fact_2048_zdvd__mono,axiom,
    ! [K: int,M: int,T: int] :
      ( ( K != zero_zero(int) )
     => ( aa(int,$o,dvd_dvd(int,M),T)
      <=> aa(int,$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),T)) ) ) ).

% zdvd_mono
tff(fact_2049_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_2050_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,J),K)),I)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)) ) ).

% le_diff_conv
tff(fact_2051_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),aa(nat,nat,minus_minus(nat,J),K))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J) ) ) ).

% Nat.le_diff_conv2
tff(fact_2052_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(nat,nat,minus_minus(nat,J),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_2053_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,J),K)),I) ) ) ).

% Nat.diff_add_assoc2
tff(fact_2054_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( ( aa(nat,nat,minus_minus(nat,J),I) = K )
      <=> ( J = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),I) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_2055_zdvd__reduce,axiom,
    ! [K: int,Na: int,M: int] :
      ( aa(int,$o,dvd_dvd(int,K),aa(int,int,aa(int,fun(int,int),plus_plus(int),Na),aa(int,int,aa(int,fun(int,int),times_times(int),K),M)))
    <=> aa(int,$o,dvd_dvd(int,K),Na) ) ).

% zdvd_reduce
tff(fact_2056_zdvd__period,axiom,
    ! [Aa2: int,D2: int,X: int,T: int,C2: int] :
      ( aa(int,$o,dvd_dvd(int,Aa2),D2)
     => ( aa(int,$o,dvd_dvd(int,Aa2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),T))
      <=> aa(int,$o,dvd_dvd(int,Aa2),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),C2),D2))),T)) ) ) ).

% zdvd_period
tff(fact_2057_mod2__eq__if,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Aa2: A] :
          aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2))) = $ite(aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2),zero_zero(A),one_one(A)) ) ).

% mod2_eq_if
tff(fact_2058_parity__cases,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Aa2: A] :
          ( ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2)
           => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),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))),Aa2)
             => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2))) != one_one(A) ) ) ) ) ).

% parity_cases
tff(fact_2059_even__set__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Aa2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),M),Aa2))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2)
            & ( M != zero_zero(nat) ) ) ) ) ).

% even_set_bit_iff
tff(fact_2060_even__unset__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Aa2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),M),Aa2))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2)
            | ( M = zero_zero(nat) ) ) ) ) ).

% even_unset_bit_iff
tff(fact_2061_pochhammer__absorb__comp,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [R: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,R),aa(nat,A,semiring_1_of_nat(A),K))),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),R),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),R),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),R)),one_one(A)),K)) ) ).

% pochhammer_absorb_comp
tff(fact_2062_nat__dvd__iff,axiom,
    ! [Z: int,M: nat] :
      ( aa(nat,$o,dvd_dvd(nat,aa(int,nat,nat2,Z)),M)
    <=> $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z),aa(int,$o,dvd_dvd(int,Z),aa(nat,int,semiring_1_of_nat(int),M)),M = zero_zero(nat)) ) ).

% nat_dvd_iff
tff(fact_2063_mult__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Na: nat,Aa2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),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),Aa2),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_2064_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
           => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),Ba) = Aa2 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_2065_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Ba)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),Ba)) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_2066_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_2067_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_2068_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_2069_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_2070_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_2071_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_2072_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_2073_div__mult1__eq,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [Aa2: A,Ba: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Ba),C2))),C2)) ) ).

% div_mult1_eq
tff(fact_2074_cancel__div__mod__rules_I2_J,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [Ba: A,Aa2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),Ba))),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),C2) ) ).

% cancel_div_mod_rules(2)
tff(fact_2075_cancel__div__mod__rules_I1_J,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [Aa2: A,Ba: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba)),Ba)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),Ba))),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),C2) ) ).

% cancel_div_mod_rules(1)
tff(fact_2076_mod__div__decomp,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [Aa2: A,Ba: A] : Aa2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba)),Ba)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),Ba)) ) ).

% mod_div_decomp
tff(fact_2077_div__mult__mod__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba)),Ba)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),Ba)) = Aa2 ) ).

% div_mult_mod_eq
tff(fact_2078_mod__div__mult__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),Ba)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba)),Ba)) = Aa2 ) ).

% mod_div_mult_eq
tff(fact_2079_mod__mult__div__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),Ba)),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba))) = Aa2 ) ).

% mod_mult_div_eq
tff(fact_2080_mult__div__mod__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [Ba: A,Aa2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),Ba)) = Aa2 ) ).

% mult_div_mod_eq
tff(fact_2081_pochhammer__minus_H,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Ba: A,K: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,Ba),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),Ba),K)) ) ).

% pochhammer_minus'
tff(fact_2082_pochhammer__minus,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Ba: A,K: nat] : comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),Ba),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,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,Ba),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K)) ) ).

% pochhammer_minus
tff(fact_2083_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_2084_unity__coeff__ex,axiom,
    ! [A: $tType] :
      ( ( dvd(A)
        & semiring_0(A) )
     => ! [P: fun(A,$o),L: A] :
          ( ? [X6: A] : aa(A,$o,P,aa(A,A,aa(A,fun(A,A),times_times(A),L),X6))
        <=> ? [X6: A] :
              ( aa(A,$o,dvd_dvd(A,L),aa(A,A,aa(A,fun(A,A),plus_plus(A),X6),zero_zero(A)))
              & aa(A,$o,P,X6) ) ) ) ).

% unity_coeff_ex
tff(fact_2085_unit__dvdE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,dvd_dvd(A,Aa2),one_one(A))
         => ~ ( ( Aa2 != zero_zero(A) )
             => ! [C4: A] : Ba != aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C4) ) ) ) ).

% unit_dvdE
tff(fact_2086_unit__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,dvd_dvd(A,Ba),one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba) = zero_zero(A) )
          <=> ( Aa2 = zero_zero(A) ) ) ) ) ).

% unit_div_eq_0_iff
tff(fact_2087_dvd__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( ( Aa2 != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,Aa2),Ba)
           => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),Aa2) = C2 )
            <=> ( Ba = aa(A,A,aa(A,fun(A,A),times_times(A),C2),Aa2) ) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_2088_div__dvd__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Ba: A,Aa2: A,C2: A] :
          ( ( Ba != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,Ba),Aa2)
           => ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba)),C2)
            <=> aa(A,$o,dvd_dvd(A,Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Ba)) ) ) ) ) ).

% div_dvd_iff_mult
tff(fact_2089_dvd__div__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,Ba: A,Aa2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,C2),Ba)
           => ( aa(A,$o,dvd_dvd(A,Aa2),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2))
            <=> aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),Ba) ) ) ) ) ).

% dvd_div_iff_mult
tff(fact_2090_dvd__div__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Aa2: A,C2: A,Ba: A,D2: A] :
          ( ( Aa2 != zero_zero(A) )
         => ( ( C2 != zero_zero(A) )
           => ( aa(A,$o,dvd_dvd(A,Aa2),Ba)
             => ( aa(A,$o,dvd_dvd(A,C2),D2)
               => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),Aa2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),D2),C2) )
                <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),D2) ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_2091_is__unit__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Ba: A,C2: A,Aa2: A] :
          ( aa(A,$o,dvd_dvd(A,Ba),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,C2),one_one(A))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba)),C2) ) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_2092_unit__div__mult__swap,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,Aa2: A,Ba: A] :
          ( aa(A,$o,dvd_dvd(A,C2),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)),C2) ) ) ) ).

% unit_div_mult_swap
tff(fact_2093_unit__div__commute,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Ba: A,Aa2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,Ba),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),Aa2),Ba)),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)),Ba) ) ) ) ).

% unit_div_commute
tff(fact_2094_div__mult__unit2,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,Ba: A,Aa2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,Ba),Aa2)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba)),C2) ) ) ) ) ).

% div_mult_unit2
tff(fact_2095_unit__eq__div2,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Ba: A,Aa2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,Ba),one_one(A))
         => ( ( Aa2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),Ba) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba) = C2 ) ) ) ) ).

% unit_eq_div2
tff(fact_2096_unit__eq__div1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Ba: A,Aa2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,Ba),one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba) = C2 )
          <=> ( Aa2 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),Ba) ) ) ) ) ).

% unit_eq_div1
tff(fact_2097_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [Aa2: A,E2: A,C2: A,Ba: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),E2)),D2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Aa2),Ba)),E2)),C2)),D2) ) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_2098_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [Aa2: A,E2: A,C2: A,Ba: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),E2)),D2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Ba),Aa2)),E2)),D2)) ) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_2099_less__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [Aa2: A,E2: A,C2: A,Ba: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),E2)),D2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Aa2),Ba)),E2)),C2)),D2) ) ) ).

% less_add_iff1
tff(fact_2100_less__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [Aa2: A,E2: A,C2: A,Ba: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),E2)),D2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Ba),Aa2)),E2)),D2)) ) ) ).

% less_add_iff2
tff(fact_2101_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_2102_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_2103_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_2104_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Aa2: A,Ba: A,Z: A] :
          aa(A,A,minus_minus(A,Aa2),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),Z)) = $ite(Z = zero_zero(A),Aa2,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),Aa2),Z)),Ba)),Z)) ) ).

% add_divide_eq_if_simps(4)
tff(fact_2105_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,aa(A,fun(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_2106_is__unit__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Aa2: A,Na: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(nat,A,power_power(A,Aa2),Na)),one_one(A))
        <=> ( aa(A,$o,dvd_dvd(A,Aa2),one_one(A))
            | ( Na = zero_zero(nat) ) ) ) ) ).

% is_unit_power_iff
tff(fact_2107_pochhammer__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),X)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),comm_s3205402744901411588hammer(A,X,Na)) ) ) ).

% pochhammer_nonneg
tff(fact_2108_dvd__imp__le,axiom,
    ! [K: nat,Na: nat] :
      ( aa(nat,$o,dvd_dvd(nat,K),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na) ) ) ).

% dvd_imp_le
tff(fact_2109_not__eq__complement,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Aa2: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),Aa2) = aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),Aa2)),one_one(A)) ) ).

% not_eq_complement
tff(fact_2110_minus__eq__not__minus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Aa2: A] : aa(A,A,uminus_uminus(A),Aa2) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,minus_minus(A,Aa2),one_one(A))) ) ).

% minus_eq_not_minus_1
tff(fact_2111_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,$o,dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Na))
      <=> aa(nat,$o,dvd_dvd(nat,M),Na) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_2112_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,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(nat,$o,dvd_dvd(nat,M),Na) ) ) ).

% dvd_mult_cancel
tff(fact_2113_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L)) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_2114_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L)),zero_zero(int)) ) ).

% neg_mod_sign
tff(fact_2115_int__mod__ge,axiom,
    ! [Aa2: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Aa2),Na)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Na)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Aa2),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Aa2),Na)) ) ) ).

% int_mod_ge
tff(fact_2116_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),I),K) = I )
    <=> ( ( K = zero_zero(int) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I)
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),K) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),zero_zero(int))
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),I) ) ) ) ).

% zmod_trivial_iff
tff(fact_2117_pos__mod__conj,axiom,
    ! [Ba: int,Aa2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Ba)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Aa2),Ba))
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Aa2),Ba)),Ba) ) ) ).

% pos_mod_conj
tff(fact_2118_neg__mod__conj,axiom,
    ! [Ba: int,Aa2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ba),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Aa2),Ba)),zero_zero(int))
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ba),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Aa2),Ba)) ) ) ).

% neg_mod_conj
tff(fact_2119_int__mod__lem,axiom,
    ! [Na: int,Ba: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Na)
     => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Ba)
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ba),Na) )
      <=> ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Ba),Na) = Ba ) ) ) ).

% int_mod_lem
tff(fact_2120_int__mod__eq,axiom,
    ! [Ba: int,Na: int,Aa2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Ba)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ba),Na)
       => ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Aa2),Na) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Ba),Na) )
         => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Aa2),Na) = Ba ) ) ) ) ).

% int_mod_eq
tff(fact_2121_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_2122_nat__diff__split__asm,axiom,
    ! [P: fun(nat,$o),Aa2: nat,Ba: nat] :
      ( aa(nat,$o,P,aa(nat,nat,minus_minus(nat,Aa2),Ba))
    <=> ~ ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Aa2),Ba)
            & ~ aa(nat,$o,P,zero_zero(nat)) )
          | ? [D5: nat] :
              ( ( Aa2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ba),D5) )
              & ~ aa(nat,$o,P,D5) ) ) ) ).

% nat_diff_split_asm
tff(fact_2123_nat__diff__split,axiom,
    ! [P: fun(nat,$o),Aa2: nat,Ba: nat] :
      ( aa(nat,$o,P,aa(nat,nat,minus_minus(nat,Aa2),Ba))
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Aa2),Ba)
         => aa(nat,$o,P,zero_zero(nat)) )
        & ! [D5: nat] :
            ( ( Aa2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ba),D5) )
           => aa(nat,$o,P,D5) ) ) ) ).

% nat_diff_split
tff(fact_2124_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,minus_minus(nat,J),K)),I)
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)) ) ) ).

% less_diff_conv2
tff(fact_2125_nonneg__mod__div,axiom,
    ! [Aa2: int,Ba: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Aa2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Ba)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Aa2),Ba))
          & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),Aa2),Ba)) ) ) ) ).

% nonneg_mod_div
tff(fact_2126_zdiv__mono__strict,axiom,
    ! [A2: int,B2: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),B2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Na)
       => ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),Na) = zero_zero(int) )
         => ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B2),Na) = zero_zero(int) )
           => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),Na)),aa(int,int,aa(int,fun(int,int),divide_divide(int),B2),Na)) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_2127_zdvd__imp__le,axiom,
    ! [Z: int,Na: int] :
      ( aa(int,$o,dvd_dvd(int,Z),Na)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Na)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),Na) ) ) ).

% zdvd_imp_le
tff(fact_2128_divmod__digit__1_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Ba)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Ba)))
             => ( aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Ba))),Ba) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),Ba) ) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_2129_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Na) )
      <=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,I),J)),U)),M) = Na ) ) ) ).

% nat_eq_add_iff1
tff(fact_2130_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Na) )
      <=> ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,J),I)),U)),Na) ) ) ) ).

% nat_eq_add_iff2
tff(fact_2131_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Na))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,I),J)),U)),M)),Na) ) ) ).

% nat_le_add_iff1
tff(fact_2132_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Na))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,J),I)),U)),Na)) ) ) ).

% nat_le_add_iff2
tff(fact_2133_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Na)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,I),J)),U)),M)),Na) ) ) ).

% nat_diff_add_eq1
tff(fact_2134_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Na)) = aa(nat,nat,minus_minus(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,J),I)),U)),Na)) ) ) ).

% nat_diff_add_eq2
tff(fact_2135_div__mod__decomp__int,axiom,
    ! [A2: int,Na: int] : A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),Na)),Na)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),Na)) ).

% div_mod_decomp_int
tff(fact_2136_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_2137_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,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ).

% even_mask_div_iff'
tff(fact_2138_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,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),D2)
         => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Na),one_one(int))),D2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Na),D2)),one_one(int)) ) ) ) ) ).

% emep1
tff(fact_2139_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,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),D2)
         => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Na)),D2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Na),D2)) ) ) ) ) ).

% eme1p
tff(fact_2140_even__nat__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(int,nat,nat2,K))
      <=> aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K) ) ) ).

% even_nat_iff
tff(fact_2141_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,aa(A,fun(A,A),plus_plus(A),X),X)),one_one(A)) ) ).

% dbl_dec_def
tff(fact_2142_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,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ) ).

% even_mask_div_iff
tff(fact_2143_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_2144_mod__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Aa2: A,M: nat,Na: nat] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),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),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),Aa2),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),Aa2),aa(nat,A,semiring_1_of_nat(A),M))) ) ).

% mod_mult2_eq'
tff(fact_2145_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_2146_odd__even__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Aa2: A,Ba: A] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2)
         => ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Ba)
           => aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)) ) ) ) ).

% odd_even_add
tff(fact_2147_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_2148_evenE,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Aa2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2)
         => ~ ! [B3: A] : Aa2 != aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3) ) ) ).

% evenE
tff(fact_2149_bit__eq__rec,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Aa2: A,Ba: A] :
          ( ( Aa2 = Ba )
        <=> ( ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2)
            <=> aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Ba) )
            & ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ) ) ).

% bit_eq_rec
tff(fact_2150_is__unit__div__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Aa2: A,Ba: A] :
          ( ( Aa2 != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,Ba),one_one(A))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),Aa2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),Ba) ) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_2151_is__unit__div__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Aa2: A,Ba: A] :
          ( ( Aa2 != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,Ba),one_one(A))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),Ba) ) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_2152_is__unitE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Aa2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,Aa2),one_one(A))
         => ~ ( ( Aa2 != zero_zero(A) )
             => ! [B3: A] :
                  ( ( B3 != zero_zero(A) )
                 => ( aa(A,$o,dvd_dvd(A,B3),one_one(A))
                   => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),Aa2) = B3 )
                     => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B3) = Aa2 )
                       => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),B3) = one_one(A) )
                         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),Aa2) != aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3) ) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_2153_even__minus,axiom,
    ! [A: $tType] :
      ( ring_parity(A)
     => ! [Aa2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,uminus_uminus(A),Aa2))
        <=> aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2) ) ) ).

% even_minus
tff(fact_2154_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_2155_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,aa(A,fun(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,aa(A,fun(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_2156_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,aa(A,fun(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,aa(A,fun(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_2157_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,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ) ) ).

% dvd_power_iff
tff(fact_2158_dvd__power,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Na: nat,X: A] :
          ( ( aa(nat,$o,aa(nat,fun(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_2159_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Aa2: A,Z: A,Ba: A] :
          aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Z))),Ba) = $ite(Z = zero_zero(A),aa(A,A,uminus_uminus(A),Ba),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),Aa2)),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),Z))),Z)) ) ).

% add_divide_eq_if_simps(6)
tff(fact_2160_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Aa2: A,Z: A,Ba: A] :
          aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Z)),Ba) = $ite(Z = zero_zero(A),aa(A,A,uminus_uminus(A),Ba),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),Z))),Z)) ) ).

% add_divide_eq_if_simps(5)
tff(fact_2161_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_2162_even__and__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Aa2),Ba))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2)
            | aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Ba) ) ) ) ).

% even_and_iff
tff(fact_2163_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_2164_even__or__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Aa2),Ba))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2)
            & aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Ba) ) ) ) ).

% even_or_iff
tff(fact_2165_pos__mod__bound2,axiom,
    ! [Aa2: int] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Aa2),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% pos_mod_bound2
tff(fact_2166_even__xor__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Aa2),Ba))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2)
          <=> aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Ba) ) ) ) ).

% even_xor_iff
tff(fact_2167_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_2168_dvd__mult__cancel2,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => ( aa(nat,$o,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_2169_dvd__mult__cancel1,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => ( aa(nat,$o,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_2170_power__diff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [Aa2: A,Na: nat,M: nat] :
          ( ( Aa2 != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
           => ( aa(nat,A,power_power(A,Aa2),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,Aa2),M)),aa(nat,A,power_power(A,Aa2),Na)) ) ) ) ) ).

% power_diff
tff(fact_2171_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,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),I)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ).

% power_dvd_imp_le
tff(fact_2172_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)),zero_zero(int))
       => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_2173_int__mod__ge_H,axiom,
    ! [Ba: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ba),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Na)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Ba),Na)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Ba),Na)) ) ) ).

% int_mod_ge'
tff(fact_2174_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_2175_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_2176_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Na: nat,K: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),K)
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Na)),K) = zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_2177_pochhammer__of__nat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [Na: nat,K: nat] :
          ( ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Na)),K) = zero_zero(A) )
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),K) ) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_2178_pochhammer__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Aa2: A,Na: nat] :
          ( ( comm_s3205402744901411588hammer(A,Aa2,Na) = zero_zero(A) )
        <=> ? [K2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),Na)
              & ( Aa2 = aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),K2)) ) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_2179_pochhammer__product_H,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Z: A,Na: nat,M: nat] : comm_s3205402744901411588hammer(A,Z,aa(nat,nat,aa(nat,fun(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,aa(A,fun(A,A),plus_plus(A),Z),aa(nat,A,semiring_1_of_nat(A),Na)),M)) ) ).

% pochhammer_product'
tff(fact_2180_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Na)),K) != zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_2181_mult__eq__if,axiom,
    ! [M: nat,Na: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na) = $ite(M = zero_zero(nat),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,M),one_one(nat))),Na))) ).

% mult_eq_if
tff(fact_2182_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Na))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,I),J)),U)),M)),Na) ) ) ).

% nat_less_add_iff1
tff(fact_2183_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Na))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,J),I)),U)),Na)) ) ) ).

% nat_less_add_iff2
tff(fact_2184_nat__mult__power__less__eq,axiom,
    ! [Ba: nat,Aa2: nat,Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Ba)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Aa2),aa(nat,nat,power_power(nat,Ba),Na))),aa(nat,nat,power_power(nat,Ba),M))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Aa2),aa(nat,nat,power_power(nat,Ba),aa(nat,nat,minus_minus(nat,M),Na))) ) ) ).

% nat_mult_power_less_eq
tff(fact_2185_even__mult__exp__div__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Aa2: A,M: nat,Na: nat] :
          ( aa(A,$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),Aa2),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,$o,aa(nat,fun(nat,$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,$o,aa(nat,fun(nat,$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),Aa2),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_2186_binomial__maximum,axiom,
    ! [Na: nat,K: nat] : aa(nat,$o,aa(nat,fun(nat,$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_2187_binomial__antimono,axiom,
    ! [K: nat,K4: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),K4)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))),K)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K4),Na)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(Na),K4)),aa(nat,nat,binomial(Na),K)) ) ) ) ).

% binomial_antimono
tff(fact_2188_binomial__mono,axiom,
    ! [K: nat,K4: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),K4)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K4)),Na)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(Na),K)),aa(nat,nat,binomial(Na),K4)) ) ) ).

% binomial_mono
tff(fact_2189_binomial__maximum_H,axiom,
    ! [Na: nat,K: nat] : aa(nat,$o,aa(nat,fun(nat,$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_2190_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C2: A,Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
         => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba)),C2))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),Ba)) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_2191_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_2192_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_2193_and__one__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Aa2),one_one(A)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% and_one_eq
tff(fact_2194_one__and__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),one_one(A)),Aa2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% one_and_eq
tff(fact_2195_even__two__times__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Aa2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2)
         => ( 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),Aa2),aa(num,A,numeral_numeral(A),bit0(one2)))) = Aa2 ) ) ) ).

% even_two_times_div_two
tff(fact_2196_scaling__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,V: A,R: A,S: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),V)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),R)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),R),S)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),U),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),R),aa(A,A,minus_minus(A,V),U))),S))),V) ) ) ) ) ).

% scaling_mono
tff(fact_2197_power__mono__odd,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat,Aa2: A,Ba: A] :
          ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,Aa2),Na)),aa(nat,A,power_power(A,Ba),Na)) ) ) ) ).

% power_mono_odd
tff(fact_2198_pos__mod__sign2,axiom,
    ! [Aa2: int] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Aa2),aa(num,int,numeral_numeral(int),bit0(one2)))) ).

% pos_mod_sign2
tff(fact_2199_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_2200_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_2201_odd__pos,axiom,
    ! [Na: nat] :
      ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ).

% odd_pos
tff(fact_2202_uminus__power__if,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Aa2: A,Na: nat] :
          aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),Aa2)),Na) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na),aa(nat,A,power_power(A,Aa2),Na),aa(A,A,uminus_uminus(A),aa(nat,A,power_power(A,Aa2),Na))) ) ).

% uminus_power_if
tff(fact_2203_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_2204_mod__exp__less__eq__exp,axiom,
    ! [Aa2: int,Na: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Aa2),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_2205_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
     => ( aa(nat,$o,dvd_dvd(nat,aa(nat,nat,power_power(nat,K),M)),aa(nat,nat,power_power(nat,K),Na))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ).

% dvd_power_iff_le
tff(fact_2206_power__diff__power__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Aa2: A,M: nat,Na: nat] :
          ( ( Aa2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,power_power(A,Aa2),M)),aa(nat,A,power_power(A,Aa2),Na)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M),aa(nat,A,power_power(A,Aa2),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,Aa2),aa(nat,nat,minus_minus(nat,Na),M)))) ) ) ) ).

% power_diff_power_eq
tff(fact_2207_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_2208_power__minus__mult,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Na: nat,Aa2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,Aa2),aa(nat,nat,minus_minus(nat,Na),one_one(nat)))),Aa2) = aa(nat,A,power_power(A,Aa2),Na) ) ) ) ).

% power_minus_mult
tff(fact_2209_mod__power__lem,axiom,
    ! [Aa2: int,Na: nat,M: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),Aa2)
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(nat,int,power_power(int,Aa2),Na)),aa(nat,int,power_power(int,Aa2),M)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na),zero_zero(int),aa(nat,int,power_power(int,Aa2),Na)) ) ) ).

% mod_power_lem
tff(fact_2210_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,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_2211_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
         => ( aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),K)) = aa(nat,A,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_2212_int__mod__pos__eq,axiom,
    ! [Aa2: int,Ba: int,Q3: int,R: int] :
      ( ( Aa2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Ba),Q3)),R) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R),Ba)
         => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Aa2),Ba) = R ) ) ) ) ).

% int_mod_pos_eq
tff(fact_2213_int__mod__neg__eq,axiom,
    ! [Aa2: int,Ba: int,Q3: int,R: int] :
      ( ( Aa2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Ba),Q3)),R) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ba),R)
         => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Aa2),Ba) = R ) ) ) ) ).

% int_mod_neg_eq
tff(fact_2214_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,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
         => ! [I4: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less(int),J3),K)
                & ( Na = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
             => aa(int,$o,P,J3) ) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
         => ! [I4: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J3),zero_zero(int))
                & ( Na = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
             => aa(int,$o,P,J3) ) ) ) ) ).

% split_zmod
tff(fact_2215_zmod__zmult2__eq,axiom,
    ! [C2: int,Aa2: int,Ba: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),C2)
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Aa2),aa(int,int,aa(int,fun(int,int),times_times(int),Ba),C2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Ba),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),Aa2),Ba)),C2))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Aa2),Ba)) ) ) ).

% zmod_zmult2_eq
tff(fact_2216_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_2217_power__sub,axiom,
    ! [Na: nat,M: nat,Aa2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Aa2)
       => ( aa(nat,nat,power_power(nat,Aa2),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,Aa2),M)),aa(nat,nat,power_power(nat,Aa2),Na)) ) ) ) ).

% power_sub
tff(fact_2218_binomial__strict__mono,axiom,
    ! [K: nat,K4: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),K4)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K4)),Na)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,binomial(Na),K)),aa(nat,nat,binomial(Na),K4)) ) ) ).

% binomial_strict_mono
tff(fact_2219_binomial__strict__antimono,axiom,
    ! [K: nat,K4: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),K4)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K4),Na)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,binomial(Na),K4)),aa(nat,nat,binomial(Na),K)) ) ) ) ).

% binomial_strict_antimono
tff(fact_2220_divmod__digit__0_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Ba))),Ba)
           => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Ba)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),Ba) ) ) ) ) ).

% divmod_digit_0(2)
tff(fact_2221_bits__stable__imp__add__self,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Aa2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2))) = Aa2 )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2)))) = zero_zero(A) ) ) ) ).

% bits_stable_imp_add_self
tff(fact_2222_div__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Aa2: 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),Aa2),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),Aa2),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(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_2223_oddE,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Aa2: A] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2)
         => ~ ! [B3: A] : Aa2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3)),one_one(A)) ) ) ).

% oddE
tff(fact_2224_zero__le__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Aa2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,Aa2),Na))
        <=> ( aa(nat,$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,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2) ) ) ) ) ).

% zero_le_power_eq
tff(fact_2225_zero__le__odd__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat,Aa2: A] :
          ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,Aa2),Na))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2) ) ) ) ).

% zero_le_odd_power
tff(fact_2226_zero__le__even__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat,Aa2: A] :
          ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,Aa2),Na)) ) ) ).

% zero_le_even_power
tff(fact_2227_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_2228_axxmod2,axiom,
    ! [X: int] :
      ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),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,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),zero_zero(int)),X)),X)),aa(num,int,numeral_numeral(int),bit0(one2))) = zero_zero(int) ) ) ).

% axxmod2
tff(fact_2229_z1pmod2,axiom,
    ! [Ba: int] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Ba)),one_one(int))),aa(num,int,numeral_numeral(int),bit0(one2))) = one_one(int) ).

% z1pmod2
tff(fact_2230_nat__power__less__diff,axiom,
    ! [Na: nat,Q3: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,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,$o,aa(nat,fun(nat,$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_2231_power__minus__is__div,axiom,
    ! [Ba: nat,Aa2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ba),Aa2)
     => ( aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,Aa2),Ba)) = 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))),Aa2)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Ba)) ) ) ).

% power_minus_is_div
tff(fact_2232_nat__le__power__trans,axiom,
    ! [Na: nat,M: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,M),K)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,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_2233_verit__le__mono__div__int,axiom,
    ! [A2: int,B2: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),B2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Na)
       => aa(int,$o,
            aa(int,fun(int,$o),ord_less_eq(int),
              aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),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_2234_split__pos__lemma,axiom,
    ! [K: int,P: fun(int,fun(int,$o)),Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),P,aa(int,int,aa(int,fun(int,int),divide_divide(int),Na),K)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Na),K))
      <=> ! [I4: int,J3: int] :
            ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J3)
              & aa(int,$o,aa(int,fun(int,$o),ord_less(int),J3),K)
              & ( Na = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
           => aa(int,$o,aa(int,fun(int,$o),P,I4),J3) ) ) ) ).

% split_pos_lemma
tff(fact_2235_split__neg__lemma,axiom,
    ! [K: int,P: fun(int,fun(int,$o)),Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),P,aa(int,int,aa(int,fun(int,int),divide_divide(int),Na),K)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Na),K))
      <=> ! [I4: int,J3: int] :
            ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),J3)
              & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J3),zero_zero(int))
              & ( Na = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
           => aa(int,$o,aa(int,fun(int,$o),P,I4),J3) ) ) ) ).

% split_neg_lemma
tff(fact_2236_divmod__digit__0_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Ba))),Ba)
           => ( 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),Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Ba))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba) ) ) ) ) ).

% divmod_digit_0(1)
tff(fact_2237_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,aa(A,fun(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_2238_zero__less__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Aa2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,power_power(A,Aa2),Na))
        <=> ( ( Na = zero_zero(nat) )
            | ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
              & ( Aa2 != zero_zero(A) ) )
            | ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2) ) ) ) ) ).

% zero_less_power_eq
tff(fact_2239_p1mod22k_H,axiom,
    ! [Ba: int,Na: nat] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Ba))),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,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Ba),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)))) ).

% p1mod22k'
tff(fact_2240_p1mod22k,axiom,
    ! [Ba: int,Na: nat] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Ba)),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,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Ba),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)))),one_one(int)) ).

% p1mod22k
tff(fact_2241_less__two__pow__divD,axiom,
    ! [X: nat,Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
        & aa(nat,$o,aa(nat,fun(nat,$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_2242_less__two__pow__divI,axiom,
    ! [X: nat,Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
       => aa(nat,$o,aa(nat,fun(nat,$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_2243_nat__less__power__trans,axiom,
    ! [Na: nat,M: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,M),K)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,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_2244_mod__double__modulus,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),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,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),X),M)),M) ) ) ) ) ) ).

% mod_double_modulus
tff(fact_2245_word__less__two__pow__divD,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Na: nat,M: nat] :
          ( aa(word(A),$o,aa(word(A),fun(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,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
            & aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),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_2246_power__le__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Aa2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,Aa2),Na)),zero_zero(A))
        <=> ( aa(nat,$o,aa(nat,fun(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,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),zero_zero(A)) )
              | ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
                & ( Aa2 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_2247_pos__zmod__mult__2,axiom,
    ! [Aa2: int,Ba: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Aa2)
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Ba))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Aa2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Ba),Aa2))) ) ) ).

% pos_zmod_mult_2
tff(fact_2248_sb__dec__lem_H,axiom,
    ! [K: nat,Aa2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),K)),Aa2)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Aa2),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,aa(int,fun(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))),Aa2)) ) ).

% sb_dec_lem'
tff(fact_2249_small__powers__of__2,axiom,
    ! [X: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit1(one2))),X)
     => aa(nat,$o,aa(nat,fun(nat,$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_2250_sb__dec__lem,axiom,
    ! [K: nat,Aa2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),K))),Aa2))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Aa2),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,aa(int,fun(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))),Aa2)) ) ).

% sb_dec_lem
tff(fact_2251_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,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),X))),aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X)),Na)) ) ).

% Bernoulli_inequality_even
tff(fact_2252_divmod__digit__1_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Ba)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Ba)))
             => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Ba)))),one_one(A)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba) ) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_2253_zero__less__binomial__iff,axiom,
    ! [Na: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,binomial(Na),K))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na) ) ).

% zero_less_binomial_iff
tff(fact_2254_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_2255_binomial__n__0,axiom,
    ! [Na: nat] : aa(nat,nat,binomial(Na),zero_zero(nat)) = one_one(nat) ).

% binomial_n_0
tff(fact_2256_times__binomial__minus1__eq,axiom,
    ! [K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),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_2257_choose__reduce__nat,axiom,
    ! [Na: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => ( aa(nat,nat,binomial(Na),K) = aa(nat,nat,aa(nat,fun(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_2258_binomial__le__pow2,axiom,
    ! [Na: nat,K: nat] : aa(nat,$o,aa(nat,fun(nat,$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_2259_binomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,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_2260_diff__diff__add__mset,axiom,
    ! [A: $tType,M2: multiset(A),N3: multiset(A),P: multiset(A)] : aa(multiset(A),multiset(A),minus_minus(multiset(A),aa(multiset(A),multiset(A),minus_minus(multiset(A),M2),N3)),P) = aa(multiset(A),multiset(A),minus_minus(multiset(A),M2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),N3),P)) ).

% diff_diff_add_mset
tff(fact_2261_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_2262_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_2263_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_2264_binomial__n__n,axiom,
    ! [Na: nat] : aa(nat,nat,binomial(Na),Na) = one_one(nat) ).

% binomial_n_n
tff(fact_2265_zle__diff1__eq,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W),aa(int,int,minus_minus(int,Z),one_one(int)))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),Z) ) ).

% zle_diff1_eq
tff(fact_2266_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,aa(word(A),fun(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,aa(word(A),fun(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_2267_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,aa(word(A),fun(word(A),$o),ord_less(word(A)),one_one(word(A))),aa(num,word(A),numeral_numeral(word(A)),W))
          <=> aa(word(A),$o,aa(word(A),fun(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_2268_diff__nat__numeral,axiom,
    ! [V: num,V3: num] : aa(nat,nat,minus_minus(nat,aa(num,nat,numeral_numeral(nat),V)),aa(num,nat,numeral_numeral(nat),V3)) = aa(int,nat,nat2,aa(int,int,minus_minus(int,aa(num,int,numeral_numeral(int),V)),aa(num,int,numeral_numeral(int),V3))) ).

% diff_nat_numeral
tff(fact_2269_add__self__mod__2,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),M)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = zero_zero(nat) ).

% add_self_mod_2
tff(fact_2270_mod2__gr__0,axiom,
    ! [M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2))))
    <=> ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(nat) ) ) ).

% mod2_gr_0
tff(fact_2271_nat__numeral__diff__1,axiom,
    ! [V: num] : aa(nat,nat,minus_minus(nat,aa(num,nat,numeral_numeral(nat),V)),one_one(nat)) = aa(int,nat,nat2,aa(int,int,minus_minus(int,aa(num,int,numeral_numeral(int),V)),one_one(int))) ).

% nat_numeral_diff_1
tff(fact_2272_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_2273_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_2274_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_2275_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_2276_mod__plus__right,axiom,
    ! [Aa2: nat,X: nat,M: nat,Ba: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Aa2),X)),M) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ba),X)),M) )
    <=> ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Aa2),M) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Ba),M) ) ) ).

% mod_plus_right
tff(fact_2277_mod__less__eq__dividend,axiom,
    ! [M: nat,Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na)),M) ).

% mod_less_eq_dividend
tff(fact_2278_mod__word__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),V: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),W),V)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),W),V) = W ) ) ) ).

% mod_word_less
tff(fact_2279_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_2280_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_2281_diff__empty,axiom,
    ! [A: $tType,M2: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),minus_minus(multiset(A),M2),zero_zero(multiset(A))) = M2 )
      & ( aa(multiset(A),multiset(A),minus_minus(multiset(A),zero_zero(multiset(A))),M2) = zero_zero(multiset(A)) ) ) ).

% diff_empty
tff(fact_2282_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_2283_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_2284_Multiset_Odiff__add,axiom,
    ! [A: $tType,M2: multiset(A),N3: multiset(A),Q: multiset(A)] : aa(multiset(A),multiset(A),minus_minus(multiset(A),M2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),N3),Q)) = aa(multiset(A),multiset(A),minus_minus(multiset(A),aa(multiset(A),multiset(A),minus_minus(multiset(A),M2),N3)),Q) ).

% Multiset.diff_add
tff(fact_2285_diff__union__cancelL,axiom,
    ! [A: $tType,N3: multiset(A),M2: multiset(A)] : aa(multiset(A),multiset(A),minus_minus(multiset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),N3),M2)),N3) = M2 ).

% diff_union_cancelL
tff(fact_2286_diff__union__cancelR,axiom,
    ! [A: $tType,M2: multiset(A),N3: multiset(A)] : aa(multiset(A),multiset(A),minus_minus(multiset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M2),N3)),N3) = M2 ).

% diff_union_cancelR
tff(fact_2287_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M5: nat,N: nat] : Z != aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),M5)),aa(nat,int,semiring_1_of_nat(int),N)) ).

% int_diff_cases
tff(fact_2288_word__sub__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),X)
         => aa(word(A),$o,aa(word(A),fun(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_2289_word__sub__le__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),X),Y)),X)
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),X) ) ) ).

% word_sub_le_iff
tff(fact_2290_word__le__minus__mono,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),C2: word(A),D2: word(A),Ba: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Aa2),C2)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),D2),Ba)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),Aa2),Ba)),Aa2)
             => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),C2),D2)),C2)
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),Aa2),Ba)),aa(word(A),word(A),minus_minus(word(A),C2),D2)) ) ) ) ) ) ).

% word_le_minus_mono
tff(fact_2291_word__le__minus__cancel,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),Y),X)),aa(word(A),word(A),minus_minus(word(A),Z),X))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),Z)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),Z) ) ) ) ).

% word_le_minus_cancel
tff(fact_2292_word__le__minus__mono__left,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Z: word(A),X: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),Y)
           => aa(word(A),$o,aa(word(A),fun(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_2293_word__less__minus__mono,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),C2: word(A),D2: word(A),Ba: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Aa2),C2)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),D2),Ba)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),minus_minus(word(A),Aa2),Ba)),Aa2)
             => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),minus_minus(word(A),C2),D2)),C2)
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),minus_minus(word(A),Aa2),Ba)),aa(word(A),word(A),minus_minus(word(A),C2),D2)) ) ) ) ) ) ).

% word_less_minus_mono
tff(fact_2294_word__sub__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),X),aa(word(A),word(A),minus_minus(word(A),X),Y))
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),X),Y) ) ) ).

% word_sub_less_iff
tff(fact_2295_sub__wrap__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),X),aa(word(A),word(A),minus_minus(word(A),X),Z))
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),X),Z) ) ) ).

% sub_wrap_lt
tff(fact_2296_Word_Omod__minus__cong,axiom,
    ! [Ba: int,B4: int,X: int,X7: int,Y: int,Y6: int,Z4: int] :
      ( ( Ba = B4 )
     => ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),X),B4) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),X7),B4) )
       => ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Y),B4) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Y6),B4) )
         => ( ( aa(int,int,minus_minus(int,X7),Y6) = Z4 )
           => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,minus_minus(int,X),Y)),Ba) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Z4),B4) ) ) ) ) ) ).

% Word.mod_minus_cong
tff(fact_2297_word__rot__lem,axiom,
    ! [L: nat,K: nat,D2: nat,Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),L),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),D2),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),K),L)) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),L)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),D2),Na)),L) = Na ) ) ) ).

% word_rot_lem
tff(fact_2298_mod__eq__0D,axiom,
    ! [M: nat,D2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),D2) = zero_zero(nat) )
     => ? [Q5: nat] : M = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),D2),Q5) ) ).

% mod_eq_0D
tff(fact_2299_nat__minus__mod__plus__right,axiom,
    ! [Na: nat,X: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),X)),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Na),M))),M) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),M) ).

% nat_minus_mod_plus_right
tff(fact_2300_le__mod__geq,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,minus_minus(nat,M),Na)),Na) ) ) ).

% le_mod_geq
tff(fact_2301_mod__eq__dvd__iff__nat,axiom,
    ! [Na: nat,M: nat,Q3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Q3) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Na),Q3) )
      <=> aa(nat,$o,dvd_dvd(nat,Q3),aa(nat,nat,minus_minus(nat,M),Na)) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_2302_nat__mod__eq__iff,axiom,
    ! [X: nat,Na: nat,Y: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),Na) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Y),Na) )
    <=> ? [Q1: nat,Q22: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),Q1)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_2303_msrevs_I2_J,axiom,
    ! [K: nat,Na: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Na)),M)),Na) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na) ).

% msrevs(2)
tff(fact_2304_word__mod__less__divisor,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A),M: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),Na)
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),M),Na)),Na) ) ) ).

% word_mod_less_divisor
tff(fact_2305_subset__minus__empty,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(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_2306_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_2307_zmod__int,axiom,
    ! [Aa2: nat,Ba: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Aa2),Ba)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(nat,int,semiring_1_of_nat(int),Aa2)),aa(nat,int,semiring_1_of_nat(int),Ba)) ).

% zmod_int
tff(fact_2308_int__le__induct,axiom,
    ! [I: int,K: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),K)
     => ( aa(int,$o,P,K)
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),K)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,minus_minus(int,I2),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_le_induct
tff(fact_2309_int__less__induct,axiom,
    ! [I: int,K: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),K)
     => ( aa(int,$o,P,aa(int,int,minus_minus(int,K),one_one(int)))
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I2),K)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,minus_minus(int,I2),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_less_induct
tff(fact_2310_union__diff__assoc,axiom,
    ! [A: $tType,C5: multiset(A),B2: multiset(A),A2: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),minus_minus(multiset(A),C5),B2) = zero_zero(multiset(A)) )
     => ( aa(multiset(A),multiset(A),minus_minus(multiset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A2),B2)),C5) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A2),aa(multiset(A),multiset(A),minus_minus(multiset(A),B2),C5)) ) ) ).

% union_diff_assoc
tff(fact_2311_plus__minus__no__overflow__ab,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Ab: word(A),C2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),minus_minus(word(A),Ab),C2))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C2),Ab)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),C2)) ) ) ) ).

% plus_minus_no_overflow_ab
tff(fact_2312_le__minus_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),C2: word(A),Ba: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Aa2),C2)),Ba)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Aa2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Aa2),C2))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C2),aa(word(A),word(A),minus_minus(word(A),Ba),Aa2)) ) ) ) ).

% le_minus'
tff(fact_2313_le__plus_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A),C2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Aa2),Ba)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C2),aa(word(A),word(A),minus_minus(word(A),Ba),Aa2))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Aa2),C2)),Ba) ) ) ) ).

% le_plus'
tff(fact_2314_le__plus,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [C2: word(A),Ba: word(A),Aa2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C2),aa(word(A),word(A),minus_minus(word(A),Ba),Aa2))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Aa2),Ba)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Aa2),C2)),Ba) ) ) ) ).

% le_plus
tff(fact_2315_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,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),Y),X)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)),X))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)) ) ) ) ).

% word_diff_ls''(3)
tff(fact_2316_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,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),Y)
           => aa(word(A),$o,aa(word(A),fun(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),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)),X)) ) ) ) ).

% word_diff_ls''(4)
tff(fact_2317_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,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)),X)),aa(word(A),word(A),minus_minus(word(A),Z),X))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),Z)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)),Z) ) ) ) ).

% word_l_diffs'(3)
tff(fact_2318_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,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)),X)),aa(word(A),word(A),minus_minus(word(A),Z),X)) ) ) ) ).

% word_l_diffs'(4)
tff(fact_2319_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,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),Y),X)),W)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)) ) ) ) ).

% word_diff_ls'(3)
tff(fact_2320_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,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),Y)
           => aa(word(A),$o,aa(word(A),fun(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_2321_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,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),W),aa(word(A),word(A),minus_minus(word(A),Z),X))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),Z)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)),Z) ) ) ) ).

% More_Word.word_l_diffs(3)
tff(fact_2322_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,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X))
           => aa(word(A),$o,aa(word(A),fun(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_2323_word__plus__mcs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: word(A),Xb: word(A),X: word(A),W: word(A),Xa2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),Xb)),X)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)),X))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),Xb)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)) ) ) ) ).

% word_plus_mcs(3)
tff(fact_2324_word__plus__mcs_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: word(A),Xb: word(A),W: word(A),Xa2: word(A),X: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),Xb)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),Xb))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),Xb)),X)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)),X)) ) ) ) ).

% word_plus_mcs(4)
tff(fact_2325_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,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)),X)),aa(word(A),word(A),minus_minus(word(A),Z),X))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),Z)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)),Z) ) ) ) ).

% Word.word_l_diffs(3)
tff(fact_2326_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,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)),X)),aa(word(A),word(A),minus_minus(word(A),Z),X)) ) ) ) ).

% Word.word_l_diffs(4)
tff(fact_2327_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,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),Y),X)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)),X))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)) ) ) ) ).

% word_diff_ls(3)
tff(fact_2328_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,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),Y)
           => aa(word(A),$o,aa(word(A),fun(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),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)),X)) ) ) ) ).

% word_diff_ls(4)
tff(fact_2329_word__less__minus__mono__left,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Z: word(A),X: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),Y)
           => aa(word(A),$o,aa(word(A),fun(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_2330_word__less__minus__cancel,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),minus_minus(word(A),Y),X)),aa(word(A),word(A),minus_minus(word(A),Z),X))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),Z)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),Z) ) ) ) ).

% word_less_minus_cancel
tff(fact_2331_minus__real__def,axiom,
    ! [X: real,Y: real] : aa(real,real,minus_minus(real,X),Y) = aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,uminus_uminus(real),Y)) ).

% minus_real_def
tff(fact_2332_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y: extended_enat,X: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),Z),Y)
     => ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),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,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),X),Y)),Z) ) ) ).

% add_diff_assoc_enat
tff(fact_2333_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_2334_mod__le__divisor,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na)),Na) ) ).

% mod_le_divisor
tff(fact_2335_div__less__mono,axiom,
    ! [A2: nat,B2: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A2),Na) = zero_zero(nat) )
         => ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),B2),Na) = zero_zero(nat) )
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A2),Na)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),B2),Na)) ) ) ) ) ).

% div_less_mono
tff(fact_2336_mod__nat__add,axiom,
    ! [X: nat,Z: nat,Y: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Z)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),Z)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),Y)),Z) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),Y)),Z),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),Y),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),Y)),Z)) ) ) ) ).

% mod_nat_add
tff(fact_2337_mod__nat__eqI,axiom,
    ! [R: nat,Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),R),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),R),M)
       => ( aa(nat,$o,dvd_dvd(nat,Na),aa(nat,nat,minus_minus(nat,M),R))
         => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na) = R ) ) ) ) ).

% mod_nat_eqI
tff(fact_2338_diff__mod__le,axiom,
    ! [Aa2: nat,D2: nat,Ba: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Aa2),D2)
     => ( aa(nat,$o,dvd_dvd(nat,Ba),D2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,Aa2),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Aa2),Ba))),aa(nat,nat,minus_minus(nat,D2),Ba)) ) ) ).

% diff_mod_le
tff(fact_2339_nat__mod__eq__lemma,axiom,
    ! [X: nat,Na: nat,Y: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),Na) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Y),Na) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),X)
       => ? [Q5: nat] : X = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),Q5)) ) ) ).

% nat_mod_eq_lemma
tff(fact_2340_mod__eq__nat2E,axiom,
    ! [M: nat,Q3: nat,Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Q3) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Na),Q3) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
       => ~ ! [S2: nat] : Na != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q3),S2)) ) ) ).

% mod_eq_nat2E
tff(fact_2341_mod__eq__nat1E,axiom,
    ! [M: nat,Q3: nat,Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Q3) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Na),Q3) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
       => ~ ! [S2: nat] : M != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q3),S2)) ) ) ).

% mod_eq_nat1E
tff(fact_2342_div__mod__decomp,axiom,
    ! [A2: nat,Na: nat] : A2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A2),Na)),Na)),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A2),Na)) ).

% div_mod_decomp
tff(fact_2343_mod__mult2__eq,axiom,
    ! [M: nat,Na: nat,Q3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),Q3)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)),Q3))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na)) ).

% mod_mult2_eq
tff(fact_2344_modulo__nat__def,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na) = aa(nat,nat,minus_minus(nat,M),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)) ).

% modulo_nat_def
tff(fact_2345_Bolzano,axiom,
    ! [Aa2: real,Ba: real,P: fun(real,fun(real,$o))] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),Ba)
     => ( ! [A3: real,B3: real,C4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),P,A3),B3)
           => ( aa(real,$o,aa(real,fun(real,$o),P,B3),C4)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),B3)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B3),C4)
                 => aa(real,$o,aa(real,fun(real,$o),P,A3),C4) ) ) ) )
       => ( ! [X2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),X2)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),Ba)
               => ? [D6: real] :
                    ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
                    & ! [A3: real,B3: real] :
                        ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),X2)
                          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),B3)
                          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,minus_minus(real,B3),A3)),D6) )
                       => aa(real,$o,aa(real,fun(real,$o),P,A3),B3) ) ) ) )
         => aa(real,$o,aa(real,fun(real,$o),P,Aa2),Ba) ) ) ) ).

% Bolzano
tff(fact_2346_le__minus,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & order(A) )
     => ! [Y: A,X: A,Aa2: word(B),C2: word(B),Ba: word(B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
         => ( aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less_eq(word(B)),aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),plus_plus(word(B)),Aa2),C2)),Ba)
           => ( aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less_eq(word(B)),Aa2),aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),plus_plus(word(B)),Aa2),C2))
             => aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less_eq(word(B)),C2),aa(word(B),word(B),minus_minus(word(B),Ba),Aa2)) ) ) ) ) ).

% le_minus
tff(fact_2347_minusinfinity,axiom,
    ! [D2: int,P1: fun(int,$o),P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( ! [X2: int,K5: int] :
            ( aa(int,$o,P1,X2)
          <=> aa(int,$o,P1,aa(int,int,minus_minus(int,X2),aa(int,int,aa(int,fun(int,int),times_times(int),K5),D2))) )
       => ( ? [Z2: int] :
            ! [X2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X2),Z2)
             => ( aa(int,$o,P,X2)
              <=> aa(int,$o,P1,X2) ) )
         => ( ? [X_13: int] : aa(int,$o,P1,X_13)
           => ? [X_12: int] : aa(int,$o,P,X_12) ) ) ) ) ).

% minusinfinity
tff(fact_2348_plusinfinity,axiom,
    ! [D2: int,P3: fun(int,$o),P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( ! [X2: int,K5: int] :
            ( aa(int,$o,P3,X2)
          <=> aa(int,$o,P3,aa(int,int,minus_minus(int,X2),aa(int,int,aa(int,fun(int,int),times_times(int),K5),D2))) )
       => ( ? [Z2: int] :
            ! [X2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z2),X2)
             => ( aa(int,$o,P,X2)
              <=> aa(int,$o,P3,X2) ) )
         => ( ? [X_13: int] : aa(int,$o,P3,X_13)
           => ? [X_12: int] : aa(int,$o,P,X_12) ) ) ) ) ).

% plusinfinity
tff(fact_2349_less__1__helper,axiom,
    ! [Na: int,M: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Na),M)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,minus_minus(int,Na),one_one(int))),M) ) ).

% less_1_helper
tff(fact_2350_int__induct,axiom,
    ! [P: fun(int,$o),K: int,I: int] :
      ( aa(int,$o,P,K)
     => ( ! [I2: int] :
            ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I2)
           => ( aa(int,$o,P,I2)
             => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))) ) )
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),K)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,minus_minus(int,I2),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_induct
tff(fact_2351_int__mod__le_H,axiom,
    ! [Ba: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,minus_minus(int,Ba),Na))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Ba),Na)),aa(int,int,minus_minus(int,Ba),Na)) ) ).

% int_mod_le'
tff(fact_2352_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),aa(int,nat,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_2353_int__div__sub__1,axiom,
    ! [M: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),M)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,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_2354_word__le__sub1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( ( X != zero_zero(word(A)) )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),one_one(word(A))),X)
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),zero_zero(word(A))),aa(word(A),word(A),minus_minus(word(A),X),one_one(word(A)))) ) ) ) ).

% word_le_sub1
tff(fact_2355_diff__size__le__size__Diff,axiom,
    ! [A: $tType,M2: multiset(A),M7: multiset(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,aa(multiset(A),nat,size_size(multiset(A)),M2)),aa(multiset(A),nat,size_size(multiset(A)),M7))),aa(multiset(A),nat,size_size(multiset(A)),aa(multiset(A),multiset(A),minus_minus(multiset(A),M2),M7))) ).

% diff_size_le_size_Diff
tff(fact_2356_plus__minus__not__NULL__ab,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Ab: word(A),C2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),minus_minus(word(A),Ab),C2))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C2),Ab)
           => ( ( C2 != zero_zero(word(A)) )
             => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),C2) != zero_zero(word(A)) ) ) ) ) ) ).

% plus_minus_not_NULL_ab
tff(fact_2357_word__less__sub1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( ( X != zero_zero(word(A)) )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),one_one(word(A))),X)
          <=> aa(word(A),$o,aa(word(A),fun(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_2358_sub__wrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(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,aa(word(A),fun(word(A),$o),ord_less(word(A)),X),Z) ) ) ) ).

% sub_wrap
tff(fact_2359_plus__minus__no__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Ab: word(A),C2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),X),aa(word(A),word(A),minus_minus(word(A),Ab),C2))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C2),Ab)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),C2)) ) ) ) ).

% plus_minus_no_overflow
tff(fact_2360_word__less__sub__right,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),Z))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),X)
           => aa(word(A),$o,aa(word(A),fun(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_2361_word__less__add__right,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),X),aa(word(A),word(A),minus_minus(word(A),Y),Z))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Z),Y)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Z)),Y) ) ) ) ).

% word_less_add_right
tff(fact_2362_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,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),minus_minus(word(A),Y),X)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)),X))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)) ) ) ) ).

% word_diff_ls''(1)
tff(fact_2363_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,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),Y)
           => aa(word(A),$o,aa(word(A),fun(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),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)),X)) ) ) ) ).

% word_diff_ls''(2)
tff(fact_2364_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,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)),X)),aa(word(A),word(A),minus_minus(word(A),Z),X))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),Z)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)),Z) ) ) ) ).

% word_l_diffs'(1)
tff(fact_2365_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,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)),X)),aa(word(A),word(A),minus_minus(word(A),Z),X)) ) ) ) ).

% word_l_diffs'(2)
tff(fact_2366_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,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),minus_minus(word(A),Y),X)),W)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)) ) ) ) ).

% word_diff_ls'(1)
tff(fact_2367_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,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),Y)
           => aa(word(A),$o,aa(word(A),fun(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_2368_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,aa(word(A),fun(word(A),$o),ord_less(word(A)),W),aa(word(A),word(A),minus_minus(word(A),Z),X))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),Z)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)),Z) ) ) ) ).

% More_Word.word_l_diffs(1)
tff(fact_2369_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,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X)),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),X))
           => aa(word(A),$o,aa(word(A),fun(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_2370_word__plus__mcs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: word(A),Xb: word(A),X: word(A),W: word(A),Xa2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),Xb)),X)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)),X))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),Xb)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)) ) ) ) ).

% word_plus_mcs(1)
tff(fact_2371_word__plus__mcs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: word(A),Xb: word(A),W: word(A),Xa2: word(A),X: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),Xb)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),Xb))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),Xb)),X)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)),X)) ) ) ) ).

% word_plus_mcs(2)
tff(fact_2372_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,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)),X)),aa(word(A),word(A),minus_minus(word(A),Z),X))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),Z)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)),Z) ) ) ) ).

% Word.word_l_diffs(1)
tff(fact_2373_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,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)),X)),aa(word(A),word(A),minus_minus(word(A),Z),X)) ) ) ) ).

% Word.word_l_diffs(2)
tff(fact_2374_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,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),minus_minus(word(A),Y),X)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)),X))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)) ) ) ) ).

% word_diff_ls(1)
tff(fact_2375_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,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),Y)
           => aa(word(A),$o,aa(word(A),fun(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),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xa2)),X)) ) ) ) ).

% word_diff_ls(2)
tff(fact_2376_word__mod__div__equality,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A),Ba: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Na),Ba)),Ba)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),Na),Ba)) = Na ) ).

% word_mod_div_equality
tff(fact_2377_mod__div__equality__div__eq,axiom,
    ! [Aa2: int,Ba: int] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),Aa2),Ba)),Ba) = aa(int,int,minus_minus(int,Aa2),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Aa2),Ba)) ).

% mod_div_equality_div_eq
tff(fact_2378_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_2379_nat__mod__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X4),Xa) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(nat,int,semiring_1_of_nat(int),X4)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_mod_as_int
tff(fact_2380_nat__minus__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa(nat,nat,minus_minus(nat,X4),Xa) = aa(int,nat,nat2,aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),X4)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_minus_as_int
tff(fact_2381_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_2382_mod__lemma,axiom,
    ! [C2: nat,R: nat,Ba: nat,Q3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),C2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),R),Ba)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ba),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Q3),C2))),R)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ba),C2)) ) ) ).

% mod_lemma
tff(fact_2383_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,aa(nat,fun(nat,$o),ord_less(nat),J3),Na)
             => ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),I4)),J3) )
               => aa(nat,$o,P,J3) ) ) ) ) ) ).

% split_mod
tff(fact_2384_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,aa(int,fun(int,int),plus_plus(int),K),L)) ) ).

% even_diff_iff
tff(fact_2385_nat__mod__distrib,axiom,
    ! [X: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
       => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),X),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(int,nat,nat2,X)),aa(int,nat,nat2,Y)) ) ) ) ).

% nat_mod_distrib
tff(fact_2386_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,aa(real,fun(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_2387_int__ops_I6_J,axiom,
    ! [Aa2: nat,Ba: nat] :
      aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,minus_minus(nat,Aa2),Ba)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),Aa2)),aa(nat,int,semiring_1_of_nat(int),Ba)),zero_zero(int),aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),Aa2)),aa(nat,int,semiring_1_of_nat(int),Ba))) ).

% int_ops(6)
tff(fact_2388_nat__diff__distrib,axiom,
    ! [Z4: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z4)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z4),Z)
       => ( aa(int,nat,nat2,aa(int,int,minus_minus(int,Z),Z4)) = aa(nat,nat,minus_minus(nat,aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z4)) ) ) ) ).

% nat_diff_distrib
tff(fact_2389_nat__diff__distrib_H,axiom,
    ! [X: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
       => ( aa(int,nat,nat2,aa(int,int,minus_minus(int,X),Y)) = aa(nat,nat,minus_minus(nat,aa(int,nat,nat2,X)),aa(int,nat,nat2,Y)) ) ) ) ).

% nat_diff_distrib'
tff(fact_2390_decr__mult__lemma,axiom,
    ! [D2: int,P: fun(int,$o),K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( ! [X2: int] :
            ( aa(int,$o,P,X2)
           => aa(int,$o,P,aa(int,int,minus_minus(int,X2),D2)) )
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
         => ! [X4: int] :
              ( aa(int,$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),K),D2))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_2391_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),L),K)
       => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,minus_minus(int,K),L)),L) ) ) ) ).

% mod_pos_geq
tff(fact_2392_word__sub__plus__one__nonzero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N6: word(A),Na: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),N6),Na)
         => ( ( N6 != zero_zero(word(A)) )
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),minus_minus(word(A),Na),N6)),one_one(word(A))) != zero_zero(word(A)) ) ) ) ) ).

% word_sub_plus_one_nonzero
tff(fact_2393_word__less__nowrapI_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Z: word(A),K: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),minus_minus(word(A),Z),K))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),K),Z)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),K)
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),K)) ) ) ) ) ).

% word_less_nowrapI'
tff(fact_2394_plus__minus__not__NULL,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Ab: word(A),C2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),X),aa(word(A),word(A),minus_minus(word(A),Ab),C2))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C2),Ab)
           => ( ( C2 != zero_zero(word(A)) )
             => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),C2) != zero_zero(word(A)) ) ) ) ) ) ).

% plus_minus_not_NULL
tff(fact_2395_word__less__nowrapI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Z: word(A),K: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),X),aa(word(A),word(A),minus_minus(word(A),Z),K))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),K),Z)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),K)
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),K)) ) ) ) ) ).

% word_less_nowrapI
tff(fact_2396_ceiling__diff__floor__le__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(int,$o,aa(int,fun(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_2397_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,$o,aa(nat,fun(nat,$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,aa(nat,fun(nat,$o),ord_less(nat),X),Y)
         => aa(int,$o,P,zero_zero(int)) ) ) ) ).

% zdiff_int_split
tff(fact_2398_real__of__nat__div2,axiom,
    ! [Na: nat,X: nat] : aa(real,$o,aa(real,fun(real,$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_2399_real__of__nat__div3,axiom,
    ! [Na: nat,X: nat] : aa(real,$o,aa(real,fun(real,$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_2400_log__divide,axiom,
    ! [Aa2: real,X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Aa2)
     => ( ( Aa2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
           => ( aa(real,real,log(Aa2),aa(real,real,aa(real,fun(real,real),divide_divide(real),X),Y)) = aa(real,real,minus_minus(real,aa(real,real,log(Aa2),X)),aa(real,real,log(Aa2),Y)) ) ) ) ) ) ).

% log_divide
tff(fact_2401_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L)
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),K)),L) = aa(int,int,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_2402_zmod__minus1,axiom,
    ! [Ba: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Ba)
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),one_one(int))),Ba) = aa(int,int,minus_minus(int,Ba),one_one(int)) ) ) ).

% zmod_minus1
tff(fact_2403_mod__sub__if__z,axiom,
    ! [X: int,Z: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X),Z)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Y),Z)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
             => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,minus_minus(int,X),Y)),Z) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Y),X),aa(int,int,minus_minus(int,X),Y),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,minus_minus(int,X),Y)),Z)) ) ) ) ) ) ) ).

% mod_sub_if_z
tff(fact_2404_mod__add__if__z,axiom,
    ! [X: int,Z: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X),Z)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Y),Z)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
             => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),Y)),Z) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),Y)),Z),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),Y),aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),Y)),Z)) ) ) ) ) ) ) ).

% mod_add_if_z
tff(fact_2405_zdiv__zminus1__eq__if,axiom,
    ! [Ba: int,Aa2: int] :
      ( ( Ba != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),Aa2)),Ba) = $ite(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Aa2),Ba) = zero_zero(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),Aa2),Ba)),aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),Aa2),Ba))),one_one(int))) ) ) ).

% zdiv_zminus1_eq_if
tff(fact_2406_zdiv__zminus2__eq__if,axiom,
    ! [Ba: int,Aa2: int] :
      ( ( Ba != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),Aa2),aa(int,int,uminus_uminus(int),Ba)) = $ite(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Aa2),Ba) = zero_zero(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),Aa2),Ba)),aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),Aa2),Ba))),one_one(int))) ) ) ).

% zdiv_zminus2_eq_if
tff(fact_2407_diff__nat__eq__if,axiom,
    ! [Z: int,Z4: int] :
      aa(nat,nat,minus_minus(nat,aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z4)) = $ite(
        aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z4),zero_zero(int)),
        aa(int,nat,nat2,Z),
        $let(
          d: int,
          d:= aa(int,int,minus_minus(int,Z),Z4),
          $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),d),zero_zero(int)),zero_zero(nat),aa(int,nat,nat2,d)) ) ) ).

% diff_nat_eq_if
tff(fact_2408_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_2409_verit__le__mono__div,axiom,
    ! [A2: nat,B2: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => aa(nat,$o,
            aa(nat,fun(nat,$o),ord_less_eq(nat),
              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),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_2410_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),L),K)
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,minus_minus(int,K),L)),L)),one_one(int)) ) ) ) ).

% div_pos_geq
tff(fact_2411_minus__log__eq__powr,axiom,
    ! [Ba: real,X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Ba)
     => ( ( Ba != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
         => ( aa(real,real,minus_minus(real,Y),aa(real,real,log(Ba),X)) = aa(real,real,log(Ba),aa(real,real,aa(real,fun(real,real),divide_divide(real),powr(real,Ba,Y)),X)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_2412_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_2413_choose__one,axiom,
    ! [Na: nat] : aa(nat,nat,binomial(Na),one_one(nat)) = Na ).

% choose_one
tff(fact_2414_log__minus__eq__powr,axiom,
    ! [Ba: real,X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Ba)
     => ( ( Ba != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
         => ( aa(real,real,minus_minus(real,aa(real,real,log(Ba),X)),Y) = aa(real,real,log(Ba),aa(real,real,aa(real,fun(real,real),times_times(real),X),powr(real,Ba,aa(real,real,uminus_uminus(real),Y)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_2415_xor__nat__unfold,axiom,
    ! [M: nat,Na: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),Na) = $ite(
        M = zero_zero(nat),
        Na,
        $ite(Na = zero_zero(nat),M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) ).

% xor_nat_unfold
tff(fact_2416_and__nat__unfold,axiom,
    ! [M: nat,Na: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),Na) = $ite(
        ( ( M = zero_zero(nat) )
        | ( Na = zero_zero(nat) ) ),
        zero_zero(nat),
        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ).

% and_nat_unfold
tff(fact_2417_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_2418_neg__zmod__mult__2,axiom,
    ! [Aa2: int,Ba: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Aa2),zero_zero(int))
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Ba))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Aa2)) = 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,aa(int,fun(int,int),plus_plus(int),Ba),one_one(int))),Aa2))),one_one(int)) ) ) ).

% neg_zmod_mult_2
tff(fact_2419_binomial__symmetric,axiom,
    ! [K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
     => ( aa(nat,nat,binomial(Na),K) = aa(nat,nat,binomial(Na),aa(nat,nat,minus_minus(nat,Na),K)) ) ) ).

% binomial_symmetric
tff(fact_2420_choose__mult__lemma,axiom,
    ! [M: nat,R: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),R)),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),R)),K)),K)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),R)),M)) ).

% choose_mult_lemma
tff(fact_2421_binomial__le__pow,axiom,
    ! [R: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),R),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(Na),R)),aa(nat,nat,power_power(nat,Na),R)) ) ).

% binomial_le_pow
tff(fact_2422_zero__less__binomial,axiom,
    ! [K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,binomial(Na),K)) ) ).

% zero_less_binomial
tff(fact_2423_choose__mult,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(Na),M)),aa(nat,nat,binomial(M),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(Na),K)),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,Na),K)),aa(nat,nat,minus_minus(nat,M),K))) ) ) ) ).

% choose_mult
tff(fact_2424_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_2425_real__average__minus__second,axiom,
    ! [Ba: real,Aa2: real] : 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),plus_plus(real),Ba),Aa2)),aa(num,real,numeral_numeral(real),bit0(one2)))),Aa2) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,Ba),Aa2)),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% real_average_minus_second
tff(fact_2426_real__average__minus__first,axiom,
    ! [Aa2: real,Ba: real] : 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),plus_plus(real),Aa2),Ba)),aa(num,real,numeral_numeral(real),bit0(one2)))),Aa2) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,Ba),Aa2)),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% real_average_minus_first
tff(fact_2427_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_2428_bezout__add__strong__nat,axiom,
    ! [Aa2: nat,Ba: nat] :
      ( ( Aa2 != zero_zero(nat) )
     => ? [D4: nat,X2: nat,Y2: nat] :
          ( aa(nat,$o,dvd_dvd(nat,D4),Aa2)
          & aa(nat,$o,dvd_dvd(nat,D4),Ba)
          & ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Aa2),X2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ba),Y2)),D4) ) ) ) ).

% bezout_add_strong_nat
tff(fact_2429_signed__take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,Aa2: A] :
          aa(A,A,bit_ri4674362597316999326ke_bit(A,Na),Aa2) = $ite(Na = zero_zero(nat),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,minus_minus(nat,Na),one_one(nat))),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2))))))) ) ).

% signed_take_bit_rec
tff(fact_2430_signed__take__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat] : aa(A,A,bit_ri4674362597316999326ke_bit(A,Na),zero_zero(A)) = zero_zero(A) ) ).

% signed_take_bit_of_0
tff(fact_2431_signed__take__bit__numeral__of__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: num] : aa(A,A,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_2432_signed__take__bit__of__minus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat] : aa(A,A,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_2433_signed__take__bit__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Aa2: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,zero_zero(nat)),Aa2) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% signed_take_bit_0
tff(fact_2434_Multiset_Odiff__right__commute,axiom,
    ! [A: $tType,M2: multiset(A),N3: multiset(A),Q: multiset(A)] : aa(multiset(A),multiset(A),minus_minus(multiset(A),aa(multiset(A),multiset(A),minus_minus(multiset(A),M2),N3)),Q) = aa(multiset(A),multiset(A),minus_minus(multiset(A),aa(multiset(A),multiset(A),minus_minus(multiset(A),M2),Q)),N3) ).

% Multiset.diff_right_commute
tff(fact_2435_signed__take__bit__minus,axiom,
    ! [Na: nat,K: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),aa(int,int,uminus_uminus(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),aa(int,int,uminus_uminus(int),K)) ).

% signed_take_bit_minus
tff(fact_2436_signed__take__bit__add,axiom,
    ! [Na: nat,K: int,L: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K)),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),L))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)) ).

% signed_take_bit_add
tff(fact_2437_signed__take__bit__mult,axiom,
    ! [Na: nat,K: int,L: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K)),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),L))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),aa(int,int,aa(int,fun(int,int),times_times(int),K),L)) ).

% signed_take_bit_mult
tff(fact_2438_signed__take__bit__diff,axiom,
    ! [Na: nat,K: int,L: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),aa(int,int,minus_minus(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K)),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),L))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),aa(int,int,minus_minus(int,K),L)) ).

% signed_take_bit_diff
tff(fact_2439_eq__diff__eq_H,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( X = aa(real,real,minus_minus(real,Y),Z) )
    <=> ( Y = aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Z) ) ) ).

% eq_diff_eq'
tff(fact_2440_mult__commute__abs,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [C2: A] : aTP_Lamp_av(A,fun(A,A),C2) = aa(A,fun(A,A),times_times(A),C2) ) ).

% mult_commute_abs
tff(fact_2441_signed__take__bit__int__less__exp,axiom,
    ! [Na: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ).

% signed_take_bit_int_less_exp
tff(fact_2442_even__signed__take__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,Aa2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,bit_ri4674362597316999326ke_bit(A,M),Aa2))
        <=> aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2) ) ) ).

% even_signed_take_bit_iff
tff(fact_2443_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_2444_signed__take__bit__int__less__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K)),K)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)),K) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_2445_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [Na: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K)) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_2446_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K)),K)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))),K) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_2447_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_2448_signed__take__bit__int__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( ( aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K) = K )
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))),K)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_2449_signed__take__bit__int__eq__self,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))
       => ( aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_2450_dvd__productE,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [P2: A,Aa2: A,Ba: A] :
          ( aa(A,$o,dvd_dvd(A,P2),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba))
         => ~ ! [X2: A,Y2: A] :
                ( ( P2 = aa(A,A,aa(A,fun(A,A),times_times(A),X2),Y2) )
               => ( aa(A,$o,dvd_dvd(A,X2),Aa2)
                 => ~ aa(A,$o,dvd_dvd(A,Y2),Ba) ) ) ) ) ).

% dvd_productE
tff(fact_2451_division__decomp,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),Ba),C2))
         => ? [B5: A,C6: A] :
              ( ( Aa2 = aa(A,A,aa(A,fun(A,A),times_times(A),B5),C6) )
              & aa(A,$o,dvd_dvd(A,B5),Ba)
              & aa(A,$o,dvd_dvd(A,C6),C2) ) ) ) ).

% division_decomp
tff(fact_2452_Euclid__induct,axiom,
    ! [P: fun(nat,fun(nat,$o)),Aa2: nat,Ba: nat] :
      ( ! [A3: nat,B3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),P,A3),B3)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),P,B3),A3) )
     => ( ! [A3: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,A3),zero_zero(nat))
       => ( ! [A3: nat,B3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),P,A3),B3)
             => aa(nat,$o,aa(nat,fun(nat,$o),P,A3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),B3)) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P,Aa2),Ba) ) ) ) ).

% Euclid_induct
tff(fact_2453_bezout__add__nat,axiom,
    ! [Aa2: nat,Ba: nat] :
    ? [D4: nat,X2: nat,Y2: nat] :
      ( aa(nat,$o,dvd_dvd(nat,D4),Aa2)
      & aa(nat,$o,dvd_dvd(nat,D4),Ba)
      & ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Aa2),X2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ba),Y2)),D4) )
        | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ba),X2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Aa2),Y2)),D4) ) ) ) ).

% bezout_add_nat
tff(fact_2454_bezout__lemma__nat,axiom,
    ! [D2: nat,Aa2: nat,Ba: nat,X: nat,Y: nat] :
      ( aa(nat,$o,dvd_dvd(nat,D2),Aa2)
     => ( aa(nat,$o,dvd_dvd(nat,D2),Ba)
       => ( ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Aa2),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ba),Y)),D2) )
            | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ba),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Aa2),Y)),D2) ) )
         => ? [X2: nat,Y2: nat] :
              ( aa(nat,$o,dvd_dvd(nat,D2),Aa2)
              & aa(nat,$o,dvd_dvd(nat,D2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Aa2),Ba))
              & ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Aa2),X2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Aa2),Ba)),Y2)),D2) )
                | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Aa2),Ba)),X2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Aa2),Y2)),D2) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_2455_bezout1__nat,axiom,
    ! [Aa2: nat,Ba: nat] :
    ? [D4: nat,X2: nat,Y2: nat] :
      ( aa(nat,$o,dvd_dvd(nat,D4),Aa2)
      & aa(nat,$o,dvd_dvd(nat,D4),Ba)
      & ( ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Aa2),X2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ba),Y2)) = D4 )
        | ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ba),X2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Aa2),Y2)) = D4 ) ) ) ).

% bezout1_nat
tff(fact_2456_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,aa(int,fun(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_2457_pochhammer__code,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Aa2: A,Na: nat] :
          comm_s3205402744901411588hammer(A,Aa2,Na) = $ite(Na = zero_zero(nat),one_one(A),set_fo6178422350223883121st_nat(A,aTP_Lamp_aw(A,fun(nat,fun(A,A)),Aa2),zero_zero(nat),aa(nat,nat,minus_minus(nat,Na),one_one(nat)),one_one(A))) ) ).

% pochhammer_code
tff(fact_2458_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa(int,int,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,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,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_2459_signed__take__bit__Suc__minus__bit1,axiom,
    ! [Na: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,Na)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit1(K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),aa(int,int,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_2460_ln__one__minus__pos__lower__bound,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,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_2461_and__mask__arith_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),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,aa(word(A),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,aa(word(A),nat,size_size(word(A)),W)),Na))) ) ) ) ).

% and_mask_arith'
tff(fact_2462_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_2463_nat_Oinject,axiom,
    ! [X23: nat,Y22: nat] :
      ( ( aa(nat,nat,suc,X23) = aa(nat,nat,suc,Y22) )
    <=> ( X23 = Y22 ) ) ).

% nat.inject
tff(fact_2464_abs__idempotent,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Aa2: A] : aa(A,A,abs_abs(A),aa(A,A,abs_abs(A),Aa2)) = aa(A,A,abs_abs(A),Aa2) ) ).

% abs_idempotent
tff(fact_2465_abs__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [Aa2: A] : aa(A,A,abs_abs(A),aa(A,A,abs_abs(A),Aa2)) = aa(A,A,abs_abs(A),Aa2) ) ).

% abs_abs
tff(fact_2466_Suc__less__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ).

% Suc_less_eq
tff(fact_2467_Suc__mono,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,Na)) ) ).

% Suc_mono
tff(fact_2468_lessI,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,suc,Na)) ).

% lessI
tff(fact_2469_add__Suc__right,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,suc,Na)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)) ).

% add_Suc_right
tff(fact_2470_Suc__le__mono,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Na)),aa(nat,nat,suc,M))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M) ) ).

% Suc_le_mono
tff(fact_2471_abs__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ( aa(A,A,abs_abs(A),zero_zero(A)) = zero_zero(A) ) ) ).

% abs_zero
tff(fact_2472_abs__eq__0,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Aa2: A] :
          ( ( aa(A,A,abs_abs(A),Aa2) = zero_zero(A) )
        <=> ( Aa2 = zero_zero(A) ) ) ) ).

% abs_eq_0
tff(fact_2473_abs__0__eq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Aa2: A] :
          ( ( zero_zero(A) = aa(A,A,abs_abs(A),Aa2) )
        <=> ( Aa2 = zero_zero(A) ) ) ) ).

% abs_0_eq
tff(fact_2474_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_2475_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_2476_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_2477_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_2478_abs__add__abs,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),Aa2)),aa(A,A,abs_abs(A),Ba))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),Aa2)),aa(A,A,abs_abs(A),Ba)) ) ).

% abs_add_abs
tff(fact_2479_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_2480_abs__mult__self__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),Aa2)),aa(A,A,abs_abs(A),Aa2)) = aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Aa2) ) ).

% abs_mult_self_eq
tff(fact_2481_abs__minus__cancel,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Aa2: A] : aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),Aa2)) = aa(A,A,abs_abs(A),Aa2) ) ).

% abs_minus_cancel
tff(fact_2482_abs__minus,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [Aa2: A] : aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),Aa2)) = aa(A,A,abs_abs(A),Aa2) ) ).

% abs_minus
tff(fact_2483_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_2484_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_2485_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_2486_abs__of__nonneg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)
         => ( aa(A,A,abs_abs(A),Aa2) = Aa2 ) ) ) ).

% abs_of_nonneg
tff(fact_2487_abs__le__self__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Aa2)),Aa2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2) ) ) ).

% abs_le_self_iff
tff(fact_2488_abs__le__zero__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Aa2)),zero_zero(A))
        <=> ( Aa2 = zero_zero(A) ) ) ) ).

% abs_le_zero_iff
tff(fact_2489_zero__less__abs__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,abs_abs(A),Aa2))
        <=> ( Aa2 != zero_zero(A) ) ) ) ).

% zero_less_abs_iff
tff(fact_2490_zero__less__Suc,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,suc,Na)) ).

% zero_less_Suc
tff(fact_2491_less__Suc0,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,suc,zero_zero(nat)))
    <=> ( Na = zero_zero(nat) ) ) ).

% less_Suc0
tff(fact_2492_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_2493_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_2494_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_2495_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_2496_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_2497_mult__Suc__right,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,suc,Na)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na)) ).

% mult_Suc_right
tff(fact_2498_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_2499_mask__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se2239418461657761734s_mask(A,zero_zero(nat)) = zero_zero(A) ) ) ).

% mask_0
tff(fact_2500_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_2501_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_2502_signed__take__bit__Suc__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat] : aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,suc,Na)),one_one(A)) = one_one(A) ) ).

% signed_take_bit_Suc_1
tff(fact_2503_ln__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,ln_ln(A),one_one(A)) = zero_zero(A) ) ) ).

% ln_one
tff(fact_2504_binomial__Suc__Suc,axiom,
    ! [Na: nat,K: nat] : aa(nat,nat,binomial(aa(nat,nat,suc,Na)),aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(Na),K)),aa(nat,nat,binomial(Na),aa(nat,nat,suc,K))) ).

% binomial_Suc_Suc
tff(fact_2505_pred__numeral__simps_I1_J,axiom,
    pred_numeral(one2) = zero_zero(nat) ).

% pred_numeral_simps(1)
tff(fact_2506_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_2507_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_2508_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_2509_of__nat__Suc,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,M)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),M)) ) ).

% of_nat_Suc
tff(fact_2510_zero__le__divide__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(A,A,abs_abs(A),Ba)))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)
            | ( Ba = zero_zero(A) ) ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_2511_divide__le__0__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(A,A,abs_abs(A),Ba))),zero_zero(A))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),zero_zero(A))
            | ( Ba = zero_zero(A) ) ) ) ) ).

% divide_le_0_abs_iff
tff(fact_2512_abs__of__nonpos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),zero_zero(A))
         => ( aa(A,A,abs_abs(A),Aa2) = aa(A,A,uminus_uminus(A),Aa2) ) ) ) ).

% abs_of_nonpos
tff(fact_2513_Suc__pred,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat)))) = Na ) ) ).

% Suc_pred
tff(fact_2514_one__le__mult__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),M)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),Na) ) ) ).

% one_le_mult_iff
tff(fact_2515_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,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,aa(nat,fun(nat,nat),plus_plus(nat),K),I)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_2516_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,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,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_2517_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_2518_Suc__diff,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),M)
       => ( aa(nat,nat,suc,aa(nat,nat,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_2519_nat__1,axiom,
    aa(int,nat,nat2,one_one(int)) = aa(nat,nat,suc,zero_zero(nat)) ).

% nat_1
tff(fact_2520_Suc__mod__mult__self4,axiom,
    ! [Na: nat,K: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),K)),M))),Na) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,M)),Na) ).

% Suc_mod_mult_self4
tff(fact_2521_Suc__mod__mult__self3,axiom,
    ! [K: nat,Na: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Na)),M))),Na) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,M)),Na) ).

% Suc_mod_mult_self3
tff(fact_2522_Suc__mod__mult__self2,axiom,
    ! [M: nat,Na: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),K)))),Na) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,M)),Na) ).

% Suc_mod_mult_self2
tff(fact_2523_Suc__mod__mult__self1,axiom,
    ! [M: nat,K: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Na)))),Na) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,M)),Na) ).

% Suc_mod_mult_self1
tff(fact_2524_Suc__numeral,axiom,
    ! [Na: num] : aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),Na)) = aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),Na),one2)) ).

% Suc_numeral
tff(fact_2525_negative__zless,axiom,
    ! [Na: nat,M: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,Na)))),aa(nat,int,semiring_1_of_nat(int),M)) ).

% negative_zless
tff(fact_2526_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_2527_less__numeral__Suc,axiom,
    ! [K: num,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pred_numeral(K)),Na) ) ).

% less_numeral_Suc
tff(fact_2528_less__Suc__numeral,axiom,
    ! [Na: nat,K: num] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Na)),aa(num,nat,numeral_numeral(nat),K))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),pred_numeral(K)) ) ).

% less_Suc_numeral
tff(fact_2529_le__numeral__Suc,axiom,
    ! [K: num,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),pred_numeral(K)),Na) ) ).

% le_numeral_Suc
tff(fact_2530_le__Suc__numeral,axiom,
    ! [Na: nat,K: num] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Na)),aa(num,nat,numeral_numeral(nat),K))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),pred_numeral(K)) ) ).

% le_Suc_numeral
tff(fact_2531_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_2532_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_2533_ln__le__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),X)),aa(real,real,ln_ln(real),Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y) ) ) ) ).

% ln_le_cancel_iff
tff(fact_2534_zabs__less__one__iff,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),Z)),one_one(int))
    <=> ( Z = zero_zero(int) ) ) ).

% zabs_less_one_iff
tff(fact_2535_add__2__eq__Suc_H,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,suc,aa(nat,nat,suc,Na)) ).

% add_2_eq_Suc'
tff(fact_2536_add__2__eq__Suc,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) = aa(nat,nat,suc,aa(nat,nat,suc,Na)) ).

% add_2_eq_Suc
tff(fact_2537_Suc__1,axiom,
    aa(nat,nat,suc,one_one(nat)) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% Suc_1
tff(fact_2538_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_2539_power2__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Aa2: A] : aa(nat,A,power_power(A,aa(A,A,abs_abs(A),Aa2)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,power_power(A,Aa2),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% power2_abs
tff(fact_2540_abs__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Aa2: A] : aa(A,A,abs_abs(A),aa(nat,A,power_power(A,Aa2),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(nat,A,power_power(A,Aa2),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% abs_power2
tff(fact_2541_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_2542_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_2543_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_2544_Suc__diff__1,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,Na),one_one(nat))) = Na ) ) ).

% Suc_diff_1
tff(fact_2545_length__ge__1__conv,axiom,
    ! [A: $tType,L: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(list(A),nat,size_size(list(A)),L))
    <=> ( L != nil(A) ) ) ).

% length_ge_1_conv
tff(fact_2546_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_2547_ln__ge__zero__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,ln_ln(real),X))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),X) ) ) ).

% ln_ge_zero_iff
tff(fact_2548_ln__le__zero__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),X)),zero_zero(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),one_one(real)) ) ) ).

% ln_le_zero_iff
tff(fact_2549_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_2550_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_2551_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_2552_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_2553_nat__abs__dvd__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa(nat,$o,dvd_dvd(nat,aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),Na)
    <=> aa(int,$o,dvd_dvd(int,K),aa(nat,int,semiring_1_of_nat(int),Na)) ) ).

% nat_abs_dvd_iff
tff(fact_2554_dvd__nat__abs__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(nat,$o,dvd_dvd(nat,Na),aa(int,nat,nat2,aa(int,int,abs_abs(int),K)))
    <=> aa(int,$o,dvd_dvd(int,aa(nat,int,semiring_1_of_nat(int),Na)),K) ) ).

% dvd_nat_abs_iff
tff(fact_2555_Suc__div__eq__add3__div__numeral,axiom,
    ! [M: nat,V: num] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M)))),aa(num,nat,numeral_numeral(nat),V)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(one2))),M)),aa(num,nat,numeral_numeral(nat),V)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_2556_div__Suc__eq__div__add3,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Na)))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(one2))),Na)) ).

% div_Suc_eq_div_add3
tff(fact_2557_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_2558_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_2559_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_2560_power__even__abs__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W: num,Aa2: A] :
          ( aa(nat,$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),Aa2)),aa(num,nat,numeral_numeral(nat),W)) = aa(nat,A,power_power(A,Aa2),aa(num,nat,numeral_numeral(nat),W)) ) ) ) ).

% power_even_abs_numeral
tff(fact_2561_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M: nat,V: 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),V)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(one2))),M)),aa(num,nat,numeral_numeral(nat),V)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_2562_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Na)))) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(one2))),Na)) ).

% mod_Suc_eq_mod_add3
tff(fact_2563_signed__take__bit__Suc__bit0,axiom,
    ! [Na: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,Na)),aa(num,int,numeral_numeral(int),bit0(K))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% signed_take_bit_Suc_bit0
tff(fact_2564_one__less__nat__eq,axiom,
    ! [Z: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),aa(int,nat,nat2,Z))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),Z) ) ).

% one_less_nat_eq
tff(fact_2565_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_2566_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_2567_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_2568_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_2569_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_2570_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_2571_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_2572_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_2573_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_2574_signed__take__bit__Suc__minus__bit0,axiom,
    ! [Na: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,Na)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% signed_take_bit_Suc_minus_bit0
tff(fact_2575_signed__take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] : aa(int,int,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),aa(int,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_2576_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_2577_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_2578_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa(int,int,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),aa(int,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_2579_signed__take__bit__Suc__bit1,axiom,
    ! [Na: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,Na)),aa(num,int,numeral_numeral(int),bit1(K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).

% signed_take_bit_Suc_bit1
tff(fact_2580_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(num,int,numeral_numeral(int),bit1(K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,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_2581_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_2582_n__not__Suc__n,axiom,
    ! [Na: nat] : Na != aa(nat,nat,suc,Na) ).

% n_not_Suc_n
tff(fact_2583_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa(nat,nat,suc,X) = aa(nat,nat,suc,Y) )
     => ( X = Y ) ) ).

% Suc_inject
tff(fact_2584_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_2585_abs__eq__0__iff,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [Aa2: A] :
          ( ( aa(A,A,abs_abs(A),Aa2) = zero_zero(A) )
        <=> ( Aa2 = zero_zero(A) ) ) ) ).

% abs_eq_0_iff
tff(fact_2586_abs__ge__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Aa2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),aa(A,A,abs_abs(A),Aa2)) ) ).

% abs_ge_self
tff(fact_2587_abs__le__D1,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Aa2)),Ba)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba) ) ) ).

% abs_le_D1
tff(fact_2588_abs__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ( aa(A,A,abs_abs(A),one_one(A)) = one_one(A) ) ) ).

% abs_one
tff(fact_2589_abs__mult,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),Aa2)),aa(A,A,abs_abs(A),Ba)) ) ).

% abs_mult
tff(fact_2590_abs__minus__commute,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,Aa2),Ba)) = aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,Ba),Aa2)) ) ).

% abs_minus_commute
tff(fact_2591_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_2592_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_2593_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_2594_not0__implies__Suc,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero(nat) )
     => ? [M5: nat] : Na = aa(nat,nat,suc,M5) ) ).

% not0_implies_Suc
tff(fact_2595_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero(nat) != aa(nat,nat,suc,M) ).

% Zero_not_Suc
tff(fact_2596_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero(nat) != aa(nat,nat,suc,M) ).

% Zero_neq_Suc
tff(fact_2597_Suc__neq__Zero,axiom,
    ! [M: nat] : aa(nat,nat,suc,M) != zero_zero(nat) ).

% Suc_neq_Zero
tff(fact_2598_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_2599_diff__induct,axiom,
    ! [P: fun(nat,fun(nat,$o)),M: nat,Na: nat] :
      ( ! [X2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,X2),zero_zero(nat))
     => ( ! [Y2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,zero_zero(nat)),aa(nat,nat,suc,Y2))
       => ( ! [X2: nat,Y2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),P,X2),Y2)
             => aa(nat,$o,aa(nat,fun(nat,$o),P,aa(nat,nat,suc,X2)),aa(nat,nat,suc,Y2)) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P,M),Na) ) ) ) ).

% diff_induct
tff(fact_2600_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_2601_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero(nat) )
     => ~ ! [Nat3: nat] : Y != aa(nat,nat,suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_2602_nat_OdiscI,axiom,
    ! [Nat: nat,X23: nat] :
      ( ( Nat = aa(nat,nat,suc,X23) )
     => ( Nat != zero_zero(nat) ) ) ).

% nat.discI
tff(fact_2603_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero(nat) != aa(nat,nat,suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_2604_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa(nat,nat,suc,Nat2) != zero_zero(nat) ).

% old.nat.distinct(2)
tff(fact_2605_nat_Odistinct_I1_J,axiom,
    ! [X23: nat] : zero_zero(nat) != aa(nat,nat,suc,X23) ).

% nat.distinct(1)
tff(fact_2606_list__decode_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero(nat) )
     => ~ ! [N: nat] : X != aa(nat,nat,suc,N) ) ).

% list_decode.cases
tff(fact_2607_zdvd__antisym__abs,axiom,
    ! [Aa2: int,Ba: int] :
      ( aa(int,$o,dvd_dvd(int,Aa2),Ba)
     => ( aa(int,$o,dvd_dvd(int,Ba),Aa2)
       => ( aa(int,int,abs_abs(int),Aa2) = aa(int,int,abs_abs(int),Ba) ) ) ) ).

% zdvd_antisym_abs
tff(fact_2608_not__less__less__Suc__eq,axiom,
    ! [Na: nat,M: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,suc,M))
      <=> ( Na = M ) ) ) ).

% not_less_less_Suc_eq
tff(fact_2609_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( ! [I2: nat] :
            ( ( J = aa(nat,nat,suc,I2) )
           => aa(nat,$o,P,I2) )
       => ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J)
             => ( aa(nat,$o,P,aa(nat,nat,suc,I2))
               => aa(nat,$o,P,I2) ) )
         => aa(nat,$o,P,I) ) ) ) ).

% strict_inc_induct
tff(fact_2610_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,fun(nat,$o))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( ! [I2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,I2),aa(nat,nat,suc,I2))
       => ( ! [I2: nat,J2: nat,K5: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J2)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),K5)
               => ( aa(nat,$o,aa(nat,fun(nat,$o),P,I2),J2)
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),P,J2),K5)
                   => aa(nat,$o,aa(nat,fun(nat,$o),P,I2),K5) ) ) ) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P,I),J) ) ) ) ).

% less_Suc_induct
tff(fact_2611_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I)),K) ) ) ).

% less_trans_Suc
tff(fact_2612_Suc__less__SucD,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,Na))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ).

% Suc_less_SucD
tff(fact_2613_less__antisym,axiom,
    ! [Na: nat,M: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,suc,M))
       => ( M = Na ) ) ) ).

% less_antisym
tff(fact_2614_Suc__less__eq2,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Na)),M)
    <=> ? [M8: nat] :
          ( ( M = aa(nat,nat,suc,M8) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M8) ) ) ).

% Suc_less_eq2
tff(fact_2615_Nat_OAll__less__Suc,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( ! [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(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,aa(nat,fun(nat,$o),ord_less(nat),I4),Na)
           => aa(nat,$o,P,I4) ) ) ) ).

% Nat.All_less_Suc
tff(fact_2616_not__less__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,suc,M)) ) ).

% not_less_eq
tff(fact_2617_less__Suc__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,suc,Na))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
        | ( M = Na ) ) ) ).

% less_Suc_eq
tff(fact_2618_Ex__less__Suc,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( ? [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(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,aa(nat,fun(nat,$o),ord_less(nat),I4),Na)
            & aa(nat,$o,P,I4) ) ) ) ).

% Ex_less_Suc
tff(fact_2619_less__SucI,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,suc,Na)) ) ).

% less_SucI
tff(fact_2620_less__SucE,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,suc,Na))
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
       => ( M = Na ) ) ) ).

% less_SucE
tff(fact_2621_Suc__lessI,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => ( ( aa(nat,nat,suc,M) != Na )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,M)),Na) ) ) ).

% Suc_lessI
tff(fact_2622_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I)),K)
     => ~ ! [J2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J2)
           => ( K != aa(nat,nat,suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_2623_Suc__lessD,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,M)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ).

% Suc_lessD
tff(fact_2624_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),K)
     => ( ( K != aa(nat,nat,suc,I) )
       => ~ ! [J2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J2)
             => ( K != aa(nat,nat,suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_2625_VEBT__internal_Oeven__odd__cases,axiom,
    ! [X: nat] :
      ( ! [N: nat] : X != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N)
     => ~ ! [N: nat] : X != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,suc,N)) ) ).

% VEBT_internal.even_odd_cases
tff(fact_2626_add__Suc__shift,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,M)),Na) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,suc,Na)) ).

% add_Suc_shift
tff(fact_2627_add__Suc,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,M)),Na) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)) ).

% add_Suc
tff(fact_2628_nat__arith_Osuc1,axiom,
    ! [A2: nat,K: nat,Aa2: nat] :
      ( ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Aa2) )
     => ( aa(nat,nat,suc,A2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(nat,nat,suc,Aa2)) ) ) ).

% nat_arith.suc1
tff(fact_2629_transitive__stepwise__le,axiom,
    ! [M: nat,Na: nat,R3: fun(nat,fun(nat,$o))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( ! [X2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),R3,X2),X2)
       => ( ! [X2: nat,Y2: nat,Z3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),R3,X2),Y2)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),R3,Y2),Z3)
               => aa(nat,$o,aa(nat,fun(nat,$o),R3,X2),Z3) ) )
         => ( ! [N: nat] : aa(nat,$o,aa(nat,fun(nat,$o),R3,N),aa(nat,nat,suc,N))
           => aa(nat,$o,aa(nat,fun(nat,$o),R3,M),Na) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_2630_nat__induct__at__least,axiom,
    ! [M: nat,Na: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( aa(nat,$o,P,M)
       => ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$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_2631_full__nat__induct,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( ! [N: nat] :
          ( ! [M3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M3)),N)
             => aa(nat,$o,P,M3) )
         => aa(nat,$o,P,N) )
     => aa(nat,$o,P,Na) ) ).

% full_nat_induct
tff(fact_2632_not__less__eq__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Na)),M) ) ).

% not_less_eq_eq
tff(fact_2633_Suc__n__not__le__n,axiom,
    ! [Na: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Na)),Na) ).

% Suc_n_not_le_n
tff(fact_2634_le__Suc__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Na))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
        | ( M = aa(nat,nat,suc,Na) ) ) ) ).

% le_Suc_eq
tff(fact_2635_Suc__le__D,axiom,
    ! [Na: nat,M9: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Na)),M9)
     => ? [M5: nat] : M9 = aa(nat,nat,suc,M5) ) ).

% Suc_le_D
tff(fact_2636_le__SucI,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Na)) ) ).

% le_SucI
tff(fact_2637_le__SucE,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Na))
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
       => ( M = aa(nat,nat,suc,Na) ) ) ) ).

% le_SucE
tff(fact_2638_Suc__leD,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% Suc_leD
tff(fact_2639_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_2640_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_2641_abs__ge__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Aa2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,abs_abs(A),Aa2)) ) ).

% abs_ge_zero
tff(fact_2642_abs__of__pos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2)
         => ( aa(A,A,abs_abs(A),Aa2) = Aa2 ) ) ) ).

% abs_of_pos
tff(fact_2643_abs__not__less__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Aa2: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),Aa2)),zero_zero(A)) ) ).

% abs_not_less_zero
tff(fact_2644_abs__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Aa2: A,Ba: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),Aa2)),aa(A,A,abs_abs(A),Ba))) ) ).

% abs_triangle_ineq
tff(fact_2645_abs__mult__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Aa2: A,C2: A,Ba: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),Aa2)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),Ba)),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),Aa2)),aa(A,A,abs_abs(A),Ba))),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D2)) ) ) ) ).

% abs_mult_less
tff(fact_2646_abs__triangle__ineq2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Aa2: A,Ba: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,aa(A,A,abs_abs(A),Aa2)),aa(A,A,abs_abs(A),Ba))),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,Aa2),Ba))) ) ).

% abs_triangle_ineq2
tff(fact_2647_abs__triangle__ineq3,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Aa2: A,Ba: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,aa(A,A,abs_abs(A),Aa2)),aa(A,A,abs_abs(A),Ba)))),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,Aa2),Ba))) ) ).

% abs_triangle_ineq3
tff(fact_2648_abs__triangle__ineq2__sym,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Aa2: A,Ba: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,aa(A,A,abs_abs(A),Aa2)),aa(A,A,abs_abs(A),Ba))),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,Ba),Aa2))) ) ).

% abs_triangle_ineq2_sym
tff(fact_2649_abs__leI,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Aa2)),Ba)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Aa2)),Ba) ) ) ) ).

% abs_leI
tff(fact_2650_abs__le__D2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Aa2)),Ba)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Aa2)),Ba) ) ) ).

% abs_le_D2
tff(fact_2651_abs__le__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Aa2)),Ba)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Aa2)),Ba) ) ) ) ).

% abs_le_iff
tff(fact_2652_abs__ge__minus__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Aa2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Aa2)),aa(A,A,abs_abs(A),Aa2)) ) ).

% abs_ge_minus_self
tff(fact_2653_abs__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),Aa2)),Ba)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),Aa2)),Ba) ) ) ) ).

% abs_less_iff
tff(fact_2654_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_2655_log__def,axiom,
    ! [Aa2: real,X: real] : aa(real,real,log(Aa2),X) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),X)),aa(real,real,ln_ln(real),Aa2)) ).

% log_def
tff(fact_2656_mask__twice2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat,X: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),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_2657_mask__eqs_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Na: nat,Ba: 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)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Aa2),bit_se2239418461657761734s_mask(word(A),Na))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Ba),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)),plus_plus(word(A)),Aa2),Ba)),bit_se2239418461657761734s_mask(word(A),Na)) ) ).

% mask_eqs(7)
tff(fact_2658_mask__eqs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: 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)),plus_plus(word(A)),Aa2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Ba),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)),plus_plus(word(A)),Aa2),Ba)),bit_se2239418461657761734s_mask(word(A),Na)) ) ).

% mask_eqs(2)
tff(fact_2659_mask__eqs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Na: nat,Ba: 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)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Aa2),bit_se2239418461657761734s_mask(word(A),Na))),Ba)),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)),plus_plus(word(A)),Aa2),Ba)),bit_se2239418461657761734s_mask(word(A),Na)) ) ).

% mask_eqs(1)
tff(fact_2660_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_2661_mask__eqs_I10_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: 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)),Aa2),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)),Aa2)),bit_se2239418461657761734s_mask(word(A),Na)) ) ).

% mask_eqs(10)
tff(fact_2662_mask__eqs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Na: nat,Ba: 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)),Aa2),bit_se2239418461657761734s_mask(word(A),Na))),Ba)),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),Aa2),Ba)),bit_se2239418461657761734s_mask(word(A),Na)) ) ).

% mask_eqs(3)
tff(fact_2663_mask__eqs_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: 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),Aa2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Ba),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),Aa2),Ba)),bit_se2239418461657761734s_mask(word(A),Na)) ) ).

% mask_eqs(4)
tff(fact_2664_mask__eqs_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Na: nat,Ba: 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)),Aa2),bit_se2239418461657761734s_mask(word(A),Na))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Ba),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),Aa2),Ba)),bit_se2239418461657761734s_mask(word(A),Na)) ) ).

% mask_eqs(8)
tff(fact_2665_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_2666_lift__Suc__mono__less__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),Na: nat,M: nat] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F2,N)),aa(nat,A,F2,aa(nat,nat,suc,N)))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F2,Na)),aa(nat,A,F2,M))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M) ) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_2667_lift__Suc__mono__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),Na: nat,N6: nat] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F2,N)),aa(nat,A,F2,aa(nat,nat,suc,N)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),N6)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F2,Na)),aa(nat,A,F2,N6)) ) ) ) ).

% lift_Suc_mono_less
tff(fact_2668_power__Suc,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [Aa2: A,Na: nat] : aa(nat,A,power_power(A,Aa2),aa(nat,nat,suc,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(nat,A,power_power(A,Aa2),Na)) ) ).

% power_Suc
tff(fact_2669_power__Suc2,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Aa2: A,Na: nat] : aa(nat,A,power_power(A,Aa2),aa(nat,nat,suc,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,Aa2),Na)),Aa2) ) ).

% power_Suc2
tff(fact_2670_lift__Suc__antimono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),Na: nat,N6: nat] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,aa(nat,nat,suc,N))),aa(nat,A,F2,N))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N6)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,N6)),aa(nat,A,F2,Na)) ) ) ) ).

% lift_Suc_antimono_le
tff(fact_2671_lift__Suc__mono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),Na: nat,N6: nat] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,N)),aa(nat,A,F2,aa(nat,nat,suc,N)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N6)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,Na)),aa(nat,A,F2,N6)) ) ) ) ).

% lift_Suc_mono_le
tff(fact_2672_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_2673_less__Suc__eq__0__disj,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,suc,Na))
    <=> ( ( M = zero_zero(nat) )
        | ? [J3: nat] :
            ( ( M = aa(nat,nat,suc,J3) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),Na) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_2674_gr0__implies__Suc,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ? [M5: nat] : Na = aa(nat,nat,suc,M5) ) ).

% gr0_implies_Suc
tff(fact_2675_All__less__Suc2,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( ! [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(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,aa(nat,fun(nat,$o),ord_less(nat),I4),Na)
           => aa(nat,$o,P,aa(nat,nat,suc,I4)) ) ) ) ).

% All_less_Suc2
tff(fact_2676_gr0__conv__Suc,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
    <=> ? [M4: nat] : Na = aa(nat,nat,suc,M4) ) ).

% gr0_conv_Suc
tff(fact_2677_Ex__less__Suc2,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( ? [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(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,aa(nat,fun(nat,$o),ord_less(nat),I4),Na)
            & aa(nat,$o,P,aa(nat,nat,suc,I4)) ) ) ) ).

% Ex_less_Suc2
tff(fact_2678_mask__eqs_I9_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Na: nat,Ba: 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)),Aa2),bit_se2239418461657761734s_mask(word(A),Na))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Ba),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)),Aa2),Ba)),bit_se2239418461657761734s_mask(word(A),Na)) ) ).

% mask_eqs(9)
tff(fact_2679_mask__eqs_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Ba: word(A),Na: nat,Aa2: 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)),Ba),bit_se2239418461657761734s_mask(word(A),Na))),Aa2)),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)),Ba),Aa2)),bit_se2239418461657761734s_mask(word(A),Na)) ) ).

% mask_eqs(6)
tff(fact_2680_mask__eqs_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: 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)),Aa2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Ba),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)),Aa2),Ba)),bit_se2239418461657761734s_mask(word(A),Na)) ) ).

% mask_eqs(5)
tff(fact_2681_add__is__1,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
          & ( Na = zero_zero(nat) ) )
        | ( ( M = zero_zero(nat) )
          & ( Na = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ) ).

% add_is_1
tff(fact_2682_one__is__add,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,suc,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na) )
    <=> ( ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
          & ( Na = zero_zero(nat) ) )
        | ( ( M = zero_zero(nat) )
          & ( Na = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ) ).

% one_is_add
tff(fact_2683_nat__compl__induct,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ! [N: nat] :
            ( ! [Nn: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$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_2684_nat__compl__induct_H,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ! [N: nat] :
            ( ! [Nn: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$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_2685_One__nat__def,axiom,
    one_one(nat) = aa(nat,nat,suc,zero_zero(nat)) ).

% One_nat_def
tff(fact_2686_less__natE,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => ~ ! [Q5: nat] : Na != aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q5)) ) ).

% less_natE
tff(fact_2687_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),M))) ).

% less_add_Suc1
tff(fact_2688_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I))) ).

% less_add_Suc2
tff(fact_2689_less__iff__Suc__add,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
    <=> ? [K2: nat] : Na = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K2)) ) ).

% less_iff_Suc_add
tff(fact_2690_less__imp__Suc__add,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => ? [K5: nat] : Na = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K5)) ) ).

% less_imp_Suc_add
tff(fact_2691_Suc__leI,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M)),Na) ) ).

% Suc_leI
tff(fact_2692_Suc__le__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M)),Na)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ).

% Suc_le_eq
tff(fact_2693_dec__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,P,I)
       => ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),N)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),J)
               => ( aa(nat,$o,P,N)
                 => aa(nat,$o,P,aa(nat,nat,suc,N)) ) ) )
         => aa(nat,$o,P,J) ) ) ) ).

% dec_induct
tff(fact_2694_inc__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,P,J)
       => ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),N)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),J)
               => ( aa(nat,$o,P,aa(nat,nat,suc,N))
                 => aa(nat,$o,P,N) ) ) )
         => aa(nat,$o,P,I) ) ) ) ).

% inc_induct
tff(fact_2695_Suc__le__lessD,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ).

% Suc_le_lessD
tff(fact_2696_le__less__Suc__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,suc,M))
      <=> ( Na = M ) ) ) ).

% le_less_Suc_eq
tff(fact_2697_less__Suc__eq__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,suc,Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% less_Suc_eq_le
tff(fact_2698_less__eq__Suc__le,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Na)),M) ) ).

% less_eq_Suc_le
tff(fact_2699_le__imp__less__Suc,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,suc,Na)) ) ).

% le_imp_less_Suc
tff(fact_2700_nat__in__between__eq_I2_J,axiom,
    ! [Aa2: nat,Ba: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Aa2),Ba)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ba),aa(nat,nat,suc,Aa2)) )
    <=> ( Ba = Aa2 ) ) ).

% nat_in_between_eq(2)
tff(fact_2701_nat__in__between__eq_I1_J,axiom,
    ! [Aa2: nat,Ba: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Aa2),Ba)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ba),aa(nat,nat,suc,Aa2)) )
    <=> ( Ba = aa(nat,nat,suc,Aa2) ) ) ).

% nat_in_between_eq(1)
tff(fact_2702_length__nth__simps_I2_J,axiom,
    ! [A: $tType,X: A,Xs2: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs2)) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs2)) ).

% length_nth_simps(2)
tff(fact_2703_Suc__length__conv,axiom,
    ! [A: $tType,Na: nat,Xs2: list(A)] :
      ( ( aa(nat,nat,suc,Na) = aa(list(A),nat,size_size(list(A)),Xs2) )
    <=> ? [Y4: A,Ys2: list(A)] :
          ( ( Xs2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys2) )
          & ( aa(list(A),nat,size_size(list(A)),Ys2) = Na ) ) ) ).

% Suc_length_conv
tff(fact_2704_length__Suc__conv,axiom,
    ! [A: $tType,Xs2: list(A),Na: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(nat,nat,suc,Na) )
    <=> ? [Y4: A,Ys2: list(A)] :
          ( ( Xs2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys2) )
          & ( aa(list(A),nat,size_size(list(A)),Ys2) = Na ) ) ) ).

% length_Suc_conv
tff(fact_2705_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_2706_diff__less__Suc,axiom,
    ! [M: nat,Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,minus_minus(nat,M),Na)),aa(nat,nat,suc,M)) ).

% diff_less_Suc
tff(fact_2707_Suc__diff__Suc,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
     => ( aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,M),aa(nat,nat,suc,Na))) = aa(nat,nat,minus_minus(nat,M),Na) ) ) ).

% Suc_diff_Suc
tff(fact_2708_Suc__eq__plus1,axiom,
    ! [Na: nat] : aa(nat,nat,suc,Na) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat)) ).

% Suc_eq_plus1
tff(fact_2709_plus__1__eq__Suc,axiom,
    aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)) = suc ).

% plus_1_eq_Suc
tff(fact_2710_Suc__eq__plus1__left,axiom,
    ! [Na: nat] : aa(nat,nat,suc,Na) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),Na) ).

% Suc_eq_plus1_left
tff(fact_2711_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ).

% Suc_mult_less_cancel1
tff(fact_2712_Suc__div__le__mono,axiom,
    ! [M: nat,Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,M)),Na)) ).

% Suc_div_le_mono
tff(fact_2713_Suc__diff__le,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
     => ( aa(nat,nat,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_2714_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_2715_mult__Suc,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,M)),Na) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na)) ).

% mult_Suc
tff(fact_2716_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% Suc_mult_le_cancel1
tff(fact_2717_mod__Suc__le__divisor,axiom,
    ! [M: nat,Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(nat,nat,suc,Na))),Na) ).

% mod_Suc_le_divisor
tff(fact_2718_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_2719_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_2720_Suc__times__binomial__eq,axiom,
    ! [Na: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,Na)),aa(nat,nat,binomial(Na),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(aa(nat,nat,suc,Na)),aa(nat,nat,suc,K))),aa(nat,nat,suc,K)) ).

% Suc_times_binomial_eq
tff(fact_2721_Suc__times__binomial,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(aa(nat,nat,suc,Na)),aa(nat,nat,suc,K))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,Na)),aa(nat,nat,binomial(Na),K)) ).

% Suc_times_binomial
tff(fact_2722_mask__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se2239418461657761734s_mask(A,one_one(nat)) = one_one(A) ) ) ).

% mask_1
tff(fact_2723_nat__intermed__int__val,axiom,
    ! [M: nat,Na: nat,F2: fun(nat,int),K: int] :
      ( ! [I2: nat] :
          ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),I2)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Na) )
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,aa(nat,int,F2,aa(nat,nat,suc,I2))),aa(nat,int,F2,I2)))),one_one(int)) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F2,M)),K)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,F2,Na))
           => ? [I2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),I2)
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),Na)
                & ( aa(nat,int,F2,I2) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_2724_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_2725_dense__eq0__I,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs(A)
        & dense_linorder(A) )
     => ! [X: A] :
          ( ! [E: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),X)),E) )
         => ( X = zero_zero(A) ) ) ) ).

% dense_eq0_I
tff(fact_2726_abs__eq__mult,axiom,
    ! [A: $tType] :
      ( ordered_ring_abs(A)
     => ! [Aa2: A,Ba: A] :
          ( ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)
              | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),zero_zero(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Ba)
              | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),zero_zero(A)) ) )
         => ( aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),Aa2)),aa(A,A,abs_abs(A),Ba)) ) ) ) ).

% abs_eq_mult
tff(fact_2727_abs__mult__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(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_2728_zero__le__power__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Aa2: A,Na: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,aa(A,A,abs_abs(A),Aa2)),Na)) ) ).

% zero_le_power_abs
tff(fact_2729_abs__eq__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [Aa2: A,Ba: A] :
          ( ( aa(A,A,abs_abs(A),Aa2) = Ba )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Ba)
            & ( ( Aa2 = Ba )
              | ( Aa2 = aa(A,A,uminus_uminus(A),Ba) ) ) ) ) ) ).

% abs_eq_iff'
tff(fact_2730_eq__abs__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [Aa2: A,Ba: A] :
          ( ( Aa2 = aa(A,A,abs_abs(A),Ba) )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)
            & ( ( Ba = Aa2 )
              | ( Ba = aa(A,A,uminus_uminus(A),Aa2) ) ) ) ) ) ).

% eq_abs_iff'
tff(fact_2731_abs__minus__le__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Aa2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,abs_abs(A),Aa2))),zero_zero(A)) ) ).

% abs_minus_le_zero
tff(fact_2732_abs__if__raw,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [X4: A] :
          aa(A,A,abs_abs(A),X4) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),zero_zero(A)),aa(A,A,uminus_uminus(A),X4),X4) ) ).

% abs_if_raw
tff(fact_2733_abs__of__neg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),zero_zero(A))
         => ( aa(A,A,abs_abs(A),Aa2) = aa(A,A,uminus_uminus(A),Aa2) ) ) ) ).

% abs_of_neg
tff(fact_2734_abs__if,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [Aa2: A] :
          aa(A,A,abs_abs(A),Aa2) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),zero_zero(A)),aa(A,A,uminus_uminus(A),Aa2),Aa2) ) ).

% abs_if
tff(fact_2735_abs__diff__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Aa2: A,R: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,X),Aa2))),R)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,Aa2),R)),X)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),R)) ) ) ) ).

% abs_diff_le_iff
tff(fact_2736_abs__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Aa2: A,Ba: A,C2: A,D2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),D2)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,Aa2),C2))),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,Ba),D2)))) ) ).

% abs_diff_triangle_ineq
tff(fact_2737_abs__triangle__ineq4,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Aa2: A,Ba: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,Aa2),Ba))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),Aa2)),aa(A,A,abs_abs(A),Ba))) ) ).

% abs_triangle_ineq4
tff(fact_2738_abs__diff__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Aa2: A,R: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,X),Aa2))),R)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,Aa2),R)),X)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),R)) ) ) ) ).

% abs_diff_less_iff
tff(fact_2739_ln__bound,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),X)),X) ) ).

% ln_bound
tff(fact_2740_ln__ge__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),X)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,ln_ln(real),X)) ) ).

% ln_ge_zero
tff(fact_2741_nat__ivt__aux,axiom,
    ! [Na: nat,F2: fun(nat,int),K: int] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Na)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,aa(nat,int,F2,aa(nat,nat,suc,I2))),aa(nat,int,F2,I2)))),one_one(int)) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F2,zero_zero(nat))),K)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,F2,Na))
         => ? [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),Na)
              & ( aa(nat,int,F2,I2) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_2742_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),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),bit_se2239418461657761734s_mask(word(A),Na))),one_one(word(A))) ) ).

% mask_Suc_rec
tff(fact_2743_zabs__def,axiom,
    ! [I: int] :
      aa(int,int,abs_abs(int),I) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),zero_zero(int)),aa(int,int,uminus_uminus(int),I),I) ).

% zabs_def
tff(fact_2744_mask__lower__twice,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat,X: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),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_2745_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,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),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_2746_nat__abs__mult__distrib,axiom,
    ! [W: int,Z: int] : aa(int,nat,nat2,aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),W))),aa(int,nat,nat2,aa(int,int,abs_abs(int),Z))) ).

% nat_abs_mult_distrib
tff(fact_2747_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_2748_dvd__imp__le__int,axiom,
    ! [I: int,D2: int] :
      ( ( I != zero_zero(int) )
     => ( aa(int,$o,dvd_dvd(int,D2),I)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),D2)),aa(int,int,abs_abs(int),I)) ) ) ).

% dvd_imp_le_int
tff(fact_2749_power__le__imp__le__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Aa2: A,Na: nat,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,Aa2),aa(nat,nat,suc,Na))),aa(nat,A,power_power(A,Ba),aa(nat,nat,suc,Na)))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Ba)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba) ) ) ) ).

% power_le_imp_le_base
tff(fact_2750_power__inject__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Aa2: A,Na: nat,Ba: A] :
          ( ( aa(nat,A,power_power(A,Aa2),aa(nat,nat,suc,Na)) = aa(nat,A,power_power(A,Ba),aa(nat,nat,suc,Na)) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Ba)
             => ( Aa2 = Ba ) ) ) ) ) ).

% power_inject_base
tff(fact_2751_power__gt1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Aa2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Aa2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(nat,A,power_power(A,Aa2),aa(nat,nat,suc,Na))) ) ) ).

% power_gt1
tff(fact_2752_AND__NOT__mask__plus__AND__mask__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Na: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),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_2753_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_2754_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_2755_ex__least__nat__less,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( aa(nat,$o,P,Na)
     => ( ~ aa(nat,$o,P,zero_zero(nat))
       => ? [K5: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K5),Na)
            & ! [I3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),K5)
               => ~ aa(nat,$o,P,I3) )
            & aa(nat,$o,P,aa(nat,nat,suc,K5)) ) ) ) ).

% ex_least_nat_less
tff(fact_2756_nat__induct__non__zero,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,P,one_one(nat))
       => ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$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_2757_diff__Suc__less,axiom,
    ! [Na: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,I))),Na) ) ).

% diff_Suc_less
tff(fact_2758_one__less__mult,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na)) ) ) ).

% one_less_mult
tff(fact_2759_n__less__m__mult__n,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na)) ) ) ).

% n_less_m_mult_n
tff(fact_2760_n__less__n__mult__m,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),M)) ) ) ).

% n_less_n_mult_m
tff(fact_2761_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_2762_Suc__le__length__iff,axiom,
    ! [A: $tType,Na: nat,Xs2: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Na)),aa(list(A),nat,size_size(list(A)),Xs2))
    <=> ? [X6: A,Ys2: list(A)] :
          ( ( Xs2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X6),Ys2) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(list(A),nat,size_size(list(A)),Ys2)) ) ) ).

% Suc_le_length_iff
tff(fact_2763_nat__one__le__power,axiom,
    ! [I: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),I)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,power_power(nat,I),Na)) ) ).

% nat_one_le_power
tff(fact_2764_int__ops_I4_J,axiom,
    ! [Aa2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,Aa2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),Aa2)),one_one(int)) ).

% int_ops(4)
tff(fact_2765_int__Suc,axiom,
    ! [Na: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,Na)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),Na)),one_one(int)) ).

% int_Suc
tff(fact_2766_zless__iff__Suc__zadd,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),Z)
    <=> ? [N2: nat] : Z = aa(int,int,aa(int,fun(int,int),plus_plus(int),W),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N2))) ) ).

% zless_iff_Suc_zadd
tff(fact_2767_Suc__times__binomial__add,axiom,
    ! [Aa2: nat,Ba: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,Aa2)),aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Aa2),Ba))),aa(nat,nat,suc,Aa2))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,Ba)),aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Aa2),Ba))),Aa2)) ).

% Suc_times_binomial_add
tff(fact_2768_binomial__Suc__Suc__eq__times,axiom,
    ! [Na: nat,K: nat] : aa(nat,nat,binomial(aa(nat,nat,suc,Na)),aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,Na)),aa(nat,nat,binomial(Na),K))),aa(nat,nat,suc,K)) ).

% binomial_Suc_Suc_eq_times
tff(fact_2769_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_2770_mask__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: num] : bit_se2239418461657761734s_mask(A,aa(num,nat,numeral_numeral(nat),Na)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2239418461657761734s_mask(A,pred_numeral(Na)))) ) ).

% mask_numeral
tff(fact_2771_abs__add__one__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,abs_abs(A),X))) ) ).

% abs_add_one_gt_zero
tff(fact_2772_ln__ge__zero__imp__ge__one,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,ln_ln(real),X))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),X) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_2773_ln__add__one__self__le__self,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X))),X) ) ).

% ln_add_one_self_le_self
tff(fact_2774_ln__mult,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),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,aa(real,fun(real,real),plus_plus(real),aa(real,real,ln_ln(real),X)),aa(real,real,ln_ln(real),Y)) ) ) ) ).

% ln_mult
tff(fact_2775_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))) ).

% nat_abs_triangle_ineq
tff(fact_2776_multiple__mask__trivia,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,Na: nat,X: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),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_2777_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_2778_power__Suc__le__self,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Aa2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,Aa2),aa(nat,nat,suc,Na))),Aa2) ) ) ) ).

% power_Suc_le_self
tff(fact_2779_power__Suc__less__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Aa2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,Aa2),aa(nat,nat,suc,Na))),one_one(A)) ) ) ) ).

% power_Suc_less_one
tff(fact_2780_div__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L)) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))) ).

% div_abs_eq_div_nat
tff(fact_2781_mask__subsume,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat,X: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),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_2782_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_2783_mod__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L)) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))) ).

% mod_abs_eq_div_nat
tff(fact_2784_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_2785_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_2786_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_2787_Suc3__eq__add__3,axiom,
    ! [Na: nat] : aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Na))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(one2))),Na) ).

% Suc3_eq_add_3
tff(fact_2788_list_Osize_I4_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),X222)),aa(nat,nat,suc,zero_zero(nat))) ).

% list.size(4)
tff(fact_2789_nz__le__conv__less,axiom,
    ! [K: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,minus_minus(nat,K),aa(nat,nat,suc,zero_zero(nat)))),M) ) ) ).

% nz_le_conv_less
tff(fact_2790_Suc__diff__eq__diff__pred,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,nat,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_2791_Suc__pred_H,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( Na = aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,Na),one_one(nat))) ) ) ).

% Suc_pred'
tff(fact_2792_add__eq__if,axiom,
    ! [M: nat,Na: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na) = $ite(M = zero_zero(nat),Na,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,M),one_one(nat))),Na))) ).

% add_eq_if
tff(fact_2793_Suc__n__minus__m__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),M)
       => ( aa(nat,nat,suc,aa(nat,nat,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_2794_div__nat__eqI,axiom,
    ! [Na: nat,Q3: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),Q3)),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(nat,nat,suc,Q3)))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na) = Q3 ) ) ) ).

% div_nat_eqI
tff(fact_2795_pochhammer__rec,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Aa2: A,Na: nat] : comm_s3205402744901411588hammer(A,Aa2,aa(nat,nat,suc,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),one_one(A)),Na)) ) ).

% pochhammer_rec
tff(fact_2796_Suc__nat__number__of__add,axiom,
    ! [V: num,Na: nat] : aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V)),Na)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),V),one2))),Na) ).

% Suc_nat_number_of_add
tff(fact_2797_pochhammer__rec_H,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Z: A,Na: nat] : comm_s3205402744901411588hammer(A,Z,aa(nat,nat,suc,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(nat,A,semiring_1_of_nat(A),Na))),comm_s3205402744901411588hammer(A,Z,Na)) ) ).

% pochhammer_rec'
tff(fact_2798_pochhammer__Suc,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Aa2: A,Na: nat] : comm_s3205402744901411588hammer(A,Aa2,aa(nat,nat,suc,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Aa2,Na)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),aa(nat,A,semiring_1_of_nat(A),Na))) ) ).

% pochhammer_Suc
tff(fact_2799_not__zle__0__negative,axiom,
    ! [Na: nat] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,Na)))) ).

% not_zle_0_negative
tff(fact_2800_negD,axiom,
    ! [X: int] :
      ( aa(int,$o,aa(int,fun(int,$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_2801_negative__zless__0,axiom,
    ! [Na: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,Na)))),zero_zero(int)) ).

% negative_zless_0
tff(fact_2802_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_2803_Suc__as__int,axiom,
    ! [X4: nat] : aa(nat,nat,suc,X4) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),X4)),one_one(int))) ).

% Suc_as_int
tff(fact_2804_ln__2__less__1,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,ln_ln(real),aa(num,real,numeral_numeral(real),bit0(one2)))),one_one(real)) ).

% ln_2_less_1
tff(fact_2805_abs__le__square__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),X)),aa(A,A,abs_abs(A),Y))
        <=> aa(A,$o,aa(A,fun(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_2806_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_2807_floor__eq3,axiom,
    ! [Na: nat,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Na)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Na)))
       => ( aa(int,nat,nat2,archim6421214686448440834_floor(real,X)) = Na ) ) ) ).

% floor_eq3
tff(fact_2808_power__even__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat,Aa2: A] :
          ( aa(nat,$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),Aa2)),Na) = aa(nat,A,power_power(A,Aa2),Na) ) ) ) ).

% power_even_abs
tff(fact_2809_ln__le__minus__one,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_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_2810_ln__diff__le,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
       => aa(real,$o,aa(real,fun(real,$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_2811_ln__add__one__self__le__self2,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),X)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X))),X) ) ).

% ln_add_one_self_le_self2
tff(fact_2812_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_2813_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_2814_ln__realpow,axiom,
    ! [X: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$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_2815_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,aa(nat,fun(nat,$o),ord_less(nat),Xb),Xa2),Xc,set_fo6178422350223883121st_nat(A,X,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa2),one_one(nat)),Xb,aa(A,A,aa(nat,fun(A,A),X,Xa2),Xc))) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_2816_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [A: $tType,F2: fun(nat,fun(A,A)),Aa2: nat,Ba: nat,Acc: A] :
      set_fo6178422350223883121st_nat(A,F2,Aa2,Ba,Acc) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ba),Aa2),Acc,set_fo6178422350223883121st_nat(A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Aa2),one_one(nat)),Ba,aa(A,A,aa(nat,fun(A,A),F2,Aa2),Acc))) ).

% fold_atLeastAtMost_nat.simps
tff(fact_2817_num_Osize_I5_J,axiom,
    ! [X23: num] : aa(num,nat,size_size(num),bit0(X23)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,size_size(num),X23)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size(5)
tff(fact_2818_num_Osize_I6_J,axiom,
    ! [X32: num] : aa(num,nat,size_size(num),bit1(X32)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,size_size(num),X32)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size(6)
tff(fact_2819_even__abs__add__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),K)),L))
    <=> aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)) ) ).

% even_abs_add_iff
tff(fact_2820_even__add__abs__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(int,int,abs_abs(int),L)))
    <=> aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)) ) ).

% even_add_abs_iff
tff(fact_2821_nat__abs__int__diff,axiom,
    ! [Aa2: nat,Ba: nat] :
      aa(int,nat,nat2,aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),Aa2)),aa(nat,int,semiring_1_of_nat(int),Ba)))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Aa2),Ba),aa(nat,nat,minus_minus(nat,Ba),Aa2),aa(nat,nat,minus_minus(nat,Aa2),Ba)) ).

% nat_abs_int_diff
tff(fact_2822_nat__approx__posE,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [E2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E2)
         => ~ ! [N: nat] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N)))),E2) ) ) ).

% nat_approx_posE
tff(fact_2823_less__2__cases__iff,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))
    <=> ( ( Na = zero_zero(nat) )
        | ( Na = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% less_2_cases_iff
tff(fact_2824_less__2__cases,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))
     => ( ( Na = zero_zero(nat) )
        | ( Na = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% less_2_cases
tff(fact_2825_Suc__div__eq__add3__div,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M)))),Na) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(one2))),M)),Na) ).

% Suc_div_eq_add3_div
tff(fact_2826_nat__2,axiom,
    aa(int,nat,nat2,aa(num,int,numeral_numeral(int),bit0(one2))) = aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))) ).

% nat_2
tff(fact_2827_le__div__geq,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,M),Na)),Na)) ) ) ) ).

% le_div_geq
tff(fact_2828_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,$o,P,zero_zero(nat)) )
        | ? [Q4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),Q4)),M)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(nat,nat,suc,Q4)))
            & aa(nat,$o,P,Q4) ) ) ) ).

% split_div'
tff(fact_2829_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M)))),Na) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(one2))),M)),Na) ).

% Suc_mod_eq_add3_mod
tff(fact_2830_Suc__times__mod__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na))),M) = one_one(nat) ) ) ).

% Suc_times_mod_eq
tff(fact_2831_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => ( aa(nat,nat,suc,aa(int,nat,nat2,Z)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z)) ) ) ).

% Suc_nat_eq_nat_zadd1
tff(fact_2832_power2__le__iff__abs__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),X)),Y) ) ) ) ).

% power2_le_iff_abs_le
tff(fact_2833_abs__square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( aa(A,$o,aa(A,fun(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,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),X)),one_one(A)) ) ) ).

% abs_square_le_1
tff(fact_2834_abs__square__less__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( aa(A,$o,aa(A,fun(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,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),X)),one_one(A)) ) ) ).

% abs_square_less_1
tff(fact_2835_power__mono__even,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat,Aa2: A,Ba: A] :
          ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Aa2)),aa(A,A,abs_abs(A),Ba))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,Aa2),Na)),aa(nat,A,power_power(A,Ba),Na)) ) ) ) ).

% power_mono_even
tff(fact_2836_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_2837_ln__one__minus__pos__upper__bound,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),one_one(real))
       => aa(real,$o,aa(real,fun(real,$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_2838_ln__powr__bound,axiom,
    ! [X: real,Aa2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Aa2)
       => aa(real,$o,aa(real,fun(real,$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,Aa2)),Aa2)) ) ) ).

% ln_powr_bound
tff(fact_2839_ln__powr__bound2,axiom,
    ! [X: real,Aa2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Aa2)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,aa(real,real,ln_ln(real),X),Aa2)),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,Aa2,Aa2)),X)) ) ) ).

% ln_powr_bound2
tff(fact_2840_log__eq__div__ln__mult__log,axiom,
    ! [Aa2: real,Ba: real,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Aa2)
     => ( ( Aa2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Ba)
         => ( ( Ba != one_one(real) )
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
             => ( aa(real,real,log(Aa2),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),Ba)),aa(real,real,ln_ln(real),Aa2))),aa(real,real,log(Ba),X)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_2841_floor__eq4,axiom,
    ! [Na: nat,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),Na)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Na)))
       => ( aa(int,nat,nat2,archim6421214686448440834_floor(real,X)) = Na ) ) ) ).

% floor_eq4
tff(fact_2842_mask__plus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] : aa(word(A),word(A),aa(word(A),fun(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_2843_less__mask__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Na: nat] :
          ( aa(word(A),$o,aa(word(A),fun(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_2844_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_2845_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_2846_incr__lemma,axiom,
    ! [D2: int,Z: int,X: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,X),Z))),one_one(int))),D2))) ) ).

% incr_lemma
tff(fact_2847_decr__lemma,axiom,
    ! [D2: int,X: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,minus_minus(int,X),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,X),Z))),one_one(int))),D2))),Z) ) ).

% decr_lemma
tff(fact_2848_power__odd__eq,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Aa2: A,Na: nat] : aa(nat,A,power_power(A,Aa2),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),Aa2),aa(nat,A,power_power(A,aa(nat,A,power_power(A,Aa2),Na)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% power_odd_eq
tff(fact_2849_Suc__n__div__2__gt__zero,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,Na)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_2850_div__2__gt__zero,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% div_2_gt_zero
tff(fact_2851_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,aa(nat,fun(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_2852_binomial__less__binomial__Suc,axiom,
    ! [K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,binomial(Na),K)),aa(nat,nat,binomial(Na),aa(nat,nat,suc,K))) ) ).

% binomial_less_binomial_Suc
tff(fact_2853_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_2854_signed__take__bit__int__less__eq,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)),K)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K)),aa(int,int,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_2855_binomial__addition__formula,axiom,
    ! [Na: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,nat,binomial(Na),aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(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_2856_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_2857_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_2858_is__aligned__AND__less__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [U: word(A),Na: nat,V: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),U),bit_se2239418461657761734s_mask(word(A),Na)) = zero_zero(word(A)) )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),V),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),V) = zero_zero(word(A)) ) ) ) ) ).

% is_aligned_AND_less_0
tff(fact_2859_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),aa(word(A),fun(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),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),bit_se2239418461657761734s_mask(word(A),Na)) ) ).

% add_mask_fold
tff(fact_2860_odd__0__le__power__imp__0__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Aa2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,Aa2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2) ) ) ).

% odd_0_le_power_imp_0_le
tff(fact_2861_odd__power__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Aa2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),zero_zero(A))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,Aa2),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_2862_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_2863_nat0__intermed__int__val,axiom,
    ! [Na: nat,F2: fun(nat,int),K: int] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Na)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,aa(nat,int,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),one_one(nat)))),aa(nat,int,F2,I2)))),one_one(int)) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F2,zero_zero(nat))),K)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,F2,Na))
         => ? [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),Na)
              & ( aa(nat,int,F2,I2) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_2864_and__mask__less__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,X: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(word(A),nat,size_size(word(A)),X))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),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_2865_mask__eq__iff__w2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(word(A),nat,size_size(word(A)),W))
         => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),Na)) = W )
          <=> aa(word(A),$o,aa(word(A),fun(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_2866_word__and__mask__le__2pm1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Na: nat] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),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_2867_word__mod__2p__is__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,X: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),aa(nat,word(A),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_2868_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_2869_signed__take__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,Aa2: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,suc,Na)),Aa2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,bit_ri4674362597316999326ke_bit(A,Na),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% signed_take_bit_Suc
tff(fact_2870_nat__div__eq__Suc__0__iff,axiom,
    ! [Na: nat,M: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),M) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) ) ) ).

% nat_div_eq_Suc_0_iff
tff(fact_2871_ln__one__plus__pos__lower__bound,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),one_one(real))
       => aa(real,$o,aa(real,fun(real,$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,aa(real,fun(real,real),plus_plus(real),one_one(real)),X))) ) ) ).

% ln_one_plus_pos_lower_bound
tff(fact_2872_unset__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,Aa2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),aa(nat,nat,suc,Na)),Aa2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),Na),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% unset_bit_Suc
tff(fact_2873_set__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,Aa2: A] : aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),aa(nat,nat,suc,Na)),Aa2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),Na),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% set_bit_Suc
tff(fact_2874_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_2875_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,Na)))),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K)) ) ).

% signed_take_bit_int_greater_eq
tff(fact_2876_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,aa(word(A),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,aa(word(A),nat,size_size(word(A)),W)),Na))) ) ).

% and_mask_arith
tff(fact_2877_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_2878_sb__inc__lem,axiom,
    ! [Aa2: int,K: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Aa2),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),K))),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Aa2),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,aa(int,fun(int,int),plus_plus(int),Aa2),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_2879_sb__inc__lem_H,axiom,
    ! [Aa2: int,K: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Aa2),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),K)))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Aa2),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,aa(int,fun(int,int),plus_plus(int),Aa2),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_2880_neg__mask__add__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)),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_2881_neg__mask__is__div_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(word(A),nat,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_2882_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,aa(A,fun(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_2883_abs__sqrt__wlog,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P: fun(A,fun(A,$o)),X: A] :
          ( ! [X2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X2)
             => aa(A,$o,aa(A,fun(A,$o),P,X2),aa(nat,A,power_power(A,X2),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_2884_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,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                        $let(
                          half: nat,
                          half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit1(bit0(one2))))),vEBT_V8646137997579335489_i_l_d(half))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),vEBT_V8646137997579335489_i_l_d(half))) )),
                      $let(
                        half: nat,
                        half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit1(one2))))),vEBT_V8646137997579335489_i_l_d(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,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_2885_mask__nat__positive__iff,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),bit_se2239418461657761734s_mask(nat,Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ).

% mask_nat_positive_iff
tff(fact_2886_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_2887_nat__mask__eq,axiom,
    ! [Na: nat] : aa(int,nat,nat2,bit_se2239418461657761734s_mask(int,Na)) = bit_se2239418461657761734s_mask(nat,Na) ).

% nat_mask_eq
tff(fact_2888_less__eq__mask,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),bit_se2239418461657761734s_mask(nat,Na)) ).

% less_eq_mask
tff(fact_2889_mask__nonnegative__int,axiom,
    ! [Na: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),bit_se2239418461657761734s_mask(int,Na)) ).

% mask_nonnegative_int
tff(fact_2890_not__mask__negative__int,axiom,
    ! [Na: nat] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_se2239418461657761734s_mask(int,Na)),zero_zero(int)) ).

% not_mask_negative_int
tff(fact_2891_abs__real__def,axiom,
    ! [Aa2: real] :
      aa(real,real,abs_abs(real),Aa2) = $ite(aa(real,$o,aa(real,fun(real,$o),ord_less(real),Aa2),zero_zero(real)),aa(real,real,uminus_uminus(real),Aa2),Aa2) ).

% abs_real_def
tff(fact_2892_sin__bound__lemma,axiom,
    ! [X: real,Y: real,U: real,V: real] :
      ( ( X = Y )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),U)),V)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),U)),Y))),V) ) ) ).

% sin_bound_lemma
tff(fact_2893_less__mask,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),bit_se2239418461657761734s_mask(nat,Na)) ) ).

% less_mask
tff(fact_2894_lemma__interval,axiom,
    ! [Aa2: real,X: real,Ba: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Aa2),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),Ba)
       => ? [D4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D4)
            & ! [Y3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,X),Y3))),D4)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),Y3)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y3),Ba) ) ) ) ) ) ).

% lemma_interval
tff(fact_2895_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_2896_mask__nat__less__exp,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$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_2897_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_2898_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_2899_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_2900_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_2901_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_2902_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),one_one(real))
       => aa(real,$o,aa(real,fun(real,$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,aa(real,fun(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_2903_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$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,$o,aa(real,fun(real,$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,aa(real,fun(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_2904_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),zero_zero(real))
       => aa(real,$o,aa(real,fun(real,$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,aa(real,fun(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_2905_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,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
          $let(
            half: nat,
            half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit1(bit0(one2))))),vEBT_V8646137997579335489_i_l_d(half))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),vEBT_V8646137997579335489_i_l_d(half))) )),
        $let(
          half: nat,
          half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit1(one2))))),vEBT_V8646137997579335489_i_l_d(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,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_2906_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,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),bit1(one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(one2)))),aa(nat,int,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,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),bit1(one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(bit0(one2))))),aa(nat,int,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_2907_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,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                        $let(
                          half: nat,
                          half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(bit0(one2))))),vEBT_V8346862874174094_d_u_p(half))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) )),
                      $let(
                        half: nat,
                        half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit1(bit0(one2))))),vEBT_V8346862874174094_d_u_p(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.elims
tff(fact_2908_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,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),bit1(bit0(one2)))),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),bit1(bit0(one2)))),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,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_2909_lemma__termdiff3,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [H: A,Z: A,K3: real,Na: nat] :
          ( ( H != zero_zero(A) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z)),K3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),H))),K3)
             => aa(real,$o,aa(real,fun(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,aa(A,fun(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,K3),aa(nat,nat,minus_minus(nat,Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),real_V7770717601297561774m_norm(A,H))) ) ) ) ) ).

% lemma_termdiff3
tff(fact_2910_monoseq__arctan__series,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real))
     => topological_monoseq(real,aTP_Lamp_ax(real,fun(nat,real),X)) ) ).

% monoseq_arctan_series
tff(fact_2911_complex__mod__minus__le__complex__mod,axiom,
    ! [X: complex] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),real_V7770717601297561774m_norm(complex,X))),real_V7770717601297561774m_norm(complex,X)) ).

% complex_mod_minus_le_complex_mod
tff(fact_2912_complex__mod__triangle__ineq2,axiom,
    ! [Ba: complex,Aa2: complex] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,minus_minus(real,real_V7770717601297561774m_norm(complex,aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),Ba),Aa2))),real_V7770717601297561774m_norm(complex,Ba))),real_V7770717601297561774m_norm(complex,Aa2)) ).

% complex_mod_triangle_ineq2
tff(fact_2913_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_2914_VEBT__internal_OTb__T__vebt__buildupi_H,axiom,
    ! [Na: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),vEBT_V9176841429113362141ildupi(Na)),aa(int,int,minus_minus(int,vEBT_VEBT_Tb(Na)),aa(num,int,numeral_numeral(int),bit0(one2)))) ).

% VEBT_internal.Tb_T_vebt_buildupi'
tff(fact_2915_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_2916_lemma__NBseq__def2,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X5: fun(A,B)] :
          ( ? [K7: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K7)
              & ! [N2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X5,N2))),K7) )
        <=> ? [N7: nat] :
            ! [N2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,aa(A,B,X5,N2))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N7))) ) ) ).

% lemma_NBseq_def2
tff(fact_2917_lemma__NBseq__def,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X5: fun(A,B)] :
          ( ? [K7: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K7)
              & ! [N2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X5,N2))),K7) )
        <=> ? [N7: nat] :
            ! [N2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X5,N2))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N7))) ) ) ).

% lemma_NBseq_def
tff(fact_2918_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_2919_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_2920_monoseq__realpow,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),one_one(real))
       => topological_monoseq(real,power_power(real,X)) ) ) ).

% monoseq_realpow
tff(fact_2921_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_2922_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_2923_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,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),bit1(bit0(one2)))),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),bit1(bit0(one2)))),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,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_2924_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,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),bit1(one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(one2)))),aa(nat,int,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,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),bit1(one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(bit0(one2))))),aa(nat,int,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_2925_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,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
          $let(
            half: nat,
            half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(bit0(one2))))),vEBT_V8346862874174094_d_u_p(half))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) )),
        $let(
          half: nat,
          half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit1(bit0(one2))))),vEBT_V8346862874174094_d_u_p(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(3)
tff(fact_2926_norm__divide__numeral,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Aa2: A,W: num] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(num,A,numeral_numeral(A),W))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(A,Aa2)),aa(num,real,numeral_numeral(real),W)) ) ).

% norm_divide_numeral
tff(fact_2927_norm__mult__numeral2,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Aa2: A,W: num] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(num,A,numeral_numeral(A),W))) = aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,Aa2)),aa(num,real,numeral_numeral(real),W)) ) ).

% norm_mult_numeral2
tff(fact_2928_norm__mult__numeral1,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [W: num,Aa2: A] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),Aa2)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),W)),real_V7770717601297561774m_norm(A,Aa2)) ) ).

% norm_mult_numeral1
tff(fact_2929_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_2930_norm__le__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,X)),zero_zero(real))
        <=> ( X = zero_zero(A) ) ) ) ).

% norm_le_zero_iff
tff(fact_2931_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_2932_norm__one,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ( real_V7770717601297561774m_norm(A,one_one(A)) = one_one(real) ) ) ).

% norm_one
tff(fact_2933_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_2934_norm__mult,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: A,Y: A] : 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),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,Y)) ) ).

% norm_mult
tff(fact_2935_norm__ge__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),real_V7770717601297561774m_norm(A,X)) ) ).

% norm_ge_zero
tff(fact_2936_norm__uminus__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y: A] : real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),X)),Y)) = real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) ) ).

% norm_uminus_minus
tff(fact_2937_norm__mult__less,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [X: A,R: real,Y: A,S: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X)),R)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Y)),S)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Y))),aa(real,real,aa(real,fun(real,real),times_times(real),R),S)) ) ) ) ).

% norm_mult_less
tff(fact_2938_norm__mult__ineq,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [X: A,Y: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Y))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,Y))) ) ).

% norm_mult_ineq
tff(fact_2939_norm__triangle__lt,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y: A,E2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,Y))),E2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y))),E2) ) ) ).

% norm_triangle_lt
tff(fact_2940_norm__add__less,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,R: real,Y: A,S: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X)),R)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Y)),S)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),R),S)) ) ) ) ).

% norm_add_less
tff(fact_2941_norm__triangle__mono,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Aa2: A,R: real,Ba: A,S: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Aa2)),R)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Ba)),S)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba))),aa(real,real,aa(real,fun(real,real),plus_plus(real),R),S)) ) ) ) ).

% norm_triangle_mono
tff(fact_2942_norm__triangle__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,Y))) ) ).

% norm_triangle_ineq
tff(fact_2943_norm__triangle__le,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y: A,E2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,Y))),E2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y))),E2) ) ) ).

% norm_triangle_le
tff(fact_2944_norm__add__leD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Aa2: A,Ba: A,C2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba))),C2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Ba)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Aa2)),C2)) ) ) ).

% norm_add_leD
tff(fact_2945_norm__diff__triangle__less,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y: A,E1: real,Z: A,E22: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X),Y))),E1)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Y),Z))),E22)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X),Z))),aa(real,real,aa(real,fun(real,real),plus_plus(real),E1),E22)) ) ) ) ).

% norm_diff_triangle_less
tff(fact_2946_norm__power__ineq,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [X: A,Na: nat] : aa(real,$o,aa(real,fun(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_2947_norm__triangle__sub,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,X)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Y)),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X),Y)))) ) ).

% norm_triangle_sub
tff(fact_2948_norm__triangle__ineq4,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Aa2: A,Ba: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Aa2),Ba))),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Aa2)),real_V7770717601297561774m_norm(A,Ba))) ) ).

% norm_triangle_ineq4
tff(fact_2949_norm__diff__triangle__le,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y: A,E1: real,Z: A,E22: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X),Y))),E1)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Y),Z))),E22)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X),Z))),aa(real,real,aa(real,fun(real,real),plus_plus(real),E1),E22)) ) ) ) ).

% norm_diff_triangle_le
tff(fact_2950_norm__triangle__le__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y: A,E2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,Y))),E2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X),Y))),E2) ) ) ).

% norm_triangle_le_diff
tff(fact_2951_norm__diff__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Aa2: A,Ba: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,minus_minus(real,real_V7770717601297561774m_norm(A,Aa2)),real_V7770717601297561774m_norm(A,Ba))),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba))) ) ).

% norm_diff_ineq
tff(fact_2952_norm__triangle__ineq2,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Aa2: A,Ba: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,minus_minus(real,real_V7770717601297561774m_norm(A,Aa2)),real_V7770717601297561774m_norm(A,Ba))),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Aa2),Ba))) ) ).

% norm_triangle_ineq2
tff(fact_2953_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_2954_norm__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Aa2: A,Ba: A,C2: A,D2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),D2)))),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Aa2),C2))),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Ba),D2)))) ) ).

% norm_diff_triangle_ineq
tff(fact_2955_norm__triangle__ineq3,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Aa2: A,Ba: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,real_V7770717601297561774m_norm(A,Aa2)),real_V7770717601297561774m_norm(A,Ba)))),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Aa2),Ba))) ) ).

% norm_triangle_ineq3
tff(fact_2956_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_2957_norm__power__diff,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A,W: A,M: nat] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z)),one_one(real))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,W)),one_one(real))
           => aa(real,$o,aa(real,fun(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_2958_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,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_V8346862874174094_d_u_p(Na)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit1(bit0(bit1(one2)))))),U)) ) ).

% vebt_inst.time_vebt_buildup
tff(fact_2959_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,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(one2)))),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(one2)))),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,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_2960_ln__series,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(num,real,numeral_numeral(real),bit0(one2)))
       => ( aa(real,real,ln_ln(real),X) = suminf(real,aTP_Lamp_ay(real,fun(nat,real),X)) ) ) ) ).

% ln_series
tff(fact_2961_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_2962_VEBT__internal_Oreplicatei_Oelims,axiom,
    ! [A: $tType,X: nat,Xa2: heap_Time_Heap(A),Y: heap_Time_Heap(list(A))] :
      ( ( vEBT_VEBT_replicatei(A,X,Xa2) = Y )
     => ( ( ( X = zero_zero(nat) )
         => ( Y != aa(list(A),heap_Time_Heap(list(A)),heap_Time_return(list(A)),nil(A)) ) )
       => ~ ! [N: nat] :
              ( ( X = aa(nat,nat,suc,N) )
             => ( Y != heap_Time_bind(A,list(A),Xa2,aa(nat,fun(A,heap_Time_Heap(list(A))),aTP_Lamp_az(heap_Time_Heap(A),fun(nat,fun(A,heap_Time_Heap(list(A)))),Xa2),N)) ) ) ) ) ).

% VEBT_internal.replicatei.elims
tff(fact_2963_powser__zero,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [F2: fun(nat,A)] : suminf(A,aTP_Lamp_ba(fun(nat,A),fun(nat,A),F2)) = aa(nat,A,F2,zero_zero(nat)) ) ).

% powser_zero
tff(fact_2964_VEBT__internal_OTb__Tb_H,axiom,
    ! [T: nat] : vEBT_VEBT_Tb(T) = aa(nat,int,semiring_1_of_nat(int),vEBT_VEBT_Tb2(T)) ).

% VEBT_internal.Tb_Tb'
tff(fact_2965_VEBT__internal_Oreplicatei_Osimps_I1_J,axiom,
    ! [A: $tType,X: heap_Time_Heap(A)] : vEBT_VEBT_replicatei(A,zero_zero(nat),X) = aa(list(A),heap_Time_Heap(list(A)),heap_Time_return(list(A)),nil(A)) ).

% VEBT_internal.replicatei.simps(1)
tff(fact_2966_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_2967_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_2968_VEBT__internal_Oreplicatei_Osimps_I2_J,axiom,
    ! [A: $tType,Na: nat,X: heap_Time_Heap(A)] : vEBT_VEBT_replicatei(A,aa(nat,nat,suc,Na),X) = heap_Time_bind(A,list(A),X,aa(heap_Time_Heap(A),fun(A,heap_Time_Heap(list(A))),aTP_Lamp_bb(nat,fun(heap_Time_Heap(A),fun(A,heap_Time_Heap(list(A)))),Na),X)) ).

% VEBT_internal.replicatei.simps(2)
tff(fact_2969_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,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(one2)))),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(one2)))),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,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_2970_suminf__geometric,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( aa(real,$o,aa(real,fun(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_2971_suminf__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ( suminf(A,aTP_Lamp_bc(nat,A)) = zero_zero(A) ) ) ).

% suminf_zero
tff(fact_2972_arctan__series,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real))
     => ( aa(real,real,arctan,X) = suminf(real,aTP_Lamp_bd(real,fun(nat,real),X)) ) ) ).

% arctan_series
tff(fact_2973_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_be(nat,real)) ).

% pi_series
tff(fact_2974_summable__arctan__series,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real))
     => summable(real,aTP_Lamp_bd(real,fun(nat,real),X)) ) ).

% summable_arctan_series
tff(fact_2975_summable__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => summable(A,aTP_Lamp_bf(nat,A)) ) ).

% summable_zero
tff(fact_2976_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_bg(nat,fun(fun(nat,A),fun(nat,A)),I),F2)) ) ).

% summable_single
tff(fact_2977_summable__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),K: nat] :
          ( summable(A,aa(nat,fun(nat,A),aTP_Lamp_bh(fun(nat,A),fun(nat,fun(nat,A)),F2),K))
        <=> summable(A,F2) ) ) ).

% summable_iff_shift
tff(fact_2978_arctan__le__zero__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arctan,X)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),zero_zero(real)) ) ).

% arctan_le_zero_iff
tff(fact_2979_zero__le__arctan__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,arctan,X))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X) ) ).

% zero_le_arctan_iff
tff(fact_2980_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_bi(A,fun(fun(nat,A),fun(nat,A)),C2),F2))
        <=> ( ( C2 = zero_zero(A) )
            | summable(A,F2) ) ) ) ).

% summable_cmult_iff
tff(fact_2981_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_bj(fun(nat,A),fun(A,fun(nat,A)),F2),C2))
        <=> ( ( C2 = zero_zero(A) )
            | summable(A,F2) ) ) ) ).

% summable_divide_iff
tff(fact_2982_summable__geometric__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( summable(A,power_power(A,C2))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,C2)),one_one(real)) ) ) ).

% summable_geometric_iff
tff(fact_2983_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_2984_summable__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,aTP_Lamp_bl(fun(nat,A),fun(nat,A),F2))
        <=> summable(A,F2) ) ) ).

% summable_Suc_iff
tff(fact_2985_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_bm(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ).

% summable_mult2
tff(fact_2986_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_bn(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ).

% summable_mult
tff(fact_2987_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_bj(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ).

% summable_divide
tff(fact_2988_summable__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => summable(A,aTP_Lamp_bo(fun(nat,A),fun(nat,A),F2)) ) ) ).

% summable_minus
tff(fact_2989_summable__minus__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,aTP_Lamp_bo(fun(nat,A),fun(nat,A),F2))
        <=> summable(A,F2) ) ) ).

% summable_minus_iff
tff(fact_2990_summable__const__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [C2: A] :
          ( summable(A,aTP_Lamp_bp(A,fun(nat,A),C2))
        <=> ( C2 = zero_zero(A) ) ) ) ).

% summable_const_iff
tff(fact_2991_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_bq(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)) ) ) ) ).

% summable_diff
tff(fact_2992_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_br(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)) ) ) ) ).

% summable_add
tff(fact_2993_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_bh(fun(nat,A),fun(nat,fun(nat,A)),F2),K)) ) ) ).

% summable_ignore_initial_segment
tff(fact_2994_summable__norm__cancel,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_bs(fun(nat,A),fun(nat,real),F2))
         => summable(A,F2) ) ) ).

% summable_norm_cancel
tff(fact_2995_summable__comparison__test_H,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [G: fun(nat,real),N3: nat,F2: fun(nat,A)] :
          ( summable(real,G)
         => ( ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N3),N)
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F2,N))),aa(nat,real,G,N)) )
           => summable(A,F2) ) ) ) ).

% summable_comparison_test'
tff(fact_2996_summable__comparison__test,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( ? [N8: nat] :
            ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N8),N)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F2,N))),aa(nat,real,G,N)) )
         => ( summable(real,G)
           => summable(A,F2) ) ) ) ).

% summable_comparison_test
tff(fact_2997_arctan__le__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arctan,X)),aa(real,real,arctan,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y) ) ).

% arctan_le_iff
tff(fact_2998_arctan__monotone_H,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arctan,X)),aa(real,real,arctan,Y)) ) ).

% arctan_monotone'
tff(fact_2999_suminf__le,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),G: fun(nat,A)] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,N)),aa(nat,A,G,N))
         => ( summable(A,F2)
           => ( summable(A,G)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),suminf(A,F2)),suminf(A,G)) ) ) ) ) ).

% suminf_le
tff(fact_3000_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_bi(A,fun(fun(nat,A),fun(nat,A)),C2),F2))
         => ( ( C2 != zero_zero(A) )
           => summable(A,F2) ) ) ) ).

% summable_mult_D
tff(fact_3001_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_3002_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,aa(A,fun(A,A),plus_plus(A),suminf(A,F2)),suminf(A,G)) = suminf(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_br(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)) ) ) ) ) ).

% suminf_add
tff(fact_3003_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_bm(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ) ).

% suminf_mult2
tff(fact_3004_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_bn(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_3005_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_bq(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)) ) ) ) ) ).

% suminf_diff
tff(fact_3006_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_bj(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_3007_suminf__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( suminf(A,aTP_Lamp_bo(fun(nat,A),fun(nat,A),F2)) = aa(A,A,uminus_uminus(A),suminf(A,F2)) ) ) ) ).

% suminf_minus
tff(fact_3008_powser__insidea,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,A),X: A,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_bt(fun(nat,A),fun(A,fun(nat,A)),F2),X))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),real_V7770717601297561774m_norm(A,X))
           => summable(real,aa(A,fun(nat,real),aTP_Lamp_bu(fun(nat,A),fun(A,fun(nat,real)),F2),Z)) ) ) ) ).

% powser_insidea
tff(fact_3009_suminf__nonneg,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),suminf(A,F2)) ) ) ) ).

% suminf_nonneg
tff(fact_3010_suminf__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N))
           => ( ( suminf(A,F2) = zero_zero(A) )
            <=> ! [N2: nat] : aa(nat,A,F2,N2) = zero_zero(A) ) ) ) ) ).

% suminf_eq_zero_iff
tff(fact_3011_summable__zero__power_H,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A)] : summable(A,aTP_Lamp_bv(fun(nat,A),fun(nat,A),F2)) ) ).

% summable_zero_power'
tff(fact_3012_summable__0__powser,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,A)] : summable(A,aTP_Lamp_bw(fun(nat,A),fun(nat,A),F2)) ) ).

% summable_0_powser
tff(fact_3013_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_3014_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_bt(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) ) ) ).

% summable_powser_split_head
tff(fact_3015_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_ca(fun(nat,A),fun(nat,fun(A,fun(nat,A))),F2),M),Z))
        <=> summable(A,aa(A,fun(nat,A),aTP_Lamp_bt(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) ) ) ).

% summable_powser_ignore_initial_segment
tff(fact_3016_pi__ge__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),pi) ).

% pi_ge_zero
tff(fact_3017_summable__norm__comparison__test,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( ? [N8: nat] :
            ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N8),N)
             => aa(real,$o,aa(real,fun(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_3018_summable__rabs__comparison__test,axiom,
    ! [F2: fun(nat,real),G: fun(nat,real)] :
      ( ? [N8: nat] :
        ! [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N8),N)
         => aa(real,$o,aa(real,fun(real,$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_3019_summable__rabs,axiom,
    ! [F2: fun(nat,real)] :
      ( summable(real,aTP_Lamp_bk(fun(nat,real),fun(nat,real),F2))
     => aa(real,$o,aa(real,fun(real,$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_3020_arctan__ubound,axiom,
    ! [Y: real] : aa(real,$o,aa(real,fun(real,$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_3021_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_3022_suminf__pos2,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),I: nat] :
          ( summable(A,F2)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F2,I))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),suminf(A,F2)) ) ) ) ) ).

% suminf_pos2
tff(fact_3023_suminf__pos__iff,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),suminf(A,F2))
            <=> ? [I4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F2,I4)) ) ) ) ) ).

% suminf_pos_iff
tff(fact_3024_suminf__split__head,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( suminf(A,aTP_Lamp_bl(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_3025_complete__algebra__summable__geometric,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] :
          ( aa(real,$o,aa(real,fun(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_3026_summable__geometric,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,C2)),one_one(real))
         => summable(A,power_power(A,C2)) ) ) ).

% summable_geometric
tff(fact_3027_arctan__lbound,axiom,
    ! [Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arctan,Y)) ).

% arctan_lbound
tff(fact_3028_arctan__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arctan,Y))
      & aa(real,$o,aa(real,fun(real,$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_3029_summable__norm,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_bs(fun(nat,A),fun(nat,real),F2))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,suminf(A,F2))),suminf(real,aTP_Lamp_bs(fun(nat,A),fun(nat,real),F2))) ) ) ).

% summable_norm
tff(fact_3030_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,aa(real,fun(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_3031_machin__Euler,axiom,
    aa(real,real,aa(real,fun(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_3032_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_3033_pi__less__4,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))) ).

% pi_less_4
tff(fact_3034_pi__ge__two,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi) ).

% pi_ge_two
tff(fact_3035_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_3036_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,aa(A,fun(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_3037_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_3038_suminf__exist__split,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R: real,F2: fun(nat,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
         => ( summable(A,F2)
           => ? [N9: nat] :
              ! [N10: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N9),N10)
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,suminf(A,aa(nat,fun(nat,A),aTP_Lamp_bh(fun(nat,A),fun(nat,fun(nat,A)),F2),N10)))),R) ) ) ) ) ).

% suminf_exist_split
tff(fact_3039_summable__power__series,axiom,
    ! [F2: fun(nat,real),Z: real] :
      ( ! [I2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F2,I2)),one_one(real))
     => ( ! [I2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,F2,I2))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Z)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z),one_one(real))
           => summable(real,aa(real,fun(nat,real),aTP_Lamp_cc(fun(nat,real),fun(real,fun(nat,real)),F2),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_3040_Abel__lemma,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R: real,R0: real,Aa2: fun(nat,A),M2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),R)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),R),R0)
           => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,aa(nat,A,Aa2,N))),aa(nat,real,power_power(real,R0),N))),M2)
             => summable(real,aa(fun(nat,A),fun(nat,real),aTP_Lamp_cd(real,fun(fun(nat,A),fun(nat,real)),R),Aa2)) ) ) ) ) ).

% Abel_lemma
tff(fact_3041_summable__ratio__test,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [C2: real,N3: nat,F2: fun(nat,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),one_one(real))
         => ( ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N3),N)
               => aa(real,$o,aa(real,fun(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_3042_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_3043_pi__half__less__two,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% pi_half_less_two
tff(fact_3044_pi__half__le__two,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% pi_half_le_two
tff(fact_3045_pi__half__gt__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ).

% pi_half_gt_zero
tff(fact_3046_pi__half__ge__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ).

% pi_half_ge_zero
tff(fact_3047_m2pi__less__pi,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))),pi) ).

% m2pi_less_pi
tff(fact_3048_minus__pi__half__less__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),zero_zero(real)) ).

% minus_pi_half_less_zero
tff(fact_3049_arctan__add,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),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,aa(real,fun(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_3050_arctan__double,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$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_3051_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_3052_flip__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,Aa2: A] : bit_se8732182000553998342ip_bit(A,aa(nat,nat,suc,Na),Aa2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),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),Aa2),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% flip_bit_Suc
tff(fact_3053_tanh__ln__real,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( 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,aa(real,fun(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_3054_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,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),aa(nat,nat,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,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),aa(nat,nat,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_3055_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_3056_flip__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),bit_se8732182000553998342ip_bit(int,Na,K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_3057_flip__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_se8732182000553998342ip_bit(int,Na,K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% flip_bit_negative_int_iff
tff(fact_3058_tanh__real__le__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),tanh(real,X)),tanh(real,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y) ) ).

% tanh_real_le_iff
tff(fact_3059_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_3060_tanh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),tanh(real,X)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),zero_zero(real)) ) ).

% tanh_real_nonpos_iff
tff(fact_3061_tanh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),tanh(real,X))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X) ) ).

% tanh_real_nonneg_iff
tff(fact_3062_cos__periodic__pi,axiom,
    ! [X: real] : cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),pi)) = aa(real,real,uminus_uminus(real),cos(real,X)) ).

% cos_periodic_pi
tff(fact_3063_cos__periodic__pi2,axiom,
    ! [X: real] : cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),pi),X)) = aa(real,real,uminus_uminus(real),cos(real,X)) ).

% cos_periodic_pi2
tff(fact_3064_sin__periodic__pi2,axiom,
    ! [X: real] : sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),pi),X)) = aa(real,real,uminus_uminus(real),sin(real,X)) ).

% sin_periodic_pi2
tff(fact_3065_sin__periodic__pi,axiom,
    ! [X: real] : sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),pi)) = aa(real,real,uminus_uminus(real),sin(real,X)) ).

% sin_periodic_pi
tff(fact_3066_sin__cos__squared__add3,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,aa(A,fun(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_3067_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_3068_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_3069_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_3070_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_3071_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_3072_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_3073_cos__periodic,axiom,
    ! [X: real] : cos(real,aa(real,real,aa(real,fun(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_3074_sin__periodic,axiom,
    ! [X: real] : sin(real,aa(real,real,aa(real,fun(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_3075_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_3076_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_3077_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_3078_sin__cos__squared__add2,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,aa(A,fun(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_3079_sin__cos__squared__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,aa(A,fun(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_3080_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_3081_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_3082_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_3083_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_3084_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_3085_sin__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,X)),cos(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),sin(A,Y))) ) ).

% sin_add
tff(fact_3086_sin__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : sin(A,aa(A,A,minus_minus(A,X),Y)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,X)),cos(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),sin(A,Y))) ) ).

% sin_diff
tff(fact_3087_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_3088_cos__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),cos(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,X)),sin(A,Y))) ) ).

% cos_add
tff(fact_3089_cos__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : cos(A,aa(A,A,minus_minus(A,X),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),cos(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,X)),sin(A,Y))) ) ).

% cos_diff
tff(fact_3090_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_3091_sincos__principal__value,axiom,
    ! [X: real] :
    ? [Y2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),Y2)
      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y2),pi)
      & ( sin(real,Y2) = sin(real,X) )
      & ( cos(real,Y2) = cos(real,X) ) ) ).

% sincos_principal_value
tff(fact_3092_sin__x__le__x,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,X)),X) ) ).

% sin_x_le_x
tff(fact_3093_sin__le__one,axiom,
    ! [X: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,X)),one_one(real)) ).

% sin_le_one
tff(fact_3094_cos__le__one,axiom,
    ! [X: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cos(real,X)),one_one(real)) ).

% cos_le_one
tff(fact_3095_abs__sin__x__le__abs__x,axiom,
    ! [X: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),sin(real,X))),aa(real,real,abs_abs(real),X)) ).

% abs_sin_x_le_abs_x
tff(fact_3096_sin__cos__le1,axiom,
    ! [X: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),sin(real,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_3097_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_3098_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_3099_sin__x__ge__neg__x,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),X)),sin(real,X)) ) ).

% sin_x_ge_neg_x
tff(fact_3100_sin__ge__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),pi)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),sin(real,X)) ) ) ).

% sin_ge_zero
tff(fact_3101_sin__ge__minus__one,axiom,
    ! [X: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),sin(real,X)) ).

% sin_ge_minus_one
tff(fact_3102_cos__monotone__0__pi__le,axiom,
    ! [Y: real,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),pi)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cos(real,X)),cos(real,Y)) ) ) ) ).

% cos_monotone_0_pi_le
tff(fact_3103_cos__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),pi)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),pi)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cos(real,X)),cos(real,Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),X) ) ) ) ) ) ).

% cos_mono_le_eq
tff(fact_3104_cos__inj__pi,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),pi)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),pi)
           => ( ( cos(real,X) = cos(real,Y) )
             => ( X = Y ) ) ) ) ) ) ).

% cos_inj_pi
tff(fact_3105_cos__ge__minus__one,axiom,
    ! [X: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),cos(real,X)) ).

% cos_ge_minus_one
tff(fact_3106_abs__sin__le__one,axiom,
    ! [X: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),sin(real,X))),one_one(real)) ).

% abs_sin_le_one
tff(fact_3107_abs__cos__le__one,axiom,
    ! [X: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),cos(real,X))),one_one(real)) ).

% abs_cos_le_one
tff(fact_3108_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,aa(A,fun(A,A),plus_plus(A),W),Z)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% sin_times_sin
tff(fact_3109_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,aa(A,fun(A,A),plus_plus(A),sin(A,aa(A,A,aa(A,fun(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_3110_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,aa(A,fun(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_3111_sin__plus__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(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,aa(A,fun(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_3112_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,aa(A,fun(A,A),plus_plus(A),W),Z)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% sin_diff_sin
tff(fact_3113_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,aa(A,fun(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_3114_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_3115_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_3116_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_3117_cos__two__neq__zero,axiom,
    cos(real,aa(num,real,numeral_numeral(real),bit0(one2))) != zero_zero(real) ).

% cos_two_neq_zero
tff(fact_3118_cos__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),pi)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),pi)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),cos(real,X)),cos(real,Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),X) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_3119_cos__monotone__0__pi,axiom,
    ! [Y: real,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),pi)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),cos(real,X)),cos(real,Y)) ) ) ) ).

% cos_monotone_0_pi
tff(fact_3120_cos__monotone__minus__pi__0_H,axiom,
    ! [Y: real,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),pi)),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),zero_zero(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cos(real,Y)),cos(real,X)) ) ) ) ).

% cos_monotone_minus_pi_0'
tff(fact_3121_sincos__total__pi,axiom,
    ! [Y: real,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
     => ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,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,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T2)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T2),pi)
            & ( X = cos(real,T2) )
            & ( Y = sin(real,T2) ) ) ) ) ).

% sincos_total_pi
tff(fact_3122_sin__expansion__lemma,axiom,
    ! [X: real,M: nat] : sin(real,aa(real,real,aa(real,fun(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,aa(real,fun(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_3123_cos__expansion__lemma,axiom,
    ! [X: real,M: nat] : cos(real,aa(real,real,aa(real,fun(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,aa(real,fun(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_3124_sin__gt__zero__02,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(num,real,numeral_numeral(real),bit0(one2)))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),sin(real,X)) ) ) ).

% sin_gt_zero_02
tff(fact_3125_cos__two__less__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),cos(real,aa(num,real,numeral_numeral(real),bit0(one2)))),zero_zero(real)) ).

% cos_two_less_zero
tff(fact_3126_cos__two__le__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cos(real,aa(num,real,numeral_numeral(real),bit0(one2)))),zero_zero(real)) ).

% cos_two_le_zero
tff(fact_3127_cos__is__zero,axiom,
    ? [X2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X2)
      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),aa(num,real,numeral_numeral(real),bit0(one2)))
      & ( cos(real,X2) = zero_zero(real) )
      & ! [Y3: real] :
          ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y3)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y3),aa(num,real,numeral_numeral(real),bit0(one2)))
            & ( cos(real,Y3) = zero_zero(real) ) )
         => ( Y3 = X2 ) ) ) ).

% cos_is_zero
tff(fact_3128_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),pi)),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),zero_zero(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),cos(real,Y)),cos(real,X)) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_3129_cos__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ? [X2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X2)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),pi)
            & ( cos(real,X2) = Y )
            & ! [Y3: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y3)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y3),pi)
                  & ( cos(real,Y3) = Y ) )
               => ( Y3 = X2 ) ) ) ) ) ).

% cos_total
tff(fact_3130_sincos__total__pi__half,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( ( aa(real,real,aa(real,fun(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,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T2)
              & aa(real,$o,aa(real,fun(real,$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_3131_sincos__total__2pi__le,axiom,
    ! [X: real,Y: real] :
      ( ( aa(real,real,aa(real,fun(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,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T2)
          & aa(real,$o,aa(real,fun(real,$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_3132_sincos__total__2pi,axiom,
    ! [X: real,Y: real] :
      ( ( aa(real,real,aa(real,fun(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,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T2)
           => ( aa(real,$o,aa(real,fun(real,$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_3133_sin__pi__divide__n__ge__0,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero(nat) )
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(nat,real,semiring_1_of_nat(real),Na)))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_3134_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,aa(A,fun(A,A),plus_plus(A),cos(A,aa(A,A,minus_minus(A,W),Z))),cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% cos_times_cos
tff(fact_3135_cos__plus__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(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,aa(A,fun(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_3136_sin__gt__zero2,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),sin(real,X)) ) ) ).

% sin_gt_zero2
tff(fact_3137_sin__lt__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),pi),X)
     => ( aa(real,$o,aa(real,fun(real,$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,$o,aa(real,fun(real,$o),ord_less(real),sin(real,X)),zero_zero(real)) ) ) ).

% sin_lt_zero
tff(fact_3138_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_3139_cos__double__less__one,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(num,real,numeral_numeral(real),bit0(one2)))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),X))),one_one(real)) ) ) ).

% cos_double_less_one
tff(fact_3140_cos__gt__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),cos(real,X)) ) ) ).

% cos_gt_zero
tff(fact_3141_sin__inj__pi,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( ( sin(real,X) = sin(real,Y) )
             => ( X = Y ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_3142_sin__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,X)),sin(real,Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_3143_sin__monotone__2pi__le,axiom,
    ! [Y: real,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Y)),sin(real,X)) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_3144_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_3145_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_3146_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_3147_sin__le__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),pi),X)
     => ( aa(real,$o,aa(real,fun(real,$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,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,X)),zero_zero(real)) ) ) ).

% sin_le_zero
tff(fact_3148_sin__less__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),zero_zero(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),sin(real,X)),zero_zero(real)) ) ) ).

% sin_less_zero
tff(fact_3149_sin__monotone__2pi,axiom,
    ! [Y: real,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),sin(real,Y)),sin(real,X)) ) ) ) ).

% sin_monotone_2pi
tff(fact_3150_sin__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),sin(real,X)),sin(real,Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),Y) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_3151_sin__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ? [X2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X2)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
            & ( sin(real,X2) = Y )
            & ! [Y3: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y3)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y3),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
                  & ( sin(real,Y3) = Y ) )
               => ( Y3 = X2 ) ) ) ) ) ).

% sin_total
tff(fact_3152_cos__gt__zero__pi,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),cos(real,X)) ) ) ).

% cos_gt_zero_pi
tff(fact_3153_cos__ge__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),cos(real,X)) ) ) ).

% cos_ge_zero
tff(fact_3154_cos__one__2pi,axiom,
    ! [X: real] :
      ( ( cos(real,X) = one_one(real) )
    <=> ( ? [X6: 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),X6)),aa(num,real,numeral_numeral(real),bit0(one2)))),pi)
        | ? [X6: 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),X6)),aa(num,real,numeral_numeral(real),bit0(one2)))),pi)) ) ) ).

% cos_one_2pi
tff(fact_3155_VEBT__internal_OTb__T__vebt__buildupi_H_H,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_V441764108873111860ildupi(Na)),aa(nat,nat,minus_minus(nat,vEBT_VEBT_Tb2(Na)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% VEBT_internal.Tb_T_vebt_buildupi''
tff(fact_3156_even__flip__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Aa2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),bit_se8732182000553998342ip_bit(A,M,Aa2))
        <=> ~ ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2)
            <=> ( M = zero_zero(nat) ) ) ) ) ).

% even_flip_bit_iff
tff(fact_3157_sin__pi__divide__n__gt__0,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(nat,real,semiring_1_of_nat(real),Na)))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_3158_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,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_V441764108873111860ildupi(Na)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit1(bit0(one2))))),U)) ) ).

% VEBT_internal.T_vebt_buildupi_univ
tff(fact_3159_VEBT__internal_OTb__T__vebt__buildupi,axiom,
    ! [Na: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),vEBT_V441764108873111860ildupi(Na))),aa(int,int,minus_minus(int,vEBT_VEBT_Tb(Na)),aa(num,int,numeral_numeral(int),bit0(one2)))) ).

% VEBT_internal.Tb_T_vebt_buildupi
tff(fact_3160_sin__zero__lemma,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$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_3161_sin__zero__iff,axiom,
    ! [X: real] :
      ( ( sin(real,X) = zero_zero(real) )
    <=> ( ? [N2: nat] :
            ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N2)
            & ( X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N2)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) )
        | ? [N2: nat] :
            ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N2)
            & ( 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),N2)),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_3162_cos__zero__lemma,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$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_3163_cos__zero__iff,axiom,
    ! [X: real] :
      ( ( cos(real,X) = zero_zero(real) )
    <=> ( ? [N2: nat] :
            ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N2)
            & ( X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N2)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) )
        | ? [N2: nat] :
            ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N2)
            & ( 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),N2)),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_3164_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,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),aa(nat,nat,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,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),aa(nat,nat,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_3165_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_3166_arcosh__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [X: A] : aa(A,A,arcosh(A),X) = aa(A,A,ln_ln(A),aa(A,A,aa(A,fun(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_3167_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_3168_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( real_V7770717601297561774m_norm(complex,Z) = one_one(real) )
     => ~ ! [T2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T2)
           => ( aa(real,$o,aa(real,fun(real,$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_3169_geometric__deriv__sums,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),one_one(real))
         => aa(A,$o,sums(A,aTP_Lamp_ce(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_3170_summable__complex__of__real,axiom,
    ! [F2: fun(nat,real)] :
      ( summable(complex,aTP_Lamp_cf(fun(nat,real),fun(nat,complex),F2))
    <=> summable(real,F2) ) ).

% summable_complex_of_real
tff(fact_3171_tan__periodic__pi,axiom,
    ! [X: real] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),pi)) = aa(real,real,tan(real),X) ).

% tan_periodic_pi
tff(fact_3172_sums__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => aa(A,$o,sums(A,aTP_Lamp_bf(nat,A)),zero_zero(A)) ) ).

% sums_zero
tff(fact_3173_triangle__0,axiom,
    nat_triangle(zero_zero(nat)) = zero_zero(nat) ).

% triangle_0
tff(fact_3174_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_3175_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_3176_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_3177_of__real__mult,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [X: real,Y: real] : aa(real,A,real_Vector_of_real(A),aa(real,real,aa(real,fun(real,real),times_times(real),X),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(real,A,real_Vector_of_real(A),X)),aa(real,A,real_Vector_of_real(A),Y)) ) ).

% of_real_mult
tff(fact_3178_of__real__add,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [X: real,Y: real] : aa(real,A,real_Vector_of_real(A),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(real,A,real_Vector_of_real(A),X)),aa(real,A,real_Vector_of_real(A),Y)) ) ).

% of_real_add
tff(fact_3179_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_3180_triangle__Suc,axiom,
    ! [Na: nat] : nat_triangle(aa(nat,nat,suc,Na)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(Na)),aa(nat,nat,suc,Na)) ).

% triangle_Suc
tff(fact_3181_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_3182_tan__periodic__n,axiom,
    ! [X: real,Na: num] : aa(real,real,tan(real),aa(real,real,aa(real,fun(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_3183_tan__periodic__nat,axiom,
    ! [X: real,Na: nat] : aa(real,real,tan(real),aa(real,real,aa(real,fun(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_3184_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_3185_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_3186_powser__sums__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Aa2: fun(nat,A),X: A] :
          ( aa(A,$o,sums(A,aTP_Lamp_bw(fun(nat,A),fun(nat,A),Aa2)),X)
        <=> ( aa(nat,A,Aa2,zero_zero(nat)) = X ) ) ) ).

% powser_sums_zero_iff
tff(fact_3187_tan__periodic,axiom,
    ! [X: real] : aa(real,real,tan(real),aa(real,real,aa(real,fun(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_3188_norm__of__real__addn,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: real,Ba: num] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(real,A,real_Vector_of_real(A),X)),aa(num,A,numeral_numeral(A),Ba))) = aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(num,real,numeral_numeral(real),Ba))) ) ).

% norm_of_real_addn
tff(fact_3189_norm__of__real__add1,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: real] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(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,aa(real,fun(real,real),plus_plus(real),X),one_one(real))) ) ).

% norm_of_real_add1
tff(fact_3190_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_3191_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_3192_Complex__add__complex__of__real,axiom,
    ! [X: real,Y: real,R: real] : aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),complex2(X,Y)),aa(real,complex,real_Vector_of_real(complex),R)) = complex2(aa(real,real,aa(real,fun(real,real),plus_plus(real),X),R),Y) ).

% Complex_add_complex_of_real
tff(fact_3193_complex__of__real__add__Complex,axiom,
    ! [R: real,X: real,Y: real] : aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),aa(real,complex,real_Vector_of_real(complex),R)),complex2(X,Y)) = complex2(aa(real,real,aa(real,fun(real,real),plus_plus(real),R),X),Y) ).

% complex_of_real_add_Complex
tff(fact_3194_sums__le,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),G: fun(nat,A),S: A,T: A] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,N)),aa(nat,A,G,N))
         => ( aa(A,$o,sums(A,F2),S)
           => ( aa(A,$o,sums(A,G),T)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),S),T) ) ) ) ) ).

% sums_le
tff(fact_3195_sums__of__real__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,real),C2: real] :
          ( aa(A,$o,sums(A,aTP_Lamp_cg(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_3196_sums__of__real,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => ! [X5: fun(nat,real),Aa2: real] :
          ( aa(real,$o,sums(real,X5),Aa2)
         => aa(A,$o,sums(A,aTP_Lamp_ch(fun(nat,real),fun(nat,A),X5)),aa(real,A,real_Vector_of_real(A),Aa2)) ) ) ).

% sums_of_real
tff(fact_3197_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_bg(nat,fun(fun(nat,A),fun(nat,A)),I),F2)),aa(nat,A,F2,I)) ) ).

% sums_single
tff(fact_3198_sums__add,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F2: fun(nat,A),Aa2: A,G: fun(nat,A),Ba: A] :
          ( aa(A,$o,sums(A,F2),Aa2)
         => ( aa(A,$o,sums(A,G),Ba)
           => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_br(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)) ) ) ) ).

% sums_add
tff(fact_3199_sums__mult2,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),Aa2: A,C2: A] :
          ( aa(A,$o,sums(A,F2),Aa2)
         => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_bm(fun(nat,A),fun(A,fun(nat,A)),F2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),C2)) ) ) ).

% sums_mult2
tff(fact_3200_sums__mult,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),Aa2: A,C2: A] :
          ( aa(A,$o,sums(A,F2),Aa2)
         => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_bn(fun(nat,A),fun(A,fun(nat,A)),F2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Aa2)) ) ) ).

% sums_mult
tff(fact_3201_sums__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),Aa2: A,G: fun(nat,A),Ba: A] :
          ( aa(A,$o,sums(A,F2),Aa2)
         => ( aa(A,$o,sums(A,G),Ba)
           => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bq(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)),aa(A,A,minus_minus(A,Aa2),Ba)) ) ) ) ).

% sums_diff
tff(fact_3202_sums__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(nat,A),Aa2: A,C2: A] :
          ( aa(A,$o,sums(A,F2),Aa2)
         => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_bj(fun(nat,A),fun(A,fun(nat,A)),F2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),C2)) ) ) ).

% sums_divide
tff(fact_3203_sums__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),Aa2: A] :
          ( aa(A,$o,sums(A,F2),Aa2)
         => aa(A,$o,sums(A,aTP_Lamp_bo(fun(nat,A),fun(nat,A),F2)),aa(A,A,uminus_uminus(A),Aa2)) ) ) ).

% sums_minus
tff(fact_3204_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_ci(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_3205_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_cj(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_3206_Complex__eq__numeral,axiom,
    ! [Aa2: real,Ba: real,W: num] :
      ( ( complex2(Aa2,Ba) = aa(num,complex,numeral_numeral(complex),W) )
    <=> ( ( Aa2 = aa(num,real,numeral_numeral(real),W) )
        & ( Ba = zero_zero(real) ) ) ) ).

% Complex_eq_numeral
tff(fact_3207_complex__add,axiom,
    ! [Aa2: real,Ba: real,C2: real,D2: real] : aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),complex2(Aa2,Ba)),complex2(C2,D2)) = complex2(aa(real,real,aa(real,fun(real,real),plus_plus(real),Aa2),C2),aa(real,real,aa(real,fun(real,real),plus_plus(real),Ba),D2)) ).

% complex_add
tff(fact_3208_summable__of__real,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => ! [X5: fun(nat,real)] :
          ( summable(real,X5)
         => summable(A,aTP_Lamp_ch(fun(nat,real),fun(nat,A),X5)) ) ) ).

% summable_of_real
tff(fact_3209_sums__mult__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,F2: fun(nat,A),Aa2: A] :
          ( aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bi(A,fun(fun(nat,A),fun(nat,A)),C2),F2)),Aa2)
         => ( ( C2 != zero_zero(A) )
           => aa(A,$o,sums(A,F2),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),C2)) ) ) ) ).

% sums_mult_D
tff(fact_3210_sums__Suc__imp,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),S: A] :
          ( ( aa(nat,A,F2,zero_zero(nat)) = zero_zero(A) )
         => ( aa(A,$o,sums(A,aTP_Lamp_bl(fun(nat,A),fun(nat,A),F2)),S)
           => aa(A,$o,sums(A,F2),S) ) ) ) ).

% sums_Suc_imp
tff(fact_3211_sums__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),S: A] :
          ( aa(A,$o,sums(A,aTP_Lamp_bl(fun(nat,A),fun(nat,A),F2)),S)
        <=> aa(A,$o,sums(A,F2),aa(A,A,aa(A,fun(A,A),plus_plus(A),S),aa(nat,A,F2,zero_zero(nat)))) ) ) ).

% sums_Suc_iff
tff(fact_3212_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_ck(fun(nat,A),fun(nat,A),F2)),L)
         => aa(A,$o,sums(A,F2),aa(A,A,aa(A,fun(A,A),plus_plus(A),L),aa(nat,A,F2,zero_zero(nat)))) ) ) ).

% sums_Suc
tff(fact_3213_sums__zero__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Na: nat,F2: fun(nat,A),S: A] :
          ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Na)
             => ( aa(nat,A,F2,I2) = zero_zero(A) ) )
         => ( aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cl(nat,fun(fun(nat,A),fun(nat,A)),Na),F2)),S)
          <=> aa(A,$o,sums(A,F2),S) ) ) ) ).

% sums_zero_iff_shift
tff(fact_3214_Complex__eq__neg__numeral,axiom,
    ! [Aa2: real,Ba: real,W: num] :
      ( ( complex2(Aa2,Ba) = aa(complex,complex,uminus_uminus(complex),aa(num,complex,numeral_numeral(complex),W)) )
    <=> ( ( Aa2 = aa(real,real,uminus_uminus(real),aa(num,real,numeral_numeral(real),W)) )
        & ( Ba = zero_zero(real) ) ) ) ).

% Complex_eq_neg_numeral
tff(fact_3215_complex__mult,axiom,
    ! [Aa2: real,Ba: real,C2: real,D2: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),complex2(Aa2,Ba)),complex2(C2,D2)) = complex2(aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),times_times(real),Aa2),C2)),aa(real,real,aa(real,fun(real,real),times_times(real),Ba),D2)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),Aa2),D2)),aa(real,real,aa(real,fun(real,real),times_times(real),Ba),C2))) ).

% complex_mult
tff(fact_3216_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_ch(fun(nat,real),fun(nat,A),X5)) ) ) ) ).

% suminf_of_real
tff(fact_3217_norm__less__p1,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [X: A] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(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_3218_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_cm(nat,fun(A,fun(nat,A)),M),Z)),aa(nat,A,power_power(A,Z),M)) ) ).

% powser_sums_if
tff(fact_3219_powser__sums__zero,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Aa2: fun(nat,A)] : aa(A,$o,sums(A,aTP_Lamp_bw(fun(nat,A),fun(nat,A),Aa2)),aa(nat,A,Aa2,zero_zero(nat))) ) ).

% powser_sums_zero
tff(fact_3220_norm__of__real__diff,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Ba: real,Aa2: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(real,A,real_Vector_of_real(A),Ba)),aa(real,A,real_Vector_of_real(A),Aa2)))),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,Ba),Aa2))) ) ).

% norm_of_real_diff
tff(fact_3221_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_3222_geometric__sums,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( aa(real,$o,aa(real,fun(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_3223_power__half__series,axiom,
    aa(real,$o,sums(real,aTP_Lamp_cn(nat,real)),one_one(real)) ).

% power_half_series
tff(fact_3224_tan__gt__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,tan(real),X)) ) ) ).

% tan_gt_zero
tff(fact_3225_lemma__tan__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
     => ? [X2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X2)
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X2),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,tan(real),X2)) ) ) ).

% lemma_tan_total
tff(fact_3226_tan__total,axiom,
    ! [Y: real] :
    ? [X2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X2)
      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X2),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),X2) = Y )
      & ! [Y3: real] :
          ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y3)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y3),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),Y3) = Y ) )
         => ( Y3 = X2 ) ) ) ).

% tan_total
tff(fact_3227_tan__monotone,axiom,
    ! [Y: real,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tan(real),Y)),aa(real,real,tan(real),X)) ) ) ) ).

% tan_monotone
tff(fact_3228_tan__monotone_H,axiom,
    ! [Y: real,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),X)
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tan(real),Y)),aa(real,real,tan(real),X)) ) ) ) ) ) ).

% tan_monotone'
tff(fact_3229_tan__mono__lt__eq,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tan(real),X)),aa(real,real,tan(real),Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),Y) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_3230_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X2)
      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X2),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),X2) = Y ) ) ).

% lemma_tan_total1
tff(fact_3231_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_3232_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_3233_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,aa(A,fun(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,aa(A,fun(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_3234_sums__if_H,axiom,
    ! [G: fun(nat,real),X: real] :
      ( aa(real,$o,sums(real,G),X)
     => aa(real,$o,sums(real,aTP_Lamp_co(fun(nat,real),fun(nat,real),G)),X) ) ).

% sums_if'
tff(fact_3235_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_cp(fun(nat,real),fun(fun(nat,real),fun(nat,real)),G),F2)),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y)) ) ) ).

% sums_if
tff(fact_3236_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_3237_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_3238_tan__total__pos,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
     => ? [X2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X2)
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X2),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),X2) = Y ) ) ) ).

% tan_total_pos
tff(fact_3239_tan__pos__pi2__le,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,tan(real),X)) ) ) ).

% tan_pos_pi2_le
tff(fact_3240_tan__less__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),zero_zero(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tan(real),X)),zero_zero(real)) ) ) ).

% tan_less_zero
tff(fact_3241_tan__mono__le,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,tan(real),X)),aa(real,real,tan(real),Y)) ) ) ) ).

% tan_mono_le
tff(fact_3242_tan__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,tan(real),X)),aa(real,real,tan(real),Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_3243_tan__bound__pi2,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$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,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,tan(real),X))),one_one(real)) ) ).

% tan_bound_pi2
tff(fact_3244_arctan,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arctan,Y))
      & aa(real,$o,aa(real,fun(real,$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_3245_arctan__tan,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,real,arctan,aa(real,real,tan(real),X)) = X ) ) ) ).

% arctan_tan
tff(fact_3246_arctan__unique,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( ( aa(real,real,tan(real),X) = Y )
         => ( aa(real,real,arctan,Y) = X ) ) ) ) ).

% arctan_unique
tff(fact_3247_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,aa(A,fun(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_3248_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,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),X)),aa(A,A,tan(A),Y)))) ) ) ) ) ) ).

% tan_diff
tff(fact_3249_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,aa(A,fun(A,A),plus_plus(A),X),Y)) != zero_zero(A) )
             => ( aa(A,A,tan(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),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_3250_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,aa(A,fun(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_3251_tan__total__pi4,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),X)),one_one(real))
     => ? [Z3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))))),Z3)
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z3),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))))
          & ( aa(real,real,tan(real),Z3) = X ) ) ) ).

% tan_total_pi4
tff(fact_3252_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,aa(A,fun(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_3253_arsinh__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [X: A] : aa(A,A,arsinh(A),X) = aa(A,A,ln_ln(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),powr(A,aa(A,A,aa(A,fun(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_3254_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_3255_sin__paired,axiom,
    ! [X: real] : aa(real,$o,sums(real,aTP_Lamp_cs(real,fun(nat,real),X)),sin(real,X)) ).

% sin_paired
tff(fact_3256_sin__tan,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$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,aa(real,fun(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_3257_cos__tan,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$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,aa(real,fun(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_3258_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_3259_real__sqrt__le__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,X)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y) ) ).

% real_sqrt_le_iff
tff(fact_3260_fact__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,zero_zero(nat)) = one_one(A) ) ) ).

% fact_0
tff(fact_3261_fact__1,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,one_one(nat)) = one_one(A) ) ) ).

% fact_1
tff(fact_3262_real__sqrt__le__0__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,X)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),zero_zero(real)) ) ).

% real_sqrt_le_0_iff
tff(fact_3263_real__sqrt__ge__0__iff,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y) ) ).

% real_sqrt_ge_0_iff
tff(fact_3264_real__sqrt__le__1__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,X)),one_one(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),one_one(real)) ) ).

% real_sqrt_le_1_iff
tff(fact_3265_real__sqrt__ge__1__iff,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Y) ) ).

% real_sqrt_ge_1_iff
tff(fact_3266_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_3267_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_3268_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_3269_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_3270_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_3271_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,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X) ) ).

% real_sqrt_pow2_iff
tff(fact_3272_real__sqrt__pow2,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$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_3273_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,aa(real,fun(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,aa(real,fun(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,aa(real,fun(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,aa(real,fun(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_3274_diffs__of__real,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [F2: fun(nat,real),X4: nat] : aa(nat,A,diffs(A,aTP_Lamp_ct(fun(nat,real),fun(nat,A),F2)),X4) = aa(real,A,real_Vector_of_real(A),aa(nat,real,diffs(real,F2),X4)) ) ).

% diffs_of_real
tff(fact_3275_real__sqrt__le__mono,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,X)),aa(real,real,sqrt,Y)) ) ).

% real_sqrt_le_mono
tff(fact_3276_real__sqrt__ge__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,sqrt,X)) ) ).

% real_sqrt_ge_zero
tff(fact_3277_real__sqrt__eq__zero__cancel,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$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_3278_real__sqrt__ge__one,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),X)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(real,real,sqrt,X)) ) ).

% real_sqrt_ge_one
tff(fact_3279_fact__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),semiring_char_0_fact(A,Na)) ) ).

% fact_ge_zero
tff(fact_3280_fact__ge__1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),semiring_char_0_fact(A,Na)) ) ).

% fact_ge_1
tff(fact_3281_fact__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),semiring_char_0_fact(A,M)),semiring_char_0_fact(A,Na)) ) ) ).

% fact_mono
tff(fact_3282_diffs__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [C2: fun(nat,A),X4: nat] : aa(nat,A,diffs(A,aTP_Lamp_cu(fun(nat,A),fun(nat,A),C2)),X4) = aa(A,A,uminus_uminus(A),aa(nat,A,diffs(A,C2),X4)) ) ).

% diffs_minus
tff(fact_3283_fact__dvd,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat,M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
         => aa(A,$o,dvd_dvd(A,semiring_char_0_fact(A,Na)),semiring_char_0_fact(A,M)) ) ) ).

% fact_dvd
tff(fact_3284_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_3285_real__div__sqrt,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$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_3286_sqrt__add__le__add__sqrt,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,sqrt,X)),aa(real,real,sqrt,Y))) ) ) ).

% sqrt_add_le_add_sqrt
tff(fact_3287_le__real__sqrt__sumsq,axiom,
    ! [X: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),X),X)),aa(real,real,aa(real,fun(real,real),times_times(real),Y),Y)))) ).

% le_real_sqrt_sumsq
tff(fact_3288_fact__mod,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom(A)
        & semidom_modulo(A) )
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),semiring_char_0_fact(A,Na)),semiring_char_0_fact(A,M)) = zero_zero(A) ) ) ) ).

% fact_mod
tff(fact_3289_fact__fact__dvd__fact,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: nat,Na: nat] : aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,Na))),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Na))) ) ).

% fact_fact_dvd_fact
tff(fact_3290_fact__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),semiring_char_0_fact(A,Na)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,power_power(nat,Na),Na))) ) ).

% fact_le_power
tff(fact_3291_sqrt2__less__2,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2)))),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% sqrt2_less_2
tff(fact_3292_arsinh__real__def,axiom,
    ! [X: real] : aa(real,real,arsinh(real),X) = aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,sqrt,aa(real,real,aa(real,fun(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_3293_choose__dvd,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
         => aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,aa(nat,nat,minus_minus(nat,Na),K)))),semiring_char_0_fact(A,Na)) ) ) ).

% choose_dvd
tff(fact_3294_diffs__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [C2: fun(nat,A),X4: nat] : aa(nat,A,diffs(A,C2),X4) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,X4))),aa(nat,A,C2,aa(nat,nat,suc,X4))) ) ).

% diffs_def
tff(fact_3295_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_3296_real__less__rsqrt,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(real,real,sqrt,Y)) ) ).

% real_less_rsqrt
tff(fact_3297_real__le__rsqrt,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(real,real,sqrt,Y)) ) ).

% real_le_rsqrt
tff(fact_3298_sqrt__le__D,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,X)),Y)
     => aa(real,$o,aa(real,fun(real,$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_3299_termdiff__converges__all,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),X: A] :
          ( ! [X2: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_cv(fun(nat,A),fun(A,fun(nat,A)),C2),X2))
         => 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_3300_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_3301_square__fact__le__2__fact,axiom,
    ! [Na: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),times_times(real),semiring_char_0_fact(real,Na)),semiring_char_0_fact(real,Na))),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ).

% square_fact_le_2_fact
tff(fact_3302_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,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( aa(real,real,sqrt,X) = Y ) ) ) ).

% real_sqrt_unique
tff(fact_3303_real__le__lsqrt,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,X)),Y) ) ) ) ).

% real_le_lsqrt
tff(fact_3304_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),U)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),U),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2))))),U) ) ).

% lemma_real_divide_sqrt_less
tff(fact_3305_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [X: real,Y: real] :
      ( ( aa(real,real,sqrt,aa(real,real,aa(real,fun(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_3306_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [X: real,Y: real] :
      ( ( aa(real,real,sqrt,aa(real,real,aa(real,fun(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_3307_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [Aa2: real,C2: real,Ba: real,D2: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Aa2),C2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Ba),D2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,Aa2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Ba),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,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_3308_real__sqrt__sum__squares__ge2,axiom,
    ! [Y: real,X: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,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_3309_real__sqrt__sum__squares__ge1,axiom,
    ! [X: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(real,real,sqrt,aa(real,real,aa(real,fun(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_3310_sqrt__ge__absD,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),X)),aa(real,real,sqrt,Y))
     => aa(real,$o,aa(real,fun(real,$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_3311_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_3312_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_3313_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_3314_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_3315_fact__reduce,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( semiring_char_0_fact(A,Na) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),semiring_char_0_fact(A,aa(nat,nat,minus_minus(nat,Na),one_one(nat)))) ) ) ) ).

% fact_reduce
tff(fact_3316_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_3317_fact__binomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),aa(nat,A,semiring_1_of_nat(A),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_3318_binomial__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
         => ( aa(nat,A,semiring_1_of_nat(A),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_3319_real__less__lsqrt,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,X)),Y) ) ) ) ).

% real_less_lsqrt
tff(fact_3320_sqrt__sum__squares__le__sum,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,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,aa(real,fun(real,real),plus_plus(real),X),Y)) ) ) ).

% sqrt_sum_squares_le_sum
tff(fact_3321_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_3322_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_3323_real__sqrt__ge__abs1,axiom,
    ! [X: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),X)),aa(real,real,sqrt,aa(real,real,aa(real,fun(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_3324_real__sqrt__ge__abs2,axiom,
    ! [Y: real,X: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,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_3325_sqrt__sum__squares__le__sum__abs,axiom,
    ! [X: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,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,aa(real,fun(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_3326_ln__sqrt,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$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_3327_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_3328_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_3329_complex__norm,axiom,
    ! [X: real,Y: real] : real_V7770717601297561774m_norm(complex,complex2(X,Y)) = aa(real,real,sqrt,aa(real,real,aa(real,fun(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_3330_termdiff__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,K3: real,C2: fun(nat,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X)),K3)
         => ( ! [X2: A] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X2)),K3)
               => summable(A,aa(A,fun(nat,A),aTP_Lamp_cv(fun(nat,A),fun(A,fun(nat,A)),C2),X2)) )
           => 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_3331_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,aa(real,fun(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_3332_arsinh__real__aux,axiom,
    ! [X: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,sqrt,aa(real,real,aa(real,fun(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_3333_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X: real,Y: real,Xa2: real,Ya: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,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,aa(real,fun(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_3334_arith__geo__mean__sqrt,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),X),Y))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y)),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_3335_powr__half__sqrt,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$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_3336_cos__x__y__le__one,axiom,
    ! [X: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),X),aa(real,real,sqrt,aa(real,real,aa(real,fun(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_3337_real__sqrt__sum__squares__less,axiom,
    ! [X: real,U: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$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,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),U),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2)))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,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_3338_arcosh__real__def,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),X)
     => ( aa(real,real,arcosh(real),X) = aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(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_3339_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,aa(real,fun(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_3340_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,aa(real,fun(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_3341_sqrt__sum__squares__half__less,axiom,
    ! [X: real,U: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$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,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),U),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,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_3342_sin__cos__sqrt,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$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_3343_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,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) ).

% arctan_half
tff(fact_3344_cos__paired,axiom,
    ! [X: real] : aa(real,$o,sums(real,aTP_Lamp_cy(real,fun(nat,real),X)),cos(real,X)) ).

% cos_paired
tff(fact_3345_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_3346_sin__coeff__def,axiom,
    ! [X4: nat] :
      sin_coeff(X4) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),X4),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,X4),aa(nat,nat,suc,zero_zero(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),semiring_char_0_fact(real,X4))) ).

% sin_coeff_def
tff(fact_3347_cos__coeff__def,axiom,
    ! [X4: nat] :
      cos_coeff(X4) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),X4),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),X4),aa(num,nat,numeral_numeral(nat),bit0(one2))))),semiring_char_0_fact(real,X4)),zero_zero(real)) ).

% cos_coeff_def
tff(fact_3348_cos__arcsin,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,aa(real,fun(real,$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_3349_sin__arccos__abs,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))
     => ( sin(real,aa(real,real,arccos,Y)) = aa(real,real,sqrt,aa(real,real,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_3350_sin__arccos,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,aa(real,fun(real,$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_3351_cos__arccos,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ( cos(real,aa(real,real,arccos,Y)) = Y ) ) ) ).

% cos_arccos
tff(fact_3352_sin__arcsin,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ( sin(real,aa(real,real,arcsin,Y)) = Y ) ) ) ).

% sin_arcsin
tff(fact_3353_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_3354_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_3355_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_3356_fact__mono__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,Na)) ) ).

% fact_mono_nat
tff(fact_3357_fact__ge__self,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),semiring_char_0_fact(nat,Na)) ).

% fact_ge_self
tff(fact_3358_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_3359_fact__ge__Suc__0__nat,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),semiring_char_0_fact(nat,Na)) ).

% fact_ge_Suc_0_nat
tff(fact_3360_dvd__fact,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
       => aa(nat,$o,dvd_dvd(nat,M),semiring_char_0_fact(nat,Na)) ) ) ).

% dvd_fact
tff(fact_3361_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_3362_arccos__le__arccos,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arccos,Y)),aa(real,real,arccos,X)) ) ) ) ).

% arccos_le_arccos
tff(fact_3363_arccos__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real))
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real)) )
     => ( ( aa(real,real,arccos,X) = aa(real,real,arccos,Y) )
      <=> ( X = Y ) ) ) ).

% arccos_eq_iff
tff(fact_3364_arccos__le__mono,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arccos,X)),aa(real,real,arccos,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),X) ) ) ) ).

% arccos_le_mono
tff(fact_3365_arcsin__minus,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,aa(real,fun(real,$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_3366_arcsin__le__arcsin,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arcsin,X)),aa(real,real,arcsin,Y)) ) ) ) ).

% arcsin_le_arcsin
tff(fact_3367_arcsin__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( ( aa(real,real,arcsin,X) = aa(real,real,arcsin,Y) )
        <=> ( X = Y ) ) ) ) ).

% arcsin_eq_iff
tff(fact_3368_arcsin__le__mono,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arcsin,X)),aa(real,real,arcsin,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y) ) ) ) ).

% arcsin_le_mono
tff(fact_3369_fact__diff__Suc,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,suc,M))
     => ( semiring_char_0_fact(nat,aa(nat,nat,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_3370_fact__div__fact__le__pow,axiom,
    ! [R: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),R),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,Na)),semiring_char_0_fact(nat,aa(nat,nat,minus_minus(nat,Na),R)))),aa(nat,nat,power_power(nat,Na),R)) ) ).

% fact_div_fact_le_pow
tff(fact_3371_binomial__fact__lemma,axiom,
    ! [K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,K)),semiring_char_0_fact(nat,aa(nat,nat,minus_minus(nat,Na),K)))),aa(nat,nat,binomial(Na),K)) = semiring_char_0_fact(nat,Na) ) ) ).

% binomial_fact_lemma
tff(fact_3372_arccos__lbound,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,arccos,Y)) ) ) ).

% arccos_lbound
tff(fact_3373_arccos__less__arccos,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arccos,Y)),aa(real,real,arccos,X)) ) ) ) ).

% arccos_less_arccos
tff(fact_3374_arccos__less__mono,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arccos,X)),aa(real,real,arccos,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),X) ) ) ) ).

% arccos_less_mono
tff(fact_3375_arccos__ubound,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arccos,Y)),pi) ) ) ).

% arccos_ubound
tff(fact_3376_arccos__cos,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),pi)
       => ( aa(real,real,arccos,cos(real,X)) = X ) ) ) ).

% arccos_cos
tff(fact_3377_arcsin__less__arcsin,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arcsin,X)),aa(real,real,arcsin,Y)) ) ) ) ).

% arcsin_less_arcsin
tff(fact_3378_arcsin__less__mono,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arcsin,X)),aa(real,real,arcsin,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),Y) ) ) ) ).

% arcsin_less_mono
tff(fact_3379_cos__arccos__abs,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))
     => ( cos(real,aa(real,real,arccos,Y)) = Y ) ) ).

% cos_arccos_abs
tff(fact_3380_binomial__altdef__nat,axiom,
    ! [K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
     => ( aa(nat,nat,binomial(Na),K) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,Na)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,K)),semiring_char_0_fact(nat,aa(nat,nat,minus_minus(nat,Na),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_3381_arccos__cos__eq__abs,axiom,
    ! [Theta: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Theta)),pi)
     => ( aa(real,real,arccos,cos(real,Theta)) = aa(real,real,abs_abs(real),Theta) ) ) ).

% arccos_cos_eq_abs
tff(fact_3382_arccos__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,arccos,Y))
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arccos,Y)),pi) ) ) ) ).

% arccos_bounded
tff(fact_3383_arccos__cos2,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),zero_zero(real))
     => ( aa(real,$o,aa(real,fun(real,$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_3384_arccos__minus,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,aa(real,fun(real,$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_3385_arccos,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,arccos,Y))
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arccos,Y)),pi)
          & ( cos(real,aa(real,real,arccos,Y)) = Y ) ) ) ) ).

% arccos
tff(fact_3386_arccos__minus__abs,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$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_3387_arccos__le__pi2,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arccos,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ).

% arccos_le_pi2
tff(fact_3388_arcsin__lt__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arcsin,Y))
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arcsin,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ).

% arcsin_lt_bounded
tff(fact_3389_arcsin__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arcsin,Y))
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arcsin,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ).

% arcsin_bounded
tff(fact_3390_arcsin__ubound,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arcsin,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ).

% arcsin_ubound
tff(fact_3391_arcsin__lbound,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arcsin,Y)) ) ) ).

% arcsin_lbound
tff(fact_3392_arcsin__sin,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,real,arcsin,sin(real,X)) = X ) ) ) ).

% arcsin_sin
tff(fact_3393_binomial__code,axiom,
    ! [Na: nat,K: nat] :
      aa(nat,nat,binomial(Na),K) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),K),
        zero_zero(nat),
        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)),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,aa(nat,fun(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_3394_arcsin,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arcsin,Y))
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arcsin,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
          & ( sin(real,aa(real,real,arcsin,Y)) = Y ) ) ) ) ).

% arcsin
tff(fact_3395_arcsin__pi,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arcsin,Y))
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arcsin,Y)),pi)
          & ( sin(real,aa(real,real,arcsin,Y)) = Y ) ) ) ) ).

% arcsin_pi
tff(fact_3396_arcsin__le__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arcsin,X)),Y)
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),sin(real,Y)) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_3397_le__arcsin__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,arcsin,X))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Y)),X) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_3398_monoI1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( ! [M5: nat,N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M5),N)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,M5)),aa(nat,A,X5,N)) )
         => topological_monoseq(A,X5) ) ) ).

% monoI1
tff(fact_3399_monoI2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( ! [M5: nat,N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M5),N)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N)),aa(nat,A,X5,M5)) )
         => topological_monoseq(A,X5) ) ) ).

% monoI2
tff(fact_3400_monoseq__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( topological_monoseq(A,X5)
        <=> ( ! [M4: nat,N2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M4),N2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,M4)),aa(nat,A,X5,N2)) )
            | ! [M4: nat,N2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M4),N2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N2)),aa(nat,A,X5,M4)) ) ) ) ) ).

% monoseq_def
tff(fact_3401_monoseq__Suc,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( topological_monoseq(A,X5)
        <=> ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N2)),aa(nat,A,X5,aa(nat,nat,suc,N2)))
            | ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,aa(nat,nat,suc,N2))),aa(nat,A,X5,N2)) ) ) ) ).

% monoseq_Suc
tff(fact_3402_mono__SucI2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,aa(nat,nat,suc,N))),aa(nat,A,X5,N))
         => topological_monoseq(A,X5) ) ) ).

% mono_SucI2
tff(fact_3403_monoseq__minus,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Aa2: fun(nat,A)] :
          ( topological_monoseq(A,Aa2)
         => topological_monoseq(A,aTP_Lamp_cz(fun(nat,A),fun(nat,A),Aa2)) ) ) ).

% monoseq_minus
tff(fact_3404_mono__SucI1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N)),aa(nat,A,X5,aa(nat,nat,suc,N)))
         => topological_monoseq(A,X5) ) ) ).

% mono_SucI1
tff(fact_3405_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_3406_cot__less__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),zero_zero(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,cot(real),X)),zero_zero(real)) ) ) ).

% cot_less_zero
tff(fact_3407_sint__range__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,aa(word(A),nat,size_size(word(A)),W)),aa(nat,nat,suc,zero_zero(nat)))))),aa(word(A),int,ring_1_signed(A,int),W))
          & aa(int,$o,aa(int,fun(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,aa(word(A),nat,size_size(word(A)),W)),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).

% sint_range_size
tff(fact_3408_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,aa(int,fun(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_3409_gbinomial__code,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Aa2: A,K: nat] :
          aa(nat,A,gbinomial(A,Aa2),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_da(A,fun(nat,fun(A,A)),Aa2),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_3410_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_3411_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_3412_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_3413_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_3414_of__int__floor__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X)) = X )
        <=> ? [N2: int] : X = aa(int,A,ring_1_of_int(A),N2) ) ) ).

% of_int_floor_cancel
tff(fact_3415_of__int__ceiling__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X)) = X )
        <=> ? [N2: int] : X = aa(int,A,ring_1_of_int(A),N2) ) ) ).

% of_int_ceiling_cancel
tff(fact_3416_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_3417_gbinomial__1,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [Aa2: A] : aa(nat,A,gbinomial(A,Aa2),one_one(nat)) = Aa2 ) ).

% gbinomial_1
tff(fact_3418_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_3419_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_3420_max__number__of_I1_J,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ord(A) )
     => ! [U: num,V: 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),V)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)),aa(num,A,numeral_numeral(A),V),aa(num,A,numeral_numeral(A),U)) ) ).

% max_number_of(1)
tff(fact_3421_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_3422_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_3423_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_3424_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_3425_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_3426_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_3427_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_3428_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_3429_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_3430_of__int__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W: int,Z: int] :
          ( aa(A,$o,aa(A,fun(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,aa(int,fun(int,$o),ord_less_eq(int),W),Z) ) ) ).

% of_int_le_iff
tff(fact_3431_of__int__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W: int,Z: int] :
          ( aa(A,$o,aa(A,fun(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,aa(int,fun(int,$o),ord_less(int),W),Z) ) ) ).

% of_int_less_iff
tff(fact_3432_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_3433_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_3434_of__int__add,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [W: int,Z: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),W),Z)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),W)),aa(int,A,ring_1_of_int(A),Z)) ) ).

% of_int_add
tff(fact_3435_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_3436_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_3437_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_3438_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_3439_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_3440_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_3441_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Ba: int,W: nat,X: int] :
          ( ( aa(nat,A,power_power(A,aa(int,A,ring_1_of_int(A),Ba)),W) = aa(int,A,ring_1_of_int(A),X) )
        <=> ( aa(nat,int,power_power(int,Ba),W) = X ) ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_3442_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [X: int,Ba: 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),Ba)),W) )
        <=> ( X = aa(nat,int,power_power(int,Ba),W) ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_3443_gbinomial__0_I1_J,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [Aa2: A] : aa(nat,A,gbinomial(A,Aa2),zero_zero(nat)) = one_one(A) ) ).

% gbinomial_0(1)
tff(fact_3444_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_3445_Word_Oof__int__sint,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_1(A) )
     => ! [Aa2: word(B)] : aa(int,A,ring_1_of_int(A),aa(word(B),int,ring_1_signed(B,int),Aa2)) = aa(word(B),A,ring_1_signed(B,A),Aa2) ) ).

% Word.of_int_sint
tff(fact_3446_max__number__of_I4_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: 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),V))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))) ) ).

% max_number_of(4)
tff(fact_3447_max__number__of_I3_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: 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),V)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V)),aa(num,A,numeral_numeral(A),V),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))) ) ).

% max_number_of(3)
tff(fact_3448_max__number__of_I2_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: 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),V))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V)),aa(num,A,numeral_numeral(A),U)) ) ).

% max_number_of(2)
tff(fact_3449_ceiling__add__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z: int] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(int,A,ring_1_of_int(A),Z))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,X)),Z) ) ).

% ceiling_add_of_int
tff(fact_3450_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_3451_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_3452_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_3453_of__int__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),zero_zero(A))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),zero_zero(int)) ) ) ).

% of_int_le_0_iff
tff(fact_3454_of__int__0__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z) ) ) ).

% of_int_0_le_iff
tff(fact_3455_of__int__le__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int,Na: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),aa(num,A,numeral_numeral(A),Na))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),aa(num,int,numeral_numeral(int),Na)) ) ) ).

% of_int_le_numeral_iff
tff(fact_3456_of__int__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: num,Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Na)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),Na)),Z) ) ) ).

% of_int_numeral_le_iff
tff(fact_3457_of__int__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z)),zero_zero(A))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),zero_zero(int)) ) ) ).

% of_int_less_0_iff
tff(fact_3458_of__int__0__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z) ) ) ).

% of_int_0_less_iff
tff(fact_3459_of__int__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: num,Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),Na)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(num,int,numeral_numeral(int),Na)),Z) ) ) ).

% of_int_numeral_less_iff
tff(fact_3460_of__int__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int,Na: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z)),aa(num,A,numeral_numeral(A),Na))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),aa(num,int,numeral_numeral(int),Na)) ) ) ).

% of_int_less_numeral_iff
tff(fact_3461_of__int__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),one_one(int)) ) ) ).

% of_int_le_1_iff
tff(fact_3462_of__int__1__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),Z) ) ) ).

% of_int_1_le_iff
tff(fact_3463_of__int__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),one_one(int)) ) ) ).

% of_int_less_1_iff
tff(fact_3464_of__int__1__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),Z) ) ) ).

% of_int_1_less_iff
tff(fact_3465_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_3466_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_3467_of__int__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: int,Ba: int,W: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),X)),aa(nat,A,power_power(A,aa(int,A,ring_1_of_int(A),Ba)),W))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X),aa(nat,int,power_power(int,Ba),W)) ) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_3468_of__int__le__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Ba: int,W: nat,X: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,aa(int,A,ring_1_of_int(A),Ba)),W)),aa(int,A,ring_1_of_int(A),X))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,power_power(int,Ba),W)),X) ) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_3469_of__int__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: int,Ba: int,W: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),X)),aa(nat,A,power_power(A,aa(int,A,ring_1_of_int(A),Ba)),W))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),X),aa(nat,int,power_power(int,Ba),W)) ) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_3470_of__int__less__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Ba: int,W: nat,X: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,aa(int,A,ring_1_of_int(A),Ba)),W)),aa(int,A,ring_1_of_int(A),X))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,power_power(int,Ba),W)),X) ) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_3471_of__nat__nat,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,Z)) = aa(int,A,ring_1_of_int(A),Z) ) ) ) ).

% of_nat_nat
tff(fact_3472_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_3473_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_3474_tan__periodic__int,axiom,
    ! [X: real,I: int] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),I)),pi))) = aa(real,real,tan(real),X) ).

% tan_periodic_int
tff(fact_3475_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_3476_numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,Na: nat,Aa2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),X)),Na)),aa(int,A,ring_1_of_int(A),Aa2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),Na)),Aa2) ) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_3477_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Aa2: int,X: num,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Aa2)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),X)),Na))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Aa2),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),Na)) ) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_3478_numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,Na: nat,Aa2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),X)),Na)),aa(int,A,ring_1_of_int(A),Aa2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),Na)),Aa2) ) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_3479_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Aa2: int,X: num,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Aa2)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),X)),Na))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Aa2),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),Na)) ) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_3480_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_3481_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_3482_cot__periodic,axiom,
    ! [X: real] : aa(real,real,cot(real),aa(real,real,aa(real,fun(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_3483_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_3484_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_3485_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Aa2: int,X: num,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Aa2)),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),Na))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Aa2),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_3486_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,Na: nat,Aa2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),Na)),aa(int,A,ring_1_of_int(A),Aa2))
        <=> aa(int,$o,aa(int,fun(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)),Aa2) ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_3487_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,Na: nat,Aa2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),Na)),aa(int,A,ring_1_of_int(A),Aa2))
        <=> aa(int,$o,aa(int,fun(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)),Aa2) ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_3488_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Aa2: int,X: num,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Aa2)),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),Na))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Aa2),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_3489_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_3490_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_3491_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_3492_signed__word__eqI,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_char_0(A) )
     => ! [V: word(B),W: word(B)] :
          ( ( aa(word(B),A,ring_1_signed(B,A),V) = aa(word(B),A,ring_1_signed(B,A),W) )
         => ( V = W ) ) ) ).

% signed_word_eqI
tff(fact_3493_word__eq__iff__signed,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & ring_char_0(B) )
     => ! [V: word(A),W: word(A)] :
          ( ( V = W )
        <=> ( aa(word(A),B,ring_1_signed(A,B),V) = aa(word(A),B,ring_1_signed(A,B),W) ) ) ) ).

% word_eq_iff_signed
tff(fact_3494_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_3495_ex__of__int__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [Z3: int] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z3)),X) ) ).

% ex_of_int_less
tff(fact_3496_ex__less__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [Z3: int] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(int,A,ring_1_of_int(A),Z3)) ) ).

% ex_less_of_int
tff(fact_3497_ex__le__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [Z3: int] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),Z3)) ) ).

% ex_le_of_int
tff(fact_3498_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_3499_max__absorb2,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(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_3500_max__absorb1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(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_3501_max__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Aa2: A,Ba: A] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),Aa2),Ba) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba),Ba,Aa2) ) ).

% max_def
tff(fact_3502_max__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [X: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),Z) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),Z)) ) ).

% max_add_distrib_left
tff(fact_3503_max__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [X: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),ord_max(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z)) ) ).

% max_add_distrib_right
tff(fact_3504_max__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X4: A,Xa: A] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),X4),Xa) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Xa),Xa,X4) ) ).

% max_def_raw
tff(fact_3505_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_3506_of__int__floor__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(A,$o,aa(A,fun(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_3507_le__of__int__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(A,$o,aa(A,fun(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_3508_of__int__and__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int,L: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(int,A,ring_1_of_int(A),K)),aa(int,A,ring_1_of_int(A),L)) ) ).

% of_int_and_eq
tff(fact_3509_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_3510_of__int__not__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int] : aa(int,A,ring_1_of_int(A),aa(int,int,bit_ri4277139882892585799ns_not(int),K)) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(int,A,ring_1_of_int(A),K)) ) ).

% of_int_not_eq
tff(fact_3511_of__int__or__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int,L: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(int,A,ring_1_of_int(A),K)),aa(int,A,ring_1_of_int(A),L)) ) ).

% of_int_or_eq
tff(fact_3512_of__int__xor__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int,L: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(int,A,ring_1_of_int(A),K)),aa(int,A,ring_1_of_int(A),L)) ) ).

% of_int_xor_eq
tff(fact_3513_of__int__mask__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat] : aa(int,A,ring_1_of_int(A),bit_se2239418461657761734s_mask(int,Na)) = bit_se2239418461657761734s_mask(A,Na) ) ).

% of_int_mask_eq
tff(fact_3514_cos__int__times__real,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [M: int,X: real] : cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),M)),aa(real,A,real_Vector_of_real(A),X))) = aa(real,A,real_Vector_of_real(A),cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),M)),X))) ) ).

% cos_int_times_real
tff(fact_3515_sin__int__times__real,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [M: int,X: real] : sin(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),M)),aa(real,A,real_Vector_of_real(A),X))) = aa(real,A,real_Vector_of_real(A),sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),M)),X))) ) ).

% sin_int_times_real
tff(fact_3516_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_3517_signed__and__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_ri3973907225187159222ations(A) )
     => ! [V: 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)),V),W)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(word(B),A,ring_1_signed(B,A),V)),aa(word(B),A,ring_1_signed(B,A),W)) ) ).

% signed_and_eq
tff(fact_3518_signed__or__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_ri3973907225187159222ations(A) )
     => ! [V: 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)),V),W)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(word(B),A,ring_1_signed(B,A),V)),aa(word(B),A,ring_1_signed(B,A),W)) ) ).

% signed_or_eq
tff(fact_3519_signed__xor__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_ri3973907225187159222ations(A) )
     => ! [V: 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)),V),W)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(word(B),A,ring_1_signed(B,A),V)),aa(word(B),A,ring_1_signed(B,A),W)) ) ).

% signed_xor_eq
tff(fact_3520_gbinomial__Suc__Suc,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Aa2: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),one_one(A))),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,Aa2),K)),aa(nat,A,gbinomial(A,Aa2),aa(nat,nat,suc,K))) ) ).

% gbinomial_Suc_Suc
tff(fact_3521_le__floor__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),archim6421214686448440834_floor(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),X) ) ) ).

% le_floor_iff
tff(fact_3522_floor__less__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,X)),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(int,A,ring_1_of_int(A),Z)) ) ) ).

% floor_less_iff
tff(fact_3523_ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Aa2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),Aa2))
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,X)),Aa2) ) ) ).

% ceiling_le
tff(fact_3524_ceiling__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,X)),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),Z)) ) ) ).

% ceiling_le_iff
tff(fact_3525_less__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),archimedean_ceiling(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z)),X) ) ) ).

% less_ceiling_iff
tff(fact_3526_gbinomial__of__nat__symmetric,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
         => ( 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_3527_int__add__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] : aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),archim6421214686448440834_floor(A,X)) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z)),X)) ) ).

% int_add_floor
tff(fact_3528_floor__add__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),Z) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(int,A,ring_1_of_int(A),Z))) ) ).

% floor_add_int
tff(fact_3529_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_3530_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_3531_real__of__int__div4,axiom,
    ! [Na: int,X: int] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),divide_divide(int),Na),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_3532_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_3533_of__int__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z)) ) ) ).

% of_int_nonneg
tff(fact_3534_of__int__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z)) ) ) ).

% of_int_pos
tff(fact_3535_of__int__leD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: int,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),Na))),X)
         => ( ( Na = zero_zero(int) )
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),X) ) ) ) ).

% of_int_leD
tff(fact_3536_of__int__lessD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: int,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),Na))),X)
         => ( ( Na = zero_zero(int) )
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),X) ) ) ) ).

% of_int_lessD
tff(fact_3537_floor__exists1,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [X2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),X2)),X)
          & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),one_one(int))))
          & ! [Y3: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Y3)),X)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Y3),one_one(int)))) )
             => ( Y3 = X2 ) ) ) ) ).

% floor_exists1
tff(fact_3538_floor__exists,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [Z3: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z3)),X)
          & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z3),one_one(int)))) ) ) ).

% floor_exists
tff(fact_3539_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_3540_gbinomial__addition__formula,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Aa2: A,K: nat] : aa(nat,A,gbinomial(A,Aa2),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,Aa2),one_one(A))),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,Aa2),one_one(A))),K)) ) ).

% gbinomial_addition_formula
tff(fact_3541_of__int__ceiling__le__add__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,R))),aa(A,A,aa(A,fun(A,A),plus_plus(A),R),one_one(A))) ) ).

% of_int_ceiling_le_add_one
tff(fact_3542_of__int__ceiling__diff__one__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,R))),one_one(A))),R) ) ).

% of_int_ceiling_diff_one_le
tff(fact_3543_gbinomial__absorb__comp,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Aa2: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Aa2),aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,A,gbinomial(A,Aa2),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,Aa2),one_one(A))),K)) ) ).

% gbinomial_absorb_comp
tff(fact_3544_gbinomial__mult__1,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Aa2: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(nat,A,gbinomial(A,Aa2),K)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),K)),aa(nat,A,gbinomial(A,Aa2),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,Aa2),aa(nat,nat,suc,K)))) ) ).

% gbinomial_mult_1
tff(fact_3545_gbinomial__mult__1_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Aa2: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,Aa2),K)),Aa2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),K)),aa(nat,A,gbinomial(A,Aa2),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,Aa2),aa(nat,nat,suc,K)))) ) ).

% gbinomial_mult_1'
tff(fact_3546_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K: nat,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),K)),Aa2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(nat,A,semiring_1_of_nat(A),K))),K)),aa(nat,A,gbinomial(A,Aa2),K)) ) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_3547_of__nat__less__of__int__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat,X: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Na)),aa(int,A,ring_1_of_int(A),X))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),Na)),X) ) ) ).

% of_nat_less_of_int_iff
tff(fact_3548_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_3549_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_3550_int__le__real__less,axiom,
    ! [Na: int,M: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Na),M)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(int,real,ring_1_of_int(real),Na)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),M)),one_one(real))) ) ).

% int_le_real_less
tff(fact_3551_int__less__real__le,axiom,
    ! [Na: int,M: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Na),M)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),Na)),one_one(real))),aa(int,real,ring_1_of_int(real),M)) ) ).

% int_less_real_le
tff(fact_3552_ceiling__divide__eq__div,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Aa2: int,Ba: int] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(int,A,ring_1_of_int(A),Aa2)),aa(int,A,ring_1_of_int(A),Ba))) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),Aa2)),Ba)) ) ).

% ceiling_divide_eq_div
tff(fact_3553_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,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),one_one(int))) ) ).

% ceiling_altdef
tff(fact_3554_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,aa(real,fun(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_3555_real__of__int__floor__add__one__gt,axiom,
    ! [R: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),R),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R))),one_one(real))) ).

% real_of_int_floor_add_one_gt
tff(fact_3556_floor__eq,axiom,
    ! [Na: int,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(int,real,ring_1_of_int(real),Na)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),Na)),one_one(real)))
       => ( archim6421214686448440834_floor(real,X) = Na ) ) ) ).

% floor_eq
tff(fact_3557_real__of__int__floor__add__one__ge,axiom,
    ! [R: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),R),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R))),one_one(real))) ).

% real_of_int_floor_add_one_ge
tff(fact_3558_real__of__int__floor__gt__diff__one,axiom,
    ! [R: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,minus_minus(real,R),one_one(real))),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R))) ).

% real_of_int_floor_gt_diff_one
tff(fact_3559_real__of__int__floor__ge__diff__one,axiom,
    ! [R: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,minus_minus(real,R),one_one(real))),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R))) ).

% real_of_int_floor_ge_diff_one
tff(fact_3560_floor__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),X)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A)))
           => ( archim6421214686448440834_floor(A,X) = Z ) ) ) ) ).

% floor_unique
tff(fact_3561_floor__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Aa2: int] :
          ( ( archim6421214686448440834_floor(A,X) = Aa2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Aa2)),X)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Aa2)),one_one(A))) ) ) ) ).

% floor_eq_iff
tff(fact_3562_floor__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,$o),T: A] :
          ( aa(int,$o,P,archim6421214686448440834_floor(A,T))
        <=> ! [I4: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),I4)),T)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),T),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),I4)),one_one(A))) )
             => aa(int,$o,P,I4) ) ) ) ).

% floor_split
tff(fact_3563_ceiling__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(A,$o,aa(A,fun(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,aa(A,fun(A,$o),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X))) ) ) ).

% ceiling_correct
tff(fact_3564_ceiling__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] :
          ( aa(A,$o,aa(A,fun(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,aa(A,fun(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_3565_ceiling__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Aa2: int] :
          ( ( archimedean_ceiling(A,X) = Aa2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),Aa2)),one_one(A))),X)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),Aa2)) ) ) ) ).

% ceiling_eq_iff
tff(fact_3566_ceiling__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,$o),T: A] :
          ( aa(int,$o,P,archimedean_ceiling(A,T))
        <=> ! [I4: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),I4)),one_one(A))),T)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),T),aa(int,A,ring_1_of_int(A),I4)) )
             => aa(int,$o,P,I4) ) ) ) ).

% ceiling_split
tff(fact_3567_Suc__times__gbinomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Aa2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),one_one(A))),aa(nat,nat,suc,K))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),one_one(A))),aa(nat,A,gbinomial(A,Aa2),K)) ) ).

% Suc_times_gbinomial
tff(fact_3568_less__floor__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),archim6421214686448440834_floor(A,X))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))),X) ) ) ).

% less_floor_iff
tff(fact_3569_floor__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,X)),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))) ) ) ).

% floor_le_iff
tff(fact_3570_gbinomial__absorption,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Aa2: 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,Aa2),aa(nat,nat,suc,K))) = aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,Aa2),one_one(A))),K)) ) ).

% gbinomial_absorption
tff(fact_3571_ceiling__less__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,X)),Z)
        <=> aa(A,$o,aa(A,fun(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_3572_le__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),archimedean_ceiling(A,X))
        <=> aa(A,$o,aa(A,fun(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_3573_floor__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X))),X)
          & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),one_one(int)))) ) ) ).

% floor_correct
tff(fact_3574_gbinomial__trinomial__revision,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,M: nat,Aa2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,Aa2),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,Aa2),K)),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,Aa2),aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,nat,minus_minus(nat,M),K))) ) ) ) ).

% gbinomial_trinomial_revision
tff(fact_3575_real__of__int__div2,axiom,
    ! [Na: int,X: int] : aa(real,$o,aa(real,fun(real,$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_3576_real__of__int__div3,axiom,
    ! [Na: int,X: int] : aa(real,$o,aa(real,fun(real,$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_3577_floor__eq2,axiom,
    ! [Na: int,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(int,real,ring_1_of_int(real),Na)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),Na)),one_one(real)))
       => ( archim6421214686448440834_floor(real,X) = Na ) ) ) ).

% floor_eq2
tff(fact_3578_floor__divide__real__eq__div,axiom,
    ! [Ba: int,Aa2: real] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Ba)
     => ( archim6421214686448440834_floor(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),Aa2),aa(int,real,ring_1_of_int(real),Ba))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),archim6421214686448440834_floor(real,Aa2)),Ba) ) ) ).

% floor_divide_real_eq_div
tff(fact_3579_floor__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q3: A,P2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P2),Q3)))),Q3)),P2) ) ) ).

% floor_divide_lower
tff(fact_3580_ceiling__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q3: A,P2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),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_3581_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_3582_gbinomial__rec,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Aa2: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),one_one(A))),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,Aa2),K)),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),one_one(A))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K)))) ) ).

% gbinomial_rec
tff(fact_3583_gbinomial__factors,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Aa2: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),one_one(A))),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),one_one(A))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K)))),aa(nat,A,gbinomial(A,Aa2),K)) ) ).

% gbinomial_factors
tff(fact_3584_gbinomial__negated__upper,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Aa2: A,K: nat] : aa(nat,A,gbinomial(A,Aa2),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)),Aa2)),one_one(A))),K)) ) ).

% gbinomial_negated_upper
tff(fact_3585_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_3586_of__int__of__nat,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: int] :
          aa(int,A,ring_1_of_int(A),K) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)),aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),K)))),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,K))) ) ).

% of_int_of_nat
tff(fact_3587_floor__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q3: A,P2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),P2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P2),Q3)))),one_one(A))),Q3)) ) ) ).

% floor_divide_upper
tff(fact_3588_ceiling__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q3: A,P2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,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_3589_ceiling__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Na: int,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Na)),X)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Na)),one_one(A)))
           => ( archimedean_ceiling(A,X) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Na),one_one(int)) ) ) ) ) ).

% ceiling_eq
tff(fact_3590_gbinomial__minus,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Aa2: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,uminus_uminus(A),Aa2)),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,aa(A,fun(A,A),plus_plus(A),Aa2),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A))),K)) ) ).

% gbinomial_minus
tff(fact_3591_gbinomial__reduce__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Aa2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
         => ( aa(nat,A,gbinomial(A,Aa2),K) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,Aa2),one_one(A))),aa(nat,nat,minus_minus(nat,K),one_one(nat)))),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,Aa2),one_one(A))),K)) ) ) ) ).

% gbinomial_reduce_nat
tff(fact_3592_cos__one__2pi__int,axiom,
    ! [X: real] :
      ( ( cos(real,X) = one_one(real) )
    <=> ? [X6: 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),X6)),aa(num,real,numeral_numeral(real),bit0(one2)))),pi) ) ).

% cos_one_2pi_int
tff(fact_3593_gbinomial__pochhammer,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Aa2: A,K: nat] : aa(nat,A,gbinomial(A,Aa2),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),Aa2),K))),semiring_char_0_fact(A,K)) ) ).

% gbinomial_pochhammer
tff(fact_3594_gbinomial__pochhammer_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Aa2: A,K: nat] : aa(nat,A,gbinomial(A,Aa2),K) = aa(A,A,aa(A,fun(A,A),divide_divide(A),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,Aa2),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K)),semiring_char_0_fact(A,K)) ) ).

% gbinomial_pochhammer'
tff(fact_3595_sint__above__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),X: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,aa(word(A),nat,size_size(word(A)),W)),one_one(nat)))),X)
         => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,ring_1_signed(A,int),W)),X) ) ) ).

% sint_above_size
tff(fact_3596_sint__below__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: int,W: word(A)] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,aa(word(A),nat,size_size(word(A)),W)),one_one(nat)))))
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X),aa(word(A),int,ring_1_signed(A,int),W)) ) ) ).

% sint_below_size
tff(fact_3597_gbinomial__absorption_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Aa2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
         => ( aa(nat,A,gbinomial(A,Aa2),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,Aa2),one_one(A))),aa(nat,nat,minus_minus(nat,K),one_one(nat)))) ) ) ) ).

% gbinomial_absorption'
tff(fact_3598_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K5: 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),K5)),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_3599_floor__log__eq__powr__iff,axiom,
    ! [X: real,Ba: real,K: int] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Ba)
       => ( ( archim6421214686448440834_floor(real,aa(real,real,log(Ba),X)) = K )
        <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Ba,aa(int,real,ring_1_of_int(real),K))),X)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),powr(real,Ba,aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))))) ) ) ) ) ).

% floor_log_eq_powr_iff
tff(fact_3600_cot__gt__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,cot(real),X)) ) ) ).

% cot_gt_zero
tff(fact_3601_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_3602_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_3603_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_3604_powr__int,axiom,
    ! [X: real,I: int] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( powr(real,X,aa(int,real,ring_1_of_int(real),I)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I),aa(nat,real,power_power(real,X),aa(int,nat,nat2,I)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,power_power(real,X),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),I))))) ) ) ).

% powr_int
tff(fact_3605_of__int__code__if,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: int] :
          aa(int,A,ring_1_of_int(A),K) = $ite(
            K = zero_zero(int),
            zero_zero(A),
            $ite(
              aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)),
              aa(A,A,uminus_uminus(A),aa(int,A,ring_1_of_int(A),aa(int,int,uminus_uminus(int),K))),
              $let(
                l: A,
                l:= aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2))))),
                $ite(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),aa(num,int,numeral_numeral(int),bit0(one2))) = zero_zero(int),l,aa(A,A,aa(A,fun(A,A),plus_plus(A),l),one_one(A))) ) ) ) ) ).

% of_int_code_if
tff(fact_3606_VEBT__internal_Olog__ceil__idem,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$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_3607_max_Oabsorb1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),Aa2),Ba) = Aa2 ) ) ) ).

% max.absorb1
tff(fact_3608_max_Oabsorb2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),Aa2),Ba) = Ba ) ) ) ).

% max.absorb2
tff(fact_3609_max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ba: A,C2: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),Ba),C2)),Aa2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),Aa2) ) ) ) ).

% max.bounded_iff
tff(fact_3610_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_3611_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_3612_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_3613_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_3614_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_3615_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_3616_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_3617_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_3618_max__nat_Oright__neutral,axiom,
    ! [Aa2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Aa2),zero_zero(nat)) = Aa2 ).

% max_nat.right_neutral
tff(fact_3619_max__nat_Oneutr__eq__iff,axiom,
    ! [Aa2: nat,Ba: nat] :
      ( ( zero_zero(nat) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Aa2),Ba) )
    <=> ( ( Aa2 = zero_zero(nat) )
        & ( Ba = zero_zero(nat) ) ) ) ).

% max_nat.neutr_eq_iff
tff(fact_3620_max__nat_Oleft__neutral,axiom,
    ! [Aa2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),zero_zero(nat)),Aa2) = Aa2 ).

% max_nat.left_neutral
tff(fact_3621_max__nat_Oeq__neutr__iff,axiom,
    ! [Aa2: nat,Ba: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Aa2),Ba) = zero_zero(nat) )
    <=> ( ( Aa2 = zero_zero(nat) )
        & ( Ba = zero_zero(nat) ) ) ) ).

% max_nat.eq_neutr_iff
tff(fact_3622_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_3623_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_3624_word__no__log__defs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: 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)),Aa2)),aa(num,word(A),numeral_numeral(word(A)),Ba)) = 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),Aa2)),aa(num,int,numeral_numeral(int),Ba))) ) ).

% word_no_log_defs(3)
tff(fact_3625_word__no__log__defs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num] : aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(num,word(A),numeral_numeral(word(A)),Aa2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Aa2))) ) ).

% word_no_log_defs(1)
tff(fact_3626_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_3627_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_3628_word__no__log__defs_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: 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)),Aa2)),aa(num,word(A),numeral_numeral(word(A)),Ba)) = 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),Aa2)),aa(num,int,numeral_numeral(int),Ba))) ) ).

% word_no_log_defs(7)
tff(fact_3629_word__no__log__defs_I11_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: 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)),Aa2)),aa(num,word(A),numeral_numeral(word(A)),Ba)) = 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),Aa2)),aa(num,int,numeral_numeral(int),Ba))) ) ).

% word_no_log_defs(11)
tff(fact_3630_word__bitwise__1__simps_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Ba: 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)),Ba)) = 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),Ba))) ) ).

% word_bitwise_1_simps(2)
tff(fact_3631_word__bitwise__1__simps_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: 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)),Aa2)),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),Aa2)),one_one(int))) ) ).

% word_bitwise_1_simps(4)
tff(fact_3632_word__no__log__defs_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: 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)),Aa2))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Ba))) = 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),Aa2))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Ba)))) ) ).

% word_no_log_defs(6)
tff(fact_3633_word__no__log__defs_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: 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)),Aa2))),aa(num,word(A),numeral_numeral(word(A)),Ba)) = 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),Aa2))),aa(num,int,numeral_numeral(int),Ba))) ) ).

% word_no_log_defs(5)
tff(fact_3634_word__no__log__defs_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: 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)),Aa2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Ba))) = 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),Aa2)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Ba)))) ) ).

% word_no_log_defs(4)
tff(fact_3635_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_3636_word__bitwise__1__simps_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Ba: 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)),Ba)) = 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),Ba))) ) ).

% word_bitwise_1_simps(6)
tff(fact_3637_word__bitwise__1__simps_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: 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)),Aa2)),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),Aa2)),one_one(int))) ) ).

% word_bitwise_1_simps(8)
tff(fact_3638_word__no__log__defs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: 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)),Aa2))) = 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),Aa2)))) ) ).

% word_no_log_defs(2)
tff(fact_3639_word__no__log__defs_I10_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: 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)),Aa2))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Ba))) = 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),Aa2))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Ba)))) ) ).

% word_no_log_defs(10)
tff(fact_3640_word__no__log__defs_I9_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: 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)),Aa2))),aa(num,word(A),numeral_numeral(word(A)),Ba)) = 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),Aa2))),aa(num,int,numeral_numeral(int),Ba))) ) ).

% word_no_log_defs(9)
tff(fact_3641_word__no__log__defs_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: 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)),Aa2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Ba))) = 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),Aa2)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Ba)))) ) ).

% word_no_log_defs(8)
tff(fact_3642_word__bitwise__1__simps_I12_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: 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)),Aa2)),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),Aa2)),one_one(int))) ) ).

% word_bitwise_1_simps(12)
tff(fact_3643_word__bitwise__1__simps_I10_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Ba: 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)),Ba)) = 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),Ba))) ) ).

% word_bitwise_1_simps(10)
tff(fact_3644_word__no__log__defs_I14_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: 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)),Aa2))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Ba))) = 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),Aa2))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Ba)))) ) ).

% word_no_log_defs(14)
tff(fact_3645_word__no__log__defs_I13_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: 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)),Aa2))),aa(num,word(A),numeral_numeral(word(A)),Ba)) = 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),Aa2))),aa(num,int,numeral_numeral(int),Ba))) ) ).

% word_no_log_defs(13)
tff(fact_3646_word__no__log__defs_I12_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: 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)),Aa2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Ba))) = 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),Aa2)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Ba)))) ) ).

% word_no_log_defs(12)
tff(fact_3647_word__bitwise__1__simps_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Ba: 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)),Ba))) = 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),Ba)))) ) ).

% word_bitwise_1_simps(3)
tff(fact_3648_word__bitwise__1__simps_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: 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)),Aa2))),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),Aa2))),one_one(int))) ) ).

% word_bitwise_1_simps(5)
tff(fact_3649_word__bitwise__1__simps_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Ba: 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)),Ba))) = 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),Ba)))) ) ).

% word_bitwise_1_simps(7)
tff(fact_3650_word__bitwise__1__simps_I9_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: 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)),Aa2))),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),Aa2))),one_one(int))) ) ).

% word_bitwise_1_simps(9)
tff(fact_3651_word__bitwise__1__simps_I11_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Ba: 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)),Ba))) = 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),Ba)))) ) ).

% word_bitwise_1_simps(11)
tff(fact_3652_word__bitwise__1__simps_I13_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: 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)),Aa2))),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),Aa2))),one_one(int))) ) ).

% word_bitwise_1_simps(13)
tff(fact_3653_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_3654_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_3655_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_3656_nat__add__max__left,axiom,
    ! [M: nat,Na: nat,Q3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),Na)),Q3) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q3)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),Q3)) ).

% nat_add_max_left
tff(fact_3657_nat__add__max__right,axiom,
    ! [M: nat,Na: nat,Q3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Na),Q3)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q3)) ).

% nat_add_max_right
tff(fact_3658_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_3659_nat__minus__add__max,axiom,
    ! [Na: nat,M: nat] : aa(nat,nat,aa(nat,fun(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_3660_word__numeral__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Ba: num] : aa(num,word(A),numeral_numeral(word(A)),Ba) = aa(int,word(A),ring_1_of_int(word(A)),aa(num,int,numeral_numeral(int),Ba)) ) ).

% word_numeral_alt
tff(fact_3661_wi__hom__add,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: int,Ba: int] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(int,word(A),ring_1_of_int(word(A)),Aa2)),aa(int,word(A),ring_1_of_int(word(A)),Ba)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),plus_plus(int),Aa2),Ba)) ) ).

% wi_hom_add
tff(fact_3662_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_3663_wi__hom__neg,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: int] : aa(word(A),word(A),uminus_uminus(word(A)),aa(int,word(A),ring_1_of_int(word(A)),Aa2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,uminus_uminus(int),Aa2)) ) ).

% wi_hom_neg
tff(fact_3664_wi__hom__sub,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: int,Ba: int] : aa(word(A),word(A),minus_minus(word(A),aa(int,word(A),ring_1_of_int(word(A)),Aa2)),aa(int,word(A),ring_1_of_int(word(A)),Ba)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,minus_minus(int,Aa2),Ba)) ) ).

% wi_hom_sub
tff(fact_3665_word__of__int__power__hom,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: int,Na: nat] : aa(nat,word(A),power_power(word(A),aa(int,word(A),ring_1_of_int(word(A)),Aa2)),Na) = aa(int,word(A),ring_1_of_int(word(A)),aa(nat,int,power_power(int,Aa2),Na)) ) ).

% word_of_int_power_hom
tff(fact_3666_wi__hom__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: int,Ba: 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)),Aa2)),aa(int,word(A),ring_1_of_int(word(A)),Ba)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),times_times(int),Aa2),Ba)) ) ).

% wi_hom_mult
tff(fact_3667_word__wi__log__defs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: int,Ba: 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)),Aa2)),aa(int,word(A),ring_1_of_int(word(A)),Ba)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Aa2),Ba)) ) ).

% word_wi_log_defs(2)
tff(fact_3668_word__wi__log__defs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: int] : aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(int,word(A),ring_1_of_int(word(A)),Aa2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_ri4277139882892585799ns_not(int),Aa2)) ) ).

% word_wi_log_defs(1)
tff(fact_3669_word__wi__log__defs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: int,Ba: 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)),Aa2)),aa(int,word(A),ring_1_of_int(word(A)),Ba)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),Aa2),Ba)) ) ).

% word_wi_log_defs(3)
tff(fact_3670_word__wi__log__defs_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: int,Ba: 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)),Aa2)),aa(int,word(A),ring_1_of_int(word(A)),Ba)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),Aa2),Ba)) ) ).

% word_wi_log_defs(4)
tff(fact_3671_word__neg__numeral__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Ba: num] : aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Ba)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Ba))) ) ).

% word_neg_numeral_alt
tff(fact_3672_word__of__int__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: int] :
          ( aa(int,$o,aa(int,fun(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)),aa(int,nat,nat2,X)) ) ) ) ).

% word_of_int_nat
tff(fact_3673_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_3674_max_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,Aa2: A,D2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D2),Ba)
           => aa(A,$o,aa(A,fun(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),Aa2),Ba)) ) ) ) ).

% max.mono
tff(fact_3675_max_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2)
         => ( Aa2 = aa(A,A,aa(A,fun(A,A),ord_max(A),Aa2),Ba) ) ) ) ).

% max.orderE
tff(fact_3676_max_OorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: A,Ba: A] :
          ( ( Aa2 = aa(A,A,aa(A,fun(A,A),ord_max(A),Aa2),Ba) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2) ) ) ).

% max.orderI
tff(fact_3677_max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ba: A,C2: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),Ba),C2)),Aa2)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),Aa2) ) ) ) ).

% max.boundedE
tff(fact_3678_max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ba: A,Aa2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),Aa2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),Ba),C2)),Aa2) ) ) ) ).

% max.boundedI
tff(fact_3679_max_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2)
        <=> ( Aa2 = aa(A,A,aa(A,fun(A,A),ord_max(A),Aa2),Ba) ) ) ) ).

% max.order_iff
tff(fact_3680_max_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: A,Ba: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),aa(A,A,aa(A,fun(A,A),ord_max(A),Aa2),Ba)) ) ).

% max.cobounded1
tff(fact_3681_max_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ba: A,Aa2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),aa(A,A,aa(A,fun(A,A),ord_max(A),Aa2),Ba)) ) ).

% max.cobounded2
tff(fact_3682_le__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),X)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Y) ) ) ) ).

% le_max_iff_disj
tff(fact_3683_max_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2)
        <=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),Aa2),Ba) = Aa2 ) ) ) ).

% max.absorb_iff1
tff(fact_3684_max_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
        <=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),Aa2),Ba) = Ba ) ) ) ).

% max.absorb_iff2
tff(fact_3685_max_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),Aa2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),Aa2),Ba)) ) ) ).

% max.coboundedI1
tff(fact_3686_max_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),Ba)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),Aa2),Ba)) ) ) ).

% max.coboundedI2
tff(fact_3687_or__nat__unfold,axiom,
    ! [M: nat,Na: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),Na) = $ite(
        M = zero_zero(nat),
        Na,
        $ite(Na = zero_zero(nat),M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) ).

% or_nat_unfold
tff(fact_3688_round__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: int] :
          ( aa(A,$o,aa(A,fun(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,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),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_3689_round__unique_H,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Na: int] :
          ( aa(A,$o,aa(A,fun(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_3690_of__int__round__abs__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(A,$o,aa(A,fun(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_3691_of__int__round__gt,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(A,$o,aa(A,fun(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_3692_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_3693_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_3694_round__0,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_round(A,zero_zero(A)) = zero_zero(int) ) ) ).

% round_0
tff(fact_3695_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_3696_round__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_round(A,one_one(A)) = one_one(int) ) ) ).

% round_1
tff(fact_3697_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_3698_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_3699_word__of__int__Ex,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
        ? [Y2: int] : X = aa(int,word(A),ring_1_of_int(word(A)),Y2) ) ).

% word_of_int_Ex
tff(fact_3700_round__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_round(A,X)),archimedean_round(A,Y)) ) ) ).

% round_mono
tff(fact_3701_floor__le__round,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,X)),archimedean_round(A,X)) ) ).

% floor_le_round
tff(fact_3702_ceiling__ge__round,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_round(A,X)),archimedean_ceiling(A,X)) ) ).

% ceiling_ge_round
tff(fact_3703_round__diff__minimal,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: A,M: int] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,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_3704_round__def,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_round(A,X) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% round_def
tff(fact_3705_of__int__round__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archimedean_round(A,X))),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% of_int_round_le
tff(fact_3706_of__int__round__ge,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(A,$o,aa(A,fun(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_3707_round__altdef,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          archimedean_round(A,X) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2)))),archimedean_frac(A,X)),archimedean_ceiling(A,X),archim6421214686448440834_floor(A,X)) ) ).

% round_altdef
tff(fact_3708_log__base__10__eq1,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$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_3709_length__Cons,axiom,
    ! [A: $tType,X: A,Xs2: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs2)) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs2)) ).

% length_Cons
tff(fact_3710_option_Osize_I4_J,axiom,
    ! [A: $tType,X23: A] : aa(option(A),nat,size_size(option(A)),some(A,X23)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size(4)
tff(fact_3711_frac__frac,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_frac(A,archimedean_frac(A,X)) = archimedean_frac(A,X) ) ).

% frac_frac
tff(fact_3712_exp__le__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,exp(real),X)),aa(real,real,exp(real),Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y) ) ).

% exp_le_cancel_iff
tff(fact_3713_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_3714_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_3715_one__le__exp__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(real,real,exp(real),X))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X) ) ).

% one_le_exp_iff
tff(fact_3716_exp__le__one__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,exp(real),X)),one_one(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),zero_zero(real)) ) ).

% exp_le_one_iff
tff(fact_3717_norm__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,exp(A),X))),aa(real,real,exp(real),real_V7770717601297561774m_norm(A,X))) ) ).

% norm_exp
tff(fact_3718_exp__times__arg__commute,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,exp(A),A2)),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,exp(A),A2)) ) ).

% exp_times_arg_commute
tff(fact_3719_exp__ge__zero,axiom,
    ! [X: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,exp(real),X)) ).

% exp_ge_zero
tff(fact_3720_not__exp__le__zero,axiom,
    ! [X: real] : ~ aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,exp(real),X)),zero_zero(real)) ).

% not_exp_le_zero
tff(fact_3721_exp__add__commuting,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A,Y: 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),Y),X) )
         => ( aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,exp(A),X)),aa(A,A,exp(A),Y)) ) ) ) ).

% exp_add_commuting
tff(fact_3722_mult__exp__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,exp(A),X)),aa(A,A,exp(A),Y)) = aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) ) ).

% mult_exp_exp
tff(fact_3723_frac__ge__0,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),archimedean_frac(A,X)) ) ).

% frac_ge_0
tff(fact_3724_frac__lt__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),archimedean_frac(A,X)),one_one(A)) ) ).

% frac_lt_1
tff(fact_3725_frac__1__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),one_one(A))) = archimedean_frac(A,X) ) ).

% frac_1_eq
tff(fact_3726_exp__ge__add__one__self,axiom,
    ! [X: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X)),aa(real,real,exp(real),X)) ).

% exp_ge_add_one_self
tff(fact_3727_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_3728_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_3729_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_3730_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_3731_option_Osize__neq,axiom,
    ! [A: $tType,X: option(A)] : aa(option(A),nat,size_size(option(A)),X) != zero_zero(nat) ).

% option.size_neq
tff(fact_3732_exp__ge__add__one__self__aux,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X)),aa(real,real,exp(real),X)) ) ).

% exp_ge_add_one_self_aux
tff(fact_3733_lemma__exp__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Y)
     => ? [X2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X2)
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),aa(real,real,minus_minus(real,Y),one_one(real)))
          & ( aa(real,real,exp(real),X2) = Y ) ) ) ).

% lemma_exp_total
tff(fact_3734_ln__ge__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,ln_ln(real),X))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,exp(real),Y)),X) ) ) ).

% ln_ge_iff
tff(fact_3735_ln__x__over__x__mono,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,exp(real),one_one(real))),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),Y)),Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),X)),X)) ) ) ).

% ln_x_over_x_mono
tff(fact_3736_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_3737_powr__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [X: A,Aa2: A] :
          powr(A,X,Aa2) = $ite(X = zero_zero(A),zero_zero(A),aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(A,A,ln_ln(A),X)))) ) ).

% powr_def
tff(fact_3738_exp__le,axiom,
    aa(real,$o,aa(real,fun(real,$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_3739_exp__divide__power__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Na: nat,X: A] :
          ( aa(nat,$o,aa(nat,fun(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_3740_frac__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( archimedean_frac(A,X) = X )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),one_one(A)) ) ) ) ).

% frac_eq
tff(fact_3741_frac__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y)),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A))) ) ).

% frac_add
tff(fact_3742_tanh__altdef,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : tanh(A,X) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,exp(A),X)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),X)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,exp(A),X)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),X)))) ) ).

% tanh_altdef
tff(fact_3743_exp__half__le2,axiom,
    aa(real,$o,aa(real,fun(real,$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_3744_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_3745_exp__bound__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,exp(A),Z))),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% exp_bound_half
tff(fact_3746_option_Osel,axiom,
    ! [A: $tType,X23: A] : the2(A,some(A,X23)) = X23 ).

% option.sel
tff(fact_3747_exp__bound,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,exp(real),X)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X)),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% exp_bound
tff(fact_3748_floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y))),one_one(int))) ) ).

% floor_add
tff(fact_3749_real__exp__bound__lemma,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,exp(real),X)),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),X))) ) ) ).

% real_exp_bound_lemma
tff(fact_3750_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [Na: nat,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(nat,real,semiring_1_of_nat(real),Na))),X)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),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_3751_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(nat,real,semiring_1_of_nat(real),Na))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,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_3752_exp__bound__lemma,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,exp(A),Z))),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),real_V7770717601297561774m_norm(A,Z)))) ) ) ).

% exp_bound_lemma
tff(fact_3753_exp__lower__Taylor__quadratic,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),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_3754_log__base__10__eq2,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$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_3755_tanh__real__altdef,axiom,
    ! [X: 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,aa(real,fun(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_3756_option_Osize__gen_I2_J,axiom,
    ! [A: $tType,X: fun(A,nat),X23: A] : size_option(A,X,some(A,X23)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,X,X23)),aa(nat,nat,suc,zero_zero(nat))) ).

% option.size_gen(2)
tff(fact_3757_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_3758_flip__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A] : bit_se8732182000553998342ip_bit(A,zero_zero(nat),Aa2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2))),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),Aa2),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% flip_bit_0
tff(fact_3759_take__bit__word__Bit1__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num,M: num] : aa(word(A),word(A),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),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),pred_numeral(Na)),aa(num,word(A),numeral_numeral(word(A)),M)))) ) ).

% take_bit_word_Bit1_eq
tff(fact_3760_nat__of__bool,axiom,
    ! [P: $o] : aa(int,nat,nat2,aa($o,int,zero_neq_one_of_bool(int),(P))) = aa($o,nat,zero_neq_one_of_bool(nat),(P)) ).

% nat_of_bool
tff(fact_3761_take__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,Na),zero_zero(A)) = zero_zero(A) ) ).

% take_bit_of_0
tff(fact_3762_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_3763_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_3764_of__bool__less__eq__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [P: $o,Q: $o] :
          ( aa(A,$o,aa(A,fun(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_3765_of__bool__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [P: $o,Q: $o] :
          ( aa(A,$o,aa(A,fun(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_3766_take__bit__and,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,Aa2: A,Ba: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Na),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Aa2),Ba)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),Aa2)),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),Ba)) ) ).

% take_bit_and
tff(fact_3767_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_3768_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_3769_take__bit__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,Aa2: A,Ba: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Na),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Aa2),Ba)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),Aa2)),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),Ba)) ) ).

% take_bit_or
tff(fact_3770_take__bit__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,Aa2: A,Ba: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Na),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Aa2),Ba)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),Aa2)),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),Ba)) ) ).

% take_bit_xor
tff(fact_3771_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_3772_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_3773_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_3774_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_3775_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_3776_take__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,zero_zero(nat)),Aa2) = zero_zero(A) ) ).

% take_bit_0
tff(fact_3777_take__bit__Suc__1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Na)),one_one(A)) = one_one(A) ) ).

% take_bit_Suc_1
tff(fact_3778_take__bit__numeral__1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),L)),one_one(A)) = one_one(A) ) ).

% take_bit_numeral_1
tff(fact_3779_zero__less__of__bool__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: $o] :
          ( aa(A,$o,aa(A,fun(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_3780_of__bool__less__one__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: $o] :
          ( aa(A,$o,aa(A,fun(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_3781_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_3782_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_3783_set__decode__zero,axiom,
    nat_set_decode(zero_zero(nat)) = bot_bot(set(nat)) ).

% set_decode_zero
tff(fact_3784_take__bit__of__1__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,Na),one_one(A)) = zero_zero(A) )
        <=> ( Na = zero_zero(nat) ) ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_3785_take__bit__minus__one__eq__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat] : aa(A,A,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_3786_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_3787_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_3788_take__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,Na),one_one(A)) = aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)) ) ).

% take_bit_of_1
tff(fact_3789_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),$o,aa(word(A),fun(word(A),$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_3790_even__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,Aa2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),Aa2))
        <=> ( ( Na = zero_zero(nat) )
            | aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2) ) ) ) ).

% even_take_bit_eq
tff(fact_3791_of__bool__half__eq__0,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Ba: $o] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa($o,A,zero_neq_one_of_bool(A),(Ba))),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ).

% of_bool_half_eq_0
tff(fact_3792_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_3793_take__bit__Suc__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,zero_zero(nat))),Aa2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% take_bit_Suc_0
tff(fact_3794_take__bit__word__Bit0__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num,M: num] : aa(word(A),word(A),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))),aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),pred_numeral(Na)),aa(num,word(A),numeral_numeral(word(A)),M))) ) ).

% take_bit_word_Bit0_eq
tff(fact_3795_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_3796_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_3797_take__bit__of__exp,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: nat,Na: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$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_3798_take__bit__of__2,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,Na),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% take_bit_of_2
tff(fact_3799_take__bit__word__minus__Bit0__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num,M: num] : aa(word(A),word(A),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))),aa(word(A),word(A),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_3800_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,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)) ) ).

% one_mod_2_pow_eq
tff(fact_3801_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_3802_take__bit__not__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,Aa2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Na),aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),Aa2))) = aa(A,A,bit_se2584673776208193580ke_bit(A,Na),aa(A,A,bit_ri4277139882892585799ns_not(A),Aa2)) ) ).

% take_bit_not_take_bit
tff(fact_3803_take__bit__not__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,Aa2: A,Ba: A] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,Na),aa(A,A,bit_ri4277139882892585799ns_not(A),Aa2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,Na),aa(A,A,bit_ri4277139882892585799ns_not(A),Ba)) )
        <=> ( aa(A,A,bit_se2584673776208193580ke_bit(A,Na),Aa2) = aa(A,A,bit_se2584673776208193580ke_bit(A,Na),Ba) ) ) ) ).

% take_bit_not_iff
tff(fact_3804_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_3805_take__bit__of__nat,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,M: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,Na),aa(nat,A,semiring_1_of_nat(A),M)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Na),M)) ) ).

% take_bit_of_nat
tff(fact_3806_take__bit__tightened,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,Aa2: A,Ba: A,M: nat] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,Na),Aa2) = aa(A,A,bit_se2584673776208193580ke_bit(A,Na),Ba) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),Aa2) = aa(A,A,bit_se2584673776208193580ke_bit(A,M),Ba) ) ) ) ) ).

% take_bit_tightened
tff(fact_3807_take__bit__add,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat,Aa2: A,Ba: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Na),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),Aa2)),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),Ba))) = aa(A,A,bit_se2584673776208193580ke_bit(A,Na),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)) ) ).

% take_bit_add
tff(fact_3808_take__bit__of__int,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,K: int] : aa(A,A,bit_se2584673776208193580ke_bit(A,Na),aa(int,A,ring_1_of_int(A),K)) = aa(int,A,ring_1_of_int(A),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)) ) ).

% take_bit_of_int
tff(fact_3809_subset__decode__imp__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),nat_set_decode(M)),nat_set_decode(Na))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% subset_decode_imp_le
tff(fact_3810_zero__less__eq__of__bool,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: $o] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa($o,A,zero_neq_one_of_bool(A),(P))) ) ).

% zero_less_eq_of_bool
tff(fact_3811_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_3812_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_3813_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_3814_of__bool__less__eq__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: $o] : aa(A,$o,aa(A,fun(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_3815_signed__take__bit__eq__iff__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,Aa2: A,Ba: A] :
          ( ( aa(A,A,bit_ri4674362597316999326ke_bit(A,Na),Aa2) = aa(A,A,bit_ri4674362597316999326ke_bit(A,Na),Ba) )
        <=> ( aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Na)),Aa2) = aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Na)),Ba) ) ) ) ).

% signed_take_bit_eq_iff_take_bit_eq
tff(fact_3816_signed__take__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,Na: nat,Aa2: A] :
          aa(A,A,bit_ri4674362597316999326ke_bit(A,M),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),Aa2)) = aa(A,A,
            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M),bit_se2584673776208193580ke_bit(A,Na),bit_ri4674362597316999326ke_bit(A,M)),
            Aa2) ) ).

% signed_take_bit_take_bit
tff(fact_3817_take__bit__eq__mask,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,Aa2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Na),Aa2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Aa2),bit_se2239418461657761734s_mask(A,Na)) ) ).

% take_bit_eq_mask
tff(fact_3818_take__bit__unset__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,M: nat,Aa2: A] :
          aa(A,A,bit_se2584673776208193580ke_bit(A,Na),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),M),Aa2)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),Aa2),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),M),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),Aa2))) ) ).

% take_bit_unset_bit_eq
tff(fact_3819_take__bit__set__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,M: nat,Aa2: A] :
          aa(A,A,bit_se2584673776208193580ke_bit(A,Na),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),M),Aa2)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),Aa2),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),M),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),Aa2))) ) ).

% take_bit_set_bit_eq
tff(fact_3820_take__bit__flip__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,M: nat,Aa2: A] :
          aa(A,A,bit_se2584673776208193580ke_bit(A,Na),bit_se8732182000553998342ip_bit(A,M,Aa2)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),Aa2),bit_se8732182000553998342ip_bit(A,M,aa(A,A,bit_se2584673776208193580ke_bit(A,Na),Aa2))) ) ).

% take_bit_flip_bit_eq
tff(fact_3821_take__bit__signed__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,Na: nat,Aa2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Na))
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,bit_ri4674362597316999326ke_bit(A,Na),Aa2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,M),Aa2) ) ) ) ).

% take_bit_signed_take_bit
tff(fact_3822_take__bit__not__eq__mask__diff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,Aa2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Na),aa(A,A,bit_ri4277139882892585799ns_not(A),Aa2)) = aa(A,A,minus_minus(A,bit_se2239418461657761734s_mask(A,Na)),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),Aa2)) ) ).

% take_bit_not_eq_mask_diff
tff(fact_3823_take__bit__not__mask__eq__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(A,A,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_3824_take__bit__Suc__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat,K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Na)),aa(num,A,numeral_numeral(A),bit0(K))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,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_3825_take__bit__eq__mod,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,Aa2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Na),Aa2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) ) ).

% take_bit_eq_mod
tff(fact_3826_of__bool__odd__eq__mod__2,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Aa2: A] : aa($o,A,zero_neq_one_of_bool(A),~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% of_bool_odd_eq_mod_2
tff(fact_3827_take__bit__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,Aa2: A] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,Na),Aa2) = zero_zero(A) )
        <=> aa(A,$o,dvd_dvd(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)),Aa2) ) ) ).

% take_bit_eq_0_iff
tff(fact_3828_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))),aa(A,A,bit_se2584673776208193580ke_bit(A,L),Na))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),L)
            & ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na) ) ) ) ).

% bin_last_bintrunc
tff(fact_3829_take__bit__numeral__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),L)),aa(num,A,numeral_numeral(A),bit0(K))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,pred_numeral(L)),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% take_bit_numeral_bit0
tff(fact_3830_bits__induct,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [P: fun(A,$o),Aa2: 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,$o,P,A3) )
         => ( ! [A3: A,B3: $o] :
                ( aa(A,$o,P,A3)
               => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),(B3))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))),aa(num,A,numeral_numeral(A),bit0(one2))) = A3 )
                 => aa(A,$o,P,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),(B3))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))) ) )
           => aa(A,$o,P,Aa2) ) ) ) ).

% bits_induct
tff(fact_3831_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,$o,aa(nat,fun(nat,$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_3832_one__or__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),one_one(A)),Aa2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2))) ) ).

% one_or_eq
tff(fact_3833_or__one__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Aa2),one_one(A)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2))) ) ).

% or_one_eq
tff(fact_3834_take__bit__Suc__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat,K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Na)),aa(num,A,numeral_numeral(A),bit1(K))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,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_3835_take__bit__Suc__minus__1__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat] : aa(A,A,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_3836_take__bit__numeral__minus__1__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: num] : aa(A,A,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_3837_take__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,Aa2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Na)),Aa2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),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),Aa2),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% take_bit_Suc
tff(fact_3838_Suc__0__or__eq,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,suc,zero_zero(nat))),Na) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ).

% Suc_0_or_eq
tff(fact_3839_or__Suc__0__eq,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),Na),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ).

% or_Suc_0_eq
tff(fact_3840_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,
        aa(nat,fun(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_3841_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,
        aa(nat,fun(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_3842_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,
        aa(nat,fun(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_3843_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,
        aa(int,fun(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_3844_not__int__rec,axiom,
    ! [K: int] : aa(int,int,bit_ri4277139882892585799ns_not(int),K) = aa(int,int,aa(int,fun(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_3845_one__xor__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),one_one(A)),Aa2) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2)))),aa($o,A,zero_neq_one_of_bool(A),~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2))) ) ).

% one_xor_eq
tff(fact_3846_xor__one__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Aa2),one_one(A)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2)))),aa($o,A,zero_neq_one_of_bool(A),~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2))) ) ).

% xor_one_eq
tff(fact_3847_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,
        aa(int,fun(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_3848_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,
        aa(int,fun(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_3849_stable__imp__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A,Na: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2))) = Aa2 )
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,Na),Aa2) = $ite(aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2),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_3850_take__bit__numeral__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),L)),aa(num,A,numeral_numeral(A),bit1(K))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,pred_numeral(L)),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),bit0(one2)))),one_one(A)) ) ).

% take_bit_numeral_bit1
tff(fact_3851_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,aa(nat,fun(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_3852_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,aa(nat,fun(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_3853_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,$o,aa(nat,fun(nat,$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_3854_take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,Aa2: A] :
          aa(A,A,bit_se2584673776208193580ke_bit(A,Na),Aa2) = $ite(Na = zero_zero(nat),zero_zero(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,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),Aa2),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),Aa2),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% take_bit_rec
tff(fact_3855_set__decode__def,axiom,
    ! [X: nat] : nat_set_decode(X) = collect(nat,aTP_Lamp_db(nat,fun(nat,$o),X)) ).

% set_decode_def
tff(fact_3856_take__bit__word__minus__Bit1__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num,M: num] : aa(word(A),word(A),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),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(word(A),word(A),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_3857_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,
              aa(int,fun(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_3858_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,
          aa(int,fun(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_3859_length__subseqs,axiom,
    ! [A: $tType,Xs2: list(A)] : aa(list(list(A)),nat,size_size(list(list(A))),subseqs(A,Xs2)) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(list(A),nat,size_size(list(A)),Xs2)) ).

% length_subseqs
tff(fact_3860_list_Osimps_I15_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222)) = aa(set(A),set(A),insert(A,X21),aa(list(A),set(A),set2(A),X222)) ).

% list.simps(15)
tff(fact_3861_pred__numeral__inc,axiom,
    ! [K: num] : pred_numeral(inc(K)) = aa(num,nat,numeral_numeral(nat),K) ).

% pred_numeral_inc
tff(fact_3862_of__nat__nat__take__bit__eq,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Na: nat,K: int] : aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K))) = aa(int,A,ring_1_of_int(A),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)) ) ).

% of_nat_nat_take_bit_eq
tff(fact_3863_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_3864_add__neg__numeral__special_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Na: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(Na))) ) ).

% add_neg_numeral_special(5)
tff(fact_3865_add__neg__numeral__special_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(M))) ) ).

% add_neg_numeral_special(6)
tff(fact_3866_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_3867_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_3868_take__bit__of__Suc__0,axiom,
    ! [Na: nat] : aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Na),aa(nat,nat,suc,zero_zero(nat))) = aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)) ).

% take_bit_of_Suc_0
tff(fact_3869_take__bit__minus,axiom,
    ! [Na: nat,K: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,Na),aa(int,int,uminus_uminus(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K))) = aa(int,int,bit_se2584673776208193580ke_bit(int,Na),aa(int,int,uminus_uminus(int),K)) ).

% take_bit_minus
tff(fact_3870_take__bit__mult,axiom,
    ! [Na: nat,K: int,L: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,Na),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),L))) = aa(int,int,bit_se2584673776208193580ke_bit(int,Na),aa(int,int,aa(int,fun(int,int),times_times(int),K),L)) ).

% take_bit_mult
tff(fact_3871_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,Na: nat,Q3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,M),Q3)),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Na),Q3)) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_3872_take__bit__nat__less__eq__self,axiom,
    ! [Na: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Na),M)),M) ).

% take_bit_nat_less_eq_self
tff(fact_3873_take__bit__diff,axiom,
    ! [Na: nat,K: int,L: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,Na),aa(int,int,minus_minus(int,aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),L))) = aa(int,int,bit_se2584673776208193580ke_bit(int,Na),aa(int,int,minus_minus(int,K),L)) ).

% take_bit_diff
tff(fact_3874_num__induct,axiom,
    ! [P: fun(num,$o),X: num] :
      ( aa(num,$o,P,one2)
     => ( ! [X2: num] :
            ( aa(num,$o,P,X2)
           => aa(num,$o,P,inc(X2)) )
       => aa(num,$o,P,X) ) ) ).

% num_induct
tff(fact_3875_nat__take__bit__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(int,nat,nat2,aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)) = aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Na),aa(int,nat,nat2,K)) ) ) ).

% nat_take_bit_eq
tff(fact_3876_take__bit__nat__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Na),aa(int,nat,nat2,K)) = aa(int,nat,nat2,aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)) ) ) ).

% take_bit_nat_eq
tff(fact_3877_add__inc,axiom,
    ! [X: num,Y: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),X),inc(Y)) = inc(aa(num,num,aa(num,fun(num,num),plus_plus(num),X),Y)) ).

% add_inc
tff(fact_3878_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,Na: nat,K: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,M),K)),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)) ) ).

% take_bit_tightened_less_eq_int
tff(fact_3879_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_3880_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_3881_take__bit__int__less__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)),K)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% take_bit_int_less_eq_self_iff
tff(fact_3882_take__bit__nonnegative,axiom,
    ! [Na: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)) ).

% take_bit_nonnegative
tff(fact_3883_take__bit__int__greater__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% take_bit_int_greater_self_iff
tff(fact_3884_not__take__bit__negative,axiom,
    ! [Na: nat,K: int] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)),zero_zero(int)) ).

% not_take_bit_negative
tff(fact_3885_subseqs__refl,axiom,
    ! [A: $tType,Xs2: list(A)] : member(list(A),Xs2,aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xs2))) ).

% subseqs_refl
tff(fact_3886_inc_Osimps_I1_J,axiom,
    inc(one2) = bit0(one2) ).

% inc.simps(1)
tff(fact_3887_inc_Osimps_I2_J,axiom,
    ! [X: num] : inc(bit0(X)) = bit1(X) ).

% inc.simps(2)
tff(fact_3888_inc_Osimps_I3_J,axiom,
    ! [X: num] : inc(bit1(X)) = bit0(inc(X)) ).

% inc.simps(3)
tff(fact_3889_add__One,axiom,
    ! [X: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),X),one2) = inc(X) ).

% add_One
tff(fact_3890_mult__inc,axiom,
    ! [X: num,Y: num] : aa(num,num,aa(num,fun(num,num),times_times(num),X),inc(Y)) = aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(num,num),times_times(num),X),Y)),X) ).

% mult_inc
tff(fact_3891_take__bit__decr__eq,axiom,
    ! [Na: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K) != zero_zero(int) )
     => ( aa(int,int,bit_se2584673776208193580ke_bit(int,Na),aa(int,int,minus_minus(int,K),one_one(int))) = aa(int,int,minus_minus(int,aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)),one_one(int)) ) ) ).

% take_bit_decr_eq
tff(fact_3892_remove__subset,axiom,
    ! [A: $tType,X: A,Sa: set(A)] :
      ( member(A,X,Sa)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(set(A),set(A),minus_minus(set(A),Sa),aa(set(A),set(A),insert(A,X),bot_bot(set(A))))),Sa) ) ).

% remove_subset
tff(fact_3893_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)),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),I)) ) ).

% and_mask_wi
tff(fact_3894_numeral__inc,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [X: num] : aa(num,A,numeral_numeral(A),inc(X)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),X)),one_one(A)) ) ).

% numeral_inc
tff(fact_3895_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_3896_Cons__in__subseqsD,axiom,
    ! [A: $tType,Y: A,Ysb: list(A),Xs2: list(A)] :
      ( member(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ysb),aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xs2)))
     => member(list(A),Ysb,aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xs2))) ) ).

% Cons_in_subseqsD
tff(fact_3897_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)),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),aa(int,int,uminus_uminus(int),one_one(int)))) ) ).

% mask_bin
tff(fact_3898_take__bit__eq__mask__iff,axiom,
    ! [Na: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K) = bit_se2239418461657761734s_mask(int,Na) )
    <=> ( aa(int,int,bit_se2584673776208193580ke_bit(int,Na),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))) = zero_zero(int) ) ) ).

% take_bit_eq_mask_iff
tff(fact_3899_take__bit__nat__eq__self__iff,axiom,
    ! [Na: nat,M: nat] :
      ( ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Na),M) = M )
    <=> aa(nat,$o,aa(nat,fun(nat,$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_3900_take__bit__nat__less__exp,axiom,
    ! [Na: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,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_3901_take__bit__nat__eq__self,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
     => ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Na),M) = M ) ) ).

% take_bit_nat_eq_self
tff(fact_3902_take__bit__nat__def,axiom,
    ! [Na: nat,M: nat] : aa(nat,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_3903_take__bit__Suc__minus__bit1,axiom,
    ! [Na: nat,K: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,Na)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit1(K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,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_3904_take__bit__int__less__exp,axiom,
    ! [Na: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_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_3905_take__bit__int__def,axiom,
    ! [Na: nat,K: int] : aa(int,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_3906_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)),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),aa(num,int,numeral_numeral(int),I))) ) ).

% and_mask_no
tff(fact_3907_subseqs_Osimps_I1_J,axiom,
    ! [A: $tType] : subseqs(A,nil(A)) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),nil(list(A))) ).

% subseqs.simps(1)
tff(fact_3908_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa(int,int,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,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,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_3909_take__bit__nat__less__self__iff,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Na),M)),M)
    <=> aa(nat,$o,aa(nat,fun(nat,$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_3910_take__bit__Suc__minus__bit0,axiom,
    ! [Na: nat,K: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,Na)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,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_3911_take__bit__int__less__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)),K)
    <=> aa(int,$o,aa(int,fun(int,$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_3912_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K))
    <=> aa(int,$o,aa(int,fun(int,$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_3913_take__bit__int__eq__self,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))
       => ( aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_3914_take__bit__int__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K) = K )
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_3915_take__bit__incr__eq,axiom,
    ! [Na: nat,K: int] :
      ( ( aa(int,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)) )
     => ( aa(int,int,bit_se2584673776208193580ke_bit(int,Na),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)) ) ) ).

% take_bit_incr_eq
tff(fact_3916_take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa(int,int,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),aa(int,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_3917_take__bit__int__less__eq,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,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_3918_take__bit__int__greater__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)) ) ).

% take_bit_int_greater_eq
tff(fact_3919_signed__take__bit__eq__take__bit__shift,axiom,
    ! [Na: nat,K: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K) = aa(int,int,minus_minus(int,aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,Na)),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,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_3920_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [Na: nat,K: int] :
      ( ( aa(int,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,aa(int,fun(int,int),plus_plus(int),K),one_one(int))) ) ).

% take_bit_eq_mask_iff_exp_dvd
tff(fact_3921_take__bit__minus__small__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))
       => ( aa(int,int,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_3922_singleton__conv,axiom,
    ! [A: $tType,Aa2: A] : collect(A,aTP_Lamp_dc(A,fun(A,$o),Aa2)) = aa(set(A),set(A),insert(A,Aa2),bot_bot(set(A))) ).

% singleton_conv
tff(fact_3923_singleton__conv2,axiom,
    ! [A: $tType,Aa2: A] : collect(A,fequal(A,Aa2)) = aa(set(A),set(A),insert(A,Aa2),bot_bot(set(A))) ).

% singleton_conv2
tff(fact_3924_word__2p__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(word(A),nat,size_size(word(A)),W))
         => ( aa(word(A),$o,aa(word(A),fun(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,aa(int,fun(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_3925_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_3926_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_3927_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_3928_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_3929_uint__nonnegative,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(word(A),int,semiring_1_unsigned(A,int),W)) ) ).

% uint_nonnegative
tff(fact_3930_uint__le__0__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( aa(int,$o,aa(int,fun(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_3931_uint__ge__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(word(A),int,semiring_1_unsigned(A,int),X)) ) ).

% uint_ge_0
tff(fact_3932_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_3933_uint__lt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,semiring_1_unsigned(A,int),X)),zero_zero(int)) ) ).

% uint_lt_0
tff(fact_3934_word__le__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(num,word(A),numeral_numeral(word(A)),Aa2)),aa(num,word(A),numeral_numeral(word(A)),Ba))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),Aa2))),aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),Ba))) ) ) ).

% word_le_no
tff(fact_3935_word__less__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(num,word(A),numeral_numeral(word(A)),Aa2)),aa(num,word(A),numeral_numeral(word(A)),Ba))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),Aa2))),aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),Ba))) ) ) ).

% word_less_no
tff(fact_3936_word__div__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: 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)),Aa2)),aa(num,word(A),numeral_numeral(word(A)),Ba)) = 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)),Aa2))),aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),Ba)))) ) ).

% word_div_no
tff(fact_3937_word__mod__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: 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)),Aa2)),aa(num,word(A),numeral_numeral(word(A)),Ba)) = 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)),Aa2))),aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),Ba)))) ) ).

% word_mod_no
tff(fact_3938_unsigned__word__eqI,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & semiring_char_0(A) )
     => ! [V: word(B),W: word(B)] :
          ( ( aa(word(B),A,semiring_1_unsigned(B,A),V) = aa(word(B),A,semiring_1_unsigned(B,A),W) )
         => ( V = W ) ) ) ).

% unsigned_word_eqI
tff(fact_3939_word__eq__iff__unsigned,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & semiring_char_0(B) )
     => ! [V: word(A),W: word(A)] :
          ( ( V = W )
        <=> ( aa(word(A),B,semiring_1_unsigned(A,B),V) = aa(word(A),B,semiring_1_unsigned(A,B),W) ) ) ) ).

% word_eq_iff_unsigned
tff(fact_3940_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_3941_word__uint__eqI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] :
          ( ( aa(word(A),int,semiring_1_unsigned(A,int),Aa2) = aa(word(A),int,semiring_1_unsigned(A,int),Ba) )
         => ( Aa2 = Ba ) ) ) ).

% word_uint_eqI
tff(fact_3942_word__uint__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] :
          ( ( Aa2 = Ba )
        <=> ( aa(word(A),int,semiring_1_unsigned(A,int),Aa2) = aa(word(A),int,semiring_1_unsigned(A,int),Ba) ) ) ) ).

% word_uint_eq_iff
tff(fact_3943_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_3944_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_3945_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_3946_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_3947_word__le__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Aa2),Ba)
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,semiring_1_unsigned(A,int),Aa2)),aa(word(A),int,semiring_1_unsigned(A,int),Ba)) ) ) ).

% word_le_def
tff(fact_3948_word__less__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Aa2),Ba)
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,semiring_1_unsigned(A,int),Aa2)),aa(word(A),int,semiring_1_unsigned(A,int),Ba)) ) ) ).

% word_less_def
tff(fact_3949_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_3950_unsigned__greater__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & unique1627219031080169319umeral(A) )
     => ! [W: word(B)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(word(B),A,semiring_1_unsigned(B,A),W)) ) ).

% unsigned_greater_eq
tff(fact_3951_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_3952_uint__div__distrib,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: 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)),V),W)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(word(A),int,semiring_1_unsigned(A,int),V)),aa(word(A),int,semiring_1_unsigned(A,int),W)) ) ).

% uint_div_distrib
tff(fact_3953_uint__take__bit__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),bit_se2584673776208193580ke_bit(word(A),Na),W)) = aa(int,int,bit_se2584673776208193580ke_bit(int,Na),aa(word(A),int,semiring_1_unsigned(A,int),W)) ) ).

% uint_take_bit_eq
tff(fact_3954_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_3955_uint__mod__distrib,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: 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)),V),W)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(word(A),int,semiring_1_unsigned(A,int),V)),aa(word(A),int,semiring_1_unsigned(A,int),W)) ) ).

% uint_mod_distrib
tff(fact_3956_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_3957_word__less__eq__iff__unsigned,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & linordered_semidom(B) )
     => ! [Aa2: word(A),Ba: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Aa2),Ba)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(word(A),B,semiring_1_unsigned(A,B),Aa2)),aa(word(A),B,semiring_1_unsigned(A,B),Ba)) ) ) ).

% word_less_eq_iff_unsigned
tff(fact_3958_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_3959_word__less__iff__unsigned,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & linordered_semidom(B) )
     => ! [Aa2: word(A),Ba: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Aa2),Ba)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(word(A),B,semiring_1_unsigned(A,B),Aa2)),aa(word(A),B,semiring_1_unsigned(A,B),Ba)) ) ) ).

% word_less_iff_unsigned
tff(fact_3960_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_3961_unsigned__and__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_se359711467146920520ations(A) )
     => ! [V: 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)),V),W)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(word(B),A,semiring_1_unsigned(B,A),V)),aa(word(B),A,semiring_1_unsigned(B,A),W)) ) ).

% unsigned_and_eq
tff(fact_3962_unsigned__or__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_se359711467146920520ations(A) )
     => ! [V: 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)),V),W)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(word(B),A,semiring_1_unsigned(B,A),V)),aa(word(B),A,semiring_1_unsigned(B,A),W)) ) ).

% unsigned_or_eq
tff(fact_3963_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),aa(word(B),word(B),bit_se2584673776208193580ke_bit(word(B),Na),W)) = aa(A,A,bit_se2584673776208193580ke_bit(A,Na),aa(word(B),A,semiring_1_unsigned(B,A),W)) ) ).

% unsigned_take_bit_eq
tff(fact_3964_unsigned__xor__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_se359711467146920520ations(A) )
     => ! [V: 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)),V),W)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(word(B),A,semiring_1_unsigned(B,A),V)),aa(word(B),A,semiring_1_unsigned(B,A),W)) ) ).

% unsigned_xor_eq
tff(fact_3965_uint__add__ge0,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xa2: word(A),X: word(B)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa2)),aa(word(B),int,semiring_1_unsigned(B,int),X))) ) ).

% uint_add_ge0
tff(fact_3966_uint__mult__ge0,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xa2: word(A),X: word(B)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa2)),aa(word(B),int,semiring_1_unsigned(B,int),X))) ) ).

% uint_mult_ge0
tff(fact_3967_nat__uint__less__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Z: nat,X: word(A)] :
          ( ( aa(int,nat,nat2,aa(word(A),int,semiring_1_unsigned(A,int),Y)) = Z )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),X),Y)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(int,nat,nat2,aa(word(A),int,semiring_1_unsigned(A,int),X))),Z) ) ) ) ).

% nat_uint_less_helper
tff(fact_3968_uint__add__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y))),aa(int,int,aa(int,fun(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_3969_uint__plus__simple__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y))
        <=> ( aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y)) = aa(int,int,aa(int,fun(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_3970_uint__plus__simple,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y))
         => ( aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y)) = aa(int,int,aa(int,fun(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_3971_no__ulen__sub,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),X),Y)),X)
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,semiring_1_unsigned(A,int),Y)),aa(word(A),int,semiring_1_unsigned(A,int),X)) ) ) ).

% no_ulen_sub
tff(fact_3972_word__add__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Aa2),Ba) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Aa2)),aa(word(A),int,semiring_1_unsigned(A,int),Ba))) ) ).

% word_add_def
tff(fact_3973_uint__sub__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A)] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,semiring_1_unsigned(A,int),Y)),aa(word(A),int,semiring_1_unsigned(A,int),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_3974_uint__sub__ge,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] : aa(int,$o,aa(int,fun(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_3975_word__minus__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A)] : aa(word(A),word(A),uminus_uminus(word(A)),Aa2) = 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),Aa2))) ) ).

% word_minus_def
tff(fact_3976_uint__minus__simple__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),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_3977_uint__minus__simple__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(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_3978_word__sub__wi,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] : aa(word(A),word(A),minus_minus(word(A),Aa2),Ba) = 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),Aa2)),aa(word(A),int,semiring_1_unsigned(A,int),Ba))) ) ).

% word_sub_wi
tff(fact_3979_word__arith__power__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Na: nat] : aa(nat,word(A),power_power(word(A),Aa2),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),Aa2)),Na)) ) ).

% word_arith_power_alt
tff(fact_3980_word__mult__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),Aa2),Ba) = 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),Aa2)),aa(word(A),int,semiring_1_unsigned(A,int),Ba))) ) ).

% word_mult_def
tff(fact_3981_word__and__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Aa2),Ba) = 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),Aa2)),aa(word(A),int,semiring_1_unsigned(A,int),Ba))) ) ).

% word_and_def
tff(fact_3982_word__div__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Aa2),Ba) = 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),Aa2)),aa(word(A),int,semiring_1_unsigned(A,int),Ba))) ) ).

% word_div_def
tff(fact_3983_word__not__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A)] : aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Aa2) = 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),Aa2))) ) ).

% word_not_def
tff(fact_3984_word__or__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Aa2),Ba) = 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),Aa2)),aa(word(A),int,semiring_1_unsigned(A,int),Ba))) ) ).

% word_or_def
tff(fact_3985_word__mod__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),Aa2),Ba) = 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),Aa2)),aa(word(A),int,semiring_1_unsigned(A,int),Ba))) ) ).

% word_mod_def
tff(fact_3986_word__xor__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Aa2),Ba) = 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),Aa2)),aa(word(A),int,semiring_1_unsigned(A,int),Ba))) ) ).

% word_xor_def
tff(fact_3987_udvd__incr__lem0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Up: int,Uq: int,Na: int,K3: word(A),N6: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Up),Uq)
         => ( ( Up = aa(int,int,aa(int,fun(int,int),times_times(int),Na),aa(word(A),int,semiring_1_unsigned(A,int),K3)) )
           => ( ( Uq = aa(int,int,aa(int,fun(int,int),times_times(int),N6),aa(word(A),int,semiring_1_unsigned(A,int),K3)) )
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Up),aa(word(A),int,semiring_1_unsigned(A,int),K3))),Uq) ) ) ) ) ).

% udvd_incr_lem0
tff(fact_3988_udvd__incr__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Up: int,Uq: int,Ua: int,Na: int,K3: word(A),N6: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Up),Uq)
         => ( ( Up = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),Na),aa(word(A),int,semiring_1_unsigned(A,int),K3))) )
           => ( ( Uq = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),N6),aa(word(A),int,semiring_1_unsigned(A,int),K3))) )
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Up),aa(word(A),int,semiring_1_unsigned(A,int),K3))),Uq) ) ) ) ) ).

% udvd_incr_lem
tff(fact_3989_udvd__incr0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),Q3: word(A),Na: int,K3: word(A),N6: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),P2),Q3)
         => ( ( aa(word(A),int,semiring_1_unsigned(A,int),P2) = aa(int,int,aa(int,fun(int,int),times_times(int),Na),aa(word(A),int,semiring_1_unsigned(A,int),K3)) )
           => ( ( aa(word(A),int,semiring_1_unsigned(A,int),Q3) = aa(int,int,aa(int,fun(int,int),times_times(int),N6),aa(word(A),int,semiring_1_unsigned(A,int),K3)) )
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P2),K3)),Q3) ) ) ) ) ).

% udvd_incr0
tff(fact_3990_udvd__decr0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),Q3: word(A),Na: int,K3: word(A),N6: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),P2),Q3)
         => ( ( aa(word(A),int,semiring_1_unsigned(A,int),P2) = aa(int,int,aa(int,fun(int,int),times_times(int),Na),aa(word(A),int,semiring_1_unsigned(A,int),K3)) )
           => ( ( aa(word(A),int,semiring_1_unsigned(A,int),Q3) = aa(int,int,aa(int,fun(int,int),times_times(int),N6),aa(word(A),int,semiring_1_unsigned(A,int),K3)) )
             => ( ( aa(word(A),int,semiring_1_unsigned(A,int),Q3) = aa(int,int,aa(int,fun(int,int),times_times(int),N6),aa(word(A),int,semiring_1_unsigned(A,int),K3)) )
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),P2),aa(word(A),word(A),minus_minus(word(A),Q3),K3)) ) ) ) ) ) ).

% udvd_decr0
tff(fact_3991_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)),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),aa(word(A),int,semiring_1_unsigned(A,int),W))) ) ).

% and_mask_bintr
tff(fact_3992_Set_Oempty__def,axiom,
    ! [A: $tType] : bot_bot(set(A)) = collect(A,aTP_Lamp_dd(A,$o)) ).

% Set.empty_def
tff(fact_3993_insert__compr,axiom,
    ! [A: $tType,Aa2: A,B2: set(A)] : aa(set(A),set(A),insert(A,Aa2),B2) = collect(A,aa(set(A),fun(A,$o),aTP_Lamp_de(A,fun(set(A),fun(A,$o)),Aa2),B2)) ).

% insert_compr
tff(fact_3994_insert__Collect,axiom,
    ! [A: $tType,Aa2: A,P: fun(A,$o)] : aa(set(A),set(A),insert(A,Aa2),collect(A,P)) = collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_df(A,fun(fun(A,$o),fun(A,$o)),Aa2),P)) ).

% insert_Collect
tff(fact_3995_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_dg(set(A),fun(set(A),fun(A,$o)),A2),B2)) ).

% set_diff_eq
tff(fact_3996_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_ab(set(A),fun(A,$o),A2)),aTP_Lamp_ab(set(A),fun(A,$o),B2))) ).

% minus_set_def
tff(fact_3997_less__set__def,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A2),B2)
    <=> aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less(fun(A,$o)),aTP_Lamp_ab(set(A),fun(A,$o),A2)),aTP_Lamp_ab(set(A),fun(A,$o),B2)) ) ).

% less_set_def
tff(fact_3998_udvd__incr_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),Q3: word(A),Ua: int,Na: int,K3: word(A),N6: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),P2),Q3)
         => ( ( aa(word(A),int,semiring_1_unsigned(A,int),P2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),Na),aa(word(A),int,semiring_1_unsigned(A,int),K3))) )
           => ( ( aa(word(A),int,semiring_1_unsigned(A,int),Q3) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),N6),aa(word(A),int,semiring_1_unsigned(A,int),K3))) )
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P2),K3)),Q3) ) ) ) ) ).

% udvd_incr'
tff(fact_3999_udvd__decr_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),Q3: word(A),Ua: int,Na: int,K3: word(A),N6: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),P2),Q3)
         => ( ( aa(word(A),int,semiring_1_unsigned(A,int),P2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),Na),aa(word(A),int,semiring_1_unsigned(A,int),K3))) )
           => ( ( aa(word(A),int,semiring_1_unsigned(A,int),Q3) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),N6),aa(word(A),int,semiring_1_unsigned(A,int),K3))) )
             => ( ( aa(word(A),int,semiring_1_unsigned(A,int),Q3) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),N6),aa(word(A),int,semiring_1_unsigned(A,int),K3))) )
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),P2),aa(word(A),word(A),minus_minus(word(A),Q3),K3)) ) ) ) ) ) ).

% udvd_decr'
tff(fact_4000_set__decode__plus__power__2,axiom,
    ! [Na: nat,Z: nat] :
      ( ~ member(nat,Na,nat_set_decode(Z))
     => ( nat_set_decode(aa(nat,nat,aa(nat,fun(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_4001_and__mask__lt__2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Na: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),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_4002_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,aa(int,fun(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_4003_uint__range__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(word(A),int,semiring_1_unsigned(A,int),W))
          & aa(int,$o,aa(int,fun(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))),aa(word(A),nat,size_size(word(A)),W))) ) ) ).

% uint_range_size
tff(fact_4004_uint__2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),aa(nat,word(A),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_4005_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_4006_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_4007_Collect__conv__if2,axiom,
    ! [A: $tType,Aa2: A,P: fun(A,$o)] :
      collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_dh(A,fun(fun(A,$o),fun(A,$o)),Aa2),P)) = $ite(aa(A,$o,P,Aa2),aa(set(A),set(A),insert(A,Aa2),bot_bot(set(A))),bot_bot(set(A))) ).

% Collect_conv_if2
tff(fact_4008_Collect__conv__if,axiom,
    ! [A: $tType,Aa2: A,P: fun(A,$o)] :
      collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_di(A,fun(fun(A,$o),fun(A,$o)),Aa2),P)) = $ite(aa(A,$o,P,Aa2),aa(set(A),set(A),insert(A,Aa2),bot_bot(set(A))),bot_bot(set(A))) ).

% Collect_conv_if
tff(fact_4009_no__plus__overflow__uint__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),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))),aa(word(A),nat,size_size(word(A)),X))) ) ) ).

% no_plus_overflow_uint_size
tff(fact_4010_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),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),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))),aa(word(A),nat,size_size(word(A)),X))),aa(int,int,aa(int,fun(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,aa(int,fun(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))),aa(word(A),nat,size_size(word(A)),X)))) ) ).

% uint_plus_if_size
tff(fact_4011_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,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,semiring_1_unsigned(A,int),Y)),aa(word(A),int,semiring_1_unsigned(A,int),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,aa(int,fun(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))),aa(word(A),nat,size_size(word(A)),X)))) ) ).

% uint_sub_if_size
tff(fact_4012_vebt__deletei__rule,axiom,
    ! [Na: nat,S: set(nat),Tia: vEBT_VEBTi,X: nat] : time_htt(vEBT_VEBTi,aa(vEBT_VEBTi,assn,vEBT_Intf_vebt_assn(Na,S),Tia),vEBT_vebt_deletei(Tia,X),vEBT_Intf_vebt_assn(Na,aa(set(nat),set(nat),minus_minus(set(nat),S),aa(set(nat),set(nat),insert(nat,X),bot_bot(set(nat))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit1(bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit1(bit0(one2)))))),aa(int,nat,nat2,archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na))))))) ).

% vebt_deletei_rule
tff(fact_4013_vebt__inserti__rule,axiom,
    ! [X: nat,Na: nat,S: set(nat),Tia: vEBT_VEBTi] :
      ( aa(nat,$o,aa(nat,fun(nat,$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,S),Tia),vEBT_vebt_inserti(Tia,X),vEBT_Intf_vebt_assn(Na,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),S),aa(set(nat),set(nat),insert(nat,X),bot_bot(set(nat))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(bit1(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(bit1(one2))))),aa(int,nat,nat2,archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na))))))) ) ).

% vebt_inserti_rule
tff(fact_4014_arctan__def,axiom,
    ! [Y: real] : aa(real,real,arctan,Y) = the(real,aTP_Lamp_dj(real,fun(real,$o),Y)) ).

% arctan_def
tff(fact_4015_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,
        $ite(aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),L)),aa($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),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L))))))) ) ).

% modulo_int_def
tff(fact_4016_sgn__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [Aa2: A] : aa(A,A,sgn_sgn(A),aa(A,A,sgn_sgn(A),Aa2)) = aa(A,A,sgn_sgn(A),Aa2) ) ).

% sgn_sgn
tff(fact_4017_sup_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Ba: A,C2: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Ba),C2)),Aa2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),Aa2) ) ) ) ).

% sup.bounded_iff
tff(fact_4018_le__sup__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),Z)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ) ).

% le_sup_iff
tff(fact_4019_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_4020_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_4021_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_4022_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [Aa2: A] : aa(A,A,sgn_sgn(A),aa(A,A,uminus_uminus(A),Aa2)) = aa(A,A,uminus_uminus(A),aa(A,A,sgn_sgn(A),Aa2)) ) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_4023_sgn__greater,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,sgn_sgn(A),Aa2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2) ) ) ).

% sgn_greater
tff(fact_4024_sgn__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,sgn_sgn(A),Aa2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),zero_zero(A)) ) ) ).

% sgn_less
tff(fact_4025_divide__sgn,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(A,A,sgn_sgn(A),Ba)) = aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(A,A,sgn_sgn(A),Ba)) ) ).

% divide_sgn
tff(fact_4026_sgn__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2)
         => ( aa(A,A,sgn_sgn(A),Aa2) = one_one(A) ) ) ) ).

% sgn_pos
tff(fact_4027_abs__sgn__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Aa2: A] :
          ( ( Aa2 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),Aa2)) = one_one(A) ) ) ) ).

% abs_sgn_eq_1
tff(fact_4028_sgn__mult__self__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),Aa2)),aa(A,A,sgn_sgn(A),Aa2)) = aa($o,A,zero_neq_one_of_bool(A),Aa2 != zero_zero(A)) ) ).

% sgn_mult_self_eq
tff(fact_4029_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [Aa2: A] : aa(A,A,sgn_sgn(A),aa(A,A,abs_abs(A),Aa2)) = aa($o,A,zero_neq_one_of_bool(A),Aa2 != zero_zero(A)) ) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_4030_sgn__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [Aa2: A] : aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),Aa2)) = aa($o,A,zero_neq_one_of_bool(A),Aa2 != zero_zero(A)) ) ).

% sgn_abs
tff(fact_4031_sgn__mult__dvd__iff,axiom,
    ! [R: int,L: int,K: int] :
      ( aa(int,$o,dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),R)),L)),K)
    <=> ( aa(int,$o,dvd_dvd(int,L),K)
        & ( ( R = zero_zero(int) )
         => ( K = zero_zero(int) ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_4032_mult__sgn__dvd__iff,axiom,
    ! [L: int,R: int,K: int] :
      ( aa(int,$o,dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),L),aa(int,int,sgn_sgn(int),R))),K)
    <=> ( aa(int,$o,dvd_dvd(int,L),K)
        & ( ( R = zero_zero(int) )
         => ( K = zero_zero(int) ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_4033_dvd__sgn__mult__iff,axiom,
    ! [L: int,R: int,K: int] :
      ( aa(int,$o,dvd_dvd(int,L),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),R)),K))
    <=> ( aa(int,$o,dvd_dvd(int,L),K)
        | ( R = zero_zero(int) ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_4034_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R: int] :
      ( aa(int,$o,dvd_dvd(int,L),aa(int,int,aa(int,fun(int,int),times_times(int),K),aa(int,int,sgn_sgn(int),R)))
    <=> ( aa(int,$o,dvd_dvd(int,L),K)
        | ( R = zero_zero(int) ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_4035_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_4036_sgn__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),zero_zero(A))
         => ( aa(A,A,sgn_sgn(A),Aa2) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% sgn_neg
tff(fact_4037_sgn__of__nat,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat] : aa(A,A,sgn_sgn(A),aa(nat,A,semiring_1_of_nat(A),Na)) = aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)) ) ).

% sgn_of_nat
tff(fact_4038_Collect__imp__eq,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] : collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_dk(fun(A,$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_4039_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_4040_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_dl(set(A),fun(set(A),fun(A,$o)),A2),B2)) ).

% Un_def
tff(fact_4041_Collect__disj__eq,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] : collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_dm(fun(A,$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_4042_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_4043_word__unat__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: word(A),W: word(A)] :
          ( ( V = W )
        <=> ( aa(word(A),nat,semiring_1_unsigned(A,nat),V) = aa(word(A),nat,semiring_1_unsigned(A,nat),W) ) ) ) ).

% word_unat_eq_iff
tff(fact_4044_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_4045_set__diff__diff__left,axiom,
    ! [A: $tType,A2: set(A),B2: set(A),C5: set(A)] : aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),minus_minus(set(A),A2),B2)),C5) = 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),C5)) ).

% set_diff_diff_left
tff(fact_4046_same__sgn__sgn__add,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Ba: A,Aa2: A] :
          ( ( aa(A,A,sgn_sgn(A),Ba) = aa(A,A,sgn_sgn(A),Aa2) )
         => ( aa(A,A,sgn_sgn(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)) = aa(A,A,sgn_sgn(A),Aa2) ) ) ) ).

% same_sgn_sgn_add
tff(fact_4047_sgn__0__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Aa2: A] :
          ( ( aa(A,A,sgn_sgn(A),Aa2) = zero_zero(A) )
        <=> ( Aa2 = zero_zero(A) ) ) ) ).

% sgn_0_0
tff(fact_4048_sgn__eq__0__iff,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [Aa2: A] :
          ( ( aa(A,A,sgn_sgn(A),Aa2) = zero_zero(A) )
        <=> ( Aa2 = zero_zero(A) ) ) ) ).

% sgn_eq_0_iff
tff(fact_4049_sgn__mult,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,sgn_sgn(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),Aa2)),aa(A,A,sgn_sgn(A),Ba)) ) ).

% sgn_mult
tff(fact_4050_Real__Vector__Spaces_Osgn__mult,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: A,Y: A] : aa(A,A,sgn_sgn(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),aa(A,A,sgn_sgn(A),X)),aa(A,A,sgn_sgn(A),Y)) ) ).

% Real_Vector_Spaces.sgn_mult
tff(fact_4051_inf__sup__ord_I4_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Y: A,X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) ) ).

% inf_sup_ord(4)
tff(fact_4052_inf__sup__ord_I3_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A] : aa(A,$o,aa(A,fun(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_4053_le__supE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Aa2: A,Ba: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Aa2),Ba)),X)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),X)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),X) ) ) ) ).

% le_supE
tff(fact_4054_le__supI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Aa2: A,X: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),X)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),X)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Aa2),Ba)),X) ) ) ) ).

% le_supI
tff(fact_4055_sup__ge1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) ) ).

% sup_ge1
tff(fact_4056_sup__ge2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,X: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) ) ).

% sup_ge2
tff(fact_4057_le__supI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Aa2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Aa2),Ba)) ) ) ).

% le_supI1
tff(fact_4058_le__supI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Ba)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Aa2),Ba)) ) ) ).

% le_supI2
tff(fact_4059_sup_Omono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,Aa2: A,D2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D2),Ba)
           => aa(A,$o,aa(A,fun(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),Aa2),Ba)) ) ) ) ).

% sup.mono
tff(fact_4060_sup__mono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Aa2: A,C2: A,Ba: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Aa2),Ba)),aa(A,A,aa(A,fun(A,A),sup_sup(A),C2),D2)) ) ) ) ).

% sup_mono
tff(fact_4061_sup__least,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,X: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),X)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z)),X) ) ) ) ).

% sup_least
tff(fact_4062_le__iff__sup,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(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_4063_sup_OorderE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2)
         => ( Aa2 = aa(A,A,aa(A,fun(A,A),sup_sup(A),Aa2),Ba) ) ) ) ).

% sup.orderE
tff(fact_4064_sup_OorderI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Aa2: A,Ba: A] :
          ( ( Aa2 = aa(A,A,aa(A,fun(A,A),sup_sup(A),Aa2),Ba) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2) ) ) ).

% sup.orderI
tff(fact_4065_sup__unique,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [F2: fun(A,fun(A,A)),X: A,Y: A] :
          ( ! [X2: A,Y2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),aa(A,A,aa(A,fun(A,A),F2,X2),Y2))
         => ( ! [X2: A,Y2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y2),aa(A,A,aa(A,fun(A,A),F2,X2),Y2))
           => ( ! [X2: A,Y2: A,Z3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y2),X2)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z3),X2)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F2,Y2),Z3)),X2) ) )
             => ( 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_4066_sup_Oabsorb1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Aa2),Ba) = Aa2 ) ) ) ).

% sup.absorb1
tff(fact_4067_sup_Oabsorb2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Aa2),Ba) = Ba ) ) ) ).

% sup.absorb2
tff(fact_4068_sup__absorb1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(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_4069_sup__absorb2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(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_4070_sup_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Ba: A,C2: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Ba),C2)),Aa2)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),Aa2) ) ) ) ).

% sup.boundedE
tff(fact_4071_sup_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Ba: A,Aa2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),Aa2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Ba),C2)),Aa2) ) ) ) ).

% sup.boundedI
tff(fact_4072_sup_Oorder__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2)
        <=> ( Aa2 = aa(A,A,aa(A,fun(A,A),sup_sup(A),Aa2),Ba) ) ) ) ).

% sup.order_iff
tff(fact_4073_sup_Ocobounded1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Aa2: A,Ba: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),aa(A,A,aa(A,fun(A,A),sup_sup(A),Aa2),Ba)) ) ).

% sup.cobounded1
tff(fact_4074_sup_Ocobounded2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Ba: A,Aa2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),aa(A,A,aa(A,fun(A,A),sup_sup(A),Aa2),Ba)) ) ).

% sup.cobounded2
tff(fact_4075_sup_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2)
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Aa2),Ba) = Aa2 ) ) ) ).

% sup.absorb_iff1
tff(fact_4076_sup_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Aa2),Ba) = Ba ) ) ) ).

% sup.absorb_iff2
tff(fact_4077_sup_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),Aa2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),Aa2),Ba)) ) ) ).

% sup.coboundedI1
tff(fact_4078_sup_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),Ba)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),Aa2),Ba)) ) ) ).

% sup.coboundedI2
tff(fact_4079_insert__def,axiom,
    ! [A: $tType,Aa2: A,B2: set(A)] : aa(set(A),set(A),insert(A,Aa2),B2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),collect(A,aTP_Lamp_dc(A,fun(A,$o),Aa2))),B2) ).

% insert_def
tff(fact_4080_sgn__not__eq__imp,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Ba: A,Aa2: A] :
          ( ( aa(A,A,sgn_sgn(A),Ba) != aa(A,A,sgn_sgn(A),Aa2) )
         => ( ( aa(A,A,sgn_sgn(A),Aa2) != zero_zero(A) )
           => ( ( aa(A,A,sgn_sgn(A),Ba) != zero_zero(A) )
             => ( aa(A,A,sgn_sgn(A),Aa2) = aa(A,A,uminus_uminus(A),aa(A,A,sgn_sgn(A),Ba)) ) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_4081_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_4082_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_4083_same__sgn__abs__add,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Ba: A,Aa2: A] :
          ( ( aa(A,A,sgn_sgn(A),Ba) = aa(A,A,sgn_sgn(A),Aa2) )
         => ( aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),Aa2)),aa(A,A,abs_abs(A),Ba)) ) ) ) ).

% same_sgn_abs_add
tff(fact_4084_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_4085_abs__mult__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),Aa2)),aa(A,A,sgn_sgn(A),Aa2)) = Aa2 ) ).

% abs_mult_sgn
tff(fact_4086_sgn__mult__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [Aa2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),Aa2)),aa(A,A,abs_abs(A),Aa2)) = Aa2 ) ).

% sgn_mult_abs
tff(fact_4087_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_4088_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_4089_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_4090_unat__mono,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Aa2),Ba)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Aa2)),aa(word(A),nat,semiring_1_unsigned(A,nat),Ba)) ) ) ).

% unat_mono
tff(fact_4091_word__less__nat__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Aa2),Ba)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Aa2)),aa(word(A),nat,semiring_1_unsigned(A,nat),Ba)) ) ) ).

% word_less_nat_alt
tff(fact_4092_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_4093_word__le__nat__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Aa2),Ba)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Aa2)),aa(word(A),nat,semiring_1_unsigned(A,nat),Ba)) ) ) ).

% word_le_nat_alt
tff(fact_4094_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_4095_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_4096_unat__eq__nat__uint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),W) = aa(int,nat,nat2,aa(word(A),int,semiring_1_unsigned(A,int),W)) ) ).

% unat_eq_nat_uint
tff(fact_4097_le__unat__uoi,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: nat,Z: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),aa(word(A),nat,semiring_1_unsigned(A,nat),Z))
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(nat,word(A),semiring_1_of_nat(word(A)),Y)) = Y ) ) ) ).

% le_unat_uoi
tff(fact_4098_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_4099_max__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: 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)),Aa2),Ba)),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)),Aa2),Ba))),aa(word(A),nat,semiring_1_unsigned(A,nat),C2)) ) ).

% max_lt
tff(fact_4100_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_4101_unat__div__distrib,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: 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)),V),W)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),V)),aa(word(A),nat,semiring_1_unsigned(A,nat),W)) ) ).

% unat_div_distrib
tff(fact_4102_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_4103_unat__mod__distrib,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: 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)),V),W)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),V)),aa(word(A),nat,semiring_1_unsigned(A,nat),W)) ) ).

% unat_mod_distrib
tff(fact_4104_ln__real__def,axiom,
    ! [X: real] : aa(real,real,ln_ln(real),X) = the(real,aTP_Lamp_dn(real,fun(real,$o),X)) ).

% ln_real_def
tff(fact_4105_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_4106_sgn__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Aa2: A] :
          ( ( aa(A,A,sgn_sgn(A),Aa2) = one_one(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2) ) ) ).

% sgn_1_pos
tff(fact_4107_abs__sgn__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Aa2: A] :
          aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),Aa2)) = $ite(Aa2 = zero_zero(A),zero_zero(A),one_one(A)) ) ).

% abs_sgn_eq
tff(fact_4108_unat__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(word(A),nat,semiring_1_unsigned(A,nat),X))
        <=> ( X != zero_zero(word(A)) ) ) ) ).

% unat_gt_0
tff(fact_4109_un__ui__le,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Aa2: word(A),Ba: word(B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Aa2)),aa(word(B),nat,semiring_1_unsigned(B,nat),Ba))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,semiring_1_unsigned(A,int),Aa2)),aa(word(B),int,semiring_1_unsigned(B,int),Ba)) ) ) ).

% un_ui_le
tff(fact_4110_unat__plus__simple,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y))
        <=> ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y)) = aa(nat,nat,aa(nat,fun(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_4111_unat__sub,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Ba: word(A),Aa2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Ba),Aa2)
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),minus_minus(word(A),Aa2),Ba)) = aa(nat,nat,minus_minus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),Aa2)),aa(word(A),nat,semiring_1_unsigned(A,nat),Ba)) ) ) ) ).

% unat_sub
tff(fact_4112_word__of__nat__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,X: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(word(A),nat,semiring_1_unsigned(A,nat),X))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(nat,word(A),semiring_1_of_nat(word(A)),Na)),X) ) ) ).

% word_of_nat_le
tff(fact_4113_word__unat__less__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Aa2),aa(nat,word(A),semiring_1_of_nat(word(A)),Ba))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Aa2)),Ba) ) ) ).

% word_unat_less_le
tff(fact_4114_word__arith__nat__add,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Aa2),Ba) = aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Aa2)),aa(word(A),nat,semiring_1_unsigned(A,nat),Ba))) ) ).

% word_arith_nat_add
tff(fact_4115_word__arith__nat__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),Aa2),Ba) = 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),Aa2)),aa(word(A),nat,semiring_1_unsigned(A,nat),Ba))) ) ).

% word_arith_nat_mult
tff(fact_4116_word__arith__nat__div,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Aa2),Ba) = 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),Aa2)),aa(word(A),nat,semiring_1_unsigned(A,nat),Ba))) ) ).

% word_arith_nat_div
tff(fact_4117_word__arith__nat__mod,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),Aa2),Ba) = 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),Aa2)),aa(word(A),nat,semiring_1_unsigned(A,nat),Ba))) ) ).

% word_arith_nat_mod
tff(fact_4118_ln__neg__is__const,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),zero_zero(real))
     => ( aa(real,real,ln_ln(real),X) = the(real,aTP_Lamp_do(real,$o)) ) ) ).

% ln_neg_is_const
tff(fact_4119_sgn__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Aa2: A] :
          ( ( aa(A,A,sgn_sgn(A),Aa2) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),zero_zero(A)) ) ) ).

% sgn_1_neg
tff(fact_4120_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,aa(A,fun(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_4121_zsgn__def,axiom,
    ! [I: int] :
      aa(int,int,sgn_sgn(int),I) = $ite(
        I = zero_zero(int),
        zero_zero(int),
        $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),I),one_one(int),aa(int,int,uminus_uminus(int),one_one(int))) ) ).

% zsgn_def
tff(fact_4122_div__sgn__abs__cancel,axiom,
    ! [V: int,K: int,L: int] :
      ( ( V != 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),V)),aa(int,int,abs_abs(int),K))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),V)),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_4123_div__dvd__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,dvd_dvd(int,L),K)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(int,int,sgn_sgn(int),L))),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L))) ) ) ).

% div_dvd_sgn_abs
tff(fact_4124_unatSuc2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Na),one_one(word(A))) != zero_zero(word(A)) )
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Na),one_one(word(A)))) = aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),Na)) ) ) ) ).

% unatSuc2
tff(fact_4125_unatSuc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),Na) != zero_zero(word(A)) )
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),Na)) = aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),Na)) ) ) ) ).

% unatSuc
tff(fact_4126_unat__Suc2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A)] :
          ( ( Na != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Na),one_one(word(A)))) = aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),Na)) ) ) ) ).

% unat_Suc2
tff(fact_4127_uno__simps_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Z: word(A),M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(word(A),nat,semiring_1_unsigned(A,nat),Z))
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(word(A),nat,semiring_1_unsigned(A,nat),Z)))) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(word(A),nat,semiring_1_unsigned(A,nat),Z)) ) ) ) ).

% uno_simps(1)
tff(fact_4128_measure__unat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A)] :
          ( ( P2 != zero_zero(word(A)) )
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),minus_minus(word(A),P2),one_one(word(A))))),aa(word(A),nat,semiring_1_unsigned(A,nat),P2)) ) ) ).

% measure_unat
tff(fact_4129_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),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),one_one(word(A)))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),X)),one_one(nat)) )
          | ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),one_one(word(A))) = zero_zero(word(A)) ) ) ) ).

% word_overflow_unat
tff(fact_4130_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_4131_arccos__def,axiom,
    ! [Y: real] : aa(real,real,arccos,Y) = the(real,aTP_Lamp_dp(real,fun(real,$o),Y)) ).

% arccos_def
tff(fact_4132_lt__plus__1__le__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,MaxBound: word(A),X: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(word(A),nat,semiring_1_unsigned(A,nat),MaxBound))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),aa(nat,word(A),semiring_1_of_nat(word(A)),Na)))
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(nat,word(A),semiring_1_of_nat(word(A)),Na)) ) ) ) ).

% lt_plus_1_le_word
tff(fact_4133_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),aa(word(A),fun(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),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),one_one(word(A))))) ) ) ) ).

% even_word_imp_odd_next
tff(fact_4134_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),aa(word(A),fun(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),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),one_one(word(A))))) ) ) ) ).

% odd_word_imp_even_next
tff(fact_4135_word__div__eq__1__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A),M: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Na),M) = one_one(word(A)) )
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),M),Na)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Na)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,semiring_1_unsigned(A,nat),M))) ) ) ) ).

% word_div_eq_1_iff
tff(fact_4136_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_4137_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 )
        <=> ? [Q4: nat] : Na = aa(nat,nat,aa(nat,fun(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),Q4),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,size_size(word(A)),W)))) ) ) ).

% of_nat_eq_size
tff(fact_4138_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_dq(real,$o)) ).

% pi_half
tff(fact_4139_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_dq(real,$o))) ).

% pi_def
tff(fact_4140_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),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),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))),aa(word(A),nat,size_size(word(A)),X))),aa(nat,nat,aa(nat,fun(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,aa(nat,fun(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))),aa(word(A),nat,size_size(word(A)),X)))) ) ).

% unat_plus_if_size
tff(fact_4141_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,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),aa(word(A),nat,semiring_1_unsigned(A,nat),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,aa(nat,fun(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))),aa(word(A),nat,size_size(word(A)),X)))),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))) ) ).

% unat_sub_if_size
tff(fact_4142_no__plus__overflow__unat__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),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))),aa(word(A),nat,size_size(word(A)),X))) ) ) ).

% no_plus_overflow_unat_size
tff(fact_4143_word__unat__mask__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,W: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(word(A),nat,size_size(word(A)),W))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),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_4144_divide__int__unfold,axiom,
    ! [K: int,M: nat,L: int,Na: nat] :
      aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),M))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),Na))) = $ite(
        ( ( aa(int,int,sgn_sgn(int),L) = zero_zero(int) )
        | ( aa(int,int,sgn_sgn(int),K) = zero_zero(int) )
        | ( Na = zero_zero(nat) ) ),
        zero_zero(int),
        $ite(aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(nat,$o,dvd_dvd(nat,Na),M)))))) ) ).

% divide_int_unfold
tff(fact_4145_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_4146_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),
        $ite(aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(int,$o,dvd_dvd(int,L),K)))))) ) ).

% divide_int_def
tff(fact_4147_arcsin__def,axiom,
    ! [Y: real] : aa(real,real,arcsin,Y) = the(real,aTP_Lamp_dr(real,fun(real,$o),Y)) ).

% arcsin_def
tff(fact_4148_sgn__div__eq__sgn__mult,axiom,
    ! [Aa2: int,Ba: int] :
      ( ( aa(int,int,aa(int,fun(int,int),divide_divide(int),Aa2),Ba) != zero_zero(int) )
     => ( aa(int,int,sgn_sgn(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),Aa2),Ba)) = aa(int,int,sgn_sgn(int),aa(int,int,aa(int,fun(int,int),times_times(int),Aa2),Ba)) ) ) ).

% sgn_div_eq_sgn_mult
tff(fact_4149_set__union,axiom,
    ! [A: $tType,Xs2: list(A),Ysb: list(A)] : aa(list(A),set(A),set2(A),union(A,Xs2,Ysb)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs2)),aa(list(A),set(A),set2(A),Ysb)) ).

% set_union
tff(fact_4150_the__sym__eq__trivial,axiom,
    ! [A: $tType,X: A] : the(A,fequal(A,X)) = X ).

% the_sym_eq_trivial
tff(fact_4151_the__eq__trivial,axiom,
    ! [A: $tType,Aa2: A] : the(A,aTP_Lamp_dc(A,fun(A,$o),Aa2)) = Aa2 ).

% the_eq_trivial
tff(fact_4152_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_4153_the__equality,axiom,
    ! [A: $tType,P: fun(A,$o),Aa2: A] :
      ( aa(A,$o,P,Aa2)
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
           => ( X2 = Aa2 ) )
       => ( the(A,P) = Aa2 ) ) ) ).

% the_equality
tff(fact_4154_zero__le__sgn__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,sgn_sgn(real),X))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X) ) ).

% zero_le_sgn_iff
tff(fact_4155_sgn__le__0__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sgn_sgn(real),X)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),zero_zero(real)) ) ).

% sgn_le_0_iff
tff(fact_4156_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,$o)),aTP_Lamp_ab(set(A),fun(A,$o),A2)),aTP_Lamp_ab(set(A),fun(A,$o),B2))) ).

% sup_set_def
tff(fact_4157_sup__nat__def,axiom,
    sup_sup(nat) = ord_max(nat) ).

% sup_nat_def
tff(fact_4158_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_ab(set(A),fun(A,$o),A2))) ).

% uminus_set_def
tff(fact_4159_Collect__neg__eq,axiom,
    ! [A: $tType,P: fun(A,$o)] : collect(A,aTP_Lamp_ds(fun(A,$o),fun(A,$o),P)) = aa(set(A),set(A),uminus_uminus(set(A)),collect(A,P)) ).

% Collect_neg_eq
tff(fact_4160_Compl__eq,axiom,
    ! [A: $tType,A2: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A2) = collect(A,aTP_Lamp_dt(set(A),fun(A,$o),A2)) ).

% Compl_eq
tff(fact_4161_sup__enat__def,axiom,
    sup_sup(extended_enat) = ord_max(extended_enat) ).

% sup_enat_def
tff(fact_4162_sup__int__def,axiom,
    sup_sup(int) = ord_max(int) ).

% sup_int_def
tff(fact_4163_sgn__real__def,axiom,
    ! [Aa2: real] :
      aa(real,real,sgn_sgn(real),Aa2) = $ite(
        Aa2 = zero_zero(real),
        zero_zero(real),
        $ite(aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Aa2),one_one(real),aa(real,real,uminus_uminus(real),one_one(real))) ) ).

% sgn_real_def
tff(fact_4164_theI,axiom,
    ! [A: $tType,P: fun(A,$o),Aa2: A] :
      ( aa(A,$o,P,Aa2)
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
           => ( X2 = Aa2 ) )
       => aa(A,$o,P,the(A,P)) ) ) ).

% theI
tff(fact_4165_theI_H,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ? [X4: A] :
          ( aa(A,$o,P,X4)
          & ! [Y2: A] :
              ( aa(A,$o,P,Y2)
             => ( Y2 = X4 ) ) )
     => aa(A,$o,P,the(A,P)) ) ).

% theI'
tff(fact_4166_theI2,axiom,
    ! [A: $tType,P: fun(A,$o),Aa2: A,Q: fun(A,$o)] :
      ( aa(A,$o,P,Aa2)
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
           => ( X2 = Aa2 ) )
       => ( ! [X2: A] :
              ( aa(A,$o,P,X2)
             => aa(A,$o,Q,X2) )
         => aa(A,$o,Q,the(A,P)) ) ) ) ).

% theI2
tff(fact_4167_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,$o)),aTP_Lamp_du($o,fun(A,fun(A,fun(A,$o))),(P)),X),Y)) ).

% If_def
tff(fact_4168_the1I2,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ? [X4: A] :
          ( aa(A,$o,P,X4)
          & ! [Y2: A] :
              ( aa(A,$o,P,Y2)
             => ( Y2 = X4 ) ) )
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
           => aa(A,$o,Q,X2) )
       => aa(A,$o,Q,the(A,P)) ) ) ).

% the1I2
tff(fact_4169_the1__equality,axiom,
    ! [A: $tType,P: fun(A,$o),Aa2: A] :
      ( ? [X4: A] :
          ( aa(A,$o,P,X4)
          & ! [Y2: A] :
              ( aa(A,$o,P,Y2)
             => ( Y2 = X4 ) ) )
     => ( aa(A,$o,P,Aa2)
       => ( the(A,P) = Aa2 ) ) ) ).

% the1_equality
tff(fact_4170_floor__real__def,axiom,
    ! [X: real] : archim6421214686448440834_floor(real,X) = the(int,aTP_Lamp_dv(real,fun(int,$o),X)) ).

% floor_real_def
tff(fact_4171_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_4172_fold__list__rl,axiom,
    ! [Xs2: list(nat),Na: nat,S: set(nat),T: vEBT_VEBTi] :
      ( ! [X2: nat] :
          ( member(nat,X2,aa(list(nat),set(nat),set2(nat),Xs2))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X2),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,S),T),aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),vEBT_Example_mfold(nat,vEBT_VEBTi,aTP_Lamp_an(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),Xs2),T),vEBT_Intf_vebt_assn(Na,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),Xs2)))) ) ).

% fold_list_rl
tff(fact_4173_signed__take__bit__eq__take__bit__minus,axiom,
    ! [Na: nat,K: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K) = aa(int,int,minus_minus(int,aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,Na)),K)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,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_4174_vebt__heap__rules_I3_J,axiom,
    ! [X: nat,Na: nat,S: set(nat),Tia: vEBT_VEBTi] :
      ( aa(nat,$o,aa(nat,fun(nat,$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,S),Tia),vEBT_vebt_inserti(Tia,X),vEBT_Intf_vebt_assn(Na,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),S),aa(set(nat),set(nat),insert(nat,X),bot_bot(set(nat)))))) ) ).

% vebt_heap_rules(3)
tff(fact_4175_num_Osize__gen_I3_J,axiom,
    ! [X32: num] : size_num(bit1(X32)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_num(X32)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size_gen(3)
tff(fact_4176_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_4177_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_4178_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_4179_signed__take__bit__nonnegative__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K))
    <=> ~ aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),Na) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_4180_signed__take__bit__negative__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K)),zero_zero(int))
    <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),Na) ) ).

% signed_take_bit_negative_iff
tff(fact_4181_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_4182_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_4183_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_4184_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_4185_bit__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Aa2: A] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),zero_zero(nat))
        <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2) ) ) ).

% bit_0
tff(fact_4186_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_4187_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_4188_bit__mod__2__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Aa2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2)))),Na)
        <=> ( ( Na = zero_zero(nat) )
            & ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2) ) ) ) ).

% bit_mod_2_iff
tff(fact_4189_bin__nth__minus__Bit0,axiom,
    ! [Na: nat,W: num] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(num,int,numeral_numeral(int),bit0(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_4190_bin__nth__minus__Bit1,axiom,
    ! [Na: nat,W: num] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(num,int,numeral_numeral(int),bit1(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_4191_bit__xor__int__iff,axiom,
    ! [K: int,L: int,Na: nat] :
      ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L)),Na)
    <=> ~ ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),Na)
        <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),L),Na) ) ) ).

% bit_xor_int_iff
tff(fact_4192_bit__not__int__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(int,int,bit_ri4277139882892585799ns_not(int),K)),Na)
    <=> ~ aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),Na) ) ).

% bit_not_int_iff
tff(fact_4193_bit__or__int__iff,axiom,
    ! [K: int,L: int,Na: nat] :
      ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)),Na)
    <=> ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),Na)
        | aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),L),Na) ) ) ).

% bit_or_int_iff
tff(fact_4194_mmap__pres,axiom,
    ! [A: $tType,B: $tType,Xs2: list(A),P: assn,Fi: fun(A,heap_Time_Heap(B))] :
      ( ! [X2: A] :
          ( member(A,X2,aa(list(A),set(A),set2(A),Xs2))
         => hoare_hoare_triple(B,P,aa(A,heap_Time_Heap(B),Fi,X2),aTP_Lamp_dw(assn,fun(B,assn),P)) )
     => hoare_hoare_triple(list(B),P,vEBT_Example_mmap(A,B,Fi,Xs2),aTP_Lamp_dx(assn,fun(list(B),assn),P)) ) ).

% mmap_pres
tff(fact_4195_bit__xor__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A,Ba: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Aa2),Ba)),Na)
        <=> ~ ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),Na)
            <=> aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Ba),Na) ) ) ) ).

% bit_xor_iff
tff(fact_4196_bit__disjunctive__add__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Aa2: A,Ba: A,Na: nat] :
          ( ! [N: nat] :
              ( ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),N)
              | ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Ba),N) )
         => ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)),Na)
          <=> ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),Na)
              | aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Ba),Na) ) ) ) ) ).

% bit_disjunctive_add_iff
tff(fact_4197_bit__or__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A,Ba: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Aa2),Ba)),Na)
        <=> ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),Na)
            | aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Ba),Na) ) ) ) ).

% bit_or_iff
tff(fact_4198_bit__and__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A,Ba: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Aa2),Ba)),Na)
        <=> ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),Na)
            & aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Ba),Na) ) ) ) ).

% bit_and_iff
tff(fact_4199_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_4200_bit__and__int__iff,axiom,
    ! [K: int,L: int,Na: nat] :
      ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)),Na)
    <=> ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),Na)
        & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),L),Na) ) ) ).

% bit_and_int_iff
tff(fact_4201_bit__unset__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Aa2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),M),Aa2)),Na)
        <=> ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),Na)
            & ( M != Na ) ) ) ) ).

% bit_unset_bit_iff
tff(fact_4202_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_4203_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_4204_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_4205_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_4206_bit__take__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Aa2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,bit_se2584673776208193580ke_bit(A,M),Aa2)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
            & aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),Na) ) ) ) ).

% bit_take_bit_iff
tff(fact_4207_disjunctive__add,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A,Ba: A] :
          ( ! [N: nat] :
              ( ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),N)
              | ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Ba),N) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Aa2),Ba) ) ) ) ).

% disjunctive_add
tff(fact_4208_bit__of__bool__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Ba: $o,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa($o,A,zero_neq_one_of_bool(A),(Ba))),Na)
        <=> ( (Ba)
            & ( Na = zero_zero(nat) ) ) ) ) ).

% bit_of_bool_iff
tff(fact_4209_signed__take__bit__eq__if__positive,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Aa2: A,Na: nat] :
          ( ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),Na)
         => ( aa(A,A,bit_ri4674362597316999326ke_bit(A,Na),Aa2) = aa(A,A,bit_se2584673776208193580ke_bit(A,Na),Aa2) ) ) ) ).

% signed_take_bit_eq_if_positive
tff(fact_4210_htt__intro,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),Ba: nat] :
      ( hoare_hoare_triple(A,P,C2,Q)
     => ( simple8792101892335181110TBOUND(A,P,C2,Ba)
       => time_htt(A,P,C2,Q,Ba) ) ) ).

% htt_intro
tff(fact_4211_disjunctive__diff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Ba: A,Aa2: A] :
          ( ! [N: nat] :
              ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Ba),N)
             => aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),N) )
         => ( aa(A,A,minus_minus(A,Aa2),Ba) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Aa2),aa(A,A,bit_ri4277139882892585799ns_not(A),Ba)) ) ) ) ).

% disjunctive_diff
tff(fact_4212_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_4213_mIf__rule,axiom,
    ! [A: $tType,P: assn,Ba: heap_Time_Heap($o),T: heap_Time_Heap(A),E2: heap_Time_Heap(A),Q: fun(A,assn)] :
      ( hoare_hoare_triple(A,P,heap_Time_bind($o,A,Ba,aa(heap_Time_Heap(A),fun($o,heap_Time_Heap(A)),aTP_Lamp_af(heap_Time_Heap(A),fun(heap_Time_Heap(A),fun($o,heap_Time_Heap(A))),T),E2)),Q)
     => hoare_hoare_triple(A,P,vEBT_Example_mIf(A,Ba,T,E2),Q) ) ).

% mIf_rule
tff(fact_4214_flip__bit__eq__if,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,Aa2: A] :
          bit_se8732182000553998342ip_bit(A,Na,Aa2) = aa(A,A,
            aa(nat,fun(A,A),
              $ite(aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),Na),bit_se2638667681897837118et_bit(A),bit_se5668285175392031749et_bit(A)),
              Na),
            Aa2) ) ).

% flip_bit_eq_if
tff(fact_4215_bit__imp__take__bit__positive,axiom,
    ! [Na: nat,M: nat,K: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
     => ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),Na)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,bit_se2584673776208193580ke_bit(int,M),K)) ) ) ).

% bit_imp_take_bit_positive
tff(fact_4216_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_4217_exp__eq__0__imp__not__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Na: nat,Aa2: A] :
          ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na) = zero_zero(A) )
         => ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),Na) ) ) ).

% exp_eq_0_imp_not_bit
tff(fact_4218_bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Aa2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),aa(nat,nat,suc,Na))
        <=> aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2)))),Na) ) ) ).

% bit_Suc
tff(fact_4219_stable__imp__bit__iff__odd,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Aa2: A,Na: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2))) = Aa2 )
         => ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),Na)
          <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2) ) ) ) ).

% stable_imp_bit_iff_odd
tff(fact_4220_bit__iff__idd__imp__stable,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Aa2: A] :
          ( ! [N: nat] :
              ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),N)
            <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2))) = Aa2 ) ) ) ).

% bit_iff_idd_imp_stable
tff(fact_4221_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N: nat] :
          ( ! [M3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),M3)
             => ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),M3)
              <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),N) ) )
         => ~ ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
             => ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),aa(nat,nat,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_4222_signed__take__bit__eq__if__negative,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Aa2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),Na)
         => ( aa(A,A,bit_ri4674362597316999326ke_bit(A,Na),Aa2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),Aa2)),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,Na))) ) ) ) ).

% signed_take_bit_eq_if_negative
tff(fact_4223_num_Osize__gen_I1_J,axiom,
    size_num(one2) = zero_zero(nat) ).

% num.size_gen(1)
tff(fact_4224_vebt__buildupi__rule__basic,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => hoare_hoare_triple(vEBT_VEBTi,one_one(assn),vEBT_vebt_buildupi(Na),vEBT_Intf_vebt_assn(Na,bot_bot(set(nat)))) ) ).

% vebt_buildupi_rule_basic
tff(fact_4225_bit__iff__odd,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Aa2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),Na)
        <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na))) ) ) ).

% bit_iff_odd
tff(fact_4226_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A,Na: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Aa2),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = zero_zero(A) )
        <=> ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),Na) ) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_4227_bit__not__iff__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Aa2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Aa2)),Na)
        <=> ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na) != zero_zero(A) )
            & ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),Na) ) ) ) ).

% bit_not_iff_eq
tff(fact_4228_vebt__heap__rules_I8_J,axiom,
    ! [Na: nat,S: set(nat),Tia: vEBT_VEBTi,X: nat] : hoare_hoare_triple(vEBT_VEBTi,aa(vEBT_VEBTi,assn,vEBT_Intf_vebt_assn(Na,S),Tia),vEBT_vebt_deletei(Tia,X),vEBT_Intf_vebt_assn(Na,aa(set(nat),set(nat),minus_minus(set(nat),S),aa(set(nat),set(nat),insert(nat,X),bot_bot(set(nat)))))) ).

% vebt_heap_rules(8)
tff(fact_4229_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_4230_vebt__heap__rules_I1_J,axiom,
    ! [Na: nat] : hoare_hoare_triple(vEBT_VEBTi,pure_assn(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)),vEBT_vebt_buildupi(Na),vEBT_Intf_vebt_assn(Na,bot_bot(set(nat)))) ).

% vebt_heap_rules(1)
tff(fact_4231_even__bit__succ__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Aa2: A,Na: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2)
         => ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),Aa2)),Na)
          <=> ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),Na)
              | ( Na = zero_zero(nat) ) ) ) ) ) ).

% even_bit_succ_iff
tff(fact_4232_odd__bit__iff__bit__pred,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Aa2: A,Na: nat] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2)
         => ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),Na)
          <=> ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,minus_minus(A,Aa2),one_one(A))),Na)
              | ( Na = zero_zero(nat) ) ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_4233_signed__take__bit__def,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,Aa2: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,Na),Aa2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),Aa2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),Na))),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,Na)))) ) ).

% signed_take_bit_def
tff(fact_4234_bit__sum__mult__2__cases,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A,Ba: A,Na: nat] :
          ( ! [J2: nat] : ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),aa(nat,nat,suc,J2))
         => ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Ba))),Na)
          <=> $ite(Na = zero_zero(nat),~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2),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))),Ba)),Na)) ) ) ) ).

% bit_sum_mult_2_cases
tff(fact_4235_bit__rec,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Aa2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),Na)
        <=> $ite(Na = zero_zero(nat),~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,nat,minus_minus(nat,Na),one_one(nat)))) ) ) ).

% bit_rec
tff(fact_4236_Bit__Operations_Oset__bit__eq,axiom,
    ! [Na: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),Na),K) = aa(int,int,aa(int,fun(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_4237_unset__bit__eq,axiom,
    ! [Na: nat,K: int] : aa(int,int,aa(nat,fun(int,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_4238_take__bit__Suc__from__most,axiom,
    ! [Na: nat,K: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,Na)),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)),aa($o,int,zero_neq_one_of_bool(int),aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),Na)))),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)) ).

% take_bit_Suc_from_most
tff(fact_4239_num_Osize__gen_I2_J,axiom,
    ! [X23: num] : size_num(bit0(X23)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_num(X23)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size_gen(2)
tff(fact_4240_return__sp__rule,axiom,
    ! [A: $tType,P: assn,X: A] : hoare_hoare_triple(A,P,aa(A,heap_Time_Heap(A),heap_Time_return(A),X),aa(A,fun(A,assn),aTP_Lamp_dy(assn,fun(A,fun(A,assn)),P),X)) ).

% return_sp_rule
tff(fact_4241_floor__rat__def,axiom,
    ! [X: rat] : archim6421214686448440834_floor(rat,X) = the(int,aTP_Lamp_dz(rat,fun(int,$o),X)) ).

% floor_rat_def
tff(fact_4242_wait__bind__decon,axiom,
    ! [A: $tType,P: assn,M: heap_Time_Heap(A),Q: fun(A,assn),Na: nat] :
      ( hoare_hoare_triple(A,P,M,Q)
     => hoare_hoare_triple(A,P,heap_Time_bind(product_unit,A,heap_Time_wait(Na),aTP_Lamp_aj(heap_Time_Heap(A),fun(product_unit,heap_Time_Heap(A)),M)),Q) ) ).

% wait_bind_decon
tff(fact_4243_bind__rule_H,axiom,
    ! [A: $tType,B: $tType,C: $tType,P: assn,F2: heap_Time_Heap(B),R3: fun(B,assn),G: fun(B,heap_Time_Heap(C)),Q: fun(C,assn)] :
      ( ! [R5: A] : hoare_hoare_triple(B,P,F2,aa(fun(B,assn),fun(B,assn),aTP_Lamp_ea(assn,fun(fun(B,assn),fun(B,assn)),P),R3))
     => ( ! [X2: B] : hoare_hoare_triple(C,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,R3,X2)),P),aa(B,heap_Time_Heap(C),G,X2),Q)
       => hoare_hoare_triple(C,P,heap_Time_bind(B,C,F2,G),Q) ) ) ).

% bind_rule'
tff(fact_4244_wait__rule,axiom,
    ! [Na: nat] : hoare_hoare_triple(product_unit,one_one(assn),heap_Time_wait(Na),aTP_Lamp_eb(product_unit,assn)) ).

% wait_rule
tff(fact_4245_obtain__pos__sum,axiom,
    ! [R: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R)
     => ~ ! [S2: rat] :
            ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),S2)
           => ! [T2: rat] :
                ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),T2)
               => ( R != aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),S2),T2) ) ) ) ) ).

% obtain_pos_sum
tff(fact_4246_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_4247_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_4248_word__eqI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [U: word(A),V: word(A)] :
          ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(word(A),nat,size_size(word(A)),U))
             => ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),U),N)
              <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),V),N) ) )
         => ( U = V ) ) ) ).

% word_eqI
tff(fact_4249_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,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(word(A),nat,size_size(word(A)),W)) ) ) ).

% test_bit_size
tff(fact_4250_test__bit__over,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(A),nat,size_size(word(A)),X)),Na)
         => ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),X),Na) ) ) ).

% test_bit_over
tff(fact_4251_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_4252_lsb__this__or__next,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),one_one(word(A)))),zero_zero(nat))
         => aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),X),zero_zero(nat)) ) ) ).

% lsb_this_or_next
tff(fact_4253_word__leI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [U: word(A),V: word(A)] :
          ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(word(A),nat,size_size(word(A)),U))
             => ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),U),N)
               => aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),V),N) ) )
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),U),V) ) ) ).

% word_leI
tff(fact_4254_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,aa(nat,fun(nat,$o),ord_less(nat),I),Na)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(word(A),nat,size_size(word(A)),bit_se2239418461657761734s_mask(word(A),Na))) ) ) ) ).

% nth_mask
tff(fact_4255_bit__nat__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),bit_se5641148757651400278ts_bit(nat),aa(int,nat,nat2,K)),Na)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
        & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),Na) ) ) ).

% bit_nat_iff
tff(fact_4256_overflow__imp__lsb,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),one_one(word(A))) = zero_zero(word(A)) )
         => aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),X),zero_zero(nat)) ) ) ).

% overflow_imp_lsb
tff(fact_4257_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,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(word(A),nat,size_size(word(A)),W))
            & aa(nat,$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_4258_vebt__heap__rules_I2_J,axiom,
    ! [Na: nat,S: set(nat),Tia: vEBT_VEBTi,X: nat] : hoare_hoare_triple($o,aa(vEBT_VEBTi,assn,vEBT_Intf_vebt_assn(Na,S),Tia),vEBT_vebt_memberi(Tia,X),aa(nat,fun($o,assn),aa(vEBT_VEBTi,fun(nat,fun($o,assn)),aa(set(nat),fun(vEBT_VEBTi,fun(nat,fun($o,assn))),aTP_Lamp_ak(nat,fun(set(nat),fun(vEBT_VEBTi,fun(nat,fun($o,assn)))),Na),S),Tia),X)) ).

% vebt_heap_rules(2)
tff(fact_4259_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,aa(word(A),fun(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_4260_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_4261_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_4262_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_4263_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_4264_word__ops__nth__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,X: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(word(A),nat,size_size(word(A)),X))
         => ( ( aa(nat,$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_4265_vebt__heap__rules_I7_J,axiom,
    ! [Na: nat,S: set(nat),Tia: vEBT_VEBTi,X: nat,Y: nat] : hoare_hoare_triple(option(nat),aa(vEBT_VEBTi,assn,vEBT_Intf_vebt_assn(Na,S),Tia),vEBT_vebt_predi(Tia,X),aa(nat,fun(option(nat),assn),aa(nat,fun(nat,fun(option(nat),assn)),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(option(nat),assn))),aa(set(nat),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(option(nat),assn)))),aTP_Lamp_al(nat,fun(set(nat),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(option(nat),assn))))),Na),S),Tia),X),Y)) ).

% vebt_heap_rules(7)
tff(fact_4266_vebt__heap__rules_I6_J,axiom,
    ! [Na: nat,S: set(nat),Tia: vEBT_VEBTi,X: nat,Y: nat] : hoare_hoare_triple(option(nat),aa(vEBT_VEBTi,assn,vEBT_Intf_vebt_assn(Na,S),Tia),vEBT_vebt_succi(Tia,X),aa(nat,fun(option(nat),assn),aa(nat,fun(nat,fun(option(nat),assn)),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(option(nat),assn))),aa(set(nat),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(option(nat),assn)))),aTP_Lamp_ao(nat,fun(set(nat),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(option(nat),assn))))),Na),S),Tia),X),Y)) ).

% vebt_heap_rules(6)
tff(fact_4267_bind__rule,axiom,
    ! [A: $tType,B: $tType,P: assn,F2: heap_Time_Heap(A),R3: fun(A,assn),G: fun(A,heap_Time_Heap(B)),Q: fun(B,assn)] :
      ( hoare_hoare_triple(A,P,F2,R3)
     => ( ! [X2: A] : hoare_hoare_triple(B,aa(A,assn,R3,X2),aa(A,heap_Time_Heap(B),G,X2),Q)
       => hoare_hoare_triple(B,P,heap_Time_bind(A,B,F2,G),Q) ) ) ).

% bind_rule
tff(fact_4268_return__wp__rule,axiom,
    ! [A: $tType,Q: fun(A,assn),X: A] : hoare_hoare_triple(A,aa(A,assn,Q,X),aa(A,heap_Time_Heap(A),heap_Time_return(A),X),Q) ).

% return_wp_rule
tff(fact_4269_frame__rule,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),R3: assn] :
      ( hoare_hoare_triple(A,P,C2,Q)
     => hoare_hoare_triple(A,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),R3),C2,aa(assn,fun(A,assn),aTP_Lamp_ec(fun(A,assn),fun(assn,fun(A,assn)),Q),R3)) ) ).

% frame_rule
tff(fact_4270_cond__TBOUND__mIf,axiom,
    ! [A: $tType,P: assn,Cond: heap_Time_Heap($o),B1: nat,Q: fun($o,assn),T: heap_Time_Heap(A),B22: nat,E2: heap_Time_Heap(A),B32: nat] :
      ( simple8792101892335181110TBOUND($o,P,Cond,B1)
     => ( ! [H2: heap_ext(product_unit)] : simple8792101892335181110TBOUND(A,aa($o,assn,Q,time_the_res($o,Cond,H2)),T,B22)
       => ( ! [H2: heap_ext(product_unit)] : simple8792101892335181110TBOUND(A,aa($o,assn,Q,time_the_res($o,Cond,H2)),E2,B32)
         => ( hoare_hoare_triple($o,P,Cond,Q)
           => simple8792101892335181110TBOUND(A,P,vEBT_Example_mIf(A,Cond,T,E2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B1),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),B22),B32))) ) ) ) ) ).

% cond_TBOUND_mIf
tff(fact_4271_sup__Un__eq,axiom,
    ! [A: $tType,R3: set(A),Sa: set(A),X4: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),sup_sup(fun(A,$o)),aTP_Lamp_ab(set(A),fun(A,$o),R3)),aTP_Lamp_ab(set(A),fun(A,$o),Sa)),X4)
    <=> member(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),R3),Sa)) ) ).

% sup_Un_eq
tff(fact_4272_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_4273_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,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),bit1(one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(one2)))),aa(nat,int,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,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),bit1(one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(bit0(one2))))),aa(nat,int,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_4274_diff__rat__def,axiom,
    ! [Q3: rat,R: rat] : aa(rat,rat,minus_minus(rat,Q3),R) = aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),Q3),aa(rat,rat,uminus_uminus(rat),R)) ).

% diff_rat_def
tff(fact_4275_pow_Osimps_I1_J,axiom,
    ! [X: num] : pow(X,one2) = X ).

% pow.simps(1)
tff(fact_4276_cond__TBOUND__bind,axiom,
    ! [A: $tType,B: $tType,P: assn,C2: heap_Time_Heap(A),T1: nat,Q: fun(A,assn),D2: fun(A,heap_Time_Heap(B)),T22: nat] :
      ( simple8792101892335181110TBOUND(A,P,C2,T1)
     => ( hoare_hoare_triple(A,P,C2,Q)
       => ( ! [X2: A,H2: heap_ext(product_unit)] :
              ( ( X2 = time_the_res(A,C2,H2) )
             => simple8792101892335181110TBOUND(B,aa(A,assn,Q,X2),aa(A,heap_Time_Heap(B),D2,X2),T22) )
         => simple8792101892335181110TBOUND(B,P,heap_Time_bind(A,B,C2,D2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),T1),T22)) ) ) ) ).

% cond_TBOUND_bind
tff(fact_4277_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,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                            $let(
                              half: nat,
                              half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit1(bit0(one2))))),vEBT_V8646137997579335489_i_l_d(half))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),vEBT_V8646137997579335489_i_l_d(half))) )),
                          $let(
                            half: nat,
                            half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit1(one2))))),vEBT_V8646137997579335489_i_l_d(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,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_4278_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,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                            $let(
                              half: nat,
                              half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(bit0(one2))))),vEBT_V8346862874174094_d_u_p(half))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) )),
                          $let(
                            half: nat,
                            half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit1(bit0(one2))))),vEBT_V8346862874174094_d_u_p(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) ) ) )
                   => ~ 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_4279_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,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),bit1(bit0(one2)))),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),bit1(bit0(one2)))),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,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_4280_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,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(one2)))),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(one2)))),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,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_4281_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,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),aa(nat,nat,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,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),aa(nat,nat,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_4282_VEBT__internal_OTBOUND__buildupi,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => time_TBOUND(vEBT_VEBTi,vEBT_vebt_buildupi(Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit1(bit0(one2))))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ) ).

% VEBT_internal.TBOUND_buildupi
tff(fact_4283_powr__real__of__int,axiom,
    ! [X: real,Na: int] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( powr(real,X,aa(int,real,ring_1_of_int(real),Na)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Na),aa(nat,real,power_power(real,X),aa(int,nat,nat2,Na)),aa(real,real,inverse_inverse(real),aa(nat,real,power_power(real,X),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),Na))))) ) ) ).

% powr_real_of_int
tff(fact_4284_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_4285_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_4286_inverse__1,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ( aa(A,A,inverse_inverse(A),one_one(A)) = one_one(A) ) ) ).

% inverse_1
tff(fact_4287_inverse__mult__distrib,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,inverse_inverse(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),Aa2)),aa(A,A,inverse_inverse(A),Ba)) ) ).

% inverse_mult_distrib
tff(fact_4288_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),Aa2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),zero_zero(A)) ) ) ).

% inverse_nonpositive_iff_nonpositive
tff(fact_4289_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,inverse_inverse(A),Aa2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2) ) ) ).

% inverse_nonnegative_iff_nonnegative
tff(fact_4290_inverse__le__iff__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),zero_zero(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),Aa2)),aa(A,A,inverse_inverse(A),Ba))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2) ) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_4291_inverse__le__iff__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Ba)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),Aa2)),aa(A,A,inverse_inverse(A),Ba))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2) ) ) ) ) ).

% inverse_le_iff_le
tff(fact_4292_left__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Aa2: A] :
          ( ( Aa2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),Aa2)),Aa2) = one_one(A) ) ) ) ).

% left_inverse
tff(fact_4293_right__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Aa2: A] :
          ( ( Aa2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(A,A,inverse_inverse(A),Aa2)) = one_one(A) ) ) ) ).

% right_inverse
tff(fact_4294_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_4295_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_4296_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_4297_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_4298_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_4299_mult__commute__imp__mult__inverse__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Y: A,X: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Y),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),Y) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),Y)),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(A,A,inverse_inverse(A),Y)) ) ) ) ).

% mult_commute_imp_mult_inverse_commute
tff(fact_4300_norm__inverse__le__norm,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [R: real,X: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),R),real_V7770717601297561774m_norm(A,X))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,inverse_inverse(A),X))),aa(real,real,inverse_inverse(real),R)) ) ) ) ).

% norm_inverse_le_norm
tff(fact_4301_nonzero__inverse__mult__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Aa2: A,Ba: A] :
          ( ( Aa2 != zero_zero(A) )
         => ( ( Ba != zero_zero(A) )
           => ( aa(A,A,inverse_inverse(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),Ba)),aa(A,A,inverse_inverse(A),Aa2)) ) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_4302_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_4303_inverse__unique,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Aa2: A,Ba: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba) = one_one(A) )
         => ( aa(A,A,inverse_inverse(A),Aa2) = Ba ) ) ) ).

% inverse_unique
tff(fact_4304_inverse__eq__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Aa2: A] : aa(A,A,inverse_inverse(A),Aa2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),Aa2) ) ).

% inverse_eq_divide
tff(fact_4305_field__class_Ofield__divide__inverse,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba) = aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(A,A,inverse_inverse(A),Ba)) ) ).

% field_class.field_divide_inverse
tff(fact_4306_divide__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba) = aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(A,A,inverse_inverse(A),Ba)) ) ).

% divide_inverse
tff(fact_4307_divide__inverse__commute,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Aa2: A,Ba: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),Ba) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),Ba)),Aa2) ) ).

% divide_inverse_commute
tff(fact_4308_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_4309_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_4310_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_4311_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_4312_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_4313_VEBT__internal_OTBOUND__replicate,axiom,
    ! [A: $tType,X: heap_Time_Heap(A),C2: nat,Na: nat] :
      ( time_TBOUND(A,X,C2)
     => time_TBOUND(list(A),vEBT_VEBT_replicatei(A,Na,X),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),C2)),Na))) ) ).

% VEBT_internal.TBOUND_replicate
tff(fact_4314_exp__fdiffs,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X4: nat] : aa(nat,A,diffs(A,aTP_Lamp_ed(nat,A)),X4) = aa(A,A,inverse_inverse(A),semiring_char_0_fact(A,X4)) ) ).

% exp_fdiffs
tff(fact_4315_le__imp__inverse__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),Ba)),aa(A,A,inverse_inverse(A),Aa2)) ) ) ) ).

% le_imp_inverse_le_neg
tff(fact_4316_inverse__le__imp__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),Aa2)),aa(A,A,inverse_inverse(A),Ba))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2) ) ) ) ).

% inverse_le_imp_le_neg
tff(fact_4317_le__imp__inverse__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),Ba)),aa(A,A,inverse_inverse(A),Aa2)) ) ) ) ).

% le_imp_inverse_le
tff(fact_4318_inverse__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),Aa2)),aa(A,A,inverse_inverse(A),Ba))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2) ) ) ) ).

% inverse_le_imp_le
tff(fact_4319_inverse__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),X)),one_one(A))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),zero_zero(A))
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),X) ) ) ) ).

% inverse_le_1_iff
tff(fact_4320_one__less__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,inverse_inverse(A),Aa2)) ) ) ) ).

% one_less_inverse
tff(fact_4321_one__less__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,inverse_inverse(A),X))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),X)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),one_one(A)) ) ) ) ).

% one_less_inverse_iff
tff(fact_4322_division__ring__inverse__add,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Aa2: A,Ba: A] :
          ( ( Aa2 != zero_zero(A) )
         => ( ( Ba != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,inverse_inverse(A),Aa2)),aa(A,A,inverse_inverse(A),Ba)) = 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),Aa2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba))),aa(A,A,inverse_inverse(A),Ba)) ) ) ) ) ).

% division_ring_inverse_add
tff(fact_4323_inverse__add,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Aa2: A,Ba: A] :
          ( ( Aa2 != zero_zero(A) )
         => ( ( Ba != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,inverse_inverse(A),Aa2)),aa(A,A,inverse_inverse(A),Ba)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)),aa(A,A,inverse_inverse(A),Aa2))),aa(A,A,inverse_inverse(A),Ba)) ) ) ) ) ).

% inverse_add
tff(fact_4324_field__class_Ofield__inverse,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Aa2: A] :
          ( ( Aa2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),Aa2)),Aa2) = one_one(A) ) ) ) ).

% field_class.field_inverse
tff(fact_4325_division__ring__inverse__diff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Aa2: A,Ba: A] :
          ( ( Aa2 != zero_zero(A) )
         => ( ( Ba != zero_zero(A) )
           => ( aa(A,A,minus_minus(A,aa(A,A,inverse_inverse(A),Aa2)),aa(A,A,inverse_inverse(A),Ba)) = 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),Aa2)),aa(A,A,minus_minus(A,Ba),Aa2))),aa(A,A,inverse_inverse(A),Ba)) ) ) ) ) ).

% division_ring_inverse_diff
tff(fact_4326_nonzero__inverse__eq__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Aa2: A] :
          ( ( Aa2 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),Aa2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),Aa2) ) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_4327_inverse__powr,axiom,
    ! [Y: real,Aa2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
     => ( powr(real,aa(real,real,inverse_inverse(real),Y),Aa2) = aa(real,real,inverse_inverse(real),powr(real,Y,Aa2)) ) ) ).

% inverse_powr
tff(fact_4328_one__le__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,inverse_inverse(A),X))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),X)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),one_one(A)) ) ) ) ).

% one_le_inverse_iff
tff(fact_4329_inverse__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),X)),one_one(A))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),zero_zero(A))
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),X) ) ) ) ).

% inverse_less_1_iff
tff(fact_4330_one__le__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,inverse_inverse(A),Aa2)) ) ) ) ).

% one_le_inverse
tff(fact_4331_inverse__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),Aa2)),aa(A,A,inverse_inverse(A),Ba))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba) ) ) ) ) ).

% inverse_le_iff
tff(fact_4332_inverse__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),Aa2)),aa(A,A,inverse_inverse(A),Ba))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),Aa2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba) ) ) ) ) ).

% inverse_less_iff
tff(fact_4333_inverse__diff__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Aa2: A,Ba: A] :
          ( ( Aa2 != zero_zero(A) )
         => ( ( Ba != zero_zero(A) )
           => ( aa(A,A,minus_minus(A,aa(A,A,inverse_inverse(A),Aa2)),aa(A,A,inverse_inverse(A),Ba)) = 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),Aa2)),aa(A,A,minus_minus(A,Aa2),Ba))),aa(A,A,inverse_inverse(A),Ba))) ) ) ) ) ).

% inverse_diff_inverse
tff(fact_4334_reals__Archimedean,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),X)
         => ? [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N)))),X) ) ) ).

% reals_Archimedean
tff(fact_4335_forall__pos__mono__1,axiom,
    ! [P: fun(real,$o),E2: real] :
      ( ! [D4: real,E: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),D4),E)
         => ( aa(real,$o,P,D4)
           => 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,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
         => aa(real,$o,P,E2) ) ) ) ).

% forall_pos_mono_1
tff(fact_4336_forall__pos__mono,axiom,
    ! [P: fun(real,$o),E2: real] :
      ( ! [D4: real,E: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),D4),E)
         => ( aa(real,$o,P,D4)
           => 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,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
         => aa(real,$o,P,E2) ) ) ) ).

% forall_pos_mono
tff(fact_4337_real__arch__inverse,axiom,
    ! [E2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
    <=> ? [N2: nat] :
          ( ( N2 != zero_zero(nat) )
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N2)))
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N2))),E2) ) ) ).

% real_arch_inverse
tff(fact_4338_sqrt__divide__self__eq,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$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_4339_summable__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : summable(A,aTP_Lamp_ee(A,fun(nat,A),X)) ) ).

% summable_exp
tff(fact_4340_ex__inverse__of__nat__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),X)
         => ? [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),N))),X) ) ) ) ).

% ex_inverse_of_nat_less
tff(fact_4341_power__diff__conv__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: nat,Na: nat] :
          ( ( X != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(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_4342_log__inverse,axiom,
    ! [Aa2: real,X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Aa2)
     => ( ( Aa2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
         => ( aa(real,real,log(Aa2),aa(real,real,inverse_inverse(real),X)) = aa(real,real,uminus_uminus(real),aa(real,real,log(Aa2),X)) ) ) ) ) ).

% log_inverse
tff(fact_4343_Complex__eq,axiom,
    ! [Aa2: real,Ba: real] : complex2(Aa2,Ba) = aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),aa(real,complex,real_Vector_of_real(complex),Aa2)),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),aa(real,complex,real_Vector_of_real(complex),Ba))) ).

% Complex_eq
tff(fact_4344_exp__plus__inverse__exp,axiom,
    ! [X: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,exp(real),X)),aa(real,real,inverse_inverse(real),aa(real,real,exp(real),X)))) ).

% exp_plus_inverse_exp
tff(fact_4345_complex__split__polar,axiom,
    ! [Z: complex] :
    ? [R5: real,A3: real] : Z = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(real,complex,real_Vector_of_real(complex),R5)),aa(complex,complex,aa(complex,fun(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))))) ).

% complex_split_polar
tff(fact_4346_plus__inverse__ge__2,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,inverse_inverse(real),X))) ) ).

% plus_inverse_ge_2
tff(fact_4347_real__inv__sqrt__pow2,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$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_4348_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_4349_real__le__x__sinh,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(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_4350_real__le__abs__sinh,axiom,
    ! [X: real] : aa(real,$o,aa(real,fun(real,$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_4351_tan__sec,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(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_4352_cmod__unit__one,axiom,
    ! [Aa2: real] : real_V7770717601297561774m_norm(complex,aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),aa(real,complex,real_Vector_of_real(complex),cos(real,Aa2))),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),aa(real,complex,real_Vector_of_real(complex),sin(real,Aa2))))) = one_one(real) ).

% cmod_unit_one
tff(fact_4353_cmod__complex__polar,axiom,
    ! [R: real,Aa2: real] : real_V7770717601297561774m_norm(complex,aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(real,complex,real_Vector_of_real(complex),R)),aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),aa(real,complex,real_Vector_of_real(complex),cos(real,Aa2))),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),aa(real,complex,real_Vector_of_real(complex),sin(real,Aa2)))))) = aa(real,real,abs_abs(real),R) ).

% cmod_complex_polar
tff(fact_4354_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_4355_csqrt__ii,axiom,
    csqrt(imaginary_unit) = aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),aa(complex,complex,aa(complex,fun(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_4356_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_4357_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_4358_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_4359_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_4360_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_4361_DeMoivre,axiom,
    ! [Aa2: real,Na: nat] : aa(nat,complex,power_power(complex,cis(Aa2)),Na) = cis(aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),Aa2)) ).

% DeMoivre
tff(fact_4362_Arg__correct,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero(complex) )
     => ( ( aa(complex,complex,sgn_sgn(complex),Z) = cis(arg(Z)) )
        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),arg(Z))
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arg(Z)),pi) ) ) ).

% Arg_correct
tff(fact_4363_cis__Arg__unique,axiom,
    ! [Z: complex,X: real] :
      ( ( aa(complex,complex,sgn_sgn(complex),Z) = cis(X) )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),X)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),pi)
         => ( arg(Z) = X ) ) ) ) ).

% cis_Arg_unique
tff(fact_4364_cis__mult,axiom,
    ! [Aa2: real,Ba: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),cis(Aa2)),cis(Ba)) = cis(aa(real,real,aa(real,fun(real,real),plus_plus(real),Aa2),Ba)) ).

% cis_mult
tff(fact_4365_of__real__sqrt,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$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_4366_Arg__bounded,axiom,
    ! [Z: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),arg(Z))
      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arg(Z)),pi) ) ).

% Arg_bounded
tff(fact_4367_complex__inverse,axiom,
    ! [Aa2: real,Ba: real] : aa(complex,complex,inverse_inverse(complex),complex2(Aa2,Ba)) = complex2(aa(real,real,aa(real,fun(real,real),divide_divide(real),Aa2),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,Aa2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Ba),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),Ba)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,Aa2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Ba),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% complex_inverse
tff(fact_4368_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,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),X)),suminf(A,aTP_Lamp_ef(A,fun(nat,A),X))) ) ).

% exp_first_two_terms
tff(fact_4369_TBOUND__return,axiom,
    ! [A: $tType,X: A] : time_TBOUND(A,aa(A,heap_Time_Heap(A),heap_Time_return(A),X),one_one(nat)) ).

% TBOUND_return
tff(fact_4370_sinh__ln__real,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$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_4371_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_4372_mult__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V6157519004096292374lgebra(A)
     => ! [X: A,Aa2: real,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(A,A,real_V8093663219630862766scaleR(A,Aa2),Y)) = aa(A,A,real_V8093663219630862766scaleR(A,Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),X),Y)) ) ).

% mult_scaleR_right
tff(fact_4373_mult__scaleR__left,axiom,
    ! [A: $tType] :
      ( real_V6157519004096292374lgebra(A)
     => ! [Aa2: real,X: A,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,Aa2),X)),Y) = aa(A,A,real_V8093663219630862766scaleR(A,Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),X),Y)) ) ).

% mult_scaleR_left
tff(fact_4374_sinh__real__le__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sinh(real,X)),sinh(real,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y) ) ).

% sinh_real_le_iff
tff(fact_4375_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_4376_scaleR__eq__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Ba: A,U: real,Aa2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),aa(A,A,real_V8093663219630862766scaleR(A,U),Aa2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),aa(A,A,real_V8093663219630862766scaleR(A,U),Ba)) )
        <=> ( ( Aa2 = Ba )
            | ( U = one_one(real) ) ) ) ) ).

% scaleR_eq_iff
tff(fact_4377_sinh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sinh(real,X)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),zero_zero(real)) ) ).

% sinh_real_nonpos_iff
tff(fact_4378_sinh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),sinh(real,X))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X) ) ).

% sinh_real_nonneg_iff
tff(fact_4379_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_4380_scaleR__collapse,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [U: real,Aa2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,minus_minus(real,one_one(real)),U)),Aa2)),aa(A,A,real_V8093663219630862766scaleR(A,U),Aa2)) = Aa2 ) ).

% scaleR_collapse
tff(fact_4381_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,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y)) ) ) ) ).

% floor_add2
tff(fact_4382_frac__gt__0__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( aa(A,$o,aa(A,fun(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_4383_scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [U: num,W: num,Aa2: 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)),Aa2)) = 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))),Aa2) ) ).

% scaleR_times
tff(fact_4384_inverse__scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [V: num,W: num,Aa2: 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),V))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),Aa2)) = 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),V))),Aa2) ) ).

% inverse_scaleR_times
tff(fact_4385_fraction__scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [U: num,V: num,W: num,Aa2: 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),V))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),Aa2)) = 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),V))),Aa2) ) ).

% fraction_scaleR_times
tff(fact_4386_scaleR__half__double,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Aa2: 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,aa(A,fun(A,A),plus_plus(A),Aa2),Aa2)) = Aa2 ) ).

% scaleR_half_double
tff(fact_4387_Ints__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Aa2: A,Na: nat] :
          ( member(A,Aa2,ring_1_Ints(A))
         => member(A,aa(nat,A,power_power(A,Aa2),Na),ring_1_Ints(A)) ) ) ).

% Ints_power
tff(fact_4388_sums__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),Aa2: A,R: real] :
          ( aa(A,$o,sums(A,X5),Aa2)
         => aa(A,$o,sums(A,aa(real,fun(nat,A),aTP_Lamp_eg(fun(nat,A),fun(real,fun(nat,A)),X5),R)),aa(A,A,real_V8093663219630862766scaleR(A,R),Aa2)) ) ) ).

% sums_scaleR_right
tff(fact_4389_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_4390_Ints__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Aa2: A] :
          ( member(A,Aa2,ring_1_Ints(A))
         => member(A,aa(A,A,uminus_uminus(A),Aa2),ring_1_Ints(A)) ) ) ).

% Ints_minus
tff(fact_4391_Ints__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Aa2: A,Ba: A] :
          ( member(A,Aa2,ring_1_Ints(A))
         => ( member(A,Ba,ring_1_Ints(A))
           => member(A,aa(A,A,minus_minus(A,Aa2),Ba),ring_1_Ints(A)) ) ) ) ).

% Ints_diff
tff(fact_4392_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_4393_Ints__induct,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Q3: A,P: fun(A,$o)] :
          ( member(A,Q3,ring_1_Ints(A))
         => ( ! [Z3: int] : aa(A,$o,P,aa(int,A,ring_1_of_int(A),Z3))
           => aa(A,$o,P,Q3) ) ) ) ).

% Ints_induct
tff(fact_4394_Ints__cases,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Q3: A] :
          ( member(A,Q3,ring_1_Ints(A))
         => ~ ! [Z3: int] : Q3 != aa(int,A,ring_1_of_int(A),Z3) ) ) ).

% Ints_cases
tff(fact_4395_summable__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),R: real] :
          ( summable(A,X5)
         => summable(A,aa(real,fun(nat,A),aTP_Lamp_eg(fun(nat,A),fun(real,fun(nat,A)),X5),R)) ) ) ).

% summable_scaleR_right
tff(fact_4396_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_4397_Ints__mult,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Aa2: A,Ba: A] :
          ( member(A,Aa2,ring_1_Ints(A))
         => ( member(A,Ba,ring_1_Ints(A))
           => member(A,aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba),ring_1_Ints(A)) ) ) ) ).

% Ints_mult
tff(fact_4398_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_4399_Ints__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Aa2: A] :
          ( member(A,Aa2,ring_1_Ints(A))
         => member(A,aa(A,A,abs_abs(A),Aa2),ring_1_Ints(A)) ) ) ).

% Ints_abs
tff(fact_4400_Ints__add,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Aa2: A,Ba: A] :
          ( member(A,Aa2,ring_1_Ints(A))
         => ( member(A,Ba,ring_1_Ints(A))
           => member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba),ring_1_Ints(A)) ) ) ) ).

% Ints_add
tff(fact_4401_scaleR__right__distrib,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Aa2: real,X: A,Y: A] : aa(A,A,real_V8093663219630862766scaleR(A,Aa2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,Aa2),X)),aa(A,A,real_V8093663219630862766scaleR(A,Aa2),Y)) ) ).

% scaleR_right_distrib
tff(fact_4402_Ints__0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => member(A,zero_zero(A),ring_1_Ints(A)) ) ).

% Ints_0
tff(fact_4403_Ints__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => member(A,one_one(A),ring_1_Ints(A)) ) ).

% Ints_1
tff(fact_4404_scaleR__left__distrib,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Aa2: real,Ba: real,X: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),plus_plus(real),Aa2),Ba)),X) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,Aa2),X)),aa(A,A,real_V8093663219630862766scaleR(A,Ba),X)) ) ).

% scaleR_left_distrib
tff(fact_4405_scaleR__left_Oadd,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: real,Y: real,Xa2: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y)),Xa2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,X),Xa2)),aa(A,A,real_V8093663219630862766scaleR(A,Y),Xa2)) ) ).

% scaleR_left.add
tff(fact_4406_of__real__def,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [R: real] : aa(real,A,real_Vector_of_real(A),R) = aa(A,A,real_V8093663219630862766scaleR(A,R),one_one(A)) ) ).

% of_real_def
tff(fact_4407_scaleR__conv__of__real,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [R: real,X: A] : aa(A,A,real_V8093663219630862766scaleR(A,R),X) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(real,A,real_Vector_of_real(A),R)),X) ) ).

% scaleR_conv_of_real
tff(fact_4408_Ints__double__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Aa2: A] :
          ( member(A,Aa2,ring_1_Ints(A))
         => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Aa2) = zero_zero(A) )
          <=> ( Aa2 = zero_zero(A) ) ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_4409_suminf__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),R: real] :
          ( summable(A,X5)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,R),suminf(A,X5)) = suminf(A,aa(real,fun(nat,A),aTP_Lamp_eg(fun(nat,A),fun(real,fun(nat,A)),X5),R)) ) ) ) ).

% suminf_scaleR_right
tff(fact_4410_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_eh(fun(nat,real),fun(A,fun(nat,A)),X5),X)) ) ) ).

% summable_scaleR_left
tff(fact_4411_sums__scaleR__left,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,real),Aa2: real,X: A] :
          ( aa(real,$o,sums(real,X5),Aa2)
         => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_eh(fun(nat,real),fun(A,fun(nat,A)),X5),X)),aa(A,A,real_V8093663219630862766scaleR(A,Aa2),X)) ) ) ).

% sums_scaleR_left
tff(fact_4412_scaleR__right__mono__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [Ba: real,Aa2: real,C2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ba),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,Aa2),C2)),aa(A,A,real_V8093663219630862766scaleR(A,Ba),C2)) ) ) ) ).

% scaleR_right_mono_neg
tff(fact_4413_scaleR__right__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [Aa2: real,Ba: real,X: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,Aa2),X)),aa(A,A,real_V8093663219630862766scaleR(A,Ba),X)) ) ) ) ).

% scaleR_right_mono
tff(fact_4414_scaleR__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,Aa2: A,Ba: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),Aa2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),Ba))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba) ) ) ) ).

% scaleR_le_cancel_left_pos
tff(fact_4415_scaleR__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,Aa2: A,Ba: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),Aa2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),Ba))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2) ) ) ) ).

% scaleR_le_cancel_left_neg
tff(fact_4416_scaleR__le__cancel__left,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),Aa2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),Ba))
        <=> ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba) )
            & ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2) ) ) ) ) ).

% scaleR_le_cancel_left
tff(fact_4417_scaleR__left__mono__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [Ba: A,Aa2: A,C2: real] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),C2),zero_zero(real))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),Aa2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),Ba)) ) ) ) ).

% scaleR_left_mono_neg
tff(fact_4418_scaleR__left__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [X: A,Y: A,Aa2: real] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Aa2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,Aa2),X)),aa(A,A,real_V8093663219630862766scaleR(A,Aa2),Y)) ) ) ) ).

% scaleR_left_mono
tff(fact_4419_Real__Vector__Spaces_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [Aa2: real,E2: A,C2: A,Ba: real,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,Aa2),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,Ba),E2)),D2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,minus_minus(real,Ba),Aa2)),E2)),D2)) ) ) ).

% Real_Vector_Spaces.le_add_iff2
tff(fact_4420_Real__Vector__Spaces_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [Aa2: real,E2: A,C2: A,Ba: real,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,Aa2),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,Ba),E2)),D2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,minus_minus(real,Aa2),Ba)),E2)),C2)),D2) ) ) ).

% Real_Vector_Spaces.le_add_iff1
tff(fact_4421_Ints__odd__nonzero,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Aa2: A] :
          ( member(A,Aa2,ring_1_Ints(A))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),Aa2)),Aa2) != zero_zero(A) ) ) ) ).

% Ints_odd_nonzero
tff(fact_4422_of__int__divide__in__Ints,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [Ba: int,Aa2: int] :
          ( aa(int,$o,dvd_dvd(int,Ba),Aa2)
         => member(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(int,A,ring_1_of_int(A),Aa2)),aa(int,A,ring_1_of_int(A),Ba)),ring_1_Ints(A)) ) ) ).

% of_int_divide_in_Ints
tff(fact_4423_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_4424_scaleR__le__0__iff,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [Aa2: real,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,Aa2),Ba)),zero_zero(A))
        <=> ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Aa2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),zero_zero(A)) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Aa2),zero_zero(real))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Ba) )
            | ( Aa2 = zero_zero(real) ) ) ) ) ).

% scaleR_le_0_iff
tff(fact_4425_zero__le__scaleR__iff,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [Aa2: real,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,Aa2),Ba))
        <=> ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Aa2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Ba) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Aa2),zero_zero(real))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),zero_zero(A)) )
            | ( Aa2 = zero_zero(real) ) ) ) ) ).

% zero_le_scaleR_iff
tff(fact_4426_scaleR__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [Aa2: real,Ba: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,Aa2),Ba)) ) ) ) ).

% scaleR_nonpos_nonpos
tff(fact_4427_scaleR__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [Aa2: real,X: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,Aa2),X)),zero_zero(A)) ) ) ) ).

% scaleR_nonpos_nonneg
tff(fact_4428_scaleR__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [Aa2: real,X: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,Aa2),X)),zero_zero(A)) ) ) ) ).

% scaleR_nonneg_nonpos
tff(fact_4429_scaleR__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [Aa2: real,X: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Aa2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,Aa2),X)) ) ) ) ).

% scaleR_nonneg_nonneg
tff(fact_4430_split__scaleR__pos__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [Aa2: real,Ba: A] :
          ( ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Aa2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Ba) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),zero_zero(real))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),zero_zero(A)) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,Aa2),Ba)) ) ) ).

% split_scaleR_pos_le
tff(fact_4431_split__scaleR__neg__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [Aa2: real,X: A] :
          ( ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Aa2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),zero_zero(A)) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),zero_zero(real))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,Aa2),X)),zero_zero(A)) ) ) ).

% split_scaleR_neg_le
tff(fact_4432_scaleR__mono_H,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [Aa2: real,Ba: real,C2: A,D2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Aa2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,Aa2),C2)),aa(A,A,real_V8093663219630862766scaleR(A,Ba),D2)) ) ) ) ) ) ).

% scaleR_mono'
tff(fact_4433_scaleR__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [Aa2: real,Ba: real,X: A,Y: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Ba)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,Aa2),X)),aa(A,A,real_V8093663219630862766scaleR(A,Ba),Y)) ) ) ) ) ) ).

% scaleR_mono
tff(fact_4434_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_eh(fun(nat,real),fun(A,fun(nat,A)),X5),X)) ) ) ) ).

% suminf_scaleR_left
tff(fact_4435_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,aa(A,fun(A,A),plus_plus(A),X),X) ) ).

% scaleR_2
tff(fact_4436_scaleR__left__le__one__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [X: A,Aa2: real] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),one_one(real))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,Aa2),X)),X) ) ) ) ).

% scaleR_left_le_one_le
tff(fact_4437_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,aa(A,fun(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_4438_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,aa(A,fun(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_4439_neg__le__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,Aa2: A,Ba: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),Ba))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),aa(A,A,real_V8093663219630862766scaleR(A,C2),Aa2)) ) ) ) ).

% neg_le_divideR_eq
tff(fact_4440_neg__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,Ba: A,Aa2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),Ba)),Aa2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),Aa2)),Ba) ) ) ) ).

% neg_divideR_le_eq
tff(fact_4441_pos__le__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,Aa2: A,Ba: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),Ba))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),Aa2)),Ba) ) ) ) ).

% pos_le_divideR_eq
tff(fact_4442_pos__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,Ba: A,Aa2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),Ba)),Aa2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),aa(A,A,real_V8093663219630862766scaleR(A,C2),Aa2)) ) ) ) ).

% pos_divideR_le_eq
tff(fact_4443_Ints__odd__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Aa2: A] :
          ( member(A,Aa2,ring_1_Ints(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),Aa2)),Aa2)),zero_zero(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),zero_zero(A)) ) ) ) ).

% Ints_odd_less_0
tff(fact_4444_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,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,abs_abs(A),X)) ) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_4445_Ints__nonzero__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( member(A,X,ring_1_Ints(A))
         => ( aa(A,$o,aa(A,fun(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_4446_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,aa(A,fun(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_4447_summable__exp__generic,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : summable(A,aTP_Lamp_ei(A,fun(nat,A),X)) ) ).

% summable_exp_generic
tff(fact_4448_sin__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,$o,sums(A,aTP_Lamp_ej(A,fun(nat,A),X)),sin(A,X)) ) ).

% sin_converges
tff(fact_4449_sin__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X4: A] : sin(A,X4) = suminf(A,aTP_Lamp_ej(A,fun(nat,A),X4)) ) ).

% sin_def
tff(fact_4450_cos__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,$o,sums(A,aTP_Lamp_ek(A,fun(nat,A),X)),cos(A,X)) ) ).

% cos_converges
tff(fact_4451_cos__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X4: A] : cos(A,X4) = suminf(A,aTP_Lamp_ek(A,fun(nat,A),X4)) ) ).

% cos_def
tff(fact_4452_summable__norm__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : summable(real,aTP_Lamp_el(A,fun(nat,real),X)) ) ).

% summable_norm_sin
tff(fact_4453_summable__norm__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : summable(real,aTP_Lamp_em(A,fun(nat,real),X)) ) ).

% summable_norm_cos
tff(fact_4454_neg__minus__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,Ba: A,Aa2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),Ba))),Aa2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),Aa2)),aa(A,A,uminus_uminus(A),Ba)) ) ) ) ).

% neg_minus_divideR_le_eq
tff(fact_4455_neg__le__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,Aa2: A,Ba: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),Ba)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Ba)),aa(A,A,real_V8093663219630862766scaleR(A,C2),Aa2)) ) ) ) ).

% neg_le_minus_divideR_eq
tff(fact_4456_pos__minus__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,Ba: A,Aa2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),Ba))),Aa2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Ba)),aa(A,A,real_V8093663219630862766scaleR(A,C2),Aa2)) ) ) ) ).

% pos_minus_divideR_le_eq
tff(fact_4457_pos__le__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,Aa2: A,Ba: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),Ba)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),Aa2)),aa(A,A,uminus_uminus(A),Ba)) ) ) ) ).

% pos_le_minus_divideR_eq
tff(fact_4458_sinh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,$o,sums(A,aTP_Lamp_en(A,fun(nat,A),X)),sinh(A,X)) ) ).

% sinh_converges
tff(fact_4459_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_4460_exp__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,$o,sums(A,aTP_Lamp_ei(A,fun(nat,A),X)),aa(A,A,exp(A),X)) ) ).

% exp_converges
tff(fact_4461_exp__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X4: A] : aa(A,A,exp(A),X4) = suminf(A,aTP_Lamp_ei(A,fun(nat,A),X4)) ) ).

% exp_def
tff(fact_4462_summable__norm__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : summable(real,aTP_Lamp_eo(A,fun(nat,real),X)) ) ).

% summable_norm_exp
tff(fact_4463_sin__minus__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,$o,sums(A,aTP_Lamp_ep(A,fun(nat,A),X)),sin(A,X)) ) ).

% sin_minus_converges
tff(fact_4464_cos__minus__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,$o,sums(A,aTP_Lamp_eq(A,fun(nat,A),X)),cos(A,X)) ) ).

% cos_minus_converges
tff(fact_4465_le__mult__floor__Ints,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & linordered_idom(B) )
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)
         => ( member(A,Aa2,ring_1_Ints(A))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(int,B,ring_1_of_int(B),aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(A,Aa2)),archim6421214686448440834_floor(A,Ba)))),aa(int,B,ring_1_of_int(B),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)))) ) ) ) ).

% le_mult_floor_Ints
tff(fact_4466_mult__ceiling__le__Ints,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & linordered_idom(B) )
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)
         => ( member(A,Aa2,ring_1_Ints(A))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(int,B,ring_1_of_int(B),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba)))),aa(int,B,ring_1_of_int(B),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(A,Aa2)),archimedean_ceiling(A,Ba)))) ) ) ) ).

% mult_ceiling_le_Ints
tff(fact_4467_frac__unique__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Aa2: A] :
          ( ( archimedean_frac(A,X) = Aa2 )
        <=> ( member(A,aa(A,A,minus_minus(A,X),Aa2),ring_1_Ints(A))
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Aa2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),one_one(A)) ) ) ) ).

% frac_unique_iff
tff(fact_4468_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_4469_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_4470_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_4471_TBOUND__mono,axiom,
    ! [A: $tType,C2: heap_Time_Heap(A),T: nat,T3: nat] :
      ( time_TBOUND(A,C2,T)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),T),T3)
       => time_TBOUND(A,C2,T3) ) ) ).

% TBOUND_mono
tff(fact_4472_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_4473_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,aa(A,fun(A,A),plus_plus(A),one_one(A)),suminf(A,aTP_Lamp_er(A,fun(nat,A),X))) ) ).

% exp_first_term
tff(fact_4474_cosh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,$o,sums(A,aTP_Lamp_es(A,fun(nat,A),X)),cosh(A,X)) ) ).

% cosh_converges
tff(fact_4475_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_eu(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_4476_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_ew(A,fun(A,fun(nat,A)),X),Y)),cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y))) ) ).

% sums_cos_x_plus_y
tff(fact_4477_Arg__def,axiom,
    ! [Z: complex] :
      arg(Z) = $ite(Z = zero_zero(complex),zero_zero(real),fChoice(real,aTP_Lamp_ex(complex,fun(real,$o),Z))) ).

% Arg_def
tff(fact_4478_atMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,K: A] :
          ( member(A,I,set_ord_atMost(A,K))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),K) ) ) ).

% atMost_iff
tff(fact_4479_Ints__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(B)
     => ! [A2: set(A),F2: fun(A,B)] :
          ( ! [X2: A] :
              ( member(A,X2,A2)
             => member(B,aa(A,B,F2,X2),ring_1_Ints(B)) )
         => member(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A2),ring_1_Ints(B)) ) ) ).

% Ints_sum
tff(fact_4480_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_ey(fun(B,nat),fun(B,A),F2)),A2) ) ).

% of_nat_sum
tff(fact_4481_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_ez(fun(B,int),fun(B,A),F2)),A2) ) ).

% of_int_sum
tff(fact_4482_of__real__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [F2: fun(B,real),S: set(B)] : aa(real,A,real_Vector_of_real(A),aa(set(B),real,groups7311177749621191930dd_sum(B,real,F2),S)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_fa(fun(B,real),fun(B,A),F2)),S) ) ).

% of_real_sum
tff(fact_4483_atMost__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_ord_atMost(A,X)),set_ord_atMost(A,Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ).

% atMost_subset_iff
tff(fact_4484_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_4485_some__insert__self,axiom,
    ! [A: $tType,Sa: set(A)] :
      ( ( Sa != bot_bot(set(A)) )
     => ( aa(set(A),set(A),insert(A,fChoice(A,aTP_Lamp_ab(set(A),fun(A,$o),Sa))),Sa) = Sa ) ) ).

% some_insert_self
tff(fact_4486_sum_OatMost__Suc,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,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_ord_atMost(nat,Na))),aa(nat,A,G,aa(nat,nat,suc,Na))) ) ).

% sum.atMost_Suc
tff(fact_4487_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_fb(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_4488_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_fc(fun(B,real),fun(A,fun(B,A)),G),X)),A2) ) ).

% scaleR_left.sum
tff(fact_4489_scaleR__right_Osum,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Aa2: real,G: fun(B,A),A2: set(B)] : aa(A,A,real_V8093663219630862766scaleR(A,Aa2),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_fd(real,fun(fun(B,A),fun(B,A)),Aa2),G)),A2) ) ).

% scaleR_right.sum
tff(fact_4490_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_fc(fun(B,real),fun(A,fun(B,A)),F2),X)),A2) ) ).

% scaleR_sum_left
tff(fact_4491_scaleR__sum__right,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Aa2: real,F2: fun(B,A),A2: set(B)] : aa(A,A,real_V8093663219630862766scaleR(A,Aa2),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_fd(real,fun(fun(B,A),fun(B,A)),Aa2),F2)),A2) ) ).

% scaleR_sum_right
tff(fact_4492_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,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_fe(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,Na))) ) ).

% sum.atMost_Suc_shift
tff(fact_4493_bounded__imp__summable,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linord2810124833399127020strict(A)
        & topolo1944317154257567458pology(A) )
     => ! [Aa2: fun(nat,A),B2: A] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,Aa2,N))
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Aa2),set_ord_atMost(nat,N))),B2)
           => summable(A,Aa2) ) ) ) ).

% bounded_imp_summable
tff(fact_4494_sum__choose__upper,axiom,
    ! [M: nat,Na: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_ff(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_4495_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_4496_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_ds(fun(A,$o),fun(A,$o),P))) ) ).

% verit_sko_forall
tff(fact_4497_verit__sko__forall_H,axiom,
    ! [A: $tType,P: fun(A,$o),A2: $o] :
      ( ( aa(A,$o,P,fChoice(A,aTP_Lamp_ds(fun(A,$o),fun(A,$o),P)))
      <=> (A2) )
     => ( ! [X_1: A] : aa(A,$o,P,X_1)
      <=> (A2) ) ) ).

% verit_sko_forall'
tff(fact_4498_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_4499_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_4500_verit__sko__ex__indirect2,axiom,
    ! [A: $tType,X: A,P: fun(A,$o),P3: fun(A,$o)] :
      ( ( X = fChoice(A,P) )
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
          <=> aa(A,$o,P3,X2) )
       => ( ? [X_1: A] : aa(A,$o,P3,X_1)
        <=> aa(A,$o,P,X) ) ) ) ).

% verit_sko_ex_indirect2
tff(fact_4501_verit__sko__forall__indirect,axiom,
    ! [A: $tType,X: A,P: fun(A,$o)] :
      ( ( X = fChoice(A,aTP_Lamp_ds(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_4502_verit__sko__forall__indirect2,axiom,
    ! [A: $tType,X: A,P: fun(A,$o),P3: fun(A,$o)] :
      ( ( X = fChoice(A,aTP_Lamp_ds(fun(A,$o),fun(A,$o),P)) )
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
          <=> aa(A,$o,P3,X2) )
       => ( ! [X_1: A] : aa(A,$o,P3,X_1)
        <=> aa(A,$o,P,X) ) ) ) ).

% verit_sko_forall_indirect2
tff(fact_4503_mod__sum__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [F2: fun(B,A),Aa2: 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_fg(fun(B,A),fun(A,fun(B,A)),F2),Aa2)),A2)),Aa2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A2)),Aa2) ) ).

% mod_sum_eq
tff(fact_4504_sums__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [I5: set(A),F2: fun(A,fun(nat,B)),X: fun(A,B)] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => aa(B,$o,sums(B,aa(A,fun(nat,B),F2,I2)),aa(A,B,X,I2)) )
         => aa(B,$o,sums(B,aa(fun(A,fun(nat,B)),fun(nat,B),aTP_Lamp_fi(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_4505_summable__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [I5: set(A),F2: fun(A,fun(nat,B))] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => summable(B,aa(A,fun(nat,B),F2,I2)) )
         => summable(B,aa(fun(A,fun(nat,B)),fun(nat,B),aTP_Lamp_fi(set(A),fun(fun(A,fun(nat,B)),fun(nat,B)),I5),F2)) ) ) ).

% summable_sum
tff(fact_4506_some__elem,axiom,
    ! [A: $tType,Sa: set(A)] :
      ( ( Sa != bot_bot(set(A)) )
     => member(A,fChoice(A,aTP_Lamp_ab(set(A),fun(A,$o),Sa)),Sa) ) ).

% some_elem
tff(fact_4507_sum__norm__le,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Sa: set(A),F2: fun(A,B),G: fun(A,real)] :
          ( ! [X2: A] :
              ( member(A,X2,Sa)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,X2))),aa(A,real,G,X2)) )
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),Sa))),aa(set(A),real,groups7311177749621191930dd_sum(A,real,G),Sa)) ) ) ).

% sum_norm_le
tff(fact_4508_norm__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(B,A),A2: set(B)] : aa(real,$o,aa(real,fun(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_fj(fun(B,A),fun(B,real),F2)),A2)) ) ).

% norm_sum
tff(fact_4509_choose__rising__sum_I2_J,axiom,
    ! [Na: nat,M: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_fk(nat,fun(nat,nat),Na)),set_ord_atMost(nat,M)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M)),one_one(nat))),M) ).

% choose_rising_sum(2)
tff(fact_4510_choose__rising__sum_I1_J,axiom,
    ! [Na: nat,M: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_fk(nat,fun(nat,nat),Na)),set_ord_atMost(nat,M)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M)),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat))) ).

% choose_rising_sum(1)
tff(fact_4511_sum__choose__lower,axiom,
    ! [R: nat,Na: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_fl(nat,fun(nat,nat),R)),set_ord_atMost(nat,Na)) = aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),R),Na))),Na) ).

% sum_choose_lower
tff(fact_4512_atMost__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : set_ord_atMost(A,U) = collect(A,aTP_Lamp_fm(A,fun(A,$o),U)) ) ).

% atMost_def
tff(fact_4513_polyfun__eq__coeffs,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Na: nat,D2: fun(nat,A)] :
          ( ! [X6: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_fn(fun(nat,A),fun(A,fun(nat,A)),C2),X6)),set_ord_atMost(nat,Na)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_fn(fun(nat,A),fun(A,fun(nat,A)),D2),X6)),set_ord_atMost(nat,Na))
        <=> ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I4),Na)
             => ( aa(nat,A,C2,I4) = aa(nat,A,D2,I4) ) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_4514_zero__polynom__imp__zero__coeffs,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_mult(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [C2: fun(nat,A),Na: nat,K: nat] :
          ( ! [W3: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_fo(fun(nat,A),fun(A,fun(nat,A)),C2),W3)),set_ord_atMost(nat,Na)) = zero_zero(A)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
           => ( aa(nat,A,C2,K) = zero_zero(A) ) ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_4515_polyfun__eq__0,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Na: nat] :
          ( ! [X6: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_fn(fun(nat,A),fun(A,fun(nat,A)),C2),X6)),set_ord_atMost(nat,Na)) = zero_zero(A)
        <=> ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I4),Na)
             => ( aa(nat,A,C2,I4) = zero_zero(A) ) ) ) ) ).

% polyfun_eq_0
tff(fact_4516_gbinomial__parallel__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Aa2: A,Na: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_fp(A,fun(nat,A),Aa2)),set_ord_atMost(nat,Na)) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),aa(nat,A,semiring_1_of_nat(A),Na))),one_one(A))),Na) ) ).

% gbinomial_parallel_sum
tff(fact_4517_sum__choose__diagonal,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_fq(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_4518_vandermonde,axiom,
    ! [M: nat,Na: nat,R: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_fr(nat,fun(nat,fun(nat,fun(nat,nat))),M),Na),R)),set_ord_atMost(nat,R)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)),R) ).

% vandermonde
tff(fact_4519_cosh__real__nonpos__le__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),zero_zero(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),zero_zero(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cosh(real,X)),cosh(real,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),X) ) ) ) ).

% cosh_real_nonpos_le_iff
tff(fact_4520_cosh__real__nonneg__le__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cosh(real,X)),cosh(real,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y) ) ) ) ).

% cosh_real_nonneg_le_iff
tff(fact_4521_cosh__real__nonneg,axiom,
    ! [X: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),cosh(real,X)) ).

% cosh_real_nonneg
tff(fact_4522_cosh__real__ge__1,axiom,
    ! [X: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),cosh(real,X)) ).

% cosh_real_ge_1
tff(fact_4523_sinh__le__cosh__real,axiom,
    ! [X: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sinh(real,X)),cosh(real,X)) ).

% sinh_le_cosh_real
tff(fact_4524_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_4525_suminf__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [I5: set(A),F2: fun(A,fun(nat,B))] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => summable(B,aa(A,fun(nat,B),F2,I2)) )
         => ( suminf(B,aa(fun(A,fun(nat,B)),fun(nat,B),aTP_Lamp_fi(set(A),fun(fun(A,fun(nat,B)),fun(nat,B)),I5),F2)) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,aTP_Lamp_fs(fun(A,fun(nat,B)),fun(A,B),F2)),I5) ) ) ) ).

% suminf_sum
tff(fact_4526_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_4527_binomial,axiom,
    ! [Aa2: nat,Ba: nat,Na: nat] : aa(nat,nat,power_power(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Aa2),Ba)),Na) = aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_ft(nat,fun(nat,fun(nat,fun(nat,nat))),Aa2),Ba),Na)),set_ord_atMost(nat,Na)) ).

% binomial
tff(fact_4528_summable__Cauchy__product,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Aa2: fun(nat,A),Ba: fun(nat,A)] :
          ( summable(real,aTP_Lamp_fu(fun(nat,A),fun(nat,real),Aa2))
         => ( summable(real,aTP_Lamp_fu(fun(nat,A),fun(nat,real),Ba))
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_fw(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Aa2),Ba)) ) ) ) ).

% summable_Cauchy_product
tff(fact_4529_Cauchy__product,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Aa2: fun(nat,A),Ba: fun(nat,A)] :
          ( summable(real,aTP_Lamp_fu(fun(nat,A),fun(nat,real),Aa2))
         => ( summable(real,aTP_Lamp_fu(fun(nat,A),fun(nat,real),Ba))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,Aa2)),suminf(A,Ba)) = suminf(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_fw(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Aa2),Ba)) ) ) ) ) ).

% Cauchy_product
tff(fact_4530_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_fx(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,Na)) ) ).

% sum.in_pairs_0
tff(fact_4531_polynomial__product,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [M: nat,Aa2: fun(nat,A),Na: nat,Ba: fun(nat,A),X: A] :
          ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),I2)
             => ( aa(nat,A,Aa2,I2) = zero_zero(A) ) )
         => ( ! [J2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),J2)
               => ( aa(nat,A,Ba,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_fy(fun(nat,A),fun(A,fun(nat,A)),Aa2),X)),set_ord_atMost(nat,M))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_fy(fun(nat,A),fun(A,fun(nat,A)),Ba),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_ga(fun(nat,A),fun(fun(nat,A),fun(A,fun(nat,A))),Aa2),Ba),X)),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na))) ) ) ) ) ).

% polynomial_product
tff(fact_4532_gbinomial__sum__lower__neg,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Aa2: A,M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_gb(A,fun(nat,A),Aa2)),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,Aa2),one_one(A))),M)) ) ).

% gbinomial_sum_lower_neg
tff(fact_4533_polynomial__product__nat,axiom,
    ! [M: nat,Aa2: fun(nat,nat),Na: nat,Ba: fun(nat,nat),X: nat] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),I2)
         => ( aa(nat,nat,Aa2,I2) = zero_zero(nat) ) )
     => ( ! [J2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),J2)
           => ( aa(nat,nat,Ba,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_gc(fun(nat,nat),fun(nat,fun(nat,nat)),Aa2),X)),set_ord_atMost(nat,M))),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_gc(fun(nat,nat),fun(nat,fun(nat,nat)),Ba),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_ge(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),Aa2),Ba),X)),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na))) ) ) ) ).

% polynomial_product_nat
tff(fact_4534_binomial__ring,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Aa2: A,Ba: A,Na: nat] : aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)),Na) = 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))),Aa2),Ba),Na)),set_ord_atMost(nat,Na)) ) ).

% binomial_ring
tff(fact_4535_choose__square__sum,axiom,
    ! [Na: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_gg(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_4536_pochhammer__binomial__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Aa2: A,Ba: A,Na: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba),Na) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_gh(A,fun(A,fun(nat,fun(nat,A))),Aa2),Ba),Na)),set_ord_atMost(nat,Na)) ) ).

% pochhammer_binomial_sum
tff(fact_4537_cosh__real__nonpos__less__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),zero_zero(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),zero_zero(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),cosh(real,X)),cosh(real,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),X) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_4538_cosh__real__nonneg__less__iff,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),cosh(real,X)),cosh(real,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),Y) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_4539_cosh__real__strict__mono,axiom,
    ! [X: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),cosh(real,X)),cosh(real,Y)) ) ) ).

% cosh_real_strict_mono
tff(fact_4540_Cauchy__product__sums,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Aa2: fun(nat,A),Ba: fun(nat,A)] :
          ( summable(real,aTP_Lamp_fu(fun(nat,A),fun(nat,real),Aa2))
         => ( summable(real,aTP_Lamp_fu(fun(nat,A),fun(nat,real),Ba))
           => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_fw(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Aa2),Ba)),aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,Aa2)),suminf(A,Ba))) ) ) ) ).

% Cauchy_product_sums
tff(fact_4541_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_4542_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_gi(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_4543_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,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),P2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),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_gj(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_gk(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_4544_arcosh__cosh__real,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,real,arcosh(real),cosh(real,X)) = X ) ) ).

% arcosh_cosh_real
tff(fact_4545_gbinomial__partial__sum__poly,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat,Aa2: 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_gl(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),Aa2),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_gm(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),Aa2),X),Y)),set_ord_atMost(nat,M)) ) ).

% gbinomial_partial_sum_poly
tff(fact_4546_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,aa(A,fun(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_gn(A,fun(A,fun(nat,fun(nat,A))),X),Y),Na)),set_ord_atMost(nat,Na)) ) ) ) ).

% exp_series_add_commuting
tff(fact_4547_cosh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : cosh(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,X)),cosh(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,X)),sinh(A,Y))) ) ).

% cosh_add
tff(fact_4548_sinh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : sinh(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,X)),cosh(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,X)),sinh(A,Y))) ) ).

% sinh_add
tff(fact_4549_cosh__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : cosh(A,aa(A,A,minus_minus(A,X),Y)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,X)),cosh(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,X)),sinh(A,Y))) ) ).

% cosh_diff
tff(fact_4550_sinh__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : sinh(A,aa(A,A,minus_minus(A,X),Y)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,X)),cosh(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,X)),sinh(A,Y))) ) ).

% sinh_diff
tff(fact_4551_cosh__plus__sinh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),cosh(A,X)),sinh(A,X)) = aa(A,A,exp(A),X) ) ).

% cosh_plus_sinh
tff(fact_4552_sinh__plus__cosh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),sinh(A,X)),cosh(A,X)) = aa(A,A,exp(A),X) ) ).

% sinh_plus_cosh
tff(fact_4553_root__polyfun,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Na: nat,Z: A,Aa2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Na)
         => ( ( aa(nat,A,power_power(A,Z),Na) = Aa2 )
          <=> ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_go(nat,fun(A,fun(A,fun(nat,A))),Na),Z),Aa2)),set_ord_atMost(nat,Na)) = zero_zero(A) ) ) ) ) ).

% root_polyfun
tff(fact_4554_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,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,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_4555_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_gp(nat,fun(nat,A),Na)),set_ord_atMost(nat,Na)) = zero_zero(A) ) ) ) ).

% choose_alternating_linear_sum
tff(fact_4556_gbinomial__sum__nat__pow2,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_gq(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_4557_gbinomial__partial__sum__poly__xpos,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat,Aa2: 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_gl(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),Aa2),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_gr(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),Aa2),X),Y)),set_ord_atMost(nat,M)) ) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_4558_binomial__r__part__sum,axiom,
    ! [M: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)),one_one(nat)))),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_4559_choose__linear__sum,axiom,
    ! [Na: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_gs(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_4560_choose__alternating__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_gt(nat,fun(nat,A),Na)),set_ord_atMost(nat,Na)) = zero_zero(A) ) ) ) ).

% choose_alternating_sum
tff(fact_4561_polyfun__extremal__lemma,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [E2: real,C2: fun(nat,A),Na: nat] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
         => ? [M10: real] :
            ! [Z2: A] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),M10),real_V7770717601297561774m_norm(A,Z2))
             => aa(real,$o,aa(real,fun(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_bt(fun(nat,A),fun(A,fun(nat,A)),C2),Z2)),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,Z2)),aa(nat,nat,suc,Na)))) ) ) ) ).

% polyfun_extremal_lemma
tff(fact_4562_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,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,semiring_1_of_nat(A),M))),one_one(A)))),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_4563_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_4564_gbinomial__partial__row__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Aa2: A,M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_gu(A,fun(nat,A),Aa2)),set_ord_atMost(nat,M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),M)),one_one(A))),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,A,gbinomial(A,Aa2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),one_one(nat)))) ) ).

% gbinomial_partial_row_sum
tff(fact_4565_choose__odd__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_gv(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_4566_choose__even__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_gw(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_4567_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) )
           => ( tanh(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),tanh(A,X)),tanh(A,Y))),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),tanh(A,X)),tanh(A,Y)))) ) ) ) ) ).

% tanh_add
tff(fact_4568_mask__eq__sum__exp,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Na: nat] : aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)),one_one(A)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2)))),collect(nat,aTP_Lamp_gx(nat,fun(nat,$o),Na))) ) ).

% mask_eq_sum_exp
tff(fact_4569_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,aa(A,fun(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_4570_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,aa(A,fun(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_4571_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_4572_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_4573_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_gx(nat,fun(nat,$o),Na))) ).

% mask_eq_sum_exp_nat
tff(fact_4574_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_4575_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,aa(A,fun(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_4576_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,aa(A,fun(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_4577_cosh__ln__real,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$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,aa(real,fun(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_4578_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_gz(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_4579_sum__abs__ge__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [F2: fun(B,A),A2: set(B)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_ha(fun(B,A),fun(B,A),F2)),A2)) ) ).

% sum_abs_ge_zero
tff(fact_4580_sum__abs,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [F2: fun(B,A),A2: set(B)] : aa(A,$o,aa(A,fun(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_ha(fun(B,A),fun(B,A),F2)),A2)) ) ).

% sum_abs
tff(fact_4581_convex__sum__bound__le,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),X: fun(A,B),Aa2: fun(A,B),Ba: B,Delta: B] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,X,I2)) )
         => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,X),I5) = one_one(B) )
           => ( ! [I2: A] :
                  ( member(A,I2,I5)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,minus_minus(B,aa(A,B,Aa2,I2)),Ba))),Delta) )
             => aa(B,$o,aa(B,fun(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_hb(fun(A,B),fun(fun(A,B),fun(A,B)),X),Aa2)),I5)),Ba))),Delta) ) ) ) ) ).

% convex_sum_bound_le
tff(fact_4582_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_ha(fun(B,A),fun(B,A),F2)),A2)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_ha(fun(B,A),fun(B,A),F2)),A2) ) ).

% abs_sum_abs
tff(fact_4583_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_hc(B,A)),A2) = zero_zero(A) ) ).

% sum.neutral_const
tff(fact_4584_Complex__sum_H,axiom,
    ! [A: $tType,F2: fun(A,real),S: set(A)] : aa(set(A),complex,groups7311177749621191930dd_sum(A,complex,aTP_Lamp_hd(fun(A,real),fun(A,complex),F2)),S) = complex2(aa(set(A),real,groups7311177749621191930dd_sum(A,real,F2),S),zero_zero(real)) ).

% Complex_sum'
tff(fact_4585_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_he(fun(A,nat),fun(A,int),F2)),A2) ).

% int_sum
tff(fact_4586_sum__subtractf__nat,axiom,
    ! [A: $tType,A2: set(A),G: fun(A,nat),F2: fun(A,nat)] :
      ( ! [X2: A] :
          ( member(A,X2,A2)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,G,X2)),aa(A,nat,F2,X2)) )
     => ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_hf(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_4587_sum__nth__roots,axiom,
    ! [Na: nat,C2: complex] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),Na)
     => ( aa(set(complex),complex,groups7311177749621191930dd_sum(complex,complex,aTP_Lamp_hg(complex,complex)),collect(complex,aa(complex,fun(complex,$o),aTP_Lamp_hh(nat,fun(complex,fun(complex,$o)),Na),C2))) = zero_zero(complex) ) ) ).

% sum_nth_roots
tff(fact_4588_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_hi(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_hk(fun(B,fun(C,A)),fun(set(B),fun(C,A)),G),A2)),B2) ) ).

% sum.swap
tff(fact_4589_sum__roots__unity,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),Na)
     => ( aa(set(complex),complex,groups7311177749621191930dd_sum(complex,complex,aTP_Lamp_hg(complex,complex)),collect(complex,aTP_Lamp_hl(nat,fun(complex,$o),Na))) = zero_zero(complex) ) ) ).

% sum_roots_unity
tff(fact_4590_sum__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [K3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( ! [I2: A] :
              ( member(A,I2,K3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),aa(A,B,G,I2)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),K3)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),K3)) ) ) ).

% sum_mono
tff(fact_4591_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_hm(fun(B,A),fun(fun(B,A),fun(B,A)),G),H)),A2) = aa(A,A,aa(A,fun(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_4592_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_ho(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),F2),G),B2)),A2) ) ).

% sum_product
tff(fact_4593_sum__distrib__right,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [F2: fun(B,A),A2: set(B),R: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A2)),R) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_hp(fun(B,A),fun(A,fun(B,A)),F2),R)),A2) ) ).

% sum_distrib_right
tff(fact_4594_sum__distrib__left,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [R: A,F2: fun(B,A),A2: set(B)] : aa(A,A,aa(A,fun(A,A),times_times(A),R),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_hq(A,fun(fun(B,A),fun(B,A)),R),F2)),A2) ) ).

% sum_distrib_left
tff(fact_4595_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_hr(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_4596_sum__divide__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [F2: fun(B,A),A2: set(B),R: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A2)),R) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_hs(fun(B,A),fun(A,fun(B,A)),F2),R)),A2) ) ).

% sum_divide_distrib
tff(fact_4597_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_ht(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_4598_sum__nonneg,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A2: set(A),F2: fun(A,B)] :
          ( ! [X2: A] :
              ( member(A,X2,A2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X2)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A2)) ) ) ).

% sum_nonneg
tff(fact_4599_sum__nonpos,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A2: set(A),F2: fun(A,B)] :
          ( ! [X2: A] :
              ( member(A,X2,A2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X2)),zero_zero(B)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A2)),zero_zero(B)) ) ) ).

% sum_nonpos
tff(fact_4600_Maclaurin__cos__expansion2,axiom,
    ! [X: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => ? [T2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T2)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T2),X)
            & ( cos(real,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_hu(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,aa(real,fun(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_4601_Maclaurin__minus__cos__expansion,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),zero_zero(real))
       => ? [T2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),T2)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T2),zero_zero(real))
            & ( cos(real,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_hu(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,aa(real,fun(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_4602_Maclaurin__sin__expansion3,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
       => ? [T2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T2)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T2),X)
            & ( sin(real,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_hv(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,aa(real,fun(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_4603_lessThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_ord_lessThan(A,X)),set_ord_lessThan(A,Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ).

% lessThan_subset_iff
tff(fact_4604_sum_OlessThan__Suc,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,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_ord_lessThan(nat,Na))),aa(nat,A,G,Na)) ) ).

% sum.lessThan_Suc
tff(fact_4605_sumr__cos__zero__one,axiom,
    ! [Na: nat] : aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_hw(nat,real)),set_ord_lessThan(nat,aa(nat,nat,suc,Na))) = one_one(real) ).

% sumr_cos_zero_one
tff(fact_4606_lessThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : set_ord_lessThan(A,U) = collect(A,aTP_Lamp_hx(A,fun(A,$o),U)) ) ).

% lessThan_def
tff(fact_4607_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_4608_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_hy(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_4609_sum__diff__distrib,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Q: fun(A,nat),P: fun(A,nat),Na: A] :
          ( ! [X2: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,Q,X2)),aa(A,nat,P,X2))
         => ( 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_hz(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Q),P)),set_ord_lessThan(A,Na)) ) ) ) ).

% sum_diff_distrib
tff(fact_4610_suminf__le__const,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),X: A] :
          ( summable(A,F2)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,N))),X)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),suminf(A,F2)),X) ) ) ) ).

% suminf_le_const
tff(fact_4611_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,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_fe(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,Na))) ) ).

% sum.lessThan_Suc_shift
tff(fact_4612_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_fb(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_4613_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_ia(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_4614_sumr__diff__mult__const2,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [F2: fun(nat,A),Na: nat,R: 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)),R)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_ib(fun(nat,A),fun(A,fun(nat,A)),F2),R)),set_ord_lessThan(nat,Na)) ) ).

% sumr_diff_mult_const2
tff(fact_4615_summableI__nonneg__bounded,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),X: A] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N))
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(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_4616_sums__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),Na: nat,S: A] :
          ( aa(A,$o,sums(A,aa(nat,fun(nat,A),aTP_Lamp_bh(fun(nat,A),fun(nat,fun(nat,A)),F2),Na)),S)
        <=> aa(A,$o,sums(A,F2),aa(A,A,aa(A,fun(A,A),plus_plus(A),S),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,Na)))) ) ) ).

% sums_iff_shift
tff(fact_4617_sums__split__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),S: A,Na: nat] :
          ( aa(A,$o,sums(A,F2),S)
         => aa(A,$o,sums(A,aa(nat,fun(nat,A),aTP_Lamp_bh(fun(nat,A),fun(nat,fun(nat,A)),F2),Na)),aa(A,A,minus_minus(A,S),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,Na)))) ) ) ).

% sums_split_initial_segment
tff(fact_4618_sums__iff__shift_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),Na: nat,S: A] :
          ( aa(A,$o,sums(A,aa(nat,fun(nat,A),aTP_Lamp_bh(fun(nat,A),fun(nat,fun(nat,A)),F2),Na)),aa(A,A,minus_minus(A,S),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_ord_lessThan(nat,Na))))
        <=> aa(A,$o,sums(A,F2),S) ) ) ).

% sums_iff_shift'
tff(fact_4619_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_4620_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_4621_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_4622_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,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_fe(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,Na))) ) ).

% sum.atMost_shift
tff(fact_4623_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,aa(A,fun(A,A),plus_plus(A),suminf(A,aa(nat,fun(nat,A),aTP_Lamp_bh(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_4624_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_bh(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_4625_sum__less__suminf,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),Na: nat] :
          ( summable(A,F2)
         => ( ! [M5: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F2,M5)) )
           => aa(A,$o,aa(A,fun(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_4626_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_ic(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_id(A,fun(A,fun(nat,fun(nat,A))),Z),H),M)),set_ord_lessThan(nat,M)) ) ).

% lemma_termdiff1
tff(fact_4627_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_4628_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_ie(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_4629_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_if(A,fun(nat,fun(A,fun(nat,A))),X),Na),Y)),set_ord_lessThan(nat,Na))) ) ).

% power_diff_sumr2
tff(fact_4630_polyfun__linear__factor__root,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: fun(nat,A),Aa2: A,Na: nat] :
          ( ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_fy(fun(nat,A),fun(A,fun(nat,A)),C2),Aa2)),set_ord_atMost(nat,Na)) = zero_zero(A) )
         => ~ ! [B3: fun(nat,A)] :
                ~ ! [Z2: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_fy(fun(nat,A),fun(A,fun(nat,A)),C2),Z2)),set_ord_atMost(nat,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Z2),Aa2)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_fy(fun(nat,A),fun(A,fun(nat,A)),B3),Z2)),set_ord_lessThan(nat,Na))) ) ) ).

% polyfun_linear_factor_root
tff(fact_4631_polyfun__linear__factor,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: fun(nat,A),Na: nat,Aa2: A] :
        ? [B3: fun(nat,A)] :
        ! [Z2: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_fy(fun(nat,A),fun(A,fun(nat,A)),C2),Z2)),set_ord_atMost(nat,Na)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Z2),Aa2)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_fy(fun(nat,A),fun(A,fun(nat,A)),B3),Z2)),set_ord_lessThan(nat,Na)))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_fy(fun(nat,A),fun(A,fun(nat,A)),C2),Aa2)),set_ord_atMost(nat,Na))) ) ).

% polyfun_linear_factor
tff(fact_4632_real__sum__nat__ivl__bounded2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat,F2: fun(nat,A),K3: A,K: nat] :
          ( ! [P4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),P4),Na)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,P4)),K3) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),K3)
           => aa(A,$o,aa(A,fun(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)),K3)) ) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_4633_sum__less__suminf2,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),Na: nat,I: nat] :
          ( summable(A,F2)
         => ( ! [M5: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,M5)) )
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),I)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F2,I))
               => aa(A,$o,aa(A,fun(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_4634_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_ig(A,fun(nat,fun(nat,A)),X),Na)),set_ord_lessThan(nat,Na))) ) ).

% one_diff_power_eq'
tff(fact_4635_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_ih(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_4636_Maclaurin__lemma,axiom,
    ! [H: real,F2: fun(real,real),J: fun(nat,real),Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H)
     => ? [B6: real] : aa(real,real,F2,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_ii(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),B6),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_4637_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_ij(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,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_ik(fun(nat,real),fun(nat,real),F2)),set_ord_lessThan(nat,Na))),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_il(fun(nat,real),fun(nat,real),G)),set_ord_lessThan(nat,Na))) ).

% sum_split_even_odd
tff(fact_4638_Maclaurin__exp__le,axiom,
    ! [X: real,Na: nat] :
    ? [T2: real] :
      ( aa(real,$o,aa(real,fun(real,$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,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_im(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_4639_polyfun__diff__alt,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Na: nat,Aa2: fun(nat,A),X: A,Y: A] :
          ( aa(nat,$o,aa(nat,fun(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_fy(fun(nat,A),fun(A,fun(nat,A)),Aa2),X)),set_ord_atMost(nat,Na))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_fy(fun(nat,A),fun(A,fun(nat,A)),Aa2),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_io(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Na),Aa2),X),Y)),set_ord_lessThan(nat,Na))) ) ) ) ).

% polyfun_diff_alt
tff(fact_4640_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,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ei(A,fun(nat,A),X)),set_ord_lessThan(nat,K))),suminf(A,aa(nat,fun(nat,A),aTP_Lamp_ip(A,fun(nat,fun(nat,A)),X),K))) ) ).

% exp_first_terms
tff(fact_4641_Maclaurin__sin__bound,axiom,
    ! [X: real,Na: nat] : aa(real,$o,aa(real,fun(real,$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_hv(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_4642_sum__pos__lt__pair,axiom,
    ! [F2: fun(nat,real),K: nat] :
      ( summable(real,F2)
     => ( ! [D4: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))),D4)))),aa(nat,real,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))),D4)),one_one(nat))))))
       => aa(real,$o,aa(real,fun(real,$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_4643_Maclaurin__exp__lt,axiom,
    ! [X: real,Na: nat] :
      ( ( X != zero_zero(real) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => ? [T2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,abs_abs(real),T2))
            & aa(real,$o,aa(real,fun(real,$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,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_im(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_4644_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,aa(A,fun(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_ir(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_4645_Maclaurin__sin__expansion,axiom,
    ! [X: real,Na: nat] :
    ? [T2: real] : sin(real,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_hv(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,aa(real,fun(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_4646_Maclaurin__sin__expansion2,axiom,
    ! [X: real,Na: nat] :
    ? [T2: real] :
      ( aa(real,$o,aa(real,fun(real,$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,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_hv(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,aa(real,fun(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_4647_Maclaurin__cos__expansion,axiom,
    ! [X: real,Na: nat] :
    ? [T2: real] :
      ( aa(real,$o,aa(real,fun(real,$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,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_hu(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,aa(real,fun(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_4648_Maclaurin__sin__expansion4,axiom,
    ! [X: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => ? [T2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T2)
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T2),X)
          & ( sin(real,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_hv(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,aa(real,fun(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_4649_some__equality,axiom,
    ! [A: $tType,P: fun(A,$o),Aa2: A] :
      ( aa(A,$o,P,Aa2)
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
           => ( X2 = Aa2 ) )
       => ( fChoice(A,P) = Aa2 ) ) ) ).

% some_equality
tff(fact_4650_some__eq__trivial,axiom,
    ! [A: $tType,X: A] : fChoice(A,aTP_Lamp_dc(A,fun(A,$o),X)) = X ).

% some_eq_trivial
tff(fact_4651_some__sym__eq__trivial,axiom,
    ! [A: $tType,X: A] : fChoice(A,fequal(A,X)) = X ).

% some_sym_eq_trivial
tff(fact_4652_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),aa(word(A),fun(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_4653_some1__equality,axiom,
    ! [A: $tType,P: fun(A,$o),Aa2: A] :
      ( ? [X4: A] :
          ( aa(A,$o,P,X4)
          & ! [Y2: A] :
              ( aa(A,$o,P,Y2)
             => ( Y2 = X4 ) ) )
     => ( aa(A,$o,P,Aa2)
       => ( fChoice(A,P) = Aa2 ) ) ) ).

% some1_equality
tff(fact_4654_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_4655_someI2__bex,axiom,
    ! [A: $tType,A2: set(A),P: fun(A,$o),Q: fun(A,$o)] :
      ( ? [X4: A] :
          ( member(A,X4,A2)
          & aa(A,$o,P,X4) )
     => ( ! [X2: A] :
            ( ( member(A,X2,A2)
              & aa(A,$o,P,X2) )
           => aa(A,$o,Q,X2) )
       => aa(A,$o,Q,fChoice(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_is(set(A),fun(fun(A,$o),fun(A,$o)),A2),P))) ) ) ).

% someI2_bex
tff(fact_4656_someI2__ex,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ? [X_13: A] : aa(A,$o,P,X_13)
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
           => aa(A,$o,Q,X2) )
       => aa(A,$o,Q,fChoice(A,P)) ) ) ).

% someI2_ex
tff(fact_4657_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_4658_someI2,axiom,
    ! [A: $tType,P: fun(A,$o),Aa2: A,Q: fun(A,$o)] :
      ( aa(A,$o,P,Aa2)
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
           => aa(A,$o,Q,X2) )
       => aa(A,$o,Q,fChoice(A,P)) ) ) ).

% someI2
tff(fact_4659_some__in__eq,axiom,
    ! [A: $tType,A2: set(A)] :
      ( member(A,fChoice(A,aTP_Lamp_ab(set(A),fun(A,$o),A2)),A2)
    <=> ( A2 != bot_bot(set(A)) ) ) ).

% some_in_eq
tff(fact_4660_bij__betw__roots__unity,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => bij_betw(nat,complex,aTP_Lamp_it(nat,fun(nat,complex),Na),set_ord_lessThan(nat,Na),collect(complex,aTP_Lamp_hl(nat,fun(complex,$o),Na))) ) ).

% bij_betw_roots_unity
tff(fact_4661_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,aa(nat,fun(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,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat))),M)),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,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_4662_gchoose__row__sum__weighted,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [R: A,M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_gu(A,fun(nat,A),R)),set_or1337092689740270186AtMost(nat,zero_zero(nat),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,M))),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,A,gbinomial(A,R),aa(nat,nat,suc,M))) ) ).

% gchoose_row_sum_weighted
tff(fact_4663_atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( member(A,I,set_or1337092689740270186AtMost(A,L,U))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),I)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),U) ) ) ) ).

% atLeastAtMost_iff
tff(fact_4664_Icc__eq__Icc,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,H: A,L3: A,H3: A] :
          ( ( set_or1337092689740270186AtMost(A,L,H) = set_or1337092689740270186AtMost(A,L3,H3) )
        <=> ( ( ( L = L3 )
              & ( H = H3 ) )
            | ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),H)
              & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L3),H3) ) ) ) ) ).

% Icc_eq_Icc
tff(fact_4665_atLeastatMost__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Aa2: A,Ba: A] :
          ( ( set_or1337092689740270186AtMost(A,Aa2,Ba) = bot_bot(set(A)) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba) ) ) ).

% atLeastatMost_empty_iff
tff(fact_4666_atLeastatMost__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Aa2: A,Ba: A] :
          ( ( bot_bot(set(A)) = set_or1337092689740270186AtMost(A,Aa2,Ba) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba) ) ) ).

% atLeastatMost_empty_iff2
tff(fact_4667_atLeastatMost__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Aa2: A,Ba: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,Aa2,Ba)),set_or1337092689740270186AtMost(A,C2,D2))
        <=> ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),Aa2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),D2) ) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4668_Icc__subset__Iic__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [L: A,H: A,H3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,L,H)),set_ord_atMost(A,H3))
        <=> ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),H)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),H),H3) ) ) ) ).

% Icc_subset_Iic_iff
tff(fact_4669_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,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Na)),M),zero_zero(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,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_4670_sum_Oreindex__bij__betw,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [H: fun(A,B),Sa: set(A),T4: set(B),G: fun(B,C)] :
          ( bij_betw(A,B,H,Sa,T4)
         => ( aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_iu(fun(A,B),fun(fun(B,C),fun(A,C)),H),G)),Sa) = aa(set(B),C,groups7311177749621191930dd_sum(B,C,G),T4) ) ) ) ).

% sum.reindex_bij_betw
tff(fact_4671_ivl__disj__un__two__touch_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_4672_all__nat__less,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( ! [M4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M4),Na)
         => aa(nat,$o,P,M4) )
    <=> ! [X6: nat] :
          ( member(nat,X6,set_or1337092689740270186AtMost(nat,zero_zero(nat),Na))
         => aa(nat,$o,P,X6) ) ) ).

% all_nat_less
tff(fact_4673_ex__nat__less,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( ? [M4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M4),Na)
          & aa(nat,$o,P,M4) )
    <=> ? [X6: nat] :
          ( member(nat,X6,set_or1337092689740270186AtMost(nat,zero_zero(nat),Na))
          & aa(nat,$o,P,X6) ) ) ).

% ex_nat_less
tff(fact_4674_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_fe(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_4675_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,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),K))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_iv(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_4676_atLeastatMost__psubset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Aa2: A,Ba: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),set_or1337092689740270186AtMost(A,Aa2,Ba)),set_or1337092689740270186AtMost(A,C2,D2))
        <=> ( ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
              | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),Aa2)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),D2)
                & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),Aa2)
                  | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),D2) ) ) )
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2) ) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4677_atLeastAtMost__insertL,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( aa(set(nat),set(nat),insert(nat,M),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),Na)) = set_or1337092689740270186AtMost(nat,M,Na) ) ) ).

% atLeastAtMost_insertL
tff(fact_4678_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Na))
     => ( set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,Na)) = aa(set(nat),set(nat),insert(nat,aa(nat,nat,suc,Na)),set_or1337092689740270186AtMost(nat,M,Na)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_4679_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( set_or1337092689740270186AtMost(nat,M,Na) = aa(set(nat),set(nat),insert(nat,M),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),Na)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_4680_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_iw(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Na),M)),set_or1337092689740270186AtMost(nat,Na,M)) ) ).

% sum.atLeastAtMost_rev
tff(fact_4681_ivl__disj__un__one_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_ord_lessThan(A,L)),set_or1337092689740270186AtMost(A,L,U)) = set_ord_atMost(A,U) ) ) ) ).

% ivl_disj_un_one(4)
tff(fact_4682_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,aa(A,fun(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_4683_sum_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,Na)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,M)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),Na))) ) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_4684_sum_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Na))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,aa(nat,nat,suc,Na))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,Na))) ) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_4685_sum_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(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),plus_plus(A),aa(nat,A,G,M)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_fe(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,Na))) ) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_4686_sum__Suc__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,Na: nat,F2: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Na))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ia(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_4687_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_fe(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,Na)) ) ).

% sum.atLeast1_atMost_eq
tff(fact_4688_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_fe(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_4689_sum_Onested__swap_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Aa2: fun(nat,fun(nat,A)),Na: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ix(fun(nat,fun(nat,A)),fun(nat,A),Aa2)),set_ord_atMost(nat,Na)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_iz(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Aa2),Na)),set_ord_lessThan(nat,Na)) ) ).

% sum.nested_swap'
tff(fact_4690_sum__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),Aa2: nat,Ba: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or1337092689740270186AtMost(nat,Aa2,Ba)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_ja(fun(nat,A),fun(nat,fun(A,A)),F2),Aa2,Ba,zero_zero(A)) ) ).

% sum_atLeastAtMost_code
tff(fact_4691_sum_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Na: nat,G: fun(nat,A),P2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat)))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),P2))) = aa(A,A,aa(A,fun(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,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),P2)))) ) ) ) ).

% sum.ub_add_nat
tff(fact_4692_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,aa(nat,fun(nat,nat),plus_plus(nat),M),Na))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,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,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)))) ) ).

% sum_up_index_split
tff(fact_4693_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_jb(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,M,Na)) = $ite(aa(nat,$o,aa(nat,fun(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,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat)))),zero_zero(A)) ) ).

% sum_natinterval_diff
tff(fact_4694_sum__telescope_H_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,Na: nat,F2: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_jc(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_4695_sum__power__shift,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M: nat,Na: nat,X: A] :
          ( aa(nat,$o,aa(nat,fun(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_4696_summable__partial__sum__bound,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),E2: real] :
          ( summable(A,F2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
           => ~ ! [N9: nat] :
                  ~ ! [M3: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N9),M3)
                     => ! [N10: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or1337092689740270186AtMost(nat,M3,N10)))),E2) ) ) ) ) ).

% summable_partial_sum_bound
tff(fact_4697_sum__gp__multiplied,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M: nat,Na: nat,X: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,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_4698_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_fx(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% sum.in_pairs
tff(fact_4699_polyfun__eq__const,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Na: nat,K: A] :
          ( ! [X6: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_fn(fun(nat,A),fun(A,fun(nat,A)),C2),X6)),set_ord_atMost(nat,Na)) = K
        <=> ( ( aa(nat,A,C2,zero_zero(nat)) = K )
            & ! [X6: nat] :
                ( member(nat,X6,set_or1337092689740270186AtMost(nat,one_one(nat),Na))
               => ( aa(nat,A,C2,X6) = zero_zero(A) ) ) ) ) ) ).

% polyfun_eq_const
tff(fact_4700_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_jd(nat,fun(nat,A),K)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Na)),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),one_one(nat))) ) ).

% gbinomial_sum_up_index
tff(fact_4701_gauss__sum__nat,axiom,
    ! [Na: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_je(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_4702_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,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Na)),one_one(A))) ) ).

% double_gauss_sum
tff(fact_4703_double__arith__series,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Aa2: 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_jf(A,fun(A,fun(nat,A)),Aa2),D2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Na)),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Aa2)),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_4704_arith__series__nat,axiom,
    ! [Aa2: nat,D2: nat,Na: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_jg(nat,fun(nat,fun(nat,nat)),Aa2),D2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,Na)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Aa2)),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_4705_Sum__Icc__nat,axiom,
    ! [M: nat,Na: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_je(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,aa(nat,fun(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_4706_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,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Na)),one_one(A))) ) ).

% double_gauss_sum_from_Suc_0
tff(fact_4707_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,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Na)),one_one(A)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% gauss_sum
tff(fact_4708_arith__series,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Aa2: A,D2: A,Na: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_jh(A,fun(A,fun(nat,A)),Aa2),D2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Na)),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Aa2)),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_4709_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,aa(nat,fun(nat,nat),plus_plus(nat),M),Na))) = $ite(X = one_one(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Na)),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,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_4710_polyfun__diff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Na: nat,Aa2: fun(nat,A),X: A,Y: A] :
          ( aa(nat,$o,aa(nat,fun(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_fy(fun(nat,A),fun(A,fun(nat,A)),Aa2),X)),set_ord_atMost(nat,Na))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_fy(fun(nat,A),fun(A,fun(nat,A)),Aa2),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_jj(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Na),Aa2),X),Y)),set_ord_lessThan(nat,Na))) ) ) ) ).

% polyfun_diff
tff(fact_4711_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,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Na)),one_one(A)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% gauss_sum_from_Suc_0
tff(fact_4712_pochhammer__times__pochhammer__half,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Z: A,Na: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z,aa(nat,nat,suc,Na))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,nat,suc,Na))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_jk(A,fun(nat,A),Z),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),one_one(nat)))) ) ).

% pochhammer_times_pochhammer_half
tff(fact_4713_sofl__test,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ( aa(int,int,aa(int,fun(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),aa(word(A),fun(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,aa(word(A),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),aa(word(A),fun(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),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y)),Y))) = zero_zero(word(A)) ) ) ) ).

% sofl_test
tff(fact_4714_Sum__Icc__int,axiom,
    ! [M: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),M),Na)
     => ( aa(set(int),int,groups7311177749621191930dd_sum(int,int,aTP_Lamp_jl(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,aa(int,fun(int,int),plus_plus(int),Na),one_one(int)))),aa(int,int,aa(int,fun(int,int),times_times(int),M),aa(int,int,minus_minus(int,M),one_one(int))))),aa(num,int,numeral_numeral(int),bit0(one2))) ) ) ).

% Sum_Icc_int
tff(fact_4715_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_4716_drop__bit__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: nat,Aa2: A] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),M),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),Aa2)) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)),Aa2) ) ).

% drop_bit_drop_bit
tff(fact_4717_drop__bit__and,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,Aa2: A,Ba: A] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Aa2),Ba)) = 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),Aa2)),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),Ba)) ) ).

% drop_bit_and
tff(fact_4718_drop__bit__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,Aa2: A,Ba: A] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Aa2),Ba)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),Aa2)),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),Ba)) ) ).

% drop_bit_or
tff(fact_4719_drop__bit__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,Aa2: A,Ba: A] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Aa2),Ba)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),Aa2)),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),Ba)) ) ).

% drop_bit_xor
tff(fact_4720_Ints__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(B)
        & ring_1(B) )
     => ! [A2: set(A),F2: fun(A,B)] :
          ( ! [X2: A] :
              ( member(A,X2,A2)
             => member(B,aa(A,B,F2,X2),ring_1_Ints(B)) )
         => member(B,groups7121269368397514597t_prod(A,B,F2,A2),ring_1_Ints(B)) ) ) ).

% Ints_prod
tff(fact_4721_prod_Oneutral__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: set(B)] : groups7121269368397514597t_prod(B,A,aTP_Lamp_jm(B,A),A2) = one_one(A) ) ).

% prod.neutral_const
tff(fact_4722_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_jn(fun(B,nat),fun(B,A),F2),A2) ) ).

% of_nat_prod
tff(fact_4723_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_jo(fun(B,int),fun(B,A),F2),A2) ) ).

% of_int_prod
tff(fact_4724_of__real__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2191834092415804123ebra_1(A) )
     => ! [F2: fun(B,real),S: set(B)] : aa(real,A,real_Vector_of_real(A),groups7121269368397514597t_prod(B,real,F2,S)) = groups7121269368397514597t_prod(B,A,aTP_Lamp_jp(fun(B,real),fun(B,A),F2),S) ) ).

% of_real_prod
tff(fact_4725_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_4726_drop__bit__of__bool,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,Ba: $o] :
          aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),aa($o,A,zero_neq_one_of_bool(A),(Ba))) = aa($o,A,zero_neq_one_of_bool(A),
            ( ( Na = zero_zero(nat) )
            & (Ba) )) ) ).

% drop_bit_of_bool
tff(fact_4727_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_4728_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_4729_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_4730_prod_OlessThan__Suc,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),groups7121269368397514597t_prod(nat,A,G,set_ord_lessThan(nat,Na))),aa(nat,A,G,Na)) ) ).

% prod.lessThan_Suc
tff(fact_4731_prod_OatMost__Suc,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),groups7121269368397514597t_prod(nat,A,G,set_ord_atMost(nat,Na))),aa(nat,A,G,aa(nat,nat,suc,Na))) ) ).

% prod.atMost_Suc
tff(fact_4732_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_4733_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_4734_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,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Na)),M),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),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_4735_prod_Oreindex__bij__betw,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [H: fun(A,B),Sa: set(A),T4: set(B),G: fun(B,C)] :
          ( bij_betw(A,B,H,Sa,T4)
         => ( groups7121269368397514597t_prod(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_jq(fun(A,B),fun(fun(B,C),fun(A,C)),H),G),Sa) = groups7121269368397514597t_prod(B,C,G,T4) ) ) ) ).

% prod.reindex_bij_betw
tff(fact_4736_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,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,groups7121269368397514597t_prod(B,A,F2,A2))),groups7121269368397514597t_prod(B,real,aTP_Lamp_jr(fun(B,A),fun(B,real),F2),A2)) ) ).

% norm_prod_le
tff(fact_4737_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_js(fun(A,B),fun(A,real),F2),A2) = real_V7770717601297561774m_norm(B,groups7121269368397514597t_prod(A,B,F2,A2)) ) ).

% prod_norm
tff(fact_4738_mod__prod__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [F2: fun(B,A),Aa2: 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_fg(fun(B,A),fun(A,fun(B,A)),F2),Aa2),A2)),Aa2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),groups7121269368397514597t_prod(B,A,F2,A2)),Aa2) ) ).

% mod_prod_eq
tff(fact_4739_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) )
         => ~ ! [A3: B] :
                ( member(B,A3,A2)
               => ( aa(B,A,G,A3) = one_one(A) ) ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_4740_prod_Oneutral,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [A2: set(A),G: fun(A,B)] :
          ( ! [X2: A] :
              ( member(A,X2,A2)
             => ( aa(A,B,G,X2) = one_one(B) ) )
         => ( groups7121269368397514597t_prod(A,B,G,A2) = one_one(B) ) ) ) ).

% prod.neutral
tff(fact_4741_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_jt(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_4742_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_ju(fun(B,A),fun(B,A),F2),A2) ) ).

% abs_prod
tff(fact_4743_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_jv(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_jx(fun(B,fun(C,A)),fun(set(B),fun(C,A)),G),A2),B2) ) ).

% prod.swap
tff(fact_4744_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_4745_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_4746_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_jy(fun(B,A),fun(nat,fun(B,A)),F2),Na),A2) ) ).

% prod_power_distrib
tff(fact_4747_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_jz(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_4748_prod__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A2: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( ! [I2: A] :
              ( member(A,I2,A2)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I2))
                & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),aa(A,B,G,I2)) ) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),groups7121269368397514597t_prod(A,B,F2,A2)),groups7121269368397514597t_prod(A,B,G,A2)) ) ) ).

% prod_mono
tff(fact_4749_prod__nonneg,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A2: set(A),F2: fun(A,B)] :
          ( ! [X2: A] :
              ( member(A,X2,A2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X2)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),groups7121269368397514597t_prod(A,B,F2,A2)) ) ) ).

% prod_nonneg
tff(fact_4750_prod__ge__1,axiom,
    ! [B: $tType,A: $tType] :
      ( linord181362715937106298miring(B)
     => ! [A2: set(A),F2: fun(A,B)] :
          ( ! [X2: A] :
              ( member(A,X2,A2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F2,X2)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),groups7121269368397514597t_prod(A,B,F2,A2)) ) ) ).

% prod_ge_1
tff(fact_4751_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,Aa2: A] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,Na),Aa2) = Aa2 )
        <=> ( aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),Aa2) = zero_zero(A) ) ) ) ).

% take_bit_eq_self_iff_drop_bit_eq_0
tff(fact_4752_take__bit__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: nat,Aa2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),Aa2)) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)),Aa2)) ) ).

% take_bit_drop_bit
tff(fact_4753_drop__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: nat,Aa2: A] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),M),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),Aa2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,minus_minus(nat,Na),M)),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),M),Aa2)) ) ).

% drop_bit_take_bit
tff(fact_4754_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_ka(fun(nat,A),fun(nat,A),G),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_4755_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_kb(A,fun(fun(B,nat),fun(B,A)),C2),F2),A2) ) ).

% power_sum
tff(fact_4756_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,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),K))) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_kc(fun(nat,A),fun(nat,fun(nat,A)),G),K),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_4757_prod__le__1,axiom,
    ! [A: $tType,B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [A2: set(A),F2: fun(A,B)] :
          ( ! [X2: A] :
              ( member(A,X2,A2)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X2))
                & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X2)),one_one(B)) ) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),groups7121269368397514597t_prod(A,B,F2,A2)),one_one(B)) ) ) ).

% prod_le_1
tff(fact_4758_bset_I1_J,axiom,
    ! [D3: int,B2: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb2: int] :
                  ( member(int,Xb2,B2)
                 => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa) ) ) )
         => ( aa(int,$o,P,X2)
           => aa(int,$o,P,aa(int,int,minus_minus(int,X2),D3)) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb2: int] :
                    ( member(int,Xb2,B2)
                   => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa) ) ) )
           => ( aa(int,$o,Q,X2)
             => aa(int,$o,Q,aa(int,int,minus_minus(int,X2),D3)) ) )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B2)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
           => ( ( aa(int,$o,P,X4)
                & aa(int,$o,Q,X4) )
             => ( aa(int,$o,P,aa(int,int,minus_minus(int,X4),D3))
                & aa(int,$o,Q,aa(int,int,minus_minus(int,X4),D3)) ) ) ) ) ) ).

% bset(1)
tff(fact_4759_bset_I2_J,axiom,
    ! [D3: int,B2: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb2: int] :
                  ( member(int,Xb2,B2)
                 => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa) ) ) )
         => ( aa(int,$o,P,X2)
           => aa(int,$o,P,aa(int,int,minus_minus(int,X2),D3)) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb2: int] :
                    ( member(int,Xb2,B2)
                   => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa) ) ) )
           => ( aa(int,$o,Q,X2)
             => aa(int,$o,Q,aa(int,int,minus_minus(int,X2),D3)) ) )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B2)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
           => ( ( aa(int,$o,P,X4)
                | aa(int,$o,Q,X4) )
             => ( aa(int,$o,P,aa(int,int,minus_minus(int,X4),D3))
                | aa(int,$o,Q,aa(int,int,minus_minus(int,X4),D3)) ) ) ) ) ) ).

% bset(2)
tff(fact_4760_aset_I1_J,axiom,
    ! [D3: int,A2: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb2: int] :
                  ( member(int,Xb2,A2)
                 => ( X2 != aa(int,int,minus_minus(int,Xb2),Xa) ) ) )
         => ( aa(int,$o,P,X2)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D3)) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb2: int] :
                    ( member(int,Xb2,A2)
                   => ( X2 != aa(int,int,minus_minus(int,Xb2),Xa) ) ) )
           => ( aa(int,$o,Q,X2)
             => aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D3)) ) )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A2)
                   => ( X4 != aa(int,int,minus_minus(int,Xb3),Xa3) ) ) )
           => ( ( aa(int,$o,P,X4)
                & aa(int,$o,Q,X4) )
             => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D3))
                & aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D3)) ) ) ) ) ) ).

% aset(1)
tff(fact_4761_aset_I2_J,axiom,
    ! [D3: int,A2: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb2: int] :
                  ( member(int,Xb2,A2)
                 => ( X2 != aa(int,int,minus_minus(int,Xb2),Xa) ) ) )
         => ( aa(int,$o,P,X2)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D3)) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb2: int] :
                    ( member(int,Xb2,A2)
                   => ( X2 != aa(int,int,minus_minus(int,Xb2),Xa) ) ) )
           => ( aa(int,$o,Q,X2)
             => aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D3)) ) )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A2)
                   => ( X4 != aa(int,int,minus_minus(int,Xb3),Xa3) ) ) )
           => ( ( aa(int,$o,P,X4)
                | aa(int,$o,Q,X4) )
             => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D3))
                | aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D3)) ) ) ) ) ) ).

% aset(2)
tff(fact_4762_bit__word__iff__drop__bit__and,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Aa2),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),Aa2)),one_one(word(A))) = one_one(word(A)) ) ) ) ).

% bit_word_iff_drop_bit_and
tff(fact_4763_bit__iff__and__drop__bit__eq__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),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),Aa2)),one_one(A)) = one_one(A) ) ) ) ).

% bit_iff_and_drop_bit_eq_1
tff(fact_4764_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_kd(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_4765_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_ke(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Na),M),set_or1337092689740270186AtMost(nat,Na,M)) ) ).

% prod.atLeastAtMost_rev
tff(fact_4766_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_4767_prod_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(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_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),Na))) ) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_4768_prod_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Na))
         => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,aa(nat,nat,suc,Na))),groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,Na))) ) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_4769_word__atLeastLessThan__Suc__atLeastAtMost__union,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: word(A),L: word(A),U: word(A)] :
          ( ( M != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),L),M)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),M),U)
             => ( aa(set(word(A)),set(word(A)),aa(set(word(A)),fun(set(word(A)),set(word(A))),sup_sup(set(word(A))),set_or1337092689740270186AtMost(word(A),L,M)),set_or1337092689740270186AtMost(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),M),one_one(word(A))),U)) = set_or1337092689740270186AtMost(word(A),L,U) ) ) ) ) ) ).

% word_atLeastLessThan_Suc_atLeastAtMost_union
tff(fact_4770_aset_I10_J,axiom,
    ! [D2: int,D3: int,A2: set(int),T: int] :
      ( aa(int,$o,dvd_dvd(int,D2),D3)
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb3: int] :
                  ( member(int,Xb3,A2)
                 => ( X4 != aa(int,int,minus_minus(int,Xb3),Xa3) ) ) )
         => ( ~ aa(int,$o,dvd_dvd(int,D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),T))
           => ~ aa(int,$o,dvd_dvd(int,D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D3)),T)) ) ) ) ).

% aset(10)
tff(fact_4771_aset_I9_J,axiom,
    ! [D2: int,D3: int,A2: set(int),T: int] :
      ( aa(int,$o,dvd_dvd(int,D2),D3)
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb3: int] :
                  ( member(int,Xb3,A2)
                 => ( X4 != aa(int,int,minus_minus(int,Xb3),Xa3) ) ) )
         => ( aa(int,$o,dvd_dvd(int,D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),T))
           => aa(int,$o,dvd_dvd(int,D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D3)),T)) ) ) ) ).

% aset(9)
tff(fact_4772_bset_I10_J,axiom,
    ! [D2: int,D3: int,B2: set(int),T: int] :
      ( aa(int,$o,dvd_dvd(int,D2),D3)
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb3: int] :
                  ( member(int,Xb3,B2)
                 => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
         => ( ~ aa(int,$o,dvd_dvd(int,D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),T))
           => ~ aa(int,$o,dvd_dvd(int,D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,minus_minus(int,X4),D3)),T)) ) ) ) ).

% bset(10)
tff(fact_4773_bset_I9_J,axiom,
    ! [D2: int,D3: int,B2: set(int),T: int] :
      ( aa(int,$o,dvd_dvd(int,D2),D3)
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb3: int] :
                  ( member(int,Xb3,B2)
                 => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
         => ( aa(int,$o,dvd_dvd(int,D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),T))
           => aa(int,$o,dvd_dvd(int,D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,minus_minus(int,X4),D3)),T)) ) ) ) ).

% bset(9)
tff(fact_4774_stable__imp__drop__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A,Na: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2))) = Aa2 )
         => ( aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),Aa2) = Aa2 ) ) ) ).

% stable_imp_drop_bit_eq
tff(fact_4775_drop__bit__half,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,Aa2: 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),Aa2),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),Aa2)),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% drop_bit_half
tff(fact_4776_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_ka(fun(nat,A),fun(nat,A),G),set_ord_lessThan(nat,Na))) ) ).

% prod.lessThan_Suc_shift
tff(fact_4777_prod_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),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_ka(fun(nat,A),fun(nat,A),G),set_or1337092689740270186AtMost(nat,M,Na))) ) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_4778_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_ka(fun(nat,A),fun(nat,A),G),set_ord_atMost(nat,Na))) ) ).

% prod.atMost_Suc_shift
tff(fact_4779_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),M),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Na))
     => ( set_or1337092689740270186AtMost(int,M,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Na)) = aa(set(int),set(int),insert(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Na)),set_or1337092689740270186AtMost(int,M,Na)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_4780_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_ka(fun(nat,A),fun(nat,A),G),set_ord_lessThan(nat,Na)) ) ).

% prod.atLeast1_atMost_eq
tff(fact_4781_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_je(nat,nat),set_or1337092689740270186AtMost(nat,one_one(nat),Na))) ) ).

% fact_prod
tff(fact_4782_prod_Onested__swap_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Aa2: fun(nat,fun(nat,A)),Na: nat] : groups7121269368397514597t_prod(nat,A,aTP_Lamp_kf(fun(nat,fun(nat,A)),fun(nat,A),Aa2),set_ord_atMost(nat,Na)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_kh(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Aa2),Na),set_ord_lessThan(nat,Na)) ) ).

% prod.nested_swap'
tff(fact_4783_prod__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [F2: fun(nat,A),Aa2: nat,Ba: nat] : groups7121269368397514597t_prod(nat,A,F2,set_or1337092689740270186AtMost(nat,Aa2,Ba)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_ki(fun(nat,A),fun(nat,fun(A,A)),F2),Aa2,Ba,one_one(A)) ) ).

% prod_atLeastAtMost_code
tff(fact_4784_prod_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Na: nat,G: fun(nat,A),P2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat)))
         => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(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,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),P2)))) ) ) ) ).

% prod.ub_add_nat
tff(fact_4785_bset_I3_J,axiom,
    ! [D3: int,T: int,B2: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( member(int,aa(int,int,minus_minus(int,T),one_one(int)),B2)
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B2)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
           => ( ( X4 = T )
             => ( aa(int,int,minus_minus(int,X4),D3) = T ) ) ) ) ) ).

% bset(3)
tff(fact_4786_bset_I4_J,axiom,
    ! [D3: int,T: int,B2: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( member(int,T,B2)
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B2)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
           => ( ( X4 != T )
             => ( aa(int,int,minus_minus(int,X4),D3) != T ) ) ) ) ) ).

% bset(4)
tff(fact_4787_bset_I5_J,axiom,
    ! [D3: int,B2: set(int),T: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb3: int] :
                  ( member(int,Xb3,B2)
                 => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X4),T)
           => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,minus_minus(int,X4),D3)),T) ) ) ) ).

% bset(5)
tff(fact_4788_bset_I7_J,axiom,
    ! [D3: int,T: int,B2: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( member(int,T,B2)
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B2)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),T),X4)
             => aa(int,$o,aa(int,fun(int,$o),ord_less(int),T),aa(int,int,minus_minus(int,X4),D3)) ) ) ) ) ).

% bset(7)
tff(fact_4789_aset_I3_J,axiom,
    ! [D3: int,T: int,A2: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( member(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),T),one_one(int)),A2)
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A2)
                   => ( X4 != aa(int,int,minus_minus(int,Xb3),Xa3) ) ) )
           => ( ( X4 = T )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D3) = T ) ) ) ) ) ).

% aset(3)
tff(fact_4790_aset_I4_J,axiom,
    ! [D3: int,T: int,A2: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( member(int,T,A2)
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A2)
                   => ( X4 != aa(int,int,minus_minus(int,Xb3),Xa3) ) ) )
           => ( ( X4 != T )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D3) != T ) ) ) ) ) ).

% aset(4)
tff(fact_4791_aset_I5_J,axiom,
    ! [D3: int,T: int,A2: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( member(int,T,A2)
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A2)
                   => ( X4 != aa(int,int,minus_minus(int,Xb3),Xa3) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X4),T)
             => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D3)),T) ) ) ) ) ).

% aset(5)
tff(fact_4792_aset_I7_J,axiom,
    ! [D3: int,A2: set(int),T: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb3: int] :
                  ( member(int,Xb3,A2)
                 => ( X4 != aa(int,int,minus_minus(int,Xb3),Xa3) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),T),X4)
           => aa(int,$o,aa(int,fun(int,$o),ord_less(int),T),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D3)) ) ) ) ).

% aset(7)
tff(fact_4793_periodic__finite__ex,axiom,
    ! [D2: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( ! [X2: int,K5: int] :
            ( aa(int,$o,P,X2)
          <=> aa(int,$o,P,aa(int,int,minus_minus(int,X2),aa(int,int,aa(int,fun(int,int),times_times(int),K5),D2))) )
       => ( ? [X_1: int] : aa(int,$o,P,X_1)
        <=> ? [X6: int] :
              ( member(int,X6,set_or1337092689740270186AtMost(int,one_one(int),D2))
              & aa(int,$o,P,X6) ) ) ) ) ).

% periodic_finite_ex
tff(fact_4794_word__subset__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),R: word(A),Y: word(A),S: word(A)] :
          ( aa(set(word(A)),$o,aa(set(word(A)),fun(set(word(A)),$o),ord_less_eq(set(word(A))),set_or1337092689740270186AtMost(word(A),X,aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),R)),one_one(word(A))))),set_or1337092689740270186AtMost(word(A),Y,aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),S)),one_one(word(A)))))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),R)),one_one(word(A))))
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),S)),one_one(word(A))))
             => ( ( S != zero_zero(word(A)) )
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),R),S) ) ) ) ) ) ).

% word_subset_less
tff(fact_4795_drop__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,Aa2: A] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),aa(nat,nat,suc,Na)),Aa2) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% drop_bit_Suc
tff(fact_4796_drop__bit__eq__div,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,Aa2: A] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),Aa2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) ) ).

% drop_bit_eq_div
tff(fact_4797_norm__prod__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [I5: set(A),Z: fun(A,B),W: fun(A,B)] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,Z,I2))),one_one(real)) )
         => ( ! [I2: A] :
                ( member(A,I2,I5)
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,W,I2))),one_one(real)) )
           => aa(real,$o,aa(real,fun(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_kj(fun(A,B),fun(fun(A,B),fun(A,real)),Z),W)),I5)) ) ) ) ).

% norm_prod_diff
tff(fact_4798_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_ka(fun(nat,A),fun(nat,A),G),set_ord_lessThan(nat,Na))) ) ).

% prod.atMost_shift
tff(fact_4799_even__drop__bit__iff__not__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,Aa2: A] :
          ( aa(A,$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),Aa2))
        <=> ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),Na) ) ) ).

% even_drop_bit_iff_not_bit
tff(fact_4800_bit__iff__odd__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),Na)
        <=> ~ aa(A,$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),Aa2)) ) ) ).

% bit_iff_odd_drop_bit
tff(fact_4801_fact__eq__fact__times,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
     => ( semiring_char_0_fact(nat,M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,Na)),groups7121269368397514597t_prod(nat,nat,aTP_Lamp_je(nat,nat),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Na),M))) ) ) ).

% fact_eq_fact_times
tff(fact_4802_simp__from__to,axiom,
    ! [I: int,J: int] :
      set_or1337092689740270186AtMost(int,I,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I),bot_bot(set(int)),aa(set(int),set(int),insert(int,I),set_or1337092689740270186AtMost(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J))) ).

% simp_from_to
tff(fact_4803_bset_I6_J,axiom,
    ! [D3: int,B2: set(int),T: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb3: int] :
                  ( member(int,Xb3,B2)
                 => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X4),T)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,minus_minus(int,X4),D3)),T) ) ) ) ).

% bset(6)
tff(fact_4804_bset_I8_J,axiom,
    ! [D3: int,T: int,B2: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( member(int,aa(int,int,minus_minus(int,T),one_one(int)),B2)
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B2)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),T),X4)
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),T),aa(int,int,minus_minus(int,X4),D3)) ) ) ) ) ).

% bset(8)
tff(fact_4805_aset_I6_J,axiom,
    ! [D3: int,T: int,A2: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( member(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),T),one_one(int)),A2)
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A2)
                   => ( X4 != aa(int,int,minus_minus(int,Xb3),Xa3) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X4),T)
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D3)),T) ) ) ) ) ).

% aset(6)
tff(fact_4806_aset_I8_J,axiom,
    ! [D3: int,A2: set(int),T: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb3: int] :
                  ( member(int,Xb3,A2)
                 => ( X4 != aa(int,int,minus_minus(int,Xb3),Xa3) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),T),X4)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),T),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D3)) ) ) ) ).

% aset(8)
tff(fact_4807_cppi,axiom,
    ! [D3: int,P: fun(int,$o),P3: fun(int,$o),A2: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( ? [Z2: int] :
          ! [X2: int] :
            ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z2),X2)
           => ( aa(int,$o,P,X2)
            <=> aa(int,$o,P3,X2) ) )
       => ( ! [X2: int] :
              ( ! [Xa: int] :
                  ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D3))
                 => ! [Xb2: int] :
                      ( member(int,Xb2,A2)
                     => ( X2 != aa(int,int,minus_minus(int,Xb2),Xa) ) ) )
             => ( aa(int,$o,P,X2)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D3)) ) )
         => ( ! [X2: int,K5: int] :
                ( aa(int,$o,P3,X2)
              <=> aa(int,$o,P3,aa(int,int,minus_minus(int,X2),aa(int,int,aa(int,fun(int,int),times_times(int),K5),D3))) )
           => ( ? [X_1: int] : aa(int,$o,P,X_1)
            <=> ( ? [X6: int] :
                    ( member(int,X6,set_or1337092689740270186AtMost(int,one_one(int),D3))
                    & aa(int,$o,P3,X6) )
                | ? [X6: int] :
                    ( member(int,X6,set_or1337092689740270186AtMost(int,one_one(int),D3))
                    & ? [Xa4: int] :
                        ( member(int,Xa4,A2)
                        & aa(int,$o,P,aa(int,int,minus_minus(int,Xa4),X6)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_4808_cpmi,axiom,
    ! [D3: int,P: fun(int,$o),P3: fun(int,$o),B2: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( ? [Z2: int] :
          ! [X2: int] :
            ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X2),Z2)
           => ( aa(int,$o,P,X2)
            <=> aa(int,$o,P3,X2) ) )
       => ( ! [X2: int] :
              ( ! [Xa: int] :
                  ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D3))
                 => ! [Xb2: int] :
                      ( member(int,Xb2,B2)
                     => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa) ) ) )
             => ( aa(int,$o,P,X2)
               => aa(int,$o,P,aa(int,int,minus_minus(int,X2),D3)) ) )
         => ( ! [X2: int,K5: int] :
                ( aa(int,$o,P3,X2)
              <=> aa(int,$o,P3,aa(int,int,minus_minus(int,X2),aa(int,int,aa(int,fun(int,int),times_times(int),K5),D3))) )
           => ( ? [X_1: int] : aa(int,$o,P,X_1)
            <=> ( ? [X6: int] :
                    ( member(int,X6,set_or1337092689740270186AtMost(int,one_one(int),D3))
                    & aa(int,$o,P3,X6) )
                | ? [X6: int] :
                    ( member(int,X6,set_or1337092689740270186AtMost(int,one_one(int),D3))
                    & ? [Xa4: int] :
                        ( member(int,Xa4,B2)
                        & aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa4),X6)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_4809_pochhammer__Suc__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Aa2: A,Na: nat] : comm_s3205402744901411588hammer(A,Aa2,aa(nat,nat,suc,Na)) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_kk(A,fun(nat,A),Aa2),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) ) ).

% pochhammer_Suc_prod
tff(fact_4810_pochhammer__prod__rev,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Aa2: A,Na: nat] : comm_s3205402744901411588hammer(A,Aa2,Na) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_kl(A,fun(nat,fun(nat,A)),Aa2),Na),set_or1337092689740270186AtMost(nat,one_one(nat),Na)) ) ).

% pochhammer_prod_rev
tff(fact_4811_fact__div__fact,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,Na)) = groups7121269368397514597t_prod(nat,nat,aTP_Lamp_je(nat,nat),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat)),M)) ) ) ).

% fact_div_fact
tff(fact_4812_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_km(fun(nat,A),fun(nat,A),G),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% prod.in_pairs
tff(fact_4813_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_km(fun(nat,A),fun(nat,A),G),set_ord_atMost(nat,Na)) ) ).

% prod.in_pairs_0
tff(fact_4814_pochhammer__Suc__prod__rev,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Aa2: A,Na: nat] : comm_s3205402744901411588hammer(A,Aa2,aa(nat,nat,suc,Na)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_kl(A,fun(nat,fun(nat,A)),Aa2),Na),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) ) ).

% pochhammer_Suc_prod_rev
tff(fact_4815_drop__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,Aa2: A] :
          aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),Aa2) = $ite(Na = zero_zero(nat),Aa2,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),Aa2),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% drop_bit_rec
tff(fact_4816_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,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),P2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),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_kn(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_ko(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_4817_gbinomial__Suc,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [Aa2: A,K: nat] : aa(nat,A,gbinomial(A,Aa2),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),groups7121269368397514597t_prod(nat,A,aTP_Lamp_kp(A,fun(nat,A),Aa2),set_or1337092689740270186AtMost(nat,zero_zero(nat),K))),semiring_char_0_fact(A,aa(nat,nat,suc,K))) ) ).

% gbinomial_Suc
tff(fact_4818_bij__betw__nth__root__unity,axiom,
    ! [C2: complex,Na: nat] :
      ( ( C2 != zero_zero(complex) )
     => ( aa(nat,$o,aa(nat,fun(nat,$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_hl(nat,fun(complex,$o),Na)),collect(complex,aa(nat,fun(complex,$o),aTP_Lamp_kq(complex,fun(nat,fun(complex,$o)),C2),Na))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_4819_frame__rule__left,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),R3: assn] :
      ( hoare_hoare_triple(A,P,C2,Q)
     => hoare_hoare_triple(A,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),R3),P),C2,aa(assn,fun(A,assn),aTP_Lamp_kr(fun(A,assn),fun(assn,fun(A,assn)),Q),R3)) ) ).

% frame_rule_left
tff(fact_4820_the__elem__def,axiom,
    ! [A: $tType,X5: set(A)] : the_elem(A,X5) = the(A,aTP_Lamp_ks(set(A),fun(A,$o),X5)) ).

% the_elem_def
tff(fact_4821_drop__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),Na),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% drop_bit_nonnegative_int_iff
tff(fact_4822_drop__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),Na),K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% drop_bit_negative_int_iff
tff(fact_4823_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_4824_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_4825_real__root__le__iff,axiom,
    ! [Na: nat,X: real,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Na),X)),aa(real,real,root(Na),Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y) ) ) ).

% real_root_le_iff
tff(fact_4826_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_4827_real__root__ge__0__iff,axiom,
    ! [Na: nat,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,root(Na),Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y) ) ) ).

% real_root_ge_0_iff
tff(fact_4828_real__root__le__0__iff,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Na),X)),zero_zero(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),zero_zero(real)) ) ) ).

% real_root_le_0_iff
tff(fact_4829_real__root__ge__1__iff,axiom,
    ! [Na: nat,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(real,real,root(Na),Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Y) ) ) ).

% real_root_ge_1_iff
tff(fact_4830_real__root__le__1__iff,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Na),X)),one_one(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),one_one(real)) ) ) ).

% real_root_le_1_iff
tff(fact_4831_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_4832_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_4833_real__root__pow__pos2,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
       => ( aa(nat,real,power_power(real,aa(real,real,root(Na),X)),Na) = X ) ) ) ).

% real_root_pow_pos2
tff(fact_4834_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_4835_drop__bit__nat__eq,axiom,
    ! [Na: nat,K: int] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se4197421643247451524op_bit(nat),Na),aa(int,nat,nat2,K)) = aa(int,nat,nat2,aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),Na),K)) ).

% drop_bit_nat_eq
tff(fact_4836_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_he(fun(A,nat),fun(A,int),F2),A2) ).

% int_prod
tff(fact_4837_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_4838_real__root__pos__pos__le,axiom,
    ! [X: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,root(Na),X)) ) ).

% real_root_pos_pos_le
tff(fact_4839_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_4840_real__root__le__mono,axiom,
    ! [Na: nat,X: real,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Na),X)),aa(real,real,root(Na),Y)) ) ) ).

% real_root_le_mono
tff(fact_4841_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_jl(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_4842_sqrt__def,axiom,
    sqrt = root(aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% sqrt_def
tff(fact_4843_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,aa(nat,fun(nat,nat),plus_plus(nat),I),J))) = groups7121269368397514597t_prod(int,int,aTP_Lamp_jl(int,int),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),I),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)))) ).

% prod_int_plus_eq
tff(fact_4844_real__root__pos__pos,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,root(Na),X)) ) ) ).

% real_root_pos_pos
tff(fact_4845_real__root__decreasing,axiom,
    ! [Na: nat,N3: nat,X: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N3)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),X)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(N3),X)),aa(real,real,root(Na),X)) ) ) ) ).

% real_root_decreasing
tff(fact_4846_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_4847_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_4848_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_4849_real__root__pos__unique,axiom,
    ! [Na: nat,Y: real,X: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( ( aa(nat,real,power_power(real,Y),Na) = X )
         => ( aa(real,real,root(Na),X) = Y ) ) ) ) ).

% real_root_pos_unique
tff(fact_4850_real__root__power__cancel,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
       => ( aa(real,real,root(Na),aa(nat,real,power_power(real,X),Na)) = X ) ) ) ).

% real_root_power_cancel
tff(fact_4851_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_4852_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_4853_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_4854_real__root__increasing,axiom,
    ! [Na: nat,N3: nat,X: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N3)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),one_one(real))
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Na),X)),aa(real,real,root(N3),X)) ) ) ) ) ).

% real_root_increasing
tff(fact_4855_log__root,axiom,
    ! [Na: nat,Aa2: real,Ba: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Aa2)
       => ( aa(real,real,log(Ba),aa(real,real,root(Na),Aa2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,log(Ba),Aa2)),aa(nat,real,semiring_1_of_nat(real),Na)) ) ) ) ).

% log_root
tff(fact_4856_log__base__root,axiom,
    ! [Na: nat,Ba: real,X: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Ba)
       => ( aa(real,real,log(aa(real,real,root(Na),Ba)),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(Ba),X)) ) ) ) ).

% log_base_root
tff(fact_4857_ln__root,axiom,
    ! [Na: nat,Ba: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Ba)
       => ( aa(real,real,ln_ln(real),aa(real,real,root(Na),Ba)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),Ba)),aa(nat,real,semiring_1_of_nat(real),Na)) ) ) ) ).

% ln_root
tff(fact_4858_the__elem__set,axiom,
    ! [A: $tType,X: A] : the_elem(A,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)))) = X ).

% the_elem_set
tff(fact_4859_root__powr__inverse,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),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_4860_upto__aux__rec,axiom,
    ! [I: int,J: int,Js: list(int)] :
      upto_aux(I,J,Js) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I),Js,upto_aux(I,aa(int,int,minus_minus(int,J),one_one(int)),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),J),Js))) ).

% upto_aux_rec
tff(fact_4861_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_4862_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,aa(nat,fun(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,aa(nat,fun(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_4863_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_4864_pred__subset__eq2,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,B)),Sa: set(product_prod(A,B))] :
      ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),aTP_Lamp_kt(set(product_prod(A,B)),fun(A,fun(B,$o)),R3)),aTP_Lamp_kt(set(product_prod(A,B)),fun(A,fun(B,$o)),Sa))
    <=> aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R3),Sa) ) ).

% pred_subset_eq2
tff(fact_4865_sup__Un__eq2,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,B)),Sa: set(product_prod(A,B)),X4: A,Xa: B] :
      ( aa(B,$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,$o))),aTP_Lamp_kt(set(product_prod(A,B)),fun(A,fun(B,$o)),R3)),aTP_Lamp_kt(set(product_prod(A,B)),fun(A,fun(B,$o)),Sa)),X4),Xa)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),R3),Sa)) ) ).

% sup_Un_eq2
tff(fact_4866_bex2I,axiom,
    ! [A: $tType,B: $tType,Aa2: A,Ba: B,Sa: set(product_prod(A,B)),P: fun(A,fun(B,$o))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba),Sa)
     => ( ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba),Sa)
         => aa(B,$o,aa(A,fun(B,$o),P,Aa2),Ba) )
       => ? [A3: A,B3: B] :
            ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3),Sa)
            & aa(B,$o,aa(A,fun(B,$o),P,A3),B3) ) ) ) ).

% bex2I
tff(fact_4867_pairself_Ocases,axiom,
    ! [B: $tType,A: $tType,X: product_prod(fun(A,B),product_prod(A,A))] :
      ~ ! [F3: fun(A,B),A3: A,B3: A] : X != aa(product_prod(A,A),product_prod(fun(A,B),product_prod(A,A)),aa(fun(A,B),fun(product_prod(A,A),product_prod(fun(A,B),product_prod(A,A))),product_Pair(fun(A,B),product_prod(A,A)),F3),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3)) ).

% pairself.cases
tff(fact_4868_prod__decode__aux_Ocases,axiom,
    ! [X: product_prod(nat,nat)] :
      ~ ! [K5: nat,M5: nat] : X != aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),K5),M5) ).

% prod_decode_aux.cases
tff(fact_4869_pred__equals__eq2,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,B)),Sa: set(product_prod(A,B))] :
      ( ! [X6: 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),X6),Xa4),R3)
        <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X6),Xa4),Sa) )
    <=> ( R3 = Sa ) ) ).

% pred_equals_eq2
tff(fact_4870_bot__empty__eq2,axiom,
    ! [B: $tType,A: $tType,X4: A,Xa: B] :
      ( aa(B,$o,aa(A,fun(B,$o),bot_bot(fun(A,fun(B,$o))),X4),Xa)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa),bot_bot(set(product_prod(A,B)))) ) ).

% bot_empty_eq2
tff(fact_4871_lsb__word__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X4: word(A)] :
          ( aa(word(A),$o,least_8051144512741203767sb_lsb(word(A)),X4)
        <=> ~ aa(word(A),$o,dvd_dvd(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),X4) ) ) ).

% lsb_word_eq
tff(fact_4872_word__lsb__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A)] :
          ( aa(word(A),$o,least_8051144512741203767sb_lsb(word(A)),Aa2)
        <=> ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(word(A),int,semiring_1_unsigned(A,int),Aa2)) ) ) ).

% word_lsb_def
tff(fact_4873_lsb__odd,axiom,
    ! [A: $tType] :
      ( least_6119777620449941438nt_lsb(A)
     => ! [X4: A] :
          ( aa(A,$o,least_8051144512741203767sb_lsb(A),X4)
        <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),X4) ) ) ).

% lsb_odd
tff(fact_4874_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_4875_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_4876_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,aa(uint32,fun(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,aa(uint32,fun(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_4877_divmod__step__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [L: num,Q3: A,R: A] :
          unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),R)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),L)),R),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Q3)),one_one(A))),aa(A,A,minus_minus(A,R),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)),R)) ) ).

% divmod_step_eq
tff(fact_4878_fun__of__rel__def,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(B,A)),X: B] : fun_of_rel(B,A,R3,X) = fChoice(A,aa(B,fun(A,$o),aTP_Lamp_ku(set(product_prod(B,A)),fun(B,fun(A,$o)),R3),X)) ).

% fun_of_rel_def
tff(fact_4879_int__lsb__numeral_I2_J,axiom,
    aa(int,$o,least_8051144512741203767sb_lsb(int),one_one(int)) ).

% int_lsb_numeral(2)
tff(fact_4880_predicate2I,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o)),Q: fun(A,fun(B,$o))] :
      ( ! [X2: A,Y2: B] :
          ( aa(B,$o,aa(A,fun(B,$o),P,X2),Y2)
         => aa(B,$o,aa(A,fun(B,$o),Q,X2),Y2) )
     => aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),P),Q) ) ).

% predicate2I
tff(fact_4881_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_4882_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_4883_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_4884_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_4885_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_4886_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_4887_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_4888_mergesort__by__rel__split_Ocases,axiom,
    ! [A: $tType,X: product_prod(product_prod(list(A),list(A)),list(A))] :
      ( ! [Xs1: list(A),Xs22: list(A)] : X != aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22)),nil(A))
     => ( ! [Xs1: list(A),Xs22: list(A),X2: A] : X != aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)))
       => ~ ! [Xs1: list(A),Xs22: list(A),X1: A,X22: A,Xs: list(A)] : X != aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),Xs))) ) ) ).

% mergesort_by_rel_split.cases
tff(fact_4889_mfold_Ocases,axiom,
    ! [A: $tType,B: $tType,X: product_prod(fun(A,fun(B,heap_Time_Heap(B))),product_prod(list(A),B))] :
      ( ! [F3: fun(A,fun(B,heap_Time_Heap(B))),S2: B] : X != aa(product_prod(list(A),B),product_prod(fun(A,fun(B,heap_Time_Heap(B))),product_prod(list(A),B)),aa(fun(A,fun(B,heap_Time_Heap(B))),fun(product_prod(list(A),B),product_prod(fun(A,fun(B,heap_Time_Heap(B))),product_prod(list(A),B))),product_Pair(fun(A,fun(B,heap_Time_Heap(B))),product_prod(list(A),B)),F3),aa(B,product_prod(list(A),B),aa(list(A),fun(B,product_prod(list(A),B)),product_Pair(list(A),B),nil(A)),S2))
     => ~ ! [F3: fun(A,fun(B,heap_Time_Heap(B))),X2: A,Xs: list(A),S2: B] : X != aa(product_prod(list(A),B),product_prod(fun(A,fun(B,heap_Time_Heap(B))),product_prod(list(A),B)),aa(fun(A,fun(B,heap_Time_Heap(B))),fun(product_prod(list(A),B),product_prod(fun(A,fun(B,heap_Time_Heap(B))),product_prod(list(A),B))),product_Pair(fun(A,fun(B,heap_Time_Heap(B))),product_prod(list(A),B)),F3),aa(B,product_prod(list(A),B),aa(list(A),fun(B,product_prod(list(A),B)),product_Pair(list(A),B),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)),S2)) ) ).

% mfold.cases
tff(fact_4890_map__tailrec__rev_Ocases,axiom,
    ! [A: $tType,B: $tType,X: product_prod(fun(A,B),product_prod(list(A),list(B)))] :
      ( ! [F3: fun(A,B),Bs: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B))),aa(fun(A,B),fun(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B)))),product_Pair(fun(A,B),product_prod(list(A),list(B))),F3),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),Bs))
     => ~ ! [F3: fun(A,B),A3: A,As: list(A),Bs: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B))),aa(fun(A,B),fun(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B)))),product_Pair(fun(A,B),product_prod(list(A),list(B))),F3),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),As)),Bs)) ) ).

% map_tailrec_rev.cases
tff(fact_4891_zipf_Ocases,axiom,
    ! [C: $tType,A: $tType,B: $tType,X: product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))] :
      ( ! [F3: fun(A,fun(B,C))] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),F3),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),nil(B)))
     => ( ! [F3: fun(A,fun(B,C)),A3: A,As: list(A),B3: B,Bs: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),F3),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),As)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B3),Bs)))
       => ( ! [A3: fun(A,fun(B,C)),V4: A,Va2: list(A)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),A3),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V4),Va2)),nil(B)))
         => ~ ! [A3: fun(A,fun(B,C)),V4: B,Va2: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),A3),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V4),Va2))) ) ) ) ).

% zipf.cases
tff(fact_4892_list__assn_Ocases,axiom,
    ! [A: $tType,B: $tType,X: product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B)))] :
      ( ! [P5: fun(A,fun(B,assn))] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),aa(fun(A,fun(B,assn)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,assn)),product_prod(list(A),list(B))),P5),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),nil(B)))
     => ( ! [P5: fun(A,fun(B,assn)),A3: A,As: list(A),C4: B,Cs: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),aa(fun(A,fun(B,assn)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,assn)),product_prod(list(A),list(B))),P5),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),As)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),C4),Cs)))
       => ( ! [Uu: fun(A,fun(B,assn)),V4: A,Va2: list(A)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),aa(fun(A,fun(B,assn)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,assn)),product_prod(list(A),list(B))),Uu),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V4),Va2)),nil(B)))
         => ~ ! [Uu: fun(A,fun(B,assn)),V4: B,Va2: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),aa(fun(A,fun(B,assn)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,assn)),product_prod(list(A),list(B))),Uu),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V4),Va2))) ) ) ) ).

% list_assn.cases
tff(fact_4893_mergesort__by__rel__merge_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))] :
      ( ! [R6: fun(A,fun(A,$o)),X2: A,Xs: list(A),Y2: A,Ys: list(A)] : X != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),R6),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys)))
     => ( ! [R6: fun(A,fun(A,$o)),Xs: list(A)] : X != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),R6),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),nil(A)))
       => ~ ! [R6: fun(A,fun(A,$o)),V4: A,Va2: list(A)] : X != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),R6),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V4),Va2))) ) ) ).

% mergesort_by_rel_merge.cases
tff(fact_4894_quicksort__by__rel_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))] :
      ( ! [R6: fun(A,fun(A,$o)),Sl: list(A)] : X != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),R6),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Sl),nil(A)))
     => ~ ! [R6: fun(A,fun(A,$o)),Sl: list(A),X2: A,Xs: list(A)] : X != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),R6),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Sl),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs))) ) ).

% quicksort_by_rel.cases
tff(fact_4895_list__all__zip_Ocases,axiom,
    ! [A: $tType,B: $tType,X: product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))] :
      ( ! [P5: fun(A,fun(B,$o))] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),P5),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),nil(B)))
     => ( ! [P5: fun(A,fun(B,$o)),A3: A,As: list(A),B3: B,Bs: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),P5),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),As)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B3),Bs)))
       => ( ! [P5: fun(A,fun(B,$o)),V4: A,Va2: list(A)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),P5),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V4),Va2)),nil(B)))
         => ~ ! [P5: fun(A,fun(B,$o)),V4: B,Va2: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),P5),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V4),Va2))) ) ) ) ).

% list_all_zip.cases
tff(fact_4896_partition__rev_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A)))] :
      ( ! [P5: fun(A,$o),Yes: list(A),No: list(A)] : X != aa(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),aa(fun(A,$o),fun(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A)))),product_Pair(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),P5),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No)),nil(A)))
     => ~ ! [P5: fun(A,$o),Yes: list(A),No: list(A),X2: A,Xs: list(A)] : X != aa(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),aa(fun(A,$o),fun(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A)))),product_Pair(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),P5),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs))) ) ).

% partition_rev.cases
tff(fact_4897_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_4898_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)),$o,aa(fun(A,fun(B,$o)),fun(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_4899_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)),$o,aa(fun(A,fun(B,$o)),fun(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_4900_arg__min__list_Ocases,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [X: product_prod(fun(A,B),list(A))] :
          ( ! [F3: fun(A,B),X2: A] : X != aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),F3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)))
         => ( ! [F3: fun(A,B),X2: A,Y2: A,Zs2: list(A)] : X != aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),F3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Zs2)))
           => ~ ! [A3: fun(A,B)] : X != aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),A3),nil(A)) ) ) ) ).

% arg_min_list.cases
tff(fact_4901_mmap_Ocases,axiom,
    ! [B: $tType,A: $tType,X: product_prod(fun(A,heap_Time_Heap(B)),list(A))] :
      ( ! [F3: fun(A,heap_Time_Heap(B))] : X != aa(list(A),product_prod(fun(A,heap_Time_Heap(B)),list(A)),aa(fun(A,heap_Time_Heap(B)),fun(list(A),product_prod(fun(A,heap_Time_Heap(B)),list(A))),product_Pair(fun(A,heap_Time_Heap(B)),list(A)),F3),nil(A))
     => ~ ! [F3: fun(A,heap_Time_Heap(B)),X2: A,Xs: list(A)] : X != aa(list(A),product_prod(fun(A,heap_Time_Heap(B)),list(A)),aa(fun(A,heap_Time_Heap(B)),fun(list(A),product_prod(fun(A,heap_Time_Heap(B)),list(A))),product_Pair(fun(A,heap_Time_Heap(B)),list(A)),F3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)) ) ).

% mmap.cases
tff(fact_4902_successively_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(A,fun(A,$o)),list(A))] :
      ( ! [P5: fun(A,fun(A,$o))] : X != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),P5),nil(A))
     => ( ! [P5: fun(A,fun(A,$o)),X2: A] : X != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),P5),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)))
       => ~ ! [P5: fun(A,fun(A,$o)),X2: A,Y2: A,Xs: list(A)] : X != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),P5),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs))) ) ) ).

% successively.cases
tff(fact_4903_sorted__wrt_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(A,fun(A,$o)),list(A))] :
      ( ! [P5: fun(A,fun(A,$o))] : X != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),P5),nil(A))
     => ~ ! [P5: fun(A,fun(A,$o)),X2: A,Ys: list(A)] : X != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),P5),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Ys)) ) ).

% sorted_wrt.cases
tff(fact_4904_shuffles_Ocases,axiom,
    ! [A: $tType,X: product_prod(list(A),list(A))] :
      ( ! [Ys: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys)
     => ( ! [Xs: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),nil(A))
       => ~ ! [X2: A,Xs: list(A),Y2: A,Ys: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys)) ) ) ).

% shuffles.cases
tff(fact_4905_merge_Ocases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: product_prod(list(A),list(A))] :
          ( ! [L22: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),L22)
         => ( ! [V4: A,Va2: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V4),Va2)),nil(A))
           => ~ ! [X1: A,L1: list(A),X22: A,L22: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),L1)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),L22)) ) ) ) ).

% merge.cases
tff(fact_4906_subset__eq__mset__impl_Ocases,axiom,
    ! [A: $tType,X: product_prod(list(A),list(A))] :
      ( ! [Ys: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys)
     => ~ ! [X2: A,Xs: list(A),Ys: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)),Ys) ) ).

% subset_eq_mset_impl.cases
tff(fact_4907_merge__list_Ocases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: product_prod(list(list(A)),list(list(A)))] :
          ( ( X != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),nil(list(A))) )
         => ( ! [L2: list(A)] : X != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L2),nil(list(A))))
           => ( ! [La: list(A),Acc2: list(list(A))] : X != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2)),nil(list(A)))
             => ( ! [La: list(A),Acc2: list(list(A)),L2: list(A)] : X != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc2)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L2),nil(list(A))))
               => ~ ! [Acc2: list(list(A)),L1: list(A),L22: list(A),Ls: list(list(A))] : X != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),Acc2),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L1),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L22),Ls))) ) ) ) ) ) ).

% merge_list.cases
tff(fact_4908_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))
         => ( ! [M5: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit0(M5)),one2)
           => ( ! [M5: num,N: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit0(M5)),bit0(N))
             => ( ! [M5: num,N: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit0(M5)),bit1(N))
               => ( ! [M5: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit1(M5)),one2)
                 => ( ! [M5: num,N: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit1(M5)),bit0(N))
                   => ~ ! [M5: num,N: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit1(M5)),bit1(N)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_4909_bin__last__conv__lsb,axiom,
    aTP_Lamp_kv(int,$o) = least_8051144512741203767sb_lsb(int) ).

% bin_last_conv_lsb
tff(fact_4910_divmod__algorithm__code_I8_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Na: num] :
          unique8689654367752047608divmod(A,bit1(M),bit1(Na)) = $ite(aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),Na),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),bit1(M))),unique1321980374590559556d_step(A,bit1(Na),unique8689654367752047608divmod(A,bit1(M),bit0(bit1(Na))))) ) ).

% divmod_algorithm_code(8)
tff(fact_4911_divmod__algorithm__code_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Na: num] :
          unique8689654367752047608divmod(A,bit0(M),bit1(Na)) = $ite(aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),Na),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),bit0(M))),unique1321980374590559556d_step(A,bit1(Na),unique8689654367752047608divmod(A,bit0(M),bit0(bit1(Na))))) ) ).

% divmod_algorithm_code(7)
tff(fact_4912_neg__eucl__rel__int__mult__2,axiom,
    ! [Ba: int,Aa2: int,Q3: int,R: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ba),zero_zero(int))
     => ( eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),Aa2),one_one(int)),Ba,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R))
       => eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Aa2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Ba),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))),R)),one_one(int)))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_4913_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_4914_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_4915_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_4916_mergesort__by__rel_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(A,fun(A,$o)),list(A))] :
      ~ ! [R6: fun(A,fun(A,$o)),Xs: list(A)] : X != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),R6),Xs) ).

% mergesort_by_rel.cases
tff(fact_4917_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_4918_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_4919_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_4920_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_4921_zminus1__lemma,axiom,
    ! [Aa2: int,Ba: int,Q3: int,R: int] :
      ( eucl_rel_int(Aa2,Ba,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R))
     => ( ( Ba != zero_zero(int) )
       => eucl_rel_int(aa(int,int,uminus_uminus(int),Aa2),Ba,
            aa(int,product_prod(int,int),
              aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),
                $ite(R = 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(R = zero_zero(int),zero_zero(int),aa(int,int,minus_minus(int,Ba),R)))) ) ) ).

% zminus1_lemma
tff(fact_4922_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q3: int,R: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R))
    <=> ( ( K = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),L),Q3)),R) )
        & $ite(
            aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L),
            ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R)
            & aa(int,$o,aa(int,fun(int,$o),ord_less(int),R),L) ),
            $ite(
              aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)),
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),R)
              & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R),zero_zero(int)) ),
              Q3 = zero_zero(int) ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_4923_eucl__rel__int__remainderI,axiom,
    ! [R: int,L: int,K: int,Q3: int] :
      ( ( aa(int,int,sgn_sgn(int),R) = aa(int,int,sgn_sgn(int),L) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),R)),aa(int,int,abs_abs(int),L))
       => ( ( K = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q3),L)),R) )
         => eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_4924_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A22: int,A32: product_prod(int,int)] :
      ( eucl_rel_int(A1,A22,A32)
     => ( ( ( A22 = zero_zero(int) )
         => ( A32 != aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),A1) ) )
       => ( ! [Q5: int] :
              ( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q5),zero_zero(int)) )
             => ( ( A22 != zero_zero(int) )
               => ( A1 != aa(int,int,aa(int,fun(int,int),times_times(int),Q5),A22) ) ) )
         => ~ ! [R5: int,Q5: int] :
                ( ( 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),A22) )
                 => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),R5)),aa(int,int,abs_abs(int),A22))
                   => ( A1 != aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q5),A22)),R5) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_4925_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod(int,int)] :
      ( eucl_rel_int(A1,A22,A32)
    <=> ( ? [K2: int] :
            ( ( A1 = K2 )
            & ( 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)),K2) ) )
        | ? [L4: int,K2: int,Q4: int] :
            ( ( A1 = K2 )
            & ( A22 = L4 )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),zero_zero(int)) )
            & ( L4 != zero_zero(int) )
            & ( K2 = aa(int,int,aa(int,fun(int,int),times_times(int),Q4),L4) ) )
        | ? [R7: int,L4: int,K2: int,Q4: int] :
            ( ( A1 = K2 )
            & ( A22 = L4 )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),R7) )
            & ( aa(int,int,sgn_sgn(int),R7) = aa(int,int,sgn_sgn(int),L4) )
            & aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),R7)),aa(int,int,abs_abs(int),L4))
            & ( K2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q4),L4)),R7) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_4926_divmod__divmod__step,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Na: num] :
          unique8689654367752047608divmod(A,M,Na) = $ite(aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),Na),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),M)),unique1321980374590559556d_step(A,Na,unique8689654367752047608divmod(A,M,bit0(Na)))) ) ).

% divmod_divmod_step
tff(fact_4927_pos__eucl__rel__int__mult__2,axiom,
    ! [Ba: int,Aa2: int,Q3: int,R: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Ba)
     => ( eucl_rel_int(Aa2,Ba,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R))
       => eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Aa2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Ba),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),R)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_4928_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_4929_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_4930_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_4931_Divides_Oadjust__div__eq,axiom,
    ! [Q3: int,R: int] : adjust_div(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Q3),aa($o,int,zero_neq_one_of_bool(int),R != zero_zero(int))) ).

% Divides.adjust_div_eq
tff(fact_4932_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_4933_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_4934_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,
                    aa(int,fun(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_4935_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,
              aa(int,fun(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_4936_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,aa(word(A),fun(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),aa(word(A),fun(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_4937_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_4938_push__bit__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat,Aa2: A] :
          ( ( aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),Aa2) = zero_zero(A) )
        <=> ( Aa2 = zero_zero(A) ) ) ) ).

% push_bit_eq_0_iff
tff(fact_4939_push__bit__push__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: nat,Aa2: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),M),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),Aa2)) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)),Aa2) ) ).

% push_bit_push_bit
tff(fact_4940_push__bit__and,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,Aa2: A,Ba: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Aa2),Ba)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),Aa2)),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),Ba)) ) ).

% push_bit_and
tff(fact_4941_push__bit__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,Aa2: A,Ba: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Aa2),Ba)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),Aa2)),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),Ba)) ) ).

% push_bit_or
tff(fact_4942_push__bit__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,Aa2: A,Ba: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Aa2),Ba)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),Aa2)),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),Ba)) ) ).

% push_bit_xor
tff(fact_4943_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_4944_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_4945_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_4946_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_4947_push__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,Aa2: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(nat,nat,suc,Na)),Aa2) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% push_bit_Suc
tff(fact_4948_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_4949_even__push__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,Aa2: A] :
          ( aa(A,$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),Aa2))
        <=> ( ( Na != zero_zero(nat) )
            | aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Aa2) ) ) ) ).

% even_push_bit_iff
tff(fact_4950_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_4951_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_4952_push__bit__minus,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,Aa2: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),aa(A,A,uminus_uminus(A),Aa2)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),Aa2)) ) ).

% push_bit_minus
tff(fact_4953_push__bit__of__int,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,K: int] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),aa(int,A,ring_1_of_int(A),K)) = aa(int,A,ring_1_of_int(A),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),Na),K)) ) ).

% push_bit_of_int
tff(fact_4954_push__bit__add,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,Aa2: A,Ba: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),Aa2)),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),Ba)) ) ).

% push_bit_add
tff(fact_4955_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_4956_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_4957_push__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: nat,Aa2: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),M),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),Aa2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),M),Aa2)) ) ).

% push_bit_take_bit
tff(fact_4958_take__bit__push__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: nat,Aa2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),Aa2)) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,minus_minus(nat,M),Na)),Aa2)) ) ).

% take_bit_push_bit
tff(fact_4959_div__push__bit__of__1__eq__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A,Na: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),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),Aa2) ) ).

% div_push_bit_of_1_eq_drop_bit
tff(fact_4960_bits__ident,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,Aa2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),Aa2))),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),Aa2)) = Aa2 ) ).

% bits_ident
tff(fact_4961_set__bit__eq__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,Aa2: A] : aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),Na),Aa2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Aa2),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),one_one(A))) ) ).

% set_bit_eq_or
tff(fact_4962_flip__bit__eq__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,Aa2: A] : bit_se8732182000553998342ip_bit(A,Na,Aa2) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Aa2),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),one_one(A))) ) ).

% flip_bit_eq_xor
tff(fact_4963_push__bit__double,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,Aa2: 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),Aa2),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),Aa2)),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% push_bit_double
tff(fact_4964_bit__iff__and__push__bit__not__eq__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),Na)
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Aa2),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_4965_push__bit__mask__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,Na: nat] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),M),bit_se2239418461657761734s_mask(A,Na)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),bit_se2239418461657761734s_mask(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M))),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,M))) ) ).

% push_bit_mask_eq
tff(fact_4966_unset__bit__eq__and__not,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,Aa2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),Na),Aa2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Aa2),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_4967_push__bit__eq__mult,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,Aa2: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),Aa2) = aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) ) ).

% push_bit_eq_mult
tff(fact_4968_exp__dvdE,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,Aa2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)),Aa2)
         => ~ ! [B3: A] : Aa2 != aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),B3) ) ) ).

% exp_dvdE
tff(fact_4969_slice__eq__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,M: nat,Aa2: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),Aa2))) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Aa2),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),bit_se2239418461657761734s_mask(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na))),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,Na)))) ) ).

% slice_eq_mask
tff(fact_4970_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,aa(nat,fun(nat,nat),plus_plus(nat),Index),one_one(nat)))) ) ) ) ).

% word_and_mask_or_conv_and_mask
tff(fact_4971_signed__take__bit__code,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,Aa2: A] :
          aa(A,A,bit_ri4674362597316999326ke_bit(A,Na),Aa2) = $let(
            l: A,
            l:= aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Na)),Aa2),
            $ite(aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),l),Na),aa(A,A,aa(A,fun(A,A),plus_plus(A),l),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(nat,nat,suc,Na)),aa(A,A,uminus_uminus(A),one_one(A)))),l) ) ) ).

% signed_take_bit_code
tff(fact_4972_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun(int,fun(int,$o))] :
      ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1))
     => ( ! [K5: int,L2: int] :
            ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),K5),L2))
           => ( ( ~ ( member(int,K5,aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
                    & member(int,L2,aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) )
               => aa(int,$o,aa(int,fun(int,$o),P,aa(int,int,aa(int,fun(int,int),divide_divide(int),K5),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L2),aa(num,int,numeral_numeral(int),bit0(one2)))) )
             => aa(int,$o,aa(int,fun(int,$o),P,K5),L2) ) )
       => aa(int,$o,aa(int,fun(int,$o),P,A0),A1) ) ) ).

% and_int.pinduct
tff(fact_4973_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_4974_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun(int,fun(int,$o))] :
      ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1))
     => ( ! [I2: int,J2: int] :
            ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),I2),J2))
           => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),J2)
               => aa(int,$o,aa(int,fun(int,$o),P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))),J2) )
             => aa(int,$o,aa(int,fun(int,$o),P,I2),J2) ) )
       => aa(int,$o,aa(int,fun(int,$o),P,A0),A1) ) ) ).

% upto.pinduct
tff(fact_4975_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_4976_push__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),Na),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% push_bit_nonnegative_int_iff
tff(fact_4977_push__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),Na),K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% push_bit_negative_int_iff
tff(fact_4978_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_4979_push__bit__nat__eq,axiom,
    ! [Na: nat,K: int] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se4730199178511100633sh_bit(nat),Na),aa(int,nat,nat2,K)) = aa(int,nat,nat2,aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),Na),K)) ).

% push_bit_nat_eq
tff(fact_4980_drop__bit__push__bit__int,axiom,
    ! [M: nat,Na: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),M),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),Na),K)) = aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(nat,nat,minus_minus(nat,M),Na)),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),aa(nat,nat,minus_minus(nat,Na),M)),K)) ).

% drop_bit_push_bit_int
tff(fact_4981_set__bit__nat__def,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,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_4982_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_4983_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,Na: nat] :
      ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),M),K)),Na)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
        & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),aa(nat,nat,minus_minus(nat,Na),M)) ) ) ).

% bit_push_bit_iff_int
tff(fact_4984_Bit__Operations_Oset__bit__int__def,axiom,
    ! [Na: nat,K: int] : aa(int,int,aa(nat,fun(int,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_4985_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q3: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),bit_se5641148757651400278ts_bit(nat),aa(nat,nat,aa(nat,fun(nat,nat),bit_se4730199178511100633sh_bit(nat),M),Q3)),Na)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
        & aa(nat,$o,aa(nat,fun(nat,$o),bit_se5641148757651400278ts_bit(nat),Q3),aa(nat,nat,minus_minus(nat,Na),M)) ) ) ).

% bit_push_bit_iff_nat
tff(fact_4986_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_4987_unset__bit__int__def,axiom,
    ! [Na: nat,K: int] : aa(int,int,aa(nat,fun(int,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_4988_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_4989_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_4990_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_4991_test__bit__split__eq,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(C)
        & type_len(A) )
     => ! [C2: word(C),Aa2: word(A),Ba: word(B)] :
          ( ( word_split(C,A,B,C2) = aa(word(B),product_prod(word(A),word(B)),aa(word(A),fun(word(B),product_prod(word(A),word(B))),product_Pair(word(A),word(B)),Aa2),Ba) )
        <=> ( ! [N2: nat] :
                ( aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),Ba),N2)
              <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),aa(word(B),nat,size_size(word(B)),Ba))
                  & aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),C2),N2) ) )
            & ! [M4: nat] :
                ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Aa2),M4)
              <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),aa(word(A),nat,size_size(word(A)),Aa2))
                  & aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),C2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M4),aa(word(B),nat,size_size(word(B)),Ba))) ) ) ) ) ) ).

% test_bit_split_eq
tff(fact_4992_test__bit__split_H,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(C)
        & type_len(A) )
     => ! [C2: word(C),Aa2: word(A),Ba: word(B)] :
          ( ( word_split(C,A,B,C2) = aa(word(B),product_prod(word(A),word(B)),aa(word(A),fun(word(B),product_prod(word(A),word(B))),product_Pair(word(A),word(B)),Aa2),Ba) )
         => ! [N10: nat,M3: nat] :
              ( ( aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),Ba),N10)
              <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N10),aa(word(B),nat,size_size(word(B)),Ba))
                  & aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),C2),N10) ) )
              & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Aa2),M3)
              <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),aa(word(A),nat,size_size(word(A)),Aa2))
                  & aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),C2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M3),aa(word(B),nat,size_size(word(B)),Ba))) ) ) ) ) ) ).

% test_bit_split'
tff(fact_4993_test__bit__split,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(C)
        & type_len(A) )
     => ! [C2: word(C),Aa2: word(A),Ba: word(B)] :
          ( ( word_split(C,A,B,C2) = aa(word(B),product_prod(word(A),word(B)),aa(word(A),fun(word(B),product_prod(word(A),word(B))),product_Pair(word(A),word(B)),Aa2),Ba) )
         => ( ! [N10: nat] :
                ( aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),Ba),N10)
              <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N10),aa(word(B),nat,size_size(word(B)),Ba))
                  & aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),C2),N10) ) )
            & ! [M3: nat] :
                ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Aa2),M3)
              <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),aa(word(A),nat,size_size(word(A)),Aa2))
                  & aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),C2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M3),aa(word(B),nat,size_size(word(B)),Ba))) ) ) ) ) ) ).

% test_bit_split
tff(fact_4994_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_4995_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_kw(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M,Na)) ) ).

% divmod_algorithm_code(6)
tff(fact_4996_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_kx(num,fun(A,fun(A,product_prod(A,A))),L)),Qr) ) ).

% divmod_step_def
tff(fact_4997_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_4998_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_4999_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_ky(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M,Na)) ) ).

% divmod_algorithm_code(5)
tff(fact_5000_semiring__norm_I26_J,axiom,
    bitM(one2) = one2 ).

% semiring_norm(26)
tff(fact_5001_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_kz(fun(B,fun(C,fun(D,A))),fun(D,fun(B,fun(C,A))),F2),Y)),X) ).

% nested_case_prod_simp
tff(fact_5002_semiring__norm_I28_J,axiom,
    ! [Na: num] : bitM(bit1(Na)) = bit1(bit0(Na)) ).

% semiring_norm(28)
tff(fact_5003_semiring__norm_I27_J,axiom,
    ! [Na: num] : bitM(bit0(Na)) = bit1(bitM(Na)) ).

% semiring_norm(27)
tff(fact_5004_inc__BitM__eq,axiom,
    ! [Na: num] : inc(bitM(Na)) = bit0(Na) ).

% inc_BitM_eq
tff(fact_5005_BitM__inc__eq,axiom,
    ! [Na: num] : bitM(inc(Na)) = bit1(Na) ).

% BitM_inc_eq
tff(fact_5006_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_5007_BitM__plus__one,axiom,
    ! [Na: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),bitM(Na)),one2) = bit0(Na) ).

% BitM_plus_one
tff(fact_5008_one__plus__BitM,axiom,
    ! [Na: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),bitM(Na)) = bit0(Na) ).

% one_plus_BitM
tff(fact_5009_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_5010_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_5011_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_5012_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_la(num,fun(nat,fun(nat,product_prod(nat,nat))),L)),Qr) ).

% divmod_step_nat_def
tff(fact_5013_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_lb(num,fun(int,fun(int,product_prod(int,int))),L)),Qr) ).

% divmod_step_int_def
tff(fact_5014_case__prod__conv,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(B,fun(C,A)),Aa2: B,Ba: 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),Aa2),Ba)) = aa(C,A,aa(B,fun(C,A),F2,Aa2),Ba) ).

% case_prod_conv
tff(fact_5015_divmod__nat__if,axiom,
    ! [M: nat,Na: nat] :
      divmod_nat(M,Na) = $ite(
        ( ( Na = zero_zero(nat) )
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ),
        aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),M),
        aa(product_prod(nat,nat),product_prod(nat,nat),product_case_prod(nat,nat,product_prod(nat,nat),aTP_Lamp_lc(nat,fun(nat,product_prod(nat,nat)))),divmod_nat(aa(nat,nat,minus_minus(nat,M),Na),Na)) ) ).

% divmod_nat_if
tff(fact_5016_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_5017_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,$o)),aTP_Lamp_ld(A,fun(B,fun(A,fun(B,$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_5018_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,$o)),aTP_Lamp_ld(A,fun(B,fun(A,fun(B,$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_5019_case__prodI,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(B,$o)),Aa2: A,Ba: B] :
      ( aa(B,$o,aa(A,fun(B,$o),F2,Aa2),Ba)
     => 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),Aa2),Ba)) ) ).

% case_prodI
tff(fact_5020_case__prodI2,axiom,
    ! [B: $tType,A: $tType,P2: product_prod(A,B),C2: fun(A,fun(B,$o))] :
      ( ! [A3: A,B3: B] :
          ( ( P2 = 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),C2,A3),B3) )
     => aa(product_prod(A,B),$o,product_case_prod(A,B,$o,C2),P2) ) ).

% case_prodI2
tff(fact_5021_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] :
      ( ! [A3: A,B3: B] :
          ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3) = P2 )
         => aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),C2,A3),B3),X) )
     => aa(C,$o,aa(product_prod(A,B),fun(C,$o),product_case_prod(A,B,fun(C,$o),C2),P2),X) ) ).

% case_prodI2'
tff(fact_5022_mem__case__prodI,axiom,
    ! [A: $tType,B: $tType,C: $tType,Z: A,C2: fun(B,fun(C,set(A))),Aa2: B,Ba: C] :
      ( member(A,Z,aa(C,set(A),aa(B,fun(C,set(A)),C2,Aa2),Ba))
     => 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),Aa2),Ba))) ) ).

% mem_case_prodI
tff(fact_5023_mem__case__prodI2,axiom,
    ! [C: $tType,B: $tType,A: $tType,P2: product_prod(A,B),Z: C,C2: fun(A,fun(B,set(C)))] :
      ( ! [A3: A,B3: B] :
          ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3) )
         => member(C,Z,aa(B,set(C),aa(A,fun(B,set(C)),C2,A3),B3)) )
     => member(C,Z,aa(product_prod(A,B),set(C),product_case_prod(A,B,set(C),C2),P2)) ) ).

% mem_case_prodI2
tff(fact_5024_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_5025_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_5026_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_5027_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_5028_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_5029_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,$o,aTP_Lamp_le(fun(product_prod(A,B),$o),fun(A,fun(B,$o)),P))) ).

% split_paired_Eps
tff(fact_5030_TBOUND__prod__case,axiom,
    ! [C: $tType,B: $tType,A: $tType,T: product_prod(A,B),F2: fun(A,fun(B,heap_Time_Heap(C))),Bnd: fun(A,fun(B,nat))] :
      ( ! [A3: A,B3: B] :
          ( ( T = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3) )
         => time_TBOUND(C,aa(B,heap_Time_Heap(C),aa(A,fun(B,heap_Time_Heap(C)),F2,A3),B3),aa(B,nat,aa(A,fun(B,nat),Bnd,A3),B3)) )
     => time_TBOUND(C,aa(product_prod(A,B),heap_Time_Heap(C),product_case_prod(A,B,heap_Time_Heap(C),F2),T),aa(product_prod(A,B),nat,product_case_prod(A,B,nat,Bnd),T)) ) ).

% TBOUND_prod_case
tff(fact_5031_case__prodD,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(B,$o)),Aa2: A,Ba: 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),Aa2),Ba))
     => aa(B,$o,aa(A,fun(B,$o),F2,Aa2),Ba) ) ).

% case_prodD
tff(fact_5032_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)
     => ~ ! [X2: A,Y2: B] :
            ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2) )
           => ~ aa(B,$o,aa(A,fun(B,$o),C2,X2),Y2) ) ) ).

% case_prodE
tff(fact_5033_case__prodD_H,axiom,
    ! [B: $tType,A: $tType,C: $tType,R3: fun(A,fun(B,fun(C,$o))),Aa2: A,Ba: B,C2: C] :
      ( aa(C,$o,aa(product_prod(A,B),fun(C,$o),product_case_prod(A,B,fun(C,$o),R3),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)),C2)
     => aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),R3,Aa2),Ba),C2) ) ).

% case_prodD'
tff(fact_5034_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,$o),C2),P2),Z)
     => ~ ! [X2: A,Y2: B] :
            ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2) )
           => ~ aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),C2,X2),Y2),Z) ) ) ).

% case_prodE'
tff(fact_5035_or__not__num__neg_Osimps_I1_J,axiom,
    bit_or_not_num_neg(one2,one2) = one2 ).

% or_not_num_neg.simps(1)
tff(fact_5036_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_5037_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_5038_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_5039_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_5040_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)] :
      ( ! [A3: A,B3: B] :
          ( ( X = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3) )
         => hoare_hoare_triple(C,P,aa(B,heap_Time_Heap(C),aa(A,fun(B,heap_Time_Heap(C)),F2,A3),B3),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_5041_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_lf(nat,fun(nat,fun(nat,$o)),Na)))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_lh(fun(nat,fun(nat,A)),fun(nat,A),G)),set_ord_atMost(nat,Na)) ) ).

% sum.triangle_reindex_eq
tff(fact_5042_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_lf(nat,fun(nat,fun(nat,$o)),Na)))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_lj(fun(nat,fun(nat,A)),fun(nat,A),G),set_ord_atMost(nat,Na)) ) ).

% prod.triangle_reindex_eq
tff(fact_5043_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_5044_or__not__num__neg_Osimps_I9_J,axiom,
    ! [Na: num,M: num] : bit_or_not_num_neg(bit1(Na),bit1(M)) = bitM(bit_or_not_num_neg(Na,M)) ).

% or_not_num_neg.simps(9)
tff(fact_5045_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_lk(nat,fun(nat,fun(nat,$o)),Na)))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_lh(fun(nat,fun(nat,A)),fun(nat,A),G)),set_ord_lessThan(nat,Na)) ) ).

% sum.triangle_reindex
tff(fact_5046_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_lk(nat,fun(nat,fun(nat,$o)),Na)))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_lj(fun(nat,fun(nat,A)),fun(nat,A),G),set_ord_lessThan(nat,Na)) ) ).

% prod.triangle_reindex
tff(fact_5047_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_5048_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_5049_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_ll(fun(B,A),fun(fun(C,fun(D,B)),fun(C,fun(D,A))),H),F2)),Prod) ).

% prod.case_distrib
tff(fact_5050_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 )
           => ! [M5: num] :
                ( ( Xa2 = bit0(M5) )
               => ( Y != bit1(M5) ) ) )
         => ( ( ( X = one2 )
             => ! [M5: num] :
                  ( ( Xa2 = bit1(M5) )
                 => ( Y != bit1(M5) ) ) )
           => ( ( ? [N: num] : X = bit0(N)
               => ( ( Xa2 = one2 )
                 => ( Y != bit0(one2) ) ) )
             => ( ! [N: num] :
                    ( ( X = bit0(N) )
                   => ! [M5: num] :
                        ( ( Xa2 = bit0(M5) )
                       => ( Y != bitM(bit_or_not_num_neg(N,M5)) ) ) )
               => ( ! [N: num] :
                      ( ( X = bit0(N) )
                     => ! [M5: num] :
                          ( ( Xa2 = bit1(M5) )
                         => ( Y != bit0(bit_or_not_num_neg(N,M5)) ) ) )
                 => ( ( ? [N: num] : X = bit1(N)
                     => ( ( Xa2 = one2 )
                       => ( Y != one2 ) ) )
                   => ( ! [N: num] :
                          ( ( X = bit1(N) )
                         => ! [M5: num] :
                              ( ( Xa2 = bit0(M5) )
                             => ( Y != bitM(bit_or_not_num_neg(N,M5)) ) ) )
                     => ~ ! [N: num] :
                            ( ( X = bit1(N) )
                           => ! [M5: num] :
                                ( ( Xa2 = bit1(M5) )
                               => ( Y != bitM(bit_or_not_num_neg(N,M5)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
tff(fact_5051_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_5052_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_5053_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_5054_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_lm(int,fun(int,int))),Qr) ).

% Divides.adjust_div_def
tff(fact_5055_cond__case__prod__eta,axiom,
    ! [C: $tType,B: $tType,A: $tType,F2: fun(A,fun(B,C)),G: fun(product_prod(A,B),C)] :
      ( ! [X2: A,Y2: B] : aa(B,C,aa(A,fun(B,C),F2,X2),Y2) = 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),X2),Y2))
     => ( product_case_prod(A,B,C,F2) = G ) ) ).

% cond_case_prod_eta
tff(fact_5056_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_ln(fun(product_prod(A,B),C),fun(A,fun(B,C)),F2)) = F2 ).

% case_prod_eta
tff(fact_5057_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,$o,Q,aa(product_prod(B,C),A,product_case_prod(B,C,A,P),Z))
     => ~ ! [X2: B,Y2: C] :
            ( ( Z = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X2),Y2) )
           => ~ aa(A,$o,Q,aa(C,A,aa(B,fun(C,A),P,X2),Y2)) ) ) ).

% case_prodE2
tff(fact_5058_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_lo(nat,fun(nat,A))),divmod_nat(Na,aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% of_nat_code_if
tff(fact_5059_int__ge__less__than2__def,axiom,
    ! [D2: int] : int_ge_less_than2(D2) = collect(product_prod(int,int),product_case_prod(int,int,$o,aTP_Lamp_lp(int,fun(int,fun(int,$o)),D2))) ).

% int_ge_less_than2_def
tff(fact_5060_int__ge__less__than__def,axiom,
    ! [D2: int] : int_ge_less_than(D2) = collect(product_prod(int,int),product_case_prod(int,int,$o,aTP_Lamp_lq(int,fun(int,fun(int,$o)),D2))) ).

% int_ge_less_than_def
tff(fact_5061_split__part,axiom,
    ! [B: $tType,A: $tType,P: $o,Q: fun(A,fun(B,$o)),X4: 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_lr($o,fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),(P)),Q)),X4)
    <=> ( (P)
        & aa(product_prod(A,B),$o,product_case_prod(A,B,$o,Q),X4) ) ) ).

% split_part
tff(fact_5062_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,$o,aTP_Lamp_ls(A,fun(B,$o))),Prod) ).

% prod.disc_eq_case
tff(fact_5063_rel__of__def,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),P: fun(product_prod(A,B),$o)] : rel_of(A,B,M,P) = collect(product_prod(A,B),product_case_prod(A,B,$o,aa(fun(product_prod(A,B),$o),fun(A,fun(B,$o)),aTP_Lamp_lt(fun(A,option(B)),fun(fun(product_prod(A,B),$o),fun(A,fun(B,$o))),M),P))) ).

% rel_of_def
tff(fact_5064_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_lu(int,fun(int,product_prod(int,int)))),quotient_of(P2)) ).

% rat_inverse_code
tff(fact_5065_upto_Opelims,axiom,
    ! [X: int,Xa2: int,Y: list(int)] :
      ( ( upto(X,Xa2) = Y )
     => ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X),Xa2))
       => ~ ( ( Y = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X),Xa2),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),X),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),X),one_one(int)),Xa2)),nil(int)) )
           => ~ accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X),Xa2)) ) ) ) ).

% upto.pelims
tff(fact_5066_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_5067_upto__empty,axiom,
    ! [J: int,I: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I)
     => ( upto(I,J) = nil(int) ) ) ).

% upto_empty
tff(fact_5068_upto__Nil2,axiom,
    ! [I: int,J: int] :
      ( ( nil(int) = upto(I,J) )
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I) ) ).

% upto_Nil2
tff(fact_5069_upto__Nil,axiom,
    ! [I: int,J: int] :
      ( ( upto(I,J) = nil(int) )
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I) ) ).

% upto_Nil
tff(fact_5070_upto__single,axiom,
    ! [I: int] : upto(I,I) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I),nil(int)) ).

% upto_single
tff(fact_5071_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_5072_length__upto,axiom,
    ! [I: int,J: int] : aa(list(int),nat,size_size(list(int)),upto(I,J)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,minus_minus(int,J),I)),one_one(int))) ).

% length_upto
tff(fact_5073_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_5074_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_5075_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_5076_upto__rec__numeral_I1_J,axiom,
    ! [M: num,Na: num] :
      upto(aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),Na)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),Na)),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),aa(num,int,numeral_numeral(int),M)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),M)),one_one(int)),aa(num,int,numeral_numeral(int),Na))),nil(int)) ).

% upto_rec_numeral(1)
tff(fact_5077_upto__rec__numeral_I4_J,axiom,
    ! [M: num,Na: num] :
      upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na))) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na))),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na)))),nil(int)) ).

% upto_rec_numeral(4)
tff(fact_5078_upto__rec__numeral_I3_J,axiom,
    ! [M: num,Na: num] :
      upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),Na)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),Na)),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),one_one(int)),aa(num,int,numeral_numeral(int),Na))),nil(int)) ).

% upto_rec_numeral(3)
tff(fact_5079_upto__rec__numeral_I2_J,axiom,
    ! [M: num,Na: num] :
      upto(aa(num,int,numeral_numeral(int),M),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na))) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na))),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),aa(num,int,numeral_numeral(int),M)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),M)),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na)))),nil(int)) ).

% upto_rec_numeral(2)
tff(fact_5080_atLeastAtMost__upto,axiom,
    ! [I: int,J: int] : set_or1337092689740270186AtMost(int,I,J) = aa(list(int),set(int),set2(int),upto(I,J)) ).

% atLeastAtMost_upto
tff(fact_5081_upto__code,axiom,
    ! [I: int,J: int] : upto(I,J) = upto_aux(I,J,nil(int)) ).

% upto_code
tff(fact_5082_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_lv(int,fun(int,product_prod(int,int)))),quotient_of(P2)) ).

% rat_abs_code
tff(fact_5083_upto__rec1,axiom,
    ! [I: int,J: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( upto(I,J) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)) ) ) ).

% upto_rec1
tff(fact_5084_upto_Osimps,axiom,
    ! [I: int,J: int] :
      upto(I,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)),nil(int)) ).

% upto.simps
tff(fact_5085_upto_Oelims,axiom,
    ! [X: int,Xa2: int,Y: list(int)] :
      ( ( upto(X,Xa2) = Y )
     => ( Y = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X),Xa2),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),X),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),X),one_one(int)),Xa2)),nil(int)) ) ) ).

% upto.elims
tff(fact_5086_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_lw(int,fun(int,product_prod(int,int)))),quotient_of(P2)) ).

% rat_uminus_code
tff(fact_5087_rat__less__code,axiom,
    ! [P2: rat,Q3: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),P2),Q3)
    <=> aa(product_prod(int,int),$o,product_case_prod(int,int,$o,aTP_Lamp_ly(rat,fun(int,fun(int,$o)),Q3)),quotient_of(P2)) ) ).

% rat_less_code
tff(fact_5088_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_5089_rat__less__eq__code,axiom,
    ! [P2: rat,Q3: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),P2),Q3)
    <=> aa(product_prod(int,int),$o,product_case_prod(int,int,$o,aTP_Lamp_ma(rat,fun(int,fun(int,$o)),Q3)),quotient_of(P2)) ) ).

% rat_less_eq_code
tff(fact_5090_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),I),J))
     => ( upto(I,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)),nil(int)) ) ) ).

% upto.psimps
tff(fact_5091_quotient__of__int,axiom,
    ! [Aa2: int] : quotient_of(of_int(Aa2)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Aa2),one_one(int)) ).

% quotient_of_int
tff(fact_5092_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_mc(rat,fun(int,fun(int,product_prod(int,int))),Q3)),quotient_of(P2)) ).

% rat_minus_code
tff(fact_5093_rat__plus__code,axiom,
    ! [P2: rat,Q3: rat] : quotient_of(aa(rat,rat,aa(rat,fun(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_me(rat,fun(int,fun(int,product_prod(int,int))),Q3)),quotient_of(P2)) ).

% rat_plus_code
tff(fact_5094_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_5095_normalize__crossproduct,axiom,
    ! [Q3: int,S: int,P2: int,R: int] :
      ( ( Q3 != zero_zero(int) )
     => ( ( S != zero_zero(int) )
       => ( ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q3)) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),R),S)) )
         => ( aa(int,int,aa(int,fun(int,int),times_times(int),P2),S) = aa(int,int,aa(int,fun(int,int),times_times(int),R),Q3) ) ) ) ) ).

% normalize_crossproduct
tff(fact_5096_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_mg(rat,fun(int,fun(int,product_prod(int,int))),Q3)),quotient_of(P2)) ).

% rat_divide_code
tff(fact_5097_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_mi(rat,fun(int,fun(int,product_prod(int,int))),Q3)),quotient_of(P2)) ).

% rat_times_code
tff(fact_5098_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_5099_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_kz(fun(B,fun(C,fun(D,A))),fun(D,fun(B,fun(C,A))),F2),Y)),X) ).

% case_prod_app
tff(fact_5100_concat__bit__Suc,axiom,
    ! [Na: nat,K: int,L: int] : aa(int,int,bit_concat_bit(aa(nat,nat,suc,Na),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,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_5101_concat__bit__0,axiom,
    ! [K: int,L: int] : aa(int,int,bit_concat_bit(zero_zero(nat),K),L) = L ).

% concat_bit_0
tff(fact_5102_concat__bit__of__zero__2,axiom,
    ! [Na: nat,K: int] : aa(int,int,bit_concat_bit(Na,K),zero_zero(int)) = aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K) ).

% concat_bit_of_zero_2
tff(fact_5103_concat__bit__nonnegative__iff,axiom,
    ! [Na: nat,K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_concat_bit(Na,K),L))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) ) ).

% concat_bit_nonnegative_iff
tff(fact_5104_concat__bit__negative__iff,axiom,
    ! [Na: nat,K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_concat_bit(Na,K),L)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ).

% concat_bit_negative_iff
tff(fact_5105_concat__bit__of__zero__1,axiom,
    ! [Na: nat,L: int] : aa(int,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_5106_concat__bit__take__bit__eq,axiom,
    ! [Na: nat,Ba: int] : bit_concat_bit(Na,aa(int,int,bit_se2584673776208193580ke_bit(int,Na),Ba)) = bit_concat_bit(Na,Ba) ).

% concat_bit_take_bit_eq
tff(fact_5107_concat__bit__eq__iff,axiom,
    ! [Na: nat,K: int,L: int,R: int,S: int] :
      ( ( aa(int,int,bit_concat_bit(Na,K),L) = aa(int,int,bit_concat_bit(Na,R),S) )
    <=> ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K) = aa(int,int,bit_se2584673776208193580ke_bit(int,Na),R) )
        & ( L = S ) ) ) ).

% concat_bit_eq_iff
tff(fact_5108_concat__bit__assoc,axiom,
    ! [Na: nat,K: int,M: nat,L: int,R: int] : aa(int,int,bit_concat_bit(Na,K),aa(int,int,bit_concat_bit(M,L),R)) = aa(int,int,bit_concat_bit(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na),aa(int,int,bit_concat_bit(Na,K),L)),R) ).

% concat_bit_assoc
tff(fact_5109_concat__bit__eq,axiom,
    ! [Na: nat,K: int,L: int] : aa(int,int,bit_concat_bit(Na,K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),Na),L)) ).

% concat_bit_eq
tff(fact_5110_concat__bit__def,axiom,
    ! [Na: nat,K: int,L: int] : aa(int,int,bit_concat_bit(Na,K),L) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),Na),L)) ).

% concat_bit_def
tff(fact_5111_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),aa(int,int,bit_concat_bit(M,K),L)),Na)
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
          & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),Na) )
        | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
          & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),L),aa(nat,nat,minus_minus(nat,Na),M)) ) ) ) ).

% bit_concat_bit_iff
tff(fact_5112_fun__cong__unused__0,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( zero(B)
     => ! [F2: fun(fun(A,B),C),G: C] :
          ( ! [X2: fun(A,B)] : aa(fun(A,B),C,F2,X2) = G
         => ( aa(fun(A,B),C,F2,aTP_Lamp_mj(A,B)) = G ) ) ) ).

% fun_cong_unused_0
tff(fact_5113_signed__take__bit__eq__concat__bit,axiom,
    ! [Na: nat,K: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K) = aa(int,int,bit_concat_bit(Na,K),aa(int,int,uminus_uminus(int),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_concat_bit
tff(fact_5114_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_5115_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_5116_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_5117_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_5118_word__cat__split__alt,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B)
        & type_len(C) )
     => ! [W: word(A),U: word(B),V: word(C)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(A),nat,size_size(word(A)),W)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(B),nat,size_size(word(B)),U)),aa(word(C),nat,size_size(word(C)),V)))
         => ( ( 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),V) )
           => ( word_cat(B,C,A,U,V) = W ) ) ) ) ).

% word_cat_split_alt
tff(fact_5119_word__split__cat__alt,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(B)
        & type_len(A) )
     => ! [W: word(A),U: word(B),V: word(C)] :
          ( ( W = word_cat(B,C,A,U,V) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(B),nat,size_size(word(B)),U)),aa(word(C),nat,size_size(word(C)),V))),aa(word(A),nat,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),V) ) ) ) ) ).

% word_split_cat_alt
tff(fact_5120_word__cat__split__size,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(A)
        & type_len(B) )
     => ! [T: word(A),U: word(B),V: word(C)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(A),nat,size_size(word(A)),T)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(B),nat,size_size(word(B)),U)),aa(word(C),nat,size_size(word(C)),V)))
         => ( ( 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),V) = word_split(A,B,C,T) )
           => ( T = word_cat(B,C,A,U,V) ) ) ) ) ).

% word_cat_split_size
tff(fact_5121_word__cat__id,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Aa2: word(B),Ba: word(A)] : word_cat(B,A,A,Aa2,Ba) = Ba ) ).

% word_cat_id
tff(fact_5122_test__bit__cat,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(B)
        & type_len(A) )
     => ! [Aa2: word(B),Ba: word(C),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),word_cat(B,C,A,Aa2,Ba)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(word(A),nat,size_size(word(A)),word_cat(B,C,A,Aa2,Ba)))
            & $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(word(C),nat,size_size(word(C)),Ba)),aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),Ba),Na),aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),Aa2),aa(nat,nat,minus_minus(nat,Na),aa(word(C),nat,size_size(word(C)),Ba)))) ) ) ) ).

% test_bit_cat
tff(fact_5123_cat__slices,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A)
        & type_len(C) )
     => ! [Aa2: word(A),Na: nat,C2: word(B),Ba: word(C)] :
          ( ( Aa2 = aa(word(B),word(A),slice2(B,A,Na),C2) )
         => ( ( Ba = aa(word(B),word(C),slice2(B,C,zero_zero(nat)),C2) )
           => ( ( Na = aa(word(C),nat,size_size(word(C)),Ba) )
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(B),nat,size_size(word(B)),C2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,size_size(word(A)),Aa2)),aa(word(C),nat,size_size(word(C)),Ba)))
               => ( word_cat(A,C,B,Aa2,Ba) = C2 ) ) ) ) ) ) ).

% cat_slices
tff(fact_5124_int__set__bit__conv__ops,axiom,
    ! [I: int,Na: nat,Ba: $o] :
      aa($o,int,aa(nat,fun($o,int),aa(int,fun(nat,fun($o,int)),generi7602027413899671122et_bit(int),I),Na),(Ba)) = $ite((Ba),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_5125_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_5126_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_5127_set__decode__inverse,axiom,
    ! [Na: nat] : aa(set(nat),nat,nat_set_encode,nat_set_decode(Na)) = Na ).

% set_decode_inverse
tff(fact_5128_set__encode__empty,axiom,
    aa(set(nat),nat,nat_set_encode,bot_bot(set(nat))) = zero_zero(nat) ).

% set_encode_empty
tff(fact_5129_slice__id,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [T: word(A)] : aa(word(A),word(A),slice2(A,A,zero_zero(nat)),T) = T ) ).

% slice_id
tff(fact_5130_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_5131_slice__cat2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Aa2: word(B),T: word(A)] : aa(word(A),word(A),slice2(A,A,zero_zero(nat)),word_cat(B,A,A,Aa2,T)) = T ) ).

% slice_cat2
tff(fact_5132_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_5133_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_5134_slice__cat1,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(C)
        & type_len(B) )
     => ! [Aa2: word(A),Ba: word(B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,size_size(word(A)),Aa2)),aa(word(B),nat,size_size(word(B)),Ba))),aa(word(C),nat,size_size(word(C)),word_cat(A,B,C,Aa2,Ba)))
         => ( aa(word(C),word(A),slice2(C,A,aa(word(B),nat,size_size(word(B)),Ba)),word_cat(A,B,C,Aa2,Ba)) = Aa2 ) ) ) ).

% slice_cat1
tff(fact_5135_split__slices,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(B)
        & type_len(A) )
     => ! [W: word(C),U: word(A),V: 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),V) )
         => ( ( U = aa(word(C),word(A),slice2(C,A,aa(word(B),nat,size_size(word(B)),V)),W) )
            & ( V = aa(word(C),word(B),slice2(C,B,zero_zero(nat)),W) ) ) ) ) ).

% split_slices
tff(fact_5136_bit__set__bit__iff__2n,axiom,
    ! [A: $tType] :
      ( generic_set_set_bit(A)
     => ! [Aa2: A,M: nat,Ba: $o,Na: nat] :
          ( aa(nat,$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),Aa2),M),(Ba))),Na)
        <=> ( $ite(M = Na,(Ba),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),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_5137_set__encode__insert,axiom,
    ! [A2: set(nat),Na: nat] :
      ( aa(set(nat),$o,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,aa(nat,fun(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_5138_sdiv__word__min,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] : aa(int,$o,aa(int,fun(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,aa(word(A),nat,size_size(word(A)),Aa2)),one_one(nat))))),signed7115095781618012415divide(int,aa(word(A),int,ring_1_signed(A,int),Aa2),aa(word(A),int,ring_1_signed(A,int),Ba))) ) ).

% sdiv_word_min
tff(fact_5139_List_Ofinite__set,axiom,
    ! [A: $tType,Xs2: list(A)] : aa(set(A),$o,finite_finite2(A),aa(list(A),set(A),set2(A),Xs2)) ).

% List.finite_set
tff(fact_5140_int__sdiv__simps_I1_J,axiom,
    ! [Aa2: int] : signed7115095781618012415divide(int,Aa2,one_one(int)) = Aa2 ).

% int_sdiv_simps(1)
tff(fact_5141_prod_Oinfinite,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [A2: set(A),G: fun(A,B)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A2)
         => ( groups7121269368397514597t_prod(A,B,G,A2) = one_one(B) ) ) ) ).

% prod.infinite
tff(fact_5142_int__sdiv__same__is__1,axiom,
    ! [Aa2: int,Ba: int] :
      ( ( Aa2 != zero_zero(int) )
     => ( ( signed7115095781618012415divide(int,Aa2,Ba) = Aa2 )
      <=> ( Ba = one_one(int) ) ) ) ).

% int_sdiv_same_is_1
tff(fact_5143_int__sdiv__simps_I3_J,axiom,
    ! [Aa2: int] : signed7115095781618012415divide(int,Aa2,aa(int,int,uminus_uminus(int),one_one(int))) = aa(int,int,uminus_uminus(int),Aa2) ).

% int_sdiv_simps(3)
tff(fact_5144_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_5145_prod__eq__1__iff,axiom,
    ! [A: $tType,A2: set(A),F2: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A2)
     => ( ( groups7121269368397514597t_prod(A,nat,F2,A2) = one_one(nat) )
      <=> ! [X6: A] :
            ( member(A,X6,A2)
           => ( aa(A,nat,F2,X6) = one_one(nat) ) ) ) ) ).

% prod_eq_1_iff
tff(fact_5146_sum_Odelta,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Sa: set(A),Aa2: A,Ba: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),Sa)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_mk(A,fun(fun(A,B),fun(A,B)),Aa2),Ba)),Sa) = $ite(member(A,Aa2,Sa),aa(A,B,Ba,Aa2),zero_zero(B)) ) ) ) ).

% sum.delta
tff(fact_5147_sum_Odelta_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Sa: set(A),Aa2: A,Ba: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),Sa)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ml(A,fun(fun(A,B),fun(A,B)),Aa2),Ba)),Sa) = $ite(member(A,Aa2,Sa),aa(A,B,Ba,Aa2),zero_zero(B)) ) ) ) ).

% sum.delta'
tff(fact_5148_prod_Odelta,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Sa: set(A),Aa2: A,Ba: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),Sa)
         => ( groups7121269368397514597t_prod(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_mm(A,fun(fun(A,B),fun(A,B)),Aa2),Ba),Sa) = $ite(member(A,Aa2,Sa),aa(A,B,Ba,Aa2),one_one(B)) ) ) ) ).

% prod.delta
tff(fact_5149_prod_Odelta_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Sa: set(A),Aa2: A,Ba: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),Sa)
         => ( groups7121269368397514597t_prod(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_mn(A,fun(fun(A,B),fun(A,B)),Aa2),Ba),Sa) = $ite(member(A,Aa2,Sa),aa(A,B,Ba,Aa2),one_one(B)) ) ) ) ).

% prod.delta'
tff(fact_5150_summable__If__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [P: fun(nat,$o),F2: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),collect(nat,P))
         => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_mo(fun(nat,$o),fun(fun(nat,A),fun(nat,A)),P),F2)) ) ) ).

% summable_If_finite
tff(fact_5151_summable__If__finite__set,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [A2: set(nat),F2: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),A2)
         => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_mp(set(nat),fun(fun(nat,A),fun(nat,A)),A2),F2)) ) ) ).

% summable_If_finite_set
tff(fact_5152_set__encode__inverse,axiom,
    ! [A2: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),A2)
     => ( nat_set_decode(aa(set(nat),nat,nat_set_encode,A2)) = A2 ) ) ).

% set_encode_inverse
tff(fact_5153_sum_Oinsert,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A2: set(A),X: A,G: fun(A,B)] :
          ( aa(set(A),$o,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,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,X)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A2)) ) ) ) ) ).

% sum.insert
tff(fact_5154_prod_Oinsert,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A2: set(A),X: A,G: fun(A,B)] :
          ( aa(set(A),$o,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_5155_int__sdiv__negated__is__minus1,axiom,
    ! [Aa2: int,Ba: int] :
      ( ( Aa2 != zero_zero(int) )
     => ( ( signed7115095781618012415divide(int,Aa2,Ba) = aa(int,int,uminus_uminus(int),Aa2) )
      <=> ( Ba = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% int_sdiv_negated_is_minus1
tff(fact_5156_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_mq(fun(nat,A),fun(nat,A),C2)),A2) = $ite(
            ( aa(set(nat),$o,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_5157_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_mr(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C2),D2)),A2) = $ite(
            ( aa(set(nat),$o,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_5158_set__bit__beyond,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Na: nat,Ba: $o] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(A),nat,size_size(word(A)),X)),Na)
         => ( aa($o,word(A),aa(nat,fun($o,word(A)),aa(word(A),fun(nat,fun($o,word(A))),generi7602027413899671122et_bit(word(A)),X),Na),(Ba)) = X ) ) ) ).

% set_bit_beyond
tff(fact_5159_set__encode__eq,axiom,
    ! [A2: set(nat),B2: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),A2)
     => ( aa(set(nat),$o,finite_finite2(nat),B2)
       => ( ( aa(set(nat),nat,nat_set_encode,A2) = aa(set(nat),nat,nat_set_encode,B2) )
        <=> ( A2 = B2 ) ) ) ) ).

% set_encode_eq
tff(fact_5160_finite__set__decode,axiom,
    ! [Na: nat] : aa(set(nat),$o,finite_finite2(nat),nat_set_decode(Na)) ).

% finite_set_decode
tff(fact_5161_finite__M__bounded__by__nat,axiom,
    ! [P: fun(nat,$o),I: nat] : aa(set(nat),$o,finite_finite2(nat),collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_ms(fun(nat,$o),fun(nat,fun(nat,$o)),P),I))) ).

% finite_M_bounded_by_nat
tff(fact_5162_finite__if__eq__beyond__finite,axiom,
    ! [A: $tType,Sa: set(A),S3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),Sa)
     => aa(set(set(A)),$o,finite_finite2(set(A)),collect(set(A),aa(set(A),fun(set(A),$o),aTP_Lamp_mt(set(A),fun(set(A),fun(set(A),$o)),Sa),S3))) ) ).

% finite_if_eq_beyond_finite
tff(fact_5163_finite__bit__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(set(nat),$o,finite_finite2(nat),collect(nat,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W))) ) ).

% finite_bit_word
tff(fact_5164_finite__nat__set__iff__bounded__le,axiom,
    ! [N3: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),N3)
    <=> ? [M4: nat] :
        ! [X6: nat] :
          ( member(nat,X6,N3)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X6),M4) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_5165_finite__less__ub,axiom,
    ! [F2: fun(nat,nat),U: nat] :
      ( ! [N: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),aa(nat,nat,F2,N))
     => aa(set(nat),$o,finite_finite2(nat),collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_mu(fun(nat,nat),fun(nat,fun(nat,$o)),F2),U))) ) ).

% finite_less_ub
tff(fact_5166_sum_Oswap__restrict,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [A2: set(A),B2: set(B),G: fun(A,fun(B,C)),R3: fun(A,fun(B,$o))] :
          ( aa(set(A),$o,finite_finite2(A),A2)
         => ( aa(set(B),$o,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_mw(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),B2),G),R3)),A2) = aa(set(B),C,groups7311177749621191930dd_sum(B,C,aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_mz(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),A2),G),R3)),B2) ) ) ) ) ).

% sum.swap_restrict
tff(fact_5167_sum__eq__empty__iff,axiom,
    ! [A: $tType,B: $tType,A2: set(A),F2: fun(A,multiset(B))] :
      ( aa(set(A),$o,finite_finite2(A),A2)
     => ( ( aa(set(A),multiset(B),groups7311177749621191930dd_sum(A,multiset(B),F2),A2) = zero_zero(multiset(B)) )
      <=> ! [X6: A] :
            ( member(A,X6,A2)
           => ( aa(A,multiset(B),F2,X6) = zero_zero(multiset(B)) ) ) ) ) ).

% sum_eq_empty_iff
tff(fact_5168_prod_Oswap__restrict,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [A2: set(A),B2: set(B),G: fun(A,fun(B,C)),R3: fun(A,fun(B,$o))] :
          ( aa(set(A),$o,finite_finite2(A),A2)
         => ( aa(set(B),$o,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_na(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),B2),G),R3),A2) = groups7121269368397514597t_prod(B,C,aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_nc(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),A2),G),R3),B2) ) ) ) ) ).

% prod.swap_restrict
tff(fact_5169_finite__lists__length__eq,axiom,
    ! [A: $tType,A2: set(A),Na: nat] :
      ( aa(set(A),$o,finite_finite2(A),A2)
     => aa(set(list(A)),$o,finite_finite2(list(A)),collect(list(A),aa(nat,fun(list(A),$o),aTP_Lamp_nd(set(A),fun(nat,fun(list(A),$o)),A2),Na))) ) ).

% finite_lists_length_eq
tff(fact_5170_finite__list,axiom,
    ! [A: $tType,A2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A2)
     => ? [Xs: list(A)] : aa(list(A),set(A),set2(A),Xs) = A2 ) ).

% finite_list
tff(fact_5171_sum__mono__inv,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [F2: fun(B,A),I5: set(B),G: fun(B,A),I: B] :
          ( ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),I5) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),I5) )
         => ( ! [I2: B] :
                ( member(B,I2,I5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,I2)),aa(B,A,G,I2)) )
           => ( member(B,I,I5)
             => ( aa(set(B),$o,finite_finite2(B),I5)
               => ( aa(B,A,F2,I) = aa(B,A,G,I) ) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_5172_finite__lists__length__le,axiom,
    ! [A: $tType,A2: set(A),Na: nat] :
      ( aa(set(A),$o,finite_finite2(A),A2)
     => aa(set(list(A)),$o,finite_finite2(list(A)),collect(list(A),aa(nat,fun(list(A),$o),aTP_Lamp_ne(set(A),fun(nat,fun(list(A),$o)),A2),Na))) ) ).

% finite_lists_length_le
tff(fact_5173_sum_Ofinite__Collect__op,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),X: fun(A,B),Y: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_nf(set(A),fun(fun(A,B),fun(A,$o)),I5),X)))
         => ( aa(set(A),$o,finite_finite2(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_nf(set(A),fun(fun(A,B),fun(A,$o)),I5),Y)))
           => aa(set(A),$o,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_ng(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),I5),X),Y))) ) ) ) ).

% sum.finite_Collect_op
tff(fact_5174_prod_Ofinite__Collect__op,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I5: set(A),X: fun(A,B),Y: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_nh(set(A),fun(fun(A,B),fun(A,$o)),I5),X)))
         => ( aa(set(A),$o,finite_finite2(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_nh(set(A),fun(fun(A,B),fun(A,$o)),I5),Y)))
           => aa(set(A),$o,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_ni(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),I5),X),Y))) ) ) ) ).

% prod.finite_Collect_op
tff(fact_5175_set__encode__inf,axiom,
    ! [A2: set(nat)] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),A2)
     => ( aa(set(nat),nat,nat_set_encode,A2) = zero_zero(nat) ) ) ).

% set_encode_inf
tff(fact_5176_filter__preserves__multiset,axiom,
    ! [A: $tType,M2: fun(A,nat),P: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),collect(A,aTP_Lamp_nj(fun(A,nat),fun(A,$o),M2)))
     => aa(set(A),$o,finite_finite2(A),collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_nk(fun(A,nat),fun(fun(A,$o),fun(A,$o)),M2),P))) ) ).

% filter_preserves_multiset
tff(fact_5177_sum_Ointer__filter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A2: set(A),G: fun(A,B),P: fun(A,$o)] :
          ( aa(set(A),$o,finite_finite2(A),A2)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_is(set(A),fun(fun(A,$o),fun(A,$o)),A2),P))) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_nl(fun(A,B),fun(fun(A,$o),fun(A,B)),G),P)),A2) ) ) ) ).

% sum.inter_filter
tff(fact_5178_prod_Ointer__filter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A2: set(A),G: fun(A,B),P: fun(A,$o)] :
          ( aa(set(A),$o,finite_finite2(A),A2)
         => ( groups7121269368397514597t_prod(A,B,G,collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_is(set(A),fun(fun(A,$o),fun(A,$o)),A2),P))) = groups7121269368397514597t_prod(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_nm(fun(A,B),fun(fun(A,$o),fun(A,B)),G),P),A2) ) ) ) ).

% prod.inter_filter
tff(fact_5179_finite__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Aa2: A,Ba: A] : aa(set(A),$o,finite_finite2(A),collect(A,aa(A,fun(A,$o),aTP_Lamp_nn(A,fun(A,fun(A,$o)),Aa2),Ba))) ) ).

% finite_int_segment
tff(fact_5180_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,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(word(A),nat,size_size(word(A)),W))
            & (X) ) ) ) ).

% test_bit_set
tff(fact_5181_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,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(word(A),nat,size_size(word(A)),W))
              & (X) ),
              aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),M) ) ) ) ).

% test_bit_set_gen
tff(fact_5182_word__set__nth__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Na: nat,Ba: $o] :
          ( ( aa($o,word(A),aa(nat,fun($o,word(A)),aa(word(A),fun(nat,fun($o,word(A))),generi7602027413899671122et_bit(word(A)),W),Na),(Ba)) = W )
        <=> ( ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),Na)
            <=> (Ba) )
            | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(A),nat,size_size(word(A)),W)),Na) ) ) ) ).

% word_set_nth_iff
tff(fact_5183_sum__le__included,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(C)
     => ! [S: set(A),T: set(B),G: fun(B,C),I: fun(B,A),F2: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( aa(set(B),$o,finite_finite2(B),T)
           => ( ! [X2: B] :
                  ( member(B,X2,T)
                 => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),zero_zero(C)),aa(B,C,G,X2)) )
             => ( ! [X2: A] :
                    ( member(A,X2,S)
                   => ? [Xa: B] :
                        ( member(B,Xa,T)
                        & ( aa(B,A,I,Xa) = X2 )
                        & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F2,X2)),aa(B,C,G,Xa)) ) )
               => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(A),C,groups7311177749621191930dd_sum(A,C,F2),S)),aa(set(B),C,groups7311177749621191930dd_sum(B,C,G),T)) ) ) ) ) ) ).

% sum_le_included
tff(fact_5184_sum__nonneg__eq__0__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A2: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A2)
         => ( ! [X2: A] :
                ( member(A,X2,A2)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X2)) )
           => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A2) = zero_zero(B) )
            <=> ! [X6: A] :
                  ( member(A,X6,A2)
                 => ( aa(A,B,F2,X6) = zero_zero(B) ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_5185_sum__strict__mono__ex1,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [A2: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A2)
         => ( ! [X2: A] :
                ( member(A,X2,A2)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X2)),aa(A,B,G,X2)) )
           => ( ? [X4: A] :
                  ( member(A,X4,A2)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X4)),aa(A,B,G,X4)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A2)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A2)) ) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_5186_sum_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [R3: fun(A,fun(A,$o)),Sa: set(B),H: fun(B,A),G: fun(B,A)] :
          ( aa(A,$o,aa(A,fun(A,$o),R3,zero_zero(A)),zero_zero(A))
         => ( ! [X1: A,Y1: A,X22: A,Y23: A] :
                ( ( aa(A,$o,aa(A,fun(A,$o),R3,X1),X22)
                  & aa(A,$o,aa(A,fun(A,$o),R3,Y1),Y23) )
               => aa(A,$o,aa(A,fun(A,$o),R3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X1),Y1)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X22),Y23)) )
           => ( aa(set(B),$o,finite_finite2(B),Sa)
             => ( ! [X2: B] :
                    ( member(B,X2,Sa)
                   => aa(A,$o,aa(A,fun(A,$o),R3,aa(B,A,H,X2)),aa(B,A,G,X2)) )
               => aa(A,$o,aa(A,fun(A,$o),R3,aa(set(B),A,groups7311177749621191930dd_sum(B,A,H),Sa)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),Sa)) ) ) ) ) ) ).

% sum.related
tff(fact_5187_finite__ranking__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Sa: set(A),P: fun(set(A),$o),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),Sa)
         => ( aa(set(A),$o,P,bot_bot(set(A)))
           => ( ! [X2: A,S4: set(A)] :
                  ( aa(set(A),$o,finite_finite2(A),S4)
                 => ( ! [Y3: A] :
                        ( member(A,Y3,S4)
                       => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Y3)),aa(A,B,F2,X2)) )
                   => ( aa(set(A),$o,P,S4)
                     => aa(set(A),$o,P,aa(set(A),set(A),insert(A,X2),S4)) ) ) )
             => aa(set(A),$o,P,Sa) ) ) ) ) ).

% finite_ranking_induct
tff(fact_5188_prod_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [R3: fun(A,fun(A,$o)),Sa: set(B),H: fun(B,A),G: fun(B,A)] :
          ( aa(A,$o,aa(A,fun(A,$o),R3,one_one(A)),one_one(A))
         => ( ! [X1: A,Y1: A,X22: A,Y23: A] :
                ( ( aa(A,$o,aa(A,fun(A,$o),R3,X1),X22)
                  & aa(A,$o,aa(A,fun(A,$o),R3,Y1),Y23) )
               => aa(A,$o,aa(A,fun(A,$o),R3,aa(A,A,aa(A,fun(A,A),times_times(A),X1),Y1)),aa(A,A,aa(A,fun(A,A),times_times(A),X22),Y23)) )
           => ( aa(set(B),$o,finite_finite2(B),Sa)
             => ( ! [X2: B] :
                    ( member(B,X2,Sa)
                   => aa(A,$o,aa(A,fun(A,$o),R3,aa(B,A,H,X2)),aa(B,A,G,X2)) )
               => aa(A,$o,aa(A,fun(A,$o),R3,groups7121269368397514597t_prod(B,A,H,Sa)),groups7121269368397514597t_prod(B,A,G,Sa)) ) ) ) ) ) ).

% prod.related
tff(fact_5189_sum_Oinsert__if,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A2: set(A),G: fun(A,B),X: A] :
          ( aa(set(A),$o,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,aa(B,fun(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_5190_prod_Oinsert__if,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A2: set(A),G: fun(A,B),X: A] :
          ( aa(set(A),$o,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_5191_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [S5: set(A),T5: set(B),Sa: set(A),I: fun(B,A),J: fun(A,B),T4: set(B),G: fun(A,C),H: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),S5)
         => ( aa(set(B),$o,finite_finite2(B),T5)
           => ( ! [A3: A] :
                  ( member(A,A3,aa(set(A),set(A),minus_minus(set(A),Sa),S5))
                 => ( aa(B,A,I,aa(A,B,J,A3)) = A3 ) )
             => ( ! [A3: A] :
                    ( member(A,A3,aa(set(A),set(A),minus_minus(set(A),Sa),S5))
                   => member(B,aa(A,B,J,A3),aa(set(B),set(B),minus_minus(set(B),T4),T5)) )
               => ( ! [B3: B] :
                      ( member(B,B3,aa(set(B),set(B),minus_minus(set(B),T4),T5))
                     => ( aa(A,B,J,aa(B,A,I,B3)) = B3 ) )
                 => ( ! [B3: B] :
                        ( member(B,B3,aa(set(B),set(B),minus_minus(set(B),T4),T5))
                       => member(A,aa(B,A,I,B3),aa(set(A),set(A),minus_minus(set(A),Sa),S5)) )
                   => ( ! [A3: A] :
                          ( member(A,A3,S5)
                         => ( aa(A,C,G,A3) = one_one(C) ) )
                     => ( ! [B3: B] :
                            ( member(B,B3,T5)
                           => ( aa(B,C,H,B3) = one_one(C) ) )
                       => ( ! [A3: A] :
                              ( member(A,A3,Sa)
                             => ( aa(B,C,H,aa(A,B,J,A3)) = aa(A,C,G,A3) ) )
                         => ( groups7121269368397514597t_prod(A,C,G,Sa) = groups7121269368397514597t_prod(B,C,H,T4) ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_5192_sum__eq__1__iff,axiom,
    ! [A: $tType,A2: set(A),F2: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A2)
     => ( ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A2) = one_one(nat) )
      <=> ? [X6: A] :
            ( member(A,X6,A2)
            & ( aa(A,nat,F2,X6) = one_one(nat) )
            & ! [Xa4: A] :
                ( member(A,Xa4,A2)
               => ( ( X6 != Xa4 )
                 => ( aa(A,nat,F2,Xa4) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_5193_sum__nonneg__leq__bound,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [S: set(A),F2: fun(A,B),B2: B,I: A] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( ! [I2: A] :
                ( member(A,I2,S)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I2)) )
           => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),S) = B2 )
             => ( member(A,I,S)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I)),B2) ) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_5194_sum__nonneg__0,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [S: set(A),F2: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( ! [I2: A] :
                ( member(A,I2,S)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I2)) )
           => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),S) = zero_zero(B) )
             => ( member(A,I,S)
               => ( aa(A,B,F2,I) = zero_zero(B) ) ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_5195_add__mset__in__multiset,axiom,
    ! [A: $tType,M2: fun(A,nat),Aa2: A] :
      ( aa(set(A),$o,finite_finite2(A),collect(A,aTP_Lamp_nj(fun(A,nat),fun(A,$o),M2)))
     => aa(set(A),$o,finite_finite2(A),collect(A,aa(A,fun(A,$o),aTP_Lamp_no(fun(A,nat),fun(A,fun(A,$o)),M2),Aa2))) ) ).

% add_mset_in_multiset
tff(fact_5196_diff__preserves__multiset,axiom,
    ! [A: $tType,M2: fun(A,nat),N3: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),collect(A,aTP_Lamp_nj(fun(A,nat),fun(A,$o),M2)))
     => aa(set(A),$o,finite_finite2(A),collect(A,aa(fun(A,nat),fun(A,$o),aTP_Lamp_np(fun(A,nat),fun(fun(A,nat),fun(A,$o)),M2),N3))) ) ).

% diff_preserves_multiset
tff(fact_5197_sum_Osetdiff__irrelevant,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A2: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,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_nq(fun(A,B),fun(A,$o),G)))) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A2) ) ) ) ).

% sum.setdiff_irrelevant
tff(fact_5198_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => aa(set(nat),$o,finite_finite2(nat),collect(nat,aTP_Lamp_nr(nat,fun(nat,$o),M))) ) ).

% finite_divisors_nat
tff(fact_5199_prod_Osetdiff__irrelevant,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A2: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A2)
         => ( groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),minus_minus(set(A),A2),collect(A,aTP_Lamp_ns(fun(A,B),fun(A,$o),G)))) = groups7121269368397514597t_prod(A,B,G,A2) ) ) ) ).

% prod.setdiff_irrelevant
tff(fact_5200_sums__If__finite__set,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [A2: set(nat),F2: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),A2)
         => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_mp(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_5201_sums__If__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [P: fun(nat,$o),F2: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),collect(nat,P))
         => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_mo(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_5202_sums__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [N3: set(nat),F2: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),N3)
         => ( ! [N: nat] :
                ( ~ member(nat,N,N3)
               => ( aa(nat,A,F2,N) = zero_zero(A) ) )
           => aa(A,$o,sums(A,F2),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),N3)) ) ) ) ).

% sums_finite
tff(fact_5203_suminf__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [N3: set(nat),F2: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),N3)
         => ( ! [N: nat] :
                ( ~ member(nat,N,N3)
               => ( aa(nat,A,F2,N) = zero_zero(A) ) )
           => ( suminf(A,F2) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),N3) ) ) ) ) ).

% suminf_finite
tff(fact_5204_finite__abs__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Aa2: A] : aa(set(A),$o,finite_finite2(A),collect(A,aTP_Lamp_nt(A,fun(A,$o),Aa2))) ) ).

% finite_abs_int_segment
tff(fact_5205_exp__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_Vector_banach(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [I5: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,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_nu(fun(A,B),fun(A,B),F2),I5) ) ) ) ).

% exp_sum
tff(fact_5206_sgn__sdiv__eq__sgn__mult,axiom,
    ! [Aa2: int,Ba: int] :
      ( ( signed7115095781618012415divide(int,Aa2,Ba) != zero_zero(int) )
     => ( aa(int,int,sgn_sgn(int),signed7115095781618012415divide(int,Aa2,Ba)) = aa(int,int,sgn_sgn(int),aa(int,int,aa(int,fun(int,int),times_times(int),Aa2),Ba)) ) ) ).

% sgn_sdiv_eq_sgn_mult
tff(fact_5207_sum__pos2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(A),I: A,F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( member(A,I,I5)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F2,I))
             => ( ! [I2: A] :
                    ( member(A,I2,I5)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I2)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),I5)) ) ) ) ) ) ).

% sum_pos2
tff(fact_5208_less__1__prod2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),I: A,F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( member(A,I,I5)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(A,B,F2,I))
             => ( ! [I2: A] :
                    ( member(A,I2,I5)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F2,I2)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),groups7121269368397514597t_prod(A,B,F2,I5)) ) ) ) ) ) ).

% less_1_prod2
tff(fact_5209_less__1__prod,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( ( I5 != bot_bot(set(A)) )
           => ( ! [I2: A] :
                  ( member(A,I2,I5)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(A,B,F2,I2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),groups7121269368397514597t_prod(A,B,F2,I5)) ) ) ) ) ).

% less_1_prod
tff(fact_5210_sum_Osubset__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [B2: set(A),A2: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),A2)
         => ( aa(set(A),$o,finite_finite2(A),A2)
           => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),minus_minus(set(A),A2),B2))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),B2)) ) ) ) ) ).

% sum.subset_diff
tff(fact_5211_prod_Osame__carrier,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [C5: set(A),A2: set(A),B2: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),C5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A2),C5)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),C5)
             => ( ! [A3: A] :
                    ( member(A,A3,aa(set(A),set(A),minus_minus(set(A),C5),A2))
                   => ( aa(A,B,G,A3) = one_one(B) ) )
               => ( ! [B3: A] :
                      ( member(A,B3,aa(set(A),set(A),minus_minus(set(A),C5),B2))
                     => ( aa(A,B,H,B3) = one_one(B) ) )
                 => ( ( groups7121269368397514597t_prod(A,B,G,A2) = groups7121269368397514597t_prod(A,B,H,B2) )
                  <=> ( groups7121269368397514597t_prod(A,B,G,C5) = groups7121269368397514597t_prod(A,B,H,C5) ) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_5212_prod_Osame__carrierI,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [C5: set(A),A2: set(A),B2: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),C5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A2),C5)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),C5)
             => ( ! [A3: A] :
                    ( member(A,A3,aa(set(A),set(A),minus_minus(set(A),C5),A2))
                   => ( aa(A,B,G,A3) = one_one(B) ) )
               => ( ! [B3: A] :
                      ( member(A,B3,aa(set(A),set(A),minus_minus(set(A),C5),B2))
                     => ( aa(A,B,H,B3) = one_one(B) ) )
                 => ( ( groups7121269368397514597t_prod(A,B,G,C5) = groups7121269368397514597t_prod(A,B,H,C5) )
                   => ( groups7121269368397514597t_prod(A,B,G,A2) = groups7121269368397514597t_prod(A,B,H,B2) ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_5213_prod_Omono__neutral__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T4: set(A),Sa: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Sa),T4)
           => ( ! [X2: A] :
                  ( member(A,X2,aa(set(A),set(A),minus_minus(set(A),T4),Sa))
                 => ( aa(A,B,G,X2) = one_one(B) ) )
             => ( groups7121269368397514597t_prod(A,B,G,Sa) = groups7121269368397514597t_prod(A,B,G,T4) ) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_5214_prod_Omono__neutral__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T4: set(A),Sa: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Sa),T4)
           => ( ! [X2: A] :
                  ( member(A,X2,aa(set(A),set(A),minus_minus(set(A),T4),Sa))
                 => ( aa(A,B,G,X2) = one_one(B) ) )
             => ( groups7121269368397514597t_prod(A,B,G,T4) = groups7121269368397514597t_prod(A,B,G,Sa) ) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_5215_prod_Omono__neutral__cong__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T4: set(A),Sa: set(A),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Sa),T4)
           => ( ! [X2: A] :
                  ( member(A,X2,aa(set(A),set(A),minus_minus(set(A),T4),Sa))
                 => ( aa(A,B,H,X2) = one_one(B) ) )
             => ( ! [X2: A] :
                    ( member(A,X2,Sa)
                   => ( aa(A,B,G,X2) = aa(A,B,H,X2) ) )
               => ( groups7121269368397514597t_prod(A,B,G,Sa) = groups7121269368397514597t_prod(A,B,H,T4) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_5216_prod_Omono__neutral__cong__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T4: set(A),Sa: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Sa),T4)
           => ( ! [X2: A] :
                  ( member(A,X2,aa(set(A),set(A),minus_minus(set(A),T4),Sa))
                 => ( aa(A,B,G,X2) = one_one(B) ) )
             => ( ! [X2: A] :
                    ( member(A,X2,Sa)
                   => ( aa(A,B,G,X2) = aa(A,B,H,X2) ) )
               => ( groups7121269368397514597t_prod(A,B,G,T4) = groups7121269368397514597t_prod(A,B,H,Sa) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_5217_prod_Osubset__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [B2: set(A),A2: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),A2)
         => ( aa(set(A),$o,finite_finite2(A),A2)
           => ( groups7121269368397514597t_prod(A,B,G,A2) = aa(B,B,aa(B,fun(B,B),times_times(B),groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),minus_minus(set(A),A2),B2))),groups7121269368397514597t_prod(A,B,G,B2)) ) ) ) ) ).

% prod.subset_diff
tff(fact_5218_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_5219_finite__roots__unity,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Na)
         => aa(set(A),$o,finite_finite2(A),collect(A,aTP_Lamp_nv(nat,fun(A,$o),Na))) ) ) ).

% finite_roots_unity
tff(fact_5220_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [S5: set(A),T5: set(B),H: fun(A,B),Sa: set(A),T4: set(B),G: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),S5)
         => ( aa(set(B),$o,finite_finite2(B),T5)
           => ( bij_betw(A,B,H,aa(set(A),set(A),minus_minus(set(A),Sa),S5),aa(set(B),set(B),minus_minus(set(B),T4),T5))
             => ( ! [A3: A] :
                    ( member(A,A3,S5)
                   => ( aa(B,C,G,aa(A,B,H,A3)) = zero_zero(C) ) )
               => ( ! [B3: B] :
                      ( member(B,B3,T5)
                     => ( aa(B,C,G,B3) = zero_zero(C) ) )
                 => ( aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_iu(fun(A,B),fun(fun(B,C),fun(A,C)),H),G)),Sa) = aa(set(B),C,groups7311177749621191930dd_sum(B,C,G),T4) ) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
tff(fact_5221_sums__If__finite__set_H,axiom,
    ! [A: $tType] :
      ( ( topolo1287966508704411220up_add(A)
        & topological_t2_space(A) )
     => ! [G: fun(nat,A),Sa: A,A2: set(nat),S5: A,F2: fun(nat,A)] :
          ( aa(A,$o,sums(A,G),Sa)
         => ( aa(set(nat),$o,finite_finite2(nat),A2)
           => ( ( S5 = aa(A,A,aa(A,fun(A,A),plus_plus(A),Sa),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_nw(fun(nat,A),fun(fun(nat,A),fun(nat,A)),G),F2)),A2)) )
             => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aa(set(nat),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_nx(fun(nat,A),fun(set(nat),fun(fun(nat,A),fun(nat,A))),G),A2),F2)),S5) ) ) ) ) ).

% sums_If_finite_set'
tff(fact_5222_prod_Oreindex__bij__betw__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [S5: set(A),T5: set(B),H: fun(A,B),Sa: set(A),T4: set(B),G: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),S5)
         => ( aa(set(B),$o,finite_finite2(B),T5)
           => ( bij_betw(A,B,H,aa(set(A),set(A),minus_minus(set(A),Sa),S5),aa(set(B),set(B),minus_minus(set(B),T4),T5))
             => ( ! [A3: A] :
                    ( member(A,A3,S5)
                   => ( aa(B,C,G,aa(A,B,H,A3)) = one_one(C) ) )
               => ( ! [B3: B] :
                      ( member(B,B3,T5)
                     => ( aa(B,C,G,B3) = one_one(C) ) )
                 => ( groups7121269368397514597t_prod(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_jq(fun(A,B),fun(fun(B,C),fun(A,C)),H),G),Sa) = groups7121269368397514597t_prod(B,C,G,T4) ) ) ) ) ) ) ) ).

% prod.reindex_bij_betw_not_neutral
tff(fact_5223_signed__divide__int__def,axiom,
    ! [K: int,L: int] : signed7115095781618012415divide(int,K,L) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(int,int,sgn_sgn(int),L))),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L))) ).

% signed_divide_int_def
tff(fact_5224_prod__mono__strict,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A2: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A2)
         => ( ! [I2: A] :
                ( member(A,I2,A2)
               => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I2))
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,I2)),aa(A,B,G,I2)) ) )
           => ( ( A2 != bot_bot(set(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),groups7121269368397514597t_prod(A,B,F2,A2)),groups7121269368397514597t_prod(A,B,G,A2)) ) ) ) ) ).

% prod_mono_strict
tff(fact_5225_even__prod__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_parity(B)
     => ! [A2: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A2)
         => ( aa(B,$o,dvd_dvd(B,aa(num,B,numeral_numeral(B),bit0(one2))),groups7121269368397514597t_prod(A,B,F2,A2))
          <=> ? [X6: A] :
                ( member(A,X6,A2)
                & aa(B,$o,dvd_dvd(B,aa(num,B,numeral_numeral(B),bit0(one2))),aa(A,B,F2,X6)) ) ) ) ) ).

% even_prod_iff
tff(fact_5226_sum__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [B2: set(A),A2: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),B2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A2),B2)
           => ( ! [B3: A] :
                  ( member(A,B3,aa(set(A),set(A),minus_minus(set(A),B2),A2))
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,B3)) )
             => aa(B,$o,aa(B,fun(B,$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_5227_sum_Oremove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A2: set(A),X: A,G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A2)
         => ( member(A,X,A2)
           => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A2) = aa(B,B,aa(B,fun(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_5228_sum_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A2: set(A),G: fun(A,B),X: A] :
          ( aa(set(A),$o,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,aa(B,fun(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_5229_prod_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A2: set(A),G: fun(A,B),X: A] :
          ( aa(set(A),$o,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_5230_prod_Oremove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A2: set(A),X: A,G: fun(A,B)] :
          ( aa(set(A),$o,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_5231_sum__le__suminf,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),I5: set(nat)] :
          ( summable(A,F2)
         => ( aa(set(nat),$o,finite_finite2(nat),I5)
           => ( ! [N: nat] :
                  ( member(nat,N,aa(set(nat),set(nat),uminus_uminus(set(nat)),I5))
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),I5)),suminf(A,F2)) ) ) ) ) ).

% sum_le_suminf
tff(fact_5232_sum_Odelta__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Sa: set(A),Aa2: A,Ba: fun(A,B),C2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),Sa)
         => ( 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_ny(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Aa2),Ba),C2)),Sa) = $ite(member(A,Aa2,Sa),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Ba,Aa2)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,C2),aa(set(A),set(A),minus_minus(set(A),Sa),aa(set(A),set(A),insert(A,Aa2),bot_bot(set(A)))))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,C2),aa(set(A),set(A),minus_minus(set(A),Sa),aa(set(A),set(A),insert(A,Aa2),bot_bot(set(A)))))) ) ) ) ).

% sum.delta_remove
tff(fact_5233_prod_Odelta__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Sa: set(A),Aa2: A,Ba: fun(A,B),C2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),Sa)
         => ( groups7121269368397514597t_prod(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_nz(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Aa2),Ba),C2),Sa) = $ite(member(A,Aa2,Sa),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Ba,Aa2)),groups7121269368397514597t_prod(A,B,C2,aa(set(A),set(A),minus_minus(set(A),Sa),aa(set(A),set(A),insert(A,Aa2),bot_bot(set(A)))))),groups7121269368397514597t_prod(A,B,C2,aa(set(A),set(A),minus_minus(set(A),Sa),aa(set(A),set(A),insert(A,Aa2),bot_bot(set(A)))))) ) ) ) ).

% prod.delta_remove
tff(fact_5234_sum__strict__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [B2: set(A),A2: set(A),Ba: A,F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),B2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A2),B2)
           => ( member(A,Ba,aa(set(A),set(A),minus_minus(set(A),B2),A2))
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F2,Ba))
               => ( ! [X2: A] :
                      ( member(A,X2,B2)
                     => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X2)) )
                 => aa(B,$o,aa(B,fun(B,$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_5235_member__le__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [I: A,A2: set(A),F2: fun(A,B)] :
          ( member(A,I,A2)
         => ( ! [X2: A] :
                ( member(A,X2,aa(set(A),set(A),minus_minus(set(A),A2),aa(set(A),set(A),insert(A,I),bot_bot(set(A)))))
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X2)) )
           => ( aa(set(A),$o,finite_finite2(A),A2)
             => aa(B,$o,aa(B,fun(B,$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_5236_prod__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [B2: set(A),A2: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),B2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A2),B2)
           => ( ! [B3: A] :
                  ( member(A,B3,aa(set(A),set(A),minus_minus(set(A),B2),A2))
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F2,B3)) )
             => ( ! [A3: A] :
                    ( member(A,A3,A2)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,A3)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),groups7121269368397514597t_prod(A,B,F2,A2)),groups7121269368397514597t_prod(A,B,F2,B2)) ) ) ) ) ) ).

% prod_mono2
tff(fact_5237_even__set__encode__iff,axiom,
    ! [A2: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),A2)
     => ( aa(nat,$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_5238_ln__prod,axiom,
    ! [A: $tType,I5: set(A),F2: fun(A,real)] :
      ( aa(set(A),$o,finite_finite2(A),I5)
     => ( ! [I2: A] :
            ( member(A,I2,I5)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F2,I2)) )
       => ( aa(real,real,ln_ln(real),groups7121269368397514597t_prod(A,real,F2,I5)) = aa(set(A),real,groups7311177749621191930dd_sum(A,real,aTP_Lamp_oa(fun(A,real),fun(A,real),F2)),I5) ) ) ) ).

% ln_prod
tff(fact_5239_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,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
           => aa(set(A),$o,finite_finite2(A),collect(A,aa(nat,fun(A,$o),aTP_Lamp_ob(fun(nat,A),fun(nat,fun(A,$o)),C2),Na))) ) ) ) ).

% polyfun_roots_finite
tff(fact_5240_polyfun__finite__roots,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Na: nat] :
          ( aa(set(A),$o,finite_finite2(A),collect(A,aa(nat,fun(A,$o),aTP_Lamp_ob(fun(nat,A),fun(nat,fun(A,$o)),C2),Na)))
        <=> ? [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I4),Na)
              & ( aa(nat,A,C2,I4) != zero_zero(A) ) ) ) ) ).

% polyfun_finite_roots
tff(fact_5241_sdiv__word__max,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),signed7115095781618012415divide(int,aa(word(A),int,ring_1_signed(A,int),Aa2),aa(word(A),int,ring_1_signed(A,int),Ba))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,aa(word(A),nat,size_size(word(A)),Aa2)),aa(nat,nat,suc,zero_zero(nat))))) ) ).

% sdiv_word_max
tff(fact_5242_finite__Collect__le__nat,axiom,
    ! [K: nat] : aa(set(nat),$o,finite_finite2(nat),collect(nat,aTP_Lamp_oc(nat,fun(nat,$o),K))) ).

% finite_Collect_le_nat
tff(fact_5243_finite__Collect__less__nat,axiom,
    ! [K: nat] : aa(set(nat),$o,finite_finite2(nat),collect(nat,aTP_Lamp_gx(nat,fun(nat,$o),K))) ).

% finite_Collect_less_nat
tff(fact_5244_finite__Collect__subsets,axiom,
    ! [A: $tType,A2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A2)
     => aa(set(set(A)),$o,finite_finite2(set(A)),collect(set(A),aTP_Lamp_od(set(A),fun(set(A),$o),A2))) ) ).

% finite_Collect_subsets
tff(fact_5245_finite__Collect__disjI,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_dm(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)))
    <=> ( aa(set(A),$o,finite_finite2(A),collect(A,P))
        & aa(set(A),$o,finite_finite2(A),collect(A,Q)) ) ) ).

% finite_Collect_disjI
tff(fact_5246_finite__Collect__conjI,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ( aa(set(A),$o,finite_finite2(A),collect(A,P))
        | aa(set(A),$o,finite_finite2(A),collect(A,Q)) )
     => aa(set(A),$o,finite_finite2(A),collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_oe(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q))) ) ).

% finite_Collect_conjI
tff(fact_5247_finite__interval__int1,axiom,
    ! [Aa2: int,Ba: int] : aa(set(int),$o,finite_finite2(int),collect(int,aa(int,fun(int,$o),aTP_Lamp_of(int,fun(int,fun(int,$o)),Aa2),Ba))) ).

% finite_interval_int1
tff(fact_5248_finite__interval__int4,axiom,
    ! [Aa2: int,Ba: int] : aa(set(int),$o,finite_finite2(int),collect(int,aa(int,fun(int,$o),aTP_Lamp_og(int,fun(int,fun(int,$o)),Aa2),Ba))) ).

% finite_interval_int4
tff(fact_5249_finite__interval__int2,axiom,
    ! [Aa2: int,Ba: int] : aa(set(int),$o,finite_finite2(int),collect(int,aa(int,fun(int,$o),aTP_Lamp_oh(int,fun(int,fun(int,$o)),Aa2),Ba))) ).

% finite_interval_int2
tff(fact_5250_finite__interval__int3,axiom,
    ! [Aa2: int,Ba: int] : aa(set(int),$o,finite_finite2(int),collect(int,aa(int,fun(int,$o),aTP_Lamp_oi(int,fun(int,fun(int,$o)),Aa2),Ba))) ).

% finite_interval_int3
tff(fact_5251_finite__nth__roots,axiom,
    ! [Na: nat,C2: complex] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => aa(set(complex),$o,finite_finite2(complex),collect(complex,aa(complex,fun(complex,$o),aTP_Lamp_hh(nat,fun(complex,fun(complex,$o)),Na),C2))) ) ).

% finite_nth_roots
tff(fact_5252_finite__maxlen,axiom,
    ! [A: $tType,M2: set(list(A))] :
      ( aa(set(list(A)),$o,finite_finite2(list(A)),M2)
     => ? [N: nat] :
        ! [X4: list(A)] :
          ( member(list(A),X4,M2)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),X4)),N) ) ) ).

% finite_maxlen
tff(fact_5253_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I != zero_zero(int) )
     => aa(set(int),$o,finite_finite2(int),collect(int,aTP_Lamp_oj(int,fun(int,$o),I))) ) ).

% finite_divisors_int
tff(fact_5254_pigeonhole__infinite__rel,axiom,
    ! [A: $tType,B: $tType,A2: set(A),B2: set(B),R3: fun(A,fun(B,$o))] :
      ( ~ aa(set(A),$o,finite_finite2(A),A2)
     => ( aa(set(B),$o,finite_finite2(B),B2)
       => ( ! [X2: A] :
              ( member(A,X2,A2)
             => ? [Xa: B] :
                  ( member(B,Xa,B2)
                  & aa(B,$o,aa(A,fun(B,$o),R3,X2),Xa) ) )
         => ? [X2: B] :
              ( member(B,X2,B2)
              & ~ aa(set(A),$o,finite_finite2(A),collect(A,aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_my(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),A2),R3),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_5255_not__finite__existsD,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ~ aa(set(A),$o,finite_finite2(A),collect(A,P))
     => ? [X_12: A] : aa(A,$o,P,X_12) ) ).

% not_finite_existsD
tff(fact_5256_finite__has__minimal2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: set(A),Aa2: A] :
          ( aa(set(A),$o,finite_finite2(A),A2)
         => ( member(A,Aa2,A2)
           => ? [X2: A] :
                ( member(A,X2,A2)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Aa2)
                & ! [Xa: A] :
                    ( member(A,Xa,A2)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),X2)
                     => ( X2 = Xa ) ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_5257_finite__has__maximal2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: set(A),Aa2: A] :
          ( aa(set(A),$o,finite_finite2(A),A2)
         => ( member(A,Aa2,A2)
           => ? [X2: A] :
                ( member(A,X2,A2)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),X2)
                & ! [Xa: A] :
                    ( member(A,Xa,A2)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Xa)
                     => ( X2 = Xa ) ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_5258_finite__has__maximal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A2)
         => ( ( A2 != bot_bot(set(A)) )
           => ? [X2: A] :
                ( member(A,X2,A2)
                & ! [Xa: A] :
                    ( member(A,Xa,A2)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Xa)
                     => ( X2 = Xa ) ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_5259_finite__has__minimal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A2)
         => ( ( A2 != bot_bot(set(A)) )
           => ? [X2: A] :
                ( member(A,X2,A2)
                & ! [Xa: A] :
                    ( member(A,Xa,A2)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),X2)
                     => ( X2 = Xa ) ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_5260_sum__count__set,axiom,
    ! [A: $tType,Xs2: list(A),X5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs2)),X5)
     => ( aa(set(A),$o,finite_finite2(A),X5)
       => ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,count_list(A,Xs2)),X5) = aa(list(A),nat,size_size(list(A)),Xs2) ) ) ) ).

% sum_count_set
tff(fact_5261_sum__diff1_H__aux,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [F4: set(A),I5: set(A),F2: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite2(A),F4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_ok(set(A),fun(fun(A,B),fun(A,$o)),I5),F2))),F4)
           => ( 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_5262_slice__Cons,axiom,
    ! [A: $tType,Begin: nat,End: nat,X: A,Xs2: list(A)] :
      slice(A,Begin,End,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs2)) = $ite(
        ( ( Begin = zero_zero(nat) )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),End) ),
        aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),slice(A,Begin,aa(nat,nat,minus_minus(nat,End),one_one(nat)),Xs2)),
        slice(A,aa(nat,nat,minus_minus(nat,Begin),one_one(nat)),aa(nat,nat,minus_minus(nat,End),one_one(nat)),Xs2) ) ).

% slice_Cons
tff(fact_5263_slice__eq__bounds__empty,axiom,
    ! [A: $tType,I: nat,Xs2: list(A)] : slice(A,I,I,Xs2) = nil(A) ).

% slice_eq_bounds_empty
tff(fact_5264_slice__Nil,axiom,
    ! [A: $tType,Begin: nat,End: nat] : slice(A,Begin,End,nil(A)) = nil(A) ).

% slice_Nil
tff(fact_5265_slice__complete,axiom,
    ! [A: $tType,Xs2: list(A)] : slice(A,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs2),Xs2) = Xs2 ).

% slice_complete
tff(fact_5266_count__notin,axiom,
    ! [A: $tType,X: A,Xs2: list(A)] :
      ( ~ member(A,X,aa(list(A),set(A),set2(A),Xs2))
     => ( aa(A,nat,count_list(A,Xs2),X) = zero_zero(nat) ) ) ).

% count_notin
tff(fact_5267_slice__len,axiom,
    ! [A: $tType,From: nat,To: nat,Xs2: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),From),To)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),To),aa(list(A),nat,size_size(list(A)),Xs2))
       => ( aa(list(A),nat,size_size(list(A)),slice(A,From,To,Xs2)) = aa(nat,nat,minus_minus(nat,To),From) ) ) ) ).

% slice_len
tff(fact_5268_sum_Oinsert_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),P2: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite2(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_nf(set(A),fun(fun(A,B),fun(A,$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,aa(B,fun(B,B),plus_plus(B),aa(A,B,P2,I)),groups1027152243600224163dd_sum(A,B,P2,I5))) ) ) ) ).

% sum.insert'
tff(fact_5269_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_ol(fun(B,A),fun(set(B),fun(B,$o)),G),I5))) = groups1027152243600224163dd_sum(B,A,G,I5) ) ).

% sum.non_neutral'
tff(fact_5270_sum_Odistrib__triv_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( groups1027152243600224163dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_om(fun(A,B),fun(fun(A,B),fun(A,B)),G),H),I5) = aa(B,B,aa(B,fun(B,B),plus_plus(B),groups1027152243600224163dd_sum(A,B,G,I5)),groups1027152243600224163dd_sum(A,B,H,I5)) ) ) ) ).

% sum.distrib_triv'
tff(fact_5271_sum_Odistrib_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_nf(set(A),fun(fun(A,B),fun(A,$o)),I5),G)))
         => ( aa(set(A),$o,finite_finite2(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_nf(set(A),fun(fun(A,B),fun(A,$o)),I5),H)))
           => ( groups1027152243600224163dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_om(fun(A,B),fun(fun(A,B),fun(A,B)),G),H),I5) = aa(B,B,aa(B,fun(B,B),plus_plus(B),groups1027152243600224163dd_sum(A,B,G,I5)),groups1027152243600224163dd_sum(A,B,H,I5)) ) ) ) ) ).

% sum.distrib'
tff(fact_5272_count__list_Osimps_I1_J,axiom,
    ! [A: $tType,Y: A] : aa(A,nat,count_list(A,nil(A)),Y) = zero_zero(nat) ).

% count_list.simps(1)
tff(fact_5273_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) = $ite(aa(set(B),$o,finite_finite2(B),collect(B,aa(set(B),fun(B,$o),aTP_Lamp_ol(fun(B,A),fun(set(B),fun(B,$o)),P2),I5))),aa(set(B),A,groups7311177749621191930dd_sum(B,A,P2),collect(B,aa(set(B),fun(B,$o),aTP_Lamp_ol(fun(B,A),fun(set(B),fun(B,$o)),P2),I5))),zero_zero(A)) ) ).

% sum.G_def
tff(fact_5274_count__le__length,axiom,
    ! [A: $tType,Xs2: list(A),X: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,count_list(A,Xs2),X)),aa(list(A),nat,size_size(list(A)),Xs2)) ).

% count_le_length
tff(fact_5275_count__list_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Xs2: list(A),Y: A] :
      aa(A,nat,count_list(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs2)),Y) = $ite(X = Y,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,count_list(A,Xs2),Y)),one_one(nat)),aa(A,nat,count_list(A,Xs2),Y)) ).

% count_list.simps(2)
tff(fact_5276_sum__diff1_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [I5: set(A),F2: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite2(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_ok(set(A),fun(fun(A,B),fun(A,$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_5277_infinite__int__iff__unbounded__le,axiom,
    ! [Sa: set(int)] :
      ( ~ aa(set(int),$o,finite_finite2(int),Sa)
    <=> ! [M4: int] :
        ? [N2: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),M4),aa(int,int,abs_abs(int),N2))
          & member(int,N2,Sa) ) ) ).

% infinite_int_iff_unbounded_le
tff(fact_5278_Cauchy__iff2,axiom,
    ! [X5: fun(nat,real)] :
      ( topolo3814608138187158403Cauchy(real,X5)
    <=> ! [J3: nat] :
        ? [M11: nat] :
        ! [M4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M11),M4)
         => ! [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M11),N2)
             => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,aa(nat,real,X5,M4)),aa(nat,real,X5,N2)))),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,J3)))) ) ) ) ).

% Cauchy_iff2
tff(fact_5279_int__of__nat__def,axiom,
    code_T6385005292777649522of_nat = semiring_1_of_nat(int) ).

% int_of_nat_def
tff(fact_5280_CauchyD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),E2: real] :
          ( topolo3814608138187158403Cauchy(A,X5)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
           => ? [M10: nat] :
              ! [M3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M10),M3)
               => ! [N10: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M10),N10)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X5,M3)),aa(nat,A,X5,N10)))),E2) ) ) ) ) ) ).

% CauchyD
tff(fact_5281_CauchyI,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( ! [E: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
             => ? [M12: nat] :
                ! [M5: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M12),M5)
                 => ! [N: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M12),N)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X5,M5)),aa(nat,A,X5,N)))),E) ) ) )
         => topolo3814608138187158403Cauchy(A,X5) ) ) ).

% CauchyI
tff(fact_5282_Cauchy__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X5)
        <=> ! [E3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
             => ? [M11: nat] :
                ! [M4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M11),M4)
                 => ! [N2: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M11),N2)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X5,M4)),aa(nat,A,X5,N2)))),E3) ) ) ) ) ) ).

% Cauchy_iff
tff(fact_5283_infinite__nat__iff__unbounded__le,axiom,
    ! [Sa: set(nat)] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),Sa)
    <=> ! [M4: nat] :
        ? [N2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M4),N2)
          & member(nat,N2,Sa) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_5284_Sum__Ico__nat,axiom,
    ! [M: nat,Na: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_je(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_5285_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_5286_smod__int__range,axiom,
    ! [Ba: int,Aa2: int] :
      ( ( Ba != zero_zero(int) )
     => member(int,signed6721504322012087516modulo(int,Aa2,Ba),set_or1337092689740270186AtMost(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(int,int,abs_abs(int),Ba))),one_one(int)),aa(int,int,minus_minus(int,aa(int,int,abs_abs(int),Ba)),one_one(int)))) ) ).

% smod_int_range
tff(fact_5287_atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( member(A,I,set_or7035219750837199246ssThan(A,L,U))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),I)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),U) ) ) ) ).

% atLeastLessThan_iff
tff(fact_5288_atLeastLessThan__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2)
         => ( set_or7035219750837199246ssThan(A,Aa2,Ba) = bot_bot(set(A)) ) ) ) ).

% atLeastLessThan_empty
tff(fact_5289_ivl__subset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [I: A,J: A,M: A,Na: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,I,J)),set_or7035219750837199246ssThan(A,M,Na))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),J),I)
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),I)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),J),Na) ) ) ) ) ).

% ivl_subset
tff(fact_5290_ivl__diff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [I: A,Na: A,M: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),Na)
         => ( aa(set(A),set(A),minus_minus(set(A),set_or7035219750837199246ssThan(A,I,M)),set_or7035219750837199246ssThan(A,I,Na)) = set_or7035219750837199246ssThan(A,Na,M) ) ) ) ).

% ivl_diff
tff(fact_5291_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_5292_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,aa(nat,fun(nat,$o),ord_less(nat),Na),M),zero_zero(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,Na))),aa(nat,A,G,Na))) ) ).

% sum.op_ivl_Suc
tff(fact_5293_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,aa(nat,fun(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_5294_atLeastLessThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: A,Ba: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,Aa2,Ba)),set_or7035219750837199246ssThan(A,C2,D2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2)
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),Aa2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),D2) ) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_5295_ivl__disj__un__two_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(3)
tff(fact_5296_ex__nat__less__eq,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( ? [M4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),Na)
          & aa(nat,$o,P,M4) )
    <=> ? [X6: nat] :
          ( member(nat,X6,set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))
          & aa(nat,$o,P,X6) ) ) ).

% ex_nat_less_eq
tff(fact_5297_all__nat__less__eq,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( ! [M4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),Na)
         => aa(nat,$o,P,M4) )
    <=> ! [X6: nat] :
          ( member(nat,X6,set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))
         => aa(nat,$o,P,X6) ) ) ).

% all_nat_less_eq
tff(fact_5298_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_fe(fun(nat,A),fun(nat,A),G)),set_or7035219750837199246ssThan(nat,M,Na)) ) ).

% sum.shift_bounds_Suc_ivl
tff(fact_5299_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,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),K))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_iv(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or7035219750837199246ssThan(nat,M,Na)) ) ).

% sum.shift_bounds_nat_ivl
tff(fact_5300_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_ka(fun(nat,A),fun(nat,A),G),set_or7035219750837199246ssThan(nat,M,Na)) ) ).

% prod.shift_bounds_Suc_ivl
tff(fact_5301_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,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),K))) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_kc(fun(nat,A),fun(nat,fun(nat,A)),G),K),set_or7035219750837199246ssThan(nat,M,Na)) ) ).

% prod.shift_bounds_nat_ivl
tff(fact_5302_sum_Oivl__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & comm_monoid_add(B) )
     => ! [Aa2: A,C2: A,Ba: A,D2: A,G: fun(A,B),H: fun(A,B)] :
          ( ( Aa2 = C2 )
         => ( ( Ba = D2 )
           => ( ! [X2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),X2)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),D2)
                   => ( aa(A,B,G,X2) = aa(A,B,H,X2) ) ) )
             => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),set_or7035219750837199246ssThan(A,Aa2,Ba)) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),set_or7035219750837199246ssThan(A,C2,D2)) ) ) ) ) ) ).

% sum.ivl_cong
tff(fact_5303_prod_Oivl__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & comm_monoid_mult(B) )
     => ! [Aa2: A,C2: A,Ba: A,D2: A,G: fun(A,B),H: fun(A,B)] :
          ( ( Aa2 = C2 )
         => ( ( Ba = D2 )
           => ( ! [X2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),X2)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),D2)
                   => ( aa(A,B,G,X2) = aa(A,B,H,X2) ) ) )
             => ( groups7121269368397514597t_prod(A,B,G,set_or7035219750837199246ssThan(A,Aa2,Ba)) = groups7121269368397514597t_prod(A,B,H,set_or7035219750837199246ssThan(A,C2,D2)) ) ) ) ) ) ).

% prod.ivl_cong
tff(fact_5304_ivl__disj__un__two_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(7)
tff(fact_5305_ivl__disj__un__one_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_ord_lessThan(A,L)),set_or7035219750837199246ssThan(A,L,U)) = set_ord_lessThan(A,U) ) ) ) ).

% ivl_disj_un_one(2)
tff(fact_5306_sum_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Na: nat,P2: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),P2)
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,Na))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,Na,P2))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,P2)) ) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_5307_sum__diff__nat__ivl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,Na: nat,P2: nat,F2: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),P2)
           => ( aa(A,A,minus_minus(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or7035219750837199246ssThan(nat,M,P2))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or7035219750837199246ssThan(nat,M,Na))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or7035219750837199246ssThan(nat,Na,P2)) ) ) ) ) ).

% sum_diff_nat_ivl
tff(fact_5308_prod_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Na: nat,P2: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),P2)
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,Na))),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,Na,P2))) = groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,P2)) ) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_5309_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( set_or7035219750837199246ssThan(nat,I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),set_or7035219750837199246ssThan(nat,I,J)),set_or7035219750837199246ssThan(nat,J,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_5310_smod__int__compares_I1_J,axiom,
    ! [Aa2: int,Ba: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Aa2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Ba)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),signed6721504322012087516modulo(int,Aa2,Ba)),Ba) ) ) ).

% smod_int_compares(1)
tff(fact_5311_smod__int__compares_I2_J,axiom,
    ! [Aa2: int,Ba: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Aa2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Ba)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),signed6721504322012087516modulo(int,Aa2,Ba)) ) ) ).

% smod_int_compares(2)
tff(fact_5312_smod__int__compares_I4_J,axiom,
    ! [Aa2: int,Ba: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Aa2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Ba)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),signed6721504322012087516modulo(int,Aa2,Ba)),zero_zero(int)) ) ) ).

% smod_int_compares(4)
tff(fact_5313_smod__int__compares_I6_J,axiom,
    ! [Aa2: int,Ba: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Aa2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ba),zero_zero(int))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),signed6721504322012087516modulo(int,Aa2,Ba)) ) ) ).

% smod_int_compares(6)
tff(fact_5314_smod__int__compares_I7_J,axiom,
    ! [Aa2: int,Ba: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Aa2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ba),zero_zero(int))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),signed6721504322012087516modulo(int,Aa2,Ba)),zero_zero(int)) ) ) ).

% smod_int_compares(7)
tff(fact_5315_smod__int__compares_I8_J,axiom,
    ! [Aa2: int,Ba: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Aa2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ba),zero_zero(int))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ba),signed6721504322012087516modulo(int,Aa2,Ba)) ) ) ).

% smod_int_compares(8)
tff(fact_5316_smod__mod__positive,axiom,
    ! [Aa2: int,Ba: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Aa2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Ba)
       => ( signed6721504322012087516modulo(int,Aa2,Ba) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Aa2),Ba) ) ) ) ).

% smod_mod_positive
tff(fact_5317_signed__modulo__int__def,axiom,
    ! [K: int,L: int] : signed6721504322012087516modulo(int,K,L) = aa(int,int,minus_minus(int,K),aa(int,int,aa(int,fun(int,int),times_times(int),signed7115095781618012415divide(int,K,L)),L)) ).

% signed_modulo_int_def
tff(fact_5318_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Aa2: A,Ba: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,Aa2,Ba)),set_or7035219750837199246ssThan(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),Aa2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),D2) ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_5319_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Aa2: A,Ba: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,Aa2,Ba)),set_or1337092689740270186AtMost(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),Aa2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),D2) ) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
tff(fact_5320_ivl__disj__un__two__touch_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(2)
tff(fact_5321_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,aa(A,fun(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_5322_sum_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,Na)) = aa(A,A,aa(A,fun(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_5323_sum_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Aa2: nat,Ba: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Aa2),Ba)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,Aa2,aa(nat,nat,suc,Ba))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,Aa2,Ba))),aa(nat,A,G,Ba)) ) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_5324_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_5325_prod_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(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_5326_prod_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Aa2: nat,Ba: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Aa2),Ba)
         => ( groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,Aa2,aa(nat,nat,suc,Ba))) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,Aa2,Ba))),aa(nat,A,G,Ba)) ) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_5327_sum_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,Na)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,Na)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,Na))) ) ) ) ).

% sum.last_plus
tff(fact_5328_prod_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,Na)),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,Na))) ) ) ) ).

% prod.last_plus
tff(fact_5329_atLeastLessThanSuc,axiom,
    ! [M: nat,Na: nat] :
      set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,Na)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na),aa(set(nat),set(nat),insert(nat,Na),set_or7035219750837199246ssThan(nat,M,Na)),bot_bot(set(nat))) ).

% atLeastLessThanSuc
tff(fact_5330_le__mask__high__bits,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Na: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),W),bit_se2239418461657761734s_mask(word(A),Na))
        <=> ! [X6: nat] :
              ( member(nat,X6,set_or7035219750837199246ssThan(nat,Na,aa(word(A),nat,size_size(word(A)),W)))
             => ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),X6) ) ) ) ).

% le_mask_high_bits
tff(fact_5331_sum__Suc__diff_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,Na: nat,F2: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ia(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_5332_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_on(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Na),M)),set_or7035219750837199246ssThan(nat,Na,M)) ) ).

% sum.atLeastLessThan_rev
tff(fact_5333_sum_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Aa2: fun(nat,fun(nat,A)),Na: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_oo(fun(nat,fun(nat,A)),fun(nat,A),Aa2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_iz(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Aa2),Na)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)) ) ).

% sum.nested_swap
tff(fact_5334_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_op(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Na),M),set_or7035219750837199246ssThan(nat,Na,M)) ) ).

% prod.atLeastLessThan_rev
tff(fact_5335_prod_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Aa2: fun(nat,fun(nat,A)),Na: nat] : groups7121269368397514597t_prod(nat,A,aTP_Lamp_oq(fun(nat,fun(nat,A)),fun(nat,A),Aa2),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_kh(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Aa2),Na),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)) ) ).

% prod.nested_swap
tff(fact_5336_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_or(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_5337_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_os(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_5338_smod__int__compares_I3_J,axiom,
    ! [Aa2: int,Ba: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Aa2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Ba)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),Ba)),signed6721504322012087516modulo(int,Aa2,Ba)) ) ) ).

% smod_int_compares(3)
tff(fact_5339_smod__int__compares_I5_J,axiom,
    ! [Aa2: int,Ba: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Aa2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ba),zero_zero(int))
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),signed6721504322012087516modulo(int,Aa2,Ba)),aa(int,int,uminus_uminus(int),Ba)) ) ) ).

% smod_int_compares(5)
tff(fact_5340_ivl__disj__un__singleton_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,U)),aa(set(A),set(A),insert(A,U),bot_bot(set(A)))) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(6)
tff(fact_5341_smod__int__alt__def,axiom,
    ! [Aa2: int,Ba: int] : signed6721504322012087516modulo(int,Aa2,Ba) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),Aa2)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,abs_abs(int),Aa2)),aa(int,int,abs_abs(int),Ba))) ).

% smod_int_alt_def
tff(fact_5342_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,aa(nat,fun(nat,$o),ord_less(nat),Na),M),zero_zero(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,Na))),aa(nat,A,G,Na))) ) ).

% sum.head_if
tff(fact_5343_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,aa(nat,fun(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_5344_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      set_or7035219750837199246ssThan(nat,M,aa(num,nat,numeral_numeral(nat),K)) = $ite(aa(nat,$o,aa(nat,fun(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_5345_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_5346_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_iw(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_5347_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_ke(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_5348_pochhammer__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Aa2: A,Na: nat] : comm_s3205402744901411588hammer(A,Aa2,Na) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_kk(A,fun(nat,A),Aa2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)) ) ).

% pochhammer_prod
tff(fact_5349_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_5350_summable__Cauchy,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
        <=> ! [E3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
             => ? [N7: nat] :
                ! [M4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),M4)
                 => ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or7035219750837199246ssThan(nat,M4,N2)))),E3) ) ) ) ) ).

% summable_Cauchy
tff(fact_5351_sums__group,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A),S: A,K: nat] :
          ( aa(A,$o,sums(A,F2),S)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
           => aa(A,$o,sums(A,aa(nat,fun(nat,A),aTP_Lamp_ot(fun(nat,A),fun(nat,fun(nat,A)),F2),K)),S) ) ) ) ).

% sums_group
tff(fact_5352_take__bit__sum,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat,Aa2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Na),Aa2) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ou(A,fun(nat,A),Aa2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)) ) ).

% take_bit_sum
tff(fact_5353_fact__split,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
         => ( semiring_char_0_fact(A,Na) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),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_5354_binomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Na),K)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ov(nat,fun(nat,fun(nat,A)),K),Na),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ) ) ).

% binomial_altdef_of_nat
tff(fact_5355_gbinomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Aa2: A,K: nat] : aa(nat,A,gbinomial(A,Aa2),K) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ow(A,fun(nat,fun(nat,A)),Aa2),K),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_altdef_of_nat
tff(fact_5356_gbinomial__mult__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Aa2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),aa(nat,A,gbinomial(A,Aa2),K)) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_ox(A,fun(nat,A),Aa2),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact
tff(fact_5357_gbinomial__mult__fact_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Aa2: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,Aa2),K)),semiring_char_0_fact(A,K)) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_ox(A,fun(nat,A),Aa2),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact'
tff(fact_5358_gbinomial__prod__rev,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [Aa2: A,K: nat] : aa(nat,A,gbinomial(A,Aa2),K) = aa(A,A,aa(A,fun(A,A),divide_divide(A),groups7121269368397514597t_prod(nat,A,aTP_Lamp_kp(A,fun(nat,A),Aa2),set_or7035219750837199246ssThan(nat,zero_zero(nat),K))),semiring_char_0_fact(A,K)) ) ).

% gbinomial_prod_rev
tff(fact_5359_Chebyshev__sum__upper,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat,Aa2: fun(nat,A),Ba: fun(nat,A)] :
          ( ! [I2: nat,J2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J2)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),Na)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,Aa2,I2)),aa(nat,A,Aa2,J2)) ) )
         => ( ! [I2: nat,J2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J2)
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),Na)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,Ba,J2)),aa(nat,A,Ba,I2)) ) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_oy(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Aa2),Ba)),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,Aa2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Ba),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)))) ) ) ) ).

% Chebyshev_sum_upper
tff(fact_5360_Chebyshev__sum__upper__nat,axiom,
    ! [Na: nat,Aa2: fun(nat,nat),Ba: fun(nat,nat)] :
      ( ! [I2: nat,J2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),Na)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,Aa2,I2)),aa(nat,nat,Aa2,J2)) ) )
     => ( ! [I2: nat,J2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J2)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),Na)
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,Ba,J2)),aa(nat,nat,Ba,I2)) ) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(fun(nat,nat),fun(nat,nat),aTP_Lamp_oz(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),Aa2),Ba)),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,Aa2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,Ba),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_5361_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_5362_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : set_or7035219750837199246ssThan(int,L,aa(int,int,aa(int,fun(int,int),plus_plus(int),U),one_one(int))) = set_or1337092689740270186AtMost(int,L,U) ).

% atLeastLessThanPlusOne_atLeastAtMost_int
tff(fact_5363_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),aa(word(A),fun(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_5364_atLeastLessThan__upto,axiom,
    ! [I: int,J: int] : set_or7035219750837199246ssThan(int,I,J) = aa(list(int),set(int),set2(int),upto(I,aa(int,int,minus_minus(int,J),one_one(int)))) ).

% atLeastLessThan_upto
tff(fact_5365_mod__word__minus__1__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Ba: 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)),Ba))) = 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))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Ba))))) ) ).

% mod_word_minus_1_minus_numeral
tff(fact_5366_horner__sum__of__bool__2__less,axiom,
    ! [Bs2: list($o)] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),groups4207007520872428315er_sum($o,int,zero_neq_one_of_bool(int),aa(num,int,numeral_numeral(int),bit0(one2)),Bs2)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(list($o),nat,size_size(list($o)),Bs2))) ).

% horner_sum_of_bool_2_less
tff(fact_5367_div__word__minus__1__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Ba: 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)),Ba))) = 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))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Ba))))) ) ).

% div_word_minus_1_minus_numeral
tff(fact_5368_horner__sum__simps_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [F2: fun(B,A),Aa2: A] : groups4207007520872428315er_sum(B,A,F2,Aa2,nil(B)) = zero_zero(A) ) ).

% horner_sum_simps(1)
tff(fact_5369_take__bit__length__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),type_len0_len_of(A,type2(A))),W) = W ) ).

% take_bit_length_eq
tff(fact_5370_horner__sum__simps_I2_J,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(A)
     => ! [F2: fun(B,A),Aa2: A,X: B,Xs2: list(B)] : groups4207007520872428315er_sum(B,A,F2,Aa2,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,F2,X)),aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),groups4207007520872428315er_sum(B,A,F2,Aa2,Xs2))) ) ).

% horner_sum_simps(2)
tff(fact_5371_drop__bit__word__beyond,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Na)
         => ( aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),Na),W) = zero_zero(word(A)) ) ) ) ).

% drop_bit_word_beyond
tff(fact_5372_push__bit__word__beyond,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Na)
         => ( aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4730199178511100633sh_bit(word(A)),Na),W) = zero_zero(word(A)) ) ) ) ).

% push_bit_word_beyond
tff(fact_5373_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)) = aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),Bin)) ) ).

% uint_bintrunc
tff(fact_5374_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_5375_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_5376_less__eq__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(num,word(A),numeral_numeral(word(A)),Aa2)),aa(num,word(A),numeral_numeral(word(A)),Ba))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),Aa2))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),Ba))) ) ) ).

% less_eq_word_numeral_numeral
tff(fact_5377_less__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(num,word(A),numeral_numeral(word(A)),Aa2)),aa(num,word(A),numeral_numeral(word(A)),Ba))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),Aa2))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),Ba))) ) ) ).

% less_word_numeral_numeral
tff(fact_5378_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)) = aa(int,nat,nat2,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),Bin))) ) ).

% unat_bintrunc
tff(fact_5379_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,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(num,int,numeral_numeral(int),W)),Na) ) ) ) ).

% bit_numeral_word_iff
tff(fact_5380_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)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(B,type2(B))),aa(num,int,numeral_numeral(int),W))) ) ).

% ucast_bintr
tff(fact_5381_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),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,type_len0_len_of(B,type2(B))),aa(num,nat,numeral_numeral(nat),Na))) ) ).

% unsigned_numeral
tff(fact_5382_unat__lt2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(nat,$o,aa(nat,fun(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_5383_uint__lt2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(int,$o,aa(int,fun(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_5384_uint__bounded,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(int,$o,aa(int,fun(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_5385_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_5386_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,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Na) ) ) ).

% exp_eq_zero_iff
tff(fact_5387_signed__take__bit__word__Suc__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,K: num] : aa(word(A),word(A),bit_ri4674362597316999326ke_bit(word(A),aa(nat,nat,suc,Na)),aa(num,word(A),numeral_numeral(word(A)),K)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,Na)),aa(int,int,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_5388_signed__take__bit__word__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num,K: num] : aa(word(A),word(A),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)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),Na)),aa(int,int,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_5389_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)) = aa(int,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),Bin)) ) ).

% sint_sbintrunc
tff(fact_5390_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))) = aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Bin))) ) ).

% uint_bintrunc_neg
tff(fact_5391_div__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: 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)),Aa2)),aa(num,word(A),numeral_numeral(word(A)),Ba)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),Aa2))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),Ba)))) ) ).

% div_word_numeral_numeral
tff(fact_5392_mod__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: 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)),Aa2)),aa(num,word(A),numeral_numeral(word(A)),Ba)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),Aa2))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),Ba)))) ) ).

% mod_word_numeral_numeral
tff(fact_5393_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),aa(int,int,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_5394_less__eq__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Aa2))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Ba)))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Aa2)))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Ba)))) ) ) ).

% less_eq_word_minus_numeral_minus_numeral
tff(fact_5395_less__eq__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(num,word(A),numeral_numeral(word(A)),Aa2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Ba)))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),Aa2))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Ba)))) ) ) ).

% less_eq_word_numeral_minus_numeral
tff(fact_5396_less__eq__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Aa2))),aa(num,word(A),numeral_numeral(word(A)),Ba))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Aa2)))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),Ba))) ) ) ).

% less_eq_word_minus_numeral_numeral
tff(fact_5397_less__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Aa2))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Ba)))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Aa2)))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Ba)))) ) ) ).

% less_word_minus_numeral_minus_numeral
tff(fact_5398_less__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(num,word(A),numeral_numeral(word(A)),Aa2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Ba)))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),Aa2))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Ba)))) ) ) ).

% less_word_numeral_minus_numeral
tff(fact_5399_less__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Aa2))),aa(num,word(A),numeral_numeral(word(A)),Ba))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Aa2)))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),Ba))) ) ) ).

% less_word_minus_numeral_numeral
tff(fact_5400_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))) = aa(int,nat,nat2,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Bin)))) ) ).

% unat_bintrunc_neg
tff(fact_5401_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,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),W))),Na) ) ) ) ).

% bit_neg_numeral_word_iff
tff(fact_5402_drop__bit__word__Suc__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,K: num] : aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),aa(nat,nat,suc,Na)),aa(num,word(A),numeral_numeral(word(A)),K)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(nat,nat,suc,Na)),aa(int,int,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_5403_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)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),K)))) ) ).

% drop_bit_word_numeral
tff(fact_5404_unat__power__lower,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(nat,word(A),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_5405_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),aa(int,nat,nat2,aa(int,int,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_5406_signed__take__bit__word__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,K: num] : aa(word(A),word(A),bit_ri4674362597316999326ke_bit(word(A),aa(nat,nat,suc,Na)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),K))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,Na)),aa(int,int,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_5407_signed__take__bit__word__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num,K: num] : aa(word(A),word(A),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)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),Na)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))))) ) ).

% signed_take_bit_word_minus_numeral
tff(fact_5408_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))) = aa(int,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),Bin))) ) ).

% sint_sbintrunc_neg
tff(fact_5409_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)),aa(int,int,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_5410_drop__bit__numeral__bit0__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: num] : aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),aa(nat,nat,suc,zero_zero(nat))),aa(num,word(A),numeral_numeral(word(A)),K)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(nat,nat,suc,zero_zero(nat))),aa(int,int,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_5411_div__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: 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)),Aa2))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Ba))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Aa2)))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Ba))))) ) ).

% div_word_minus_numeral_minus_numeral
tff(fact_5412_div__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: 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)),Aa2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Ba))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),Aa2))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Ba))))) ) ).

% div_word_numeral_minus_numeral
tff(fact_5413_div__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: 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)),Aa2))),aa(num,word(A),numeral_numeral(word(A)),Ba)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Aa2)))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),Ba)))) ) ).

% div_word_minus_numeral_numeral
tff(fact_5414_mod__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: 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)),Aa2))),aa(num,word(A),numeral_numeral(word(A)),Ba)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Aa2)))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),Ba)))) ) ).

% mod_word_minus_numeral_numeral
tff(fact_5415_mod__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: 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)),Aa2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Ba))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),Aa2))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Ba))))) ) ).

% mod_word_numeral_minus_numeral
tff(fact_5416_mod__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: 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)),Aa2))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Ba))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Aa2)))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Ba))))) ) ).

% mod_word_minus_numeral_minus_numeral
tff(fact_5417_word__less__sub__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,X: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),one_one(word(A))))
          <=> aa(word(A),$o,aa(word(A),fun(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_5418_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),aa(int,int,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_5419_drop__bit__word__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,K: num] : aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),aa(nat,nat,suc,Na)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),K))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(nat,nat,suc,Na)),aa(int,int,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_5420_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)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))))) ) ).

% drop_bit_word_minus_numeral
tff(fact_5421_less__word__numeral__minus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(num,word(A),numeral_numeral(word(A)),Aa2)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),Aa2))),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_5422_less__word__minus__numeral__minus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Aa2))),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Aa2)))),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_5423_div__word__minus__1__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Ba: 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)),Ba)) = 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))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),Ba)))) ) ).

% div_word_minus_1_numeral
tff(fact_5424_mod__word__minus__1__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Ba: 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)),Ba)) = 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))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),Ba)))) ) ).

% mod_word_minus_1_numeral
tff(fact_5425_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) )
        <=> ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,type_len0_len_of(A,type2(A))),M) = aa(nat,nat,bit_se2584673776208193580ke_bit(nat,type_len0_len_of(A,type2(A))),Na) ) ) ) ).

% word_of_nat_eq_iff
tff(fact_5426_zero__word_Orsp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),zero_zero(int)) = aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),zero_zero(int)) ) ) ).

% zero_word.rsp
tff(fact_5427_size__0__same,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),V: word(A)] :
          ( ( type_len0_len_of(A,type2(A)) = zero_zero(nat) )
         => ( W = V ) ) ) ).

% size_0_same
tff(fact_5428_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,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A))) ) ) ).

% bin_nth_uint_imp
tff(fact_5429_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_5430_bit__word__eqI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] :
          ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),type_len0_len_of(A,type2(A)))
             => ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Aa2),N)
              <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Ba),N) ) )
         => ( Aa2 = Ba ) ) ) ).

% bit_word_eqI
tff(fact_5431_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,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A))) ) ) ).

% bit_imp_le_length
tff(fact_5432_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) )
        <=> ( aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),K) = aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),L) ) ) ) ).

% word_of_int_eq_iff
tff(fact_5433_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)) = aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),K) ) ).

% uint_word_of_int_eq
tff(fact_5434_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)) = aa(A,A,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_5435_uint__sint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),W) = aa(int,int,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_5436_bit__ucast__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Aa2: word(B),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Aa2)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),Aa2),Na) ) ) ) ).

% bit_ucast_iff
tff(fact_5437_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,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(B,type2(B)))
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),W),Na) ) ) ) ).

% bit_word_ucast_iff
tff(fact_5438_ucast__up__mono__le,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),Y)
           => aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less_eq(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),X)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y)) ) ) ) ).

% ucast_up_mono_le
tff(fact_5439_ucast__le__ucast,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less_eq(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),X)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y))
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),Y) ) ) ) ).

% ucast_le_ucast
tff(fact_5440_ucast__ucast__add,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [X: word(A),Y: word(B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(B),word(A),semiring_1_unsigned(B,word(A)),aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),plus_plus(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),X)),Y)) = aa(word(A),word(A),aa(word(A),fun(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_5441_wi__bintr,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: int] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Na)
         => ( aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),W)) = aa(int,word(A),ring_1_of_int(word(A)),W) ) ) ) ).

% wi_bintr
tff(fact_5442_bintr__uint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Na)
         => ( aa(int,int,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_5443_eq__ucast__ucast__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(B),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( ( 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_5444_up__ucast__inj__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( ( aa(word(A),word(B),semiring_1_unsigned(A,word(B)),X) = aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y) )
          <=> ( X = Y ) ) ) ) ).

% up_ucast_inj_eq
tff(fact_5445_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,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(B,type2(B))),type_len0_len_of(C,type2(C)))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(C,type2(C))),type_len0_len_of(A,type2(A)))
             => ( X = aa(word(C),word(B),semiring_1_unsigned(C,word(B)),Y) ) ) ) ) ) ).

% ucast_ucast_eq
tff(fact_5446_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,aa(nat,fun(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_5447_up__scast__inj__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( ( aa(word(A),word(B),ring_1_signed(A,word(B)),X) = aa(word(A),word(B),ring_1_signed(A,word(B)),Y) )
          <=> ( X = Y ) ) ) ) ).

% up_scast_inj_eq
tff(fact_5448_take__bit__word__beyond__length__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Na)
         => ( aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),Na),W) = W ) ) ) ).

% take_bit_word_beyond_length_eq
tff(fact_5449_unat__ucast__up__simp,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(B),nat,semiring_1_unsigned(B,nat),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),X)) = aa(word(A),nat,semiring_1_unsigned(A,nat),X) ) ) ) ).

% unat_ucast_up_simp
tff(fact_5450_less__ucast__ucast__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(B),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less(word(B)),X),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),X)),Y) ) ) ) ).

% less_ucast_ucast_less
tff(fact_5451_ucast__less__ucast,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),X)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y))
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),X),Y) ) ) ) ).

% ucast_less_ucast
tff(fact_5452_ucast__up__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),X),Y)
           => aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),X)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y)) ) ) ) ).

% ucast_up_mono
tff(fact_5453_ucast__drop__bit__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Na: nat,W: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(A),word(B),semiring_1_unsigned(A,word(B)),aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),Na),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_5454_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,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(B),nat,size_size(word(B)),X)),type_len0_len_of(A,type2(A)))
           => ( X = Y ) ) ) ) ).

% up_scast_inj
tff(fact_5455_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_5456_size__word_Orep__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),nat,size_size(word(A)),X) = type_len0_len_of(A,type2(A)) ) ).

% size_word.rep_eq
tff(fact_5457_word__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(word(A),nat,size_size(word(A)),W) = type_len0_len_of(A,type2(A)) ) ).

% word_size
tff(fact_5458_num__of__bintr_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: num] :
          ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),Aa2)) = aa(num,int,numeral_numeral(int),Ba) )
         => ( aa(num,word(A),numeral_numeral(word(A)),Aa2) = aa(num,word(A),numeral_numeral(word(A)),Ba) ) ) ) ).

% num_of_bintr'
tff(fact_5459_one__word_Orsp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),one_one(int)) = aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),one_one(int)) ) ) ).

% one_word.rsp
tff(fact_5460_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_5461_word__cat__inj,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(C)
        & type_len(A) )
     => ! [Aa2: word(A),Ba: word(B),C2: word(A),D2: word(B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))),type_len0_len_of(C,type2(C)))
         => ( ( word_cat(A,B,C,Aa2,Ba) = word_cat(A,B,C,C2,D2) )
          <=> ( ( Aa2 = C2 )
              & ( Ba = D2 ) ) ) ) ) ).

% word_cat_inj
tff(fact_5462_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)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),X))) ) ).

% num_abs_bintr
tff(fact_5463_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))) = aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),zero_zero(int)) ) ) ).

% uint_word_arith_bintrs(7)
tff(fact_5464_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))) = aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),one_one(int)) ) ) ).

% uint_word_arith_bintrs(8)
tff(fact_5465_word__of__int__less__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: int,L: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(int,word(A),ring_1_of_int(word(A)),K)),aa(int,word(A),ring_1_of_int(word(A)),L))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),K)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),L)) ) ) ).

% word_of_int_less_eq_iff
tff(fact_5466_uint__word__arith__bintrs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Aa2),Ba)) = aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Aa2)),aa(word(A),int,semiring_1_unsigned(A,int),Ba))) ) ).

% uint_word_arith_bintrs(1)
tff(fact_5467_word__of__int__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: int,L: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(int,word(A),ring_1_of_int(word(A)),K)),aa(int,word(A),ring_1_of_int(word(A)),L))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),K)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),L)) ) ) ).

% word_of_int_less_iff
tff(fact_5468_uint__word__arith__bintrs_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),uminus_uminus(word(A)),Aa2)) = aa(int,int,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),Aa2))) ) ).

% uint_word_arith_bintrs(4)
tff(fact_5469_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_5470_mask__over__length,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Na)
         => ( bit_se2239418461657761734s_mask(word(A),Na) = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% mask_over_length
tff(fact_5471_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)) = aa(A,A,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_5472_uint__word__arith__bintrs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),minus_minus(word(A),Aa2),Ba)) = aa(int,int,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),Aa2)),aa(word(A),int,semiring_1_unsigned(A,int),Ba))) ) ).

% uint_word_arith_bintrs(2)
tff(fact_5473_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,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),Na) ) ) ) ).

% bit_uint_iff
tff(fact_5474_ucast__sub__ucast,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Y: word(A),X: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),X)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
           => ( aa(word(A),word(B),semiring_1_unsigned(A,word(B)),aa(word(A),word(A),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_5475_bit__word__of__int__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: int,Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(int,word(A),ring_1_of_int(word(A)),K)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),Na) ) ) ) ).

% bit_word_of_int_iff
tff(fact_5476_ucast__mask__drop,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Na: nat,X: word(B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Na)
         => ( aa(word(B),word(A),semiring_1_unsigned(B,word(A)),aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),bit_se5824344872417868541ns_and(word(B)),X),bit_se2239418461657761734s_mask(word(B),Na))) = aa(word(B),word(A),semiring_1_unsigned(B,word(A)),X) ) ) ) ).

% ucast_mask_drop
tff(fact_5477_uint__word__arith__bintrs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: 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)),Aa2),Ba)) = aa(int,int,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),Aa2)),aa(word(A),int,semiring_1_unsigned(A,int),Ba))) ) ).

% uint_word_arith_bintrs(3)
tff(fact_5478_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),aa(word(B),word(B),bit_se2584673776208193580ke_bit(word(B),Na),W)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(B,type2(B))),aa(A,A,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_5479_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)) = aa(A,A,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_5480_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),aa(A,A,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_5481_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_5482_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)) = aa(A,A,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_5483_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),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,type_len0_len_of(B,type2(B))),Na)) ) ).

% unsigned_of_nat
tff(fact_5484_bit__word__cat__iff,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(B)
        & type_len(A) )
     => ! [V: word(B),W: word(C),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),word_cat(B,C,A,V,W)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
            & $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(C,type2(C))),aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),W),Na),aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),V),aa(nat,nat,minus_minus(nat,Na),type_len0_len_of(C,type2(C))))) ) ) ) ).

% bit_word_cat_iff
tff(fact_5485_word__of__nat__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,Na: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(nat,word(A),semiring_1_of_nat(word(A)),M)),aa(nat,word(A),semiring_1_of_nat(word(A)),Na))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,type_len0_len_of(A,type2(A))),M)),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,type_len0_len_of(A,type2(A))),Na)) ) ) ).

% word_of_nat_less_iff
tff(fact_5486_word__of__nat__less__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,Na: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(nat,word(A),semiring_1_of_nat(word(A)),M)),aa(nat,word(A),semiring_1_of_nat(word(A)),Na))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,type_len0_len_of(A,type2(A))),M)),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,type_len0_len_of(A,type2(A))),Na)) ) ) ).

% word_of_nat_less_eq_iff
tff(fact_5487_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,$o),bit_se5641148757651400278ts_bit(word(B)),W),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na))
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A))) ) ) ) ).

% nth_slice
tff(fact_5488_word__cat__eq_H,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(C)
        & type_len(A) )
     => ! [Aa2: word(B),Ba: word(C)] : word_cat(B,C,A,Aa2,Ba) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_concat_bit(type_len0_len_of(C,type2(C)),aa(word(C),int,semiring_1_unsigned(C,int),Ba)),aa(word(B),int,semiring_1_unsigned(B,int),Aa2))) ) ).

% word_cat_eq'
tff(fact_5489_word__cat__bin_H,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(A)
        & type_len(B) )
     => ! [V: word(B),W: word(C)] : word_cat(B,C,A,V,W) = aa(word(A),word(A),aa(word(A),fun(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)),V))),aa(word(C),word(A),semiring_1_unsigned(C,word(A)),W)) ) ).

% word_cat_bin'
tff(fact_5490_two__power__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A)))
           => ( ( aa(nat,word(A),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_5491_signed__take__bit__decr__length__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( bit_ri3973907225187159222ations(A)
        & type_len(B) )
     => ! [K: A,L: A] :
          ( ( aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,minus_minus(nat,type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat)))),K) = aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,minus_minus(nat,type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat)))),L) )
        <=> ( aa(A,A,bit_se2584673776208193580ke_bit(A,type_len0_len_of(B,type2(B))),K) = aa(A,A,bit_se2584673776208193580ke_bit(A,type_len0_len_of(B,type2(B))),L) ) ) ) ).

% signed_take_bit_decr_length_iff
tff(fact_5492_num__of__sbintr_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: num] :
          ( ( aa(int,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),Aa2)) = aa(num,int,numeral_numeral(int),Ba) )
         => ( aa(num,word(A),numeral_numeral(word(A)),Aa2) = aa(num,word(A),numeral_numeral(word(A)),Ba) ) ) ) ).

% num_of_sbintr'
tff(fact_5493_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_5494_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),aa(int,nat,nat2,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(B,type2(B))),K))) ) ).

% unsigned_of_int
tff(fact_5495_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_5496_bin__nth__sint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Na)
         => ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(word(A),int,ring_1_signed(A,int),W)),Na)
          <=> aa(nat,$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_5497_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)) = aa(int,int,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_5498_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,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A))) ) ) ) ).

% neg_mask_test_bit
tff(fact_5499_word__of__int__bin__cat__eq__iff,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(C)
        & type_len(B) )
     => ! [Ba: word(B),Aa2: word(A),D2: word(B),C2: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))),type_len0_len_of(C,type2(C)))
         => ( ( aa(int,word(C),ring_1_of_int(word(C)),aa(int,int,bit_concat_bit(type_len0_len_of(B,type2(B)),aa(word(B),int,semiring_1_unsigned(B,int),Ba)),aa(word(A),int,semiring_1_unsigned(A,int),Aa2))) = aa(int,word(C),ring_1_of_int(word(C)),aa(int,int,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))) )
          <=> ( ( Ba = D2 )
              & ( Aa2 = C2 ) ) ) ) ) ).

% word_of_int_bin_cat_eq_iff
tff(fact_5500_mask__exceed,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,X: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Na)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),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_5501_unsigned__less,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & unique1627219031080169319umeral(A) )
     => ! [W: word(B)] : aa(A,$o,aa(A,fun(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_5502_not__degenerate__imp__2__neq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),type_len0_len_of(A,type2(A)))
       => ( aa(num,word(A),numeral_numeral(word(A)),bit0(one2)) != zero_zero(word(A)) ) ) ) ).

% not_degenerate_imp_2_neq_0
tff(fact_5503_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,aa(nat,fun(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_5504_word__nchotomy,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W5: word(A)] :
        ? [N: nat] :
          ( ( W5 = aa(nat,word(A),semiring_1_of_nat(word(A)),N) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% word_nchotomy
tff(fact_5505_word__of__nat__inj,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: nat,Y: nat] :
          ( aa(nat,$o,aa(nat,fun(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,aa(nat,fun(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_5506_of__nat__inj,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: nat,Y: nat] :
          ( aa(nat,$o,aa(nat,fun(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,aa(nat,fun(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_5507_More__Word_Opower__not__zero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
         => ( aa(nat,word(A),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_5508_word__power__increasing,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: nat,Y: nat] :
          ( aa(word(A),$o,aa(word(A),fun(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,aa(nat,fun(nat,$o),ord_less(nat),X),type_len0_len_of(A,type2(A)))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),type_len0_len_of(A,type2(A)))
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Y) ) ) ) ) ).

% word_power_increasing
tff(fact_5509_power__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Na)
         => ( aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na) = zero_zero(word(A)) ) ) ) ).

% power_overflow
tff(fact_5510_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,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A))) ) ) ) ).

% nth_w2p
tff(fact_5511_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,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A))) ) ) ).

% nth_w2p_same
tff(fact_5512_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_5513_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_5514_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_5515_sint__uint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(word(A),int,ring_1_signed(A,int),W) = aa(int,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)))),aa(word(A),int,semiring_1_unsigned(A,int),W)) ) ).

% sint_uint
tff(fact_5516_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)),aa(int,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),X))) ) ).

% num_abs_sbintr
tff(fact_5517_ucast__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),X),Y)
         => ( aa(word(A),$o,aa(word(A),fun(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,aa(word(B),fun(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_5518_ucast__ucast__len,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(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_5519_sint__word__ariths_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),int,ring_1_signed(A,int),zero_zero(word(A))) = aa(int,int,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_5520_sint__word__ariths_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),int,ring_1_signed(A,int),one_one(word(A))) = aa(int,int,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_5521_sint__word__ariths_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] : aa(word(A),int,ring_1_signed(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Aa2),Ba)) = aa(int,int,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),plus_plus(int),aa(word(A),int,ring_1_signed(A,int),Aa2)),aa(word(A),int,ring_1_signed(A,int),Ba))) ) ).

% sint_word_ariths(1)
tff(fact_5522_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,aa(nat,fun(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_5523_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,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Na)
              & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))) )
            | aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),Na) ) ) ) ).

% bit_sint_iff
tff(fact_5524_sint__word__ariths_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A)] : aa(word(A),int,ring_1_signed(A,int),aa(word(A),word(A),uminus_uminus(word(A)),Aa2)) = aa(int,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(word(A),int,ring_1_signed(A,int),Aa2))) ) ).

% sint_word_ariths(4)
tff(fact_5525_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_5526_sint__word__ariths_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] : aa(word(A),int,ring_1_signed(A,int),aa(word(A),word(A),minus_minus(word(A),Aa2),Ba)) = aa(int,int,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),Aa2)),aa(word(A),int,ring_1_signed(A,int),Ba))) ) ).

% sint_word_ariths(2)
tff(fact_5527_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)) = aa(A,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)))),aa(word(C),A,ring_1_signed(C,A),W)) ) ).

% signed_scast_eq
tff(fact_5528_sint__word__ariths_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: 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)),Aa2),Ba)) = aa(int,int,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),Aa2)),aa(word(A),int,ring_1_signed(A,int),Ba))) ) ).

% sint_word_ariths(3)
tff(fact_5529_less__Suc__unat__less__bound,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,X: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),X)))
         => aa(nat,$o,aa(nat,fun(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_5530_uint__2__id,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))
       => ( aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))) = aa(num,int,numeral_numeral(int),bit0(one2)) ) ) ) ).

% uint_2_id
tff(fact_5531_lt2p__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Na)
         => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,semiring_1_unsigned(A,int),W)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ) ).

% lt2p_lem
tff(fact_5532_two__power__increasing,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A)))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(nat,word(A),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_5533_power__le__mono,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A)))
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M) ) ) ) ) ).

% power_le_mono
tff(fact_5534_unat__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Ba: num] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(num,word(A),numeral_numeral(word(A)),Ba)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(num,nat,numeral_numeral(nat),Ba)),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_5535_of__nat__mono__maybe_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: nat,Y: nat] :
          ( aa(nat,$o,aa(nat,fun(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,aa(nat,fun(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,aa(nat,fun(nat,$o),ord_less(nat),Y),X)
            <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(nat,word(A),semiring_1_of_nat(word(A)),Y)),aa(nat,word(A),semiring_1_of_nat(word(A)),X)) ) ) ) ) ).

% of_nat_mono_maybe'
tff(fact_5536_of__nat__mono__maybe,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: nat,Y: nat] :
          ( aa(nat,$o,aa(nat,fun(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,aa(nat,fun(nat,$o),ord_less(nat),Y),X)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(nat,word(A),semiring_1_of_nat(word(A)),Y)),aa(nat,word(A),semiring_1_of_nat(word(A)),X)) ) ) ) ).

% of_nat_mono_maybe
tff(fact_5537_unat__split__asm,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(nat,$o),X: word(A)] :
          ( aa(nat,$o,P,aa(word(A),nat,semiring_1_unsigned(A,nat),X))
        <=> ~ ? [N2: nat] :
                ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),N2) = X )
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
                & ~ aa(nat,$o,P,N2) ) ) ) ).

% unat_split_asm
tff(fact_5538_of__nat__inverse,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [R: nat,Aa2: word(A)] :
          ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),R) = Aa2 )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),R),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),Aa2) = R ) ) ) ) ).

% of_nat_inverse
tff(fact_5539_unat__split,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(nat,$o),X: word(A)] :
          ( aa(nat,$o,P,aa(word(A),nat,semiring_1_unsigned(A,nat),X))
        <=> ! [N2: nat] :
              ( ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),N2) = X )
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) )
             => aa(nat,$o,P,N2) ) ) ) ).

% unat_split
tff(fact_5540_unat__of__nat__len,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: nat] :
          ( aa(nat,$o,aa(nat,fun(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_5541_unat__eq__of__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,X: word(A)] :
          ( aa(nat,$o,aa(nat,fun(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_5542_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,aa(word(A),fun(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_5543_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,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A))) ) ) ) ).

% test_bit_2p
tff(fact_5544_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)) )
        <=> ? [Q4: nat] : M = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q4),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_5545_word__1__le__power,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),one_one(word(A))),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) ) ) ).

% word_1_le_power
tff(fact_5546_ucast__of__nat__small,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [X: nat] :
          ( aa(nat,$o,aa(nat,fun(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_5547_uint__sub__lt2p,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(A),Y: word(B)] : aa(int,$o,aa(int,fun(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_5548_uint__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Ba: num] : aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),Ba)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(num,int,numeral_numeral(int),Ba)),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_5549_p2__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A))) ) ) ).

% p2_gt_0
tff(fact_5550_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_5551_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_5552_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_5553_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,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,ring_1_signed(A,int),W)),zero_zero(int)) ) ) ).

% bit_last_iff
tff(fact_5554_mask__lt__2pn,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),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_5555_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_5556_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_5557_ucast__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),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_5558_signed__of__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_1(A) )
     => ! [Na: int] : aa(word(B),A,ring_1_signed(B,A),aa(int,word(B),ring_1_of_int(word(B)),Na)) = aa(int,A,ring_1_of_int(A),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat)))),Na)) ) ).

% signed_of_int
tff(fact_5559_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_5560_of__nat__n__less__equal__power__2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(nat,word(A),semiring_1_of_nat(word(A)),Na)),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) ) ) ).

% of_nat_n_less_equal_power_2
tff(fact_5561_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)) = aa(A,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)))),aa(word(C),A,semiring_1_unsigned(C,A),W)) ) ).

% signed_ucast_eq
tff(fact_5562_complement__nth__w2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N6: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(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_5563_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,aa(word(A),fun(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_5564_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,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
            & ( aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),W),Na)
              | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(B,type2(B))),Na)
                & aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),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_5565_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_5566_word__power__less__diff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,Q3: word(A),M: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),Q3)),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))
         => ( aa(word(A),$o,aa(word(A),fun(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,aa(word(A),fun(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_5567_unat__word__ariths_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: 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)),Aa2),Ba)) = 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),Aa2)),aa(word(A),nat,semiring_1_unsigned(A,nat),Ba))),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_5568_ucast__mono__le,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),Y)
         => ( aa(word(A),$o,aa(word(A),fun(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,aa(word(B),fun(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_5569_take__bit__word__eq__self__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] :
          ( ( aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),Na),W) = W )
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Na)
            | aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),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_5570_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)) = aa(A,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)))),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_5571_unat__add__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),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)),plus_plus(word(A)),X),Y)) = aa(nat,nat,aa(nat,fun(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_5572_unat__add__lem_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),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)),plus_plus(word(A)),X),Y)) = aa(nat,nat,aa(nat,fun(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_5573_Word_Oof__nat__neq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,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_5574_More__Word_Oof__nat__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),Na) = zero_zero(word(A)) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,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_5575_of__nat__mono__maybe__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: nat,Y: nat] :
          ( aa(nat,$o,aa(nat,fun(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,aa(nat,fun(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,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),X)
            <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(nat,word(A),semiring_1_of_nat(word(A)),Y)),aa(nat,word(A),semiring_1_of_nat(word(A)),X)) ) ) ) ) ).

% of_nat_mono_maybe_le
tff(fact_5576_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_5577_unat__word__ariths_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Aa2),Ba)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Aa2)),aa(word(A),nat,semiring_1_unsigned(A,nat),Ba))),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_5578_bool__mask_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),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_5579_uint__range_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(word(A),int,semiring_1_unsigned(A,int),X))
          & aa(int,$o,aa(int,fun(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_5580_sint__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(int,$o,aa(int,fun(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_5581_ucast__mono__le_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Y: word(A),X: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(B,type2(B))))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),type_len0_len_of(B,type2(B))),type_len0_len_of(A,type2(A)))
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),Y)
             => aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less_eq(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),X)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y)) ) ) ) ) ).

% ucast_mono_le'
tff(fact_5582_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 )
        <=> ? [Q4: nat] : Na = aa(nat,nat,aa(nat,fun(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),Q4),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_5583_unat__mult__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),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_5584_word__of__int__inj,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: int,Y: int] :
          ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
            & aa(int,$o,aa(int,fun(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,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
              & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Y),aa(nat,int,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_5585_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,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),N)
               => ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),N),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ) ).

% word_int_cases
tff(fact_5586_unat__ucast__no__overflow__le,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Ba: word(A),F2: word(B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Ba)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(B,type2(B))))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),type_len0_len_of(B,type2(B))),type_len0_len_of(A,type2(A)))
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),F2)),Ba)
            <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(B),nat,semiring_1_unsigned(B,nat),F2)),aa(word(A),nat,semiring_1_unsigned(A,nat),Ba)) ) ) ) ) ).

% unat_ucast_no_overflow_le
tff(fact_5587_uint__m2p__not__non__neg,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : ~ aa(int,$o,aa(int,fun(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_5588_uint__m2p__neg,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(int,$o,aa(int,fun(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_5589_unat__ucast__less__no__overflow__simp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,F2: word(A)] :
          ( aa(nat,$o,aa(nat,fun(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,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),F2)),Na)
          <=> aa(word(A),$o,aa(word(A),fun(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_5590_unat__ucast__less__no__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,F2: word(A)] :
          ( aa(nat,$o,aa(nat,fun(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,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),F2)),Na)
           => aa(word(A),$o,aa(word(A),fun(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_5591_uint__power__lower,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
         => ( aa(word(A),int,semiring_1_unsigned(A,int),aa(nat,word(A),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_5592_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,aa(word(A),fun(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,aa(nat,fun(nat,$o),ord_less_eq(nat),M),type_len0_len_of(A,type2(A)))
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M) ) ) ) ) ).

% nth_bounded
tff(fact_5593_upper__bits__unset__is__l2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,P2: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
         => ( ! [N11: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N11)
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N11),type_len0_len_of(A,type2(A)))
                 => ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),P2),N11) ) )
          <=> aa(word(A),$o,aa(word(A),fun(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_5594_less__2p__is__upper__bits__unset,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),Na: nat] :
          ( aa(word(A),$o,aa(word(A),fun(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,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
            & ! [N11: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N11)
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N11),type_len0_len_of(A,type2(A)))
                 => ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),P2),N11) ) ) ) ) ) ).

% less_2p_is_upper_bits_unset
tff(fact_5595_uint__add__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),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)),plus_plus(word(A)),X),Y)) = aa(int,int,aa(int,fun(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_5596_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_5597_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_5598_wi__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: int,M: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(int,word(A),ring_1_of_int(word(A)),Na)),aa(int,word(A),ring_1_of_int(word(A)),M))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Na),aa(nat,int,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_5599_uint__word__ariths_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Aa2),Ba)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Aa2)),aa(word(A),int,semiring_1_unsigned(A,int),Ba))),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_5600_word__2p__mult__inc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Na)),type_len0_len_of(A,type2(A)))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(nat,word(A),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_5601_unat__word__ariths_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: 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)),Aa2),Ba)) = 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),Aa2)),aa(word(A),nat,semiring_1_unsigned(A,nat),Ba))),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_5602_wi__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: int,M: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(int,word(A),ring_1_of_int(word(A)),Na)),aa(int,word(A),ring_1_of_int(word(A)),M))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Na),aa(nat,int,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_5603_unat__word__ariths_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: 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)),Aa2),Ba)) = 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),Aa2)),aa(word(A),nat,semiring_1_unsigned(A,nat),Ba))),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_5604_power__2__ge__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),one_one(word(A)))),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A))) ) ) ).

% power_2_ge_iff
tff(fact_5605_word__power__less__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Sz: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Sz),type_len0_len_of(A,type2(A)))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),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_5606_uint__word__ariths_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),uminus_uminus(word(A)),Aa2)) = 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),Aa2))),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_5607_le__mask__iff__lt__2n,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),W),bit_se2239418461657761734s_mask(word(A),Na))
          <=> aa(word(A),$o,aa(word(A),fun(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_5608_and__mask__less_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),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_5609_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,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),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_5610_sint__1__cases,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A)] :
          ( ( ( type_len0_len_of(A,type2(A)) = one_one(nat) )
           => ( ( Aa2 = zero_zero(word(A)) )
             => ( aa(word(A),int,ring_1_signed(A,int),Aa2) != zero_zero(int) ) ) )
         => ( ( ( type_len0_len_of(A,type2(A)) = one_one(nat) )
             => ( ( Aa2 = one_one(word(A)) )
               => ( aa(word(A),int,ring_1_signed(A,int),one_one(word(A))) != aa(int,int,uminus_uminus(int),one_one(int)) ) ) )
           => ~ ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),type_len0_len_of(A,type2(A)))
               => ( aa(word(A),int,ring_1_signed(A,int),one_one(word(A))) != one_one(int) ) ) ) ) ) ).

% sint_1_cases
tff(fact_5611_uint__word__ariths_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),minus_minus(word(A),Aa2),Ba)) = 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),Aa2)),aa(word(A),int,semiring_1_unsigned(A,int),Ba))),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_5612_uint__mult__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,semiring_1_unsigned(A,int),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_5613_uint__word__ariths_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: 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)),Aa2),Ba)) = 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),Aa2)),aa(word(A),int,semiring_1_unsigned(A,int),Ba))),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_5614_signed__of__nat,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_1(A) )
     => ! [Na: nat] : aa(word(B),A,ring_1_signed(B,A),aa(nat,word(B),semiring_1_of_nat(word(B)),Na)) = aa(int,A,ring_1_of_int(A),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,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_5615_word__power__mod__div,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat,X: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A)))
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),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_5616_scast__1_H,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ( aa(word(B),word(A),ring_1_signed(B,word(A)),one_one(word(B))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat)))),one_one(int))) ) ) ).

% scast_1'
tff(fact_5617_unat__plus__if_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] :
          aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Aa2),Ba)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Aa2)),aa(word(A),nat,semiring_1_unsigned(A,nat),Ba))),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),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Aa2)),aa(word(A),nat,semiring_1_unsigned(A,nat),Ba)),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Aa2)),aa(word(A),nat,semiring_1_unsigned(A,nat),Ba))),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_5618_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_5619_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,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),aa(word(A),nat,semiring_1_unsigned(A,nat),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,aa(nat,fun(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_5620_no__olen__add__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),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_5621_word__add__le__mono2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),J: word(A),K: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),J)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),K),I)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),K),J)) ) ) ) ).

% word_add_le_mono2
tff(fact_5622_word__add__le__mono1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),J: word(A),K: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),J)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K)) ) ) ) ).

% word_add_le_mono1
tff(fact_5623_word__add__le__dest,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),J) ) ) ) ) ).

% word_add_le_dest
tff(fact_5624_word__add__le__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K))
            <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),J) ) ) ) ) ).

% word_add_le_iff
tff(fact_5625_sint__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(int,$o,aa(int,fun(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_5626_word__add__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K))
            <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),J) ) ) ) ) ).

% word_add_less_iff
tff(fact_5627_word__add__less__dest,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),J) ) ) ) ) ).

% word_add_less_dest
tff(fact_5628_word__add__less__mono1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),J: word(A),K: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),J)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K)) ) ) ) ).

% word_add_less_mono1
tff(fact_5629_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_5630_sint__ge,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(int,$o,aa(int,fun(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_5631_unat__less__power,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Sz: nat,K: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Sz),type_len0_len_of(A,type2(A)))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),K),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Sz))
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),K)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Sz)) ) ) ) ).

% unat_less_power
tff(fact_5632_word__of__int__inverse,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [R: int,Aa2: word(A)] :
          ( ( aa(int,word(A),ring_1_of_int(word(A)),R) = Aa2 )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R),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),Aa2) = R ) ) ) ) ) ).

% word_of_int_inverse
tff(fact_5633_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,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I4)
                & aa(int,$o,aa(int,fun(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_5634_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,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I4)
                & aa(int,$o,aa(int,fun(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_5635_word__mult__less__dest,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),J),K))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),J) ) ) ) ) ).

% word_mult_less_dest
tff(fact_5636_div__lt_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),X: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),K),X))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),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_5637_div__lt_H_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),X: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),K),X))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),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_5638_double__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: 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))),Aa2) = zero_zero(word(A)) )
        <=> ( ( Aa2 = zero_zero(word(A)) )
            | ( Aa2 = 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_5639_word__le__exists_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),Y)
         => ? [Z3: word(A)] :
              ( ( Y = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Z3) )
              & aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),X)),aa(word(A),int,semiring_1_unsigned(A,int),Z3))),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_5640_no__olen__add_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),X))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Y)),aa(word(A),int,semiring_1_unsigned(A,int),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_5641_no__olen__add,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),X),Y))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),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_5642_More__Word_Oof__nat__power,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: nat,X: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),P2),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),X))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),type_len0_len_of(A,type2(A)))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(nat,word(A),semiring_1_of_nat(word(A)),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_5643_uint__plus__if_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] :
          aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Aa2),Ba)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Aa2)),aa(word(A),int,semiring_1_unsigned(A,int),Ba))),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),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Aa2)),aa(word(A),int,semiring_1_unsigned(A,int),Ba)),aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Aa2)),aa(word(A),int,semiring_1_unsigned(A,int),Ba))),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_5644_word__less__power__trans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A),M: nat,K: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Na),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,M),K)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A)))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(nat,word(A),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_5645_word__less__power__trans2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A),M: nat,K: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Na),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,M),K)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A)))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),Na),aa(nat,word(A),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_5646_uint__sub__if_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] :
          aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),minus_minus(word(A),Aa2),Ba)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,semiring_1_unsigned(A,int),Ba)),aa(word(A),int,semiring_1_unsigned(A,int),Aa2)),aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),Aa2)),aa(word(A),int,semiring_1_unsigned(A,int),Ba)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),Aa2)),aa(word(A),int,semiring_1_unsigned(A,int),Ba))),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_5647_word__less__two__pow__divI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Na: nat,M: nat] :
          ( aa(word(A),$o,aa(word(A),fun(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,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),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_5648_uint__neg__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Ba: 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)),Ba))) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Ba))),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_5649_word__power__nonzero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Na: nat] :
          ( aa(word(A),$o,aa(word(A),fun(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,aa(nat,fun(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_5650_mult__pow2__inj,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,Na: nat,X: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)),type_len0_len_of(A,type2(A)))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),X),bit_se2239418461657761734s_mask(word(A),M))
           => ( aa(word(A),$o,aa(word(A),fun(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_5651_div__lt__uint_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),X: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),K),X))
         => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,semiring_1_unsigned(A,int),I)),aa(word(A),int,semiring_1_unsigned(A,int),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_5652_div__lt__uint_H_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),X: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),K),X))
         => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,semiring_1_unsigned(A,int),I)),aa(word(A),int,semiring_1_unsigned(A,int),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_5653_push__bit__word__eq__nonzero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),M: nat,Na: nat] :
          ( aa(word(A),$o,aa(word(A),fun(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,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)),type_len0_len_of(A,type2(A)))
           => ( ( 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_5654_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_5655_sint__greater__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(int,$o,aa(int,fun(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_5656_int__eq__sint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: nat] :
          ( aa(nat,$o,aa(nat,fun(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_5657_word__mult__less__mono1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),J: word(A),K: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),J)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),K)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),J),K)) ) ) ) ) ).

% word_mult_less_mono1
tff(fact_5658_word__mult__less__cancel,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A),I: word(A),J: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),K)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),J),K))
              <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),J) ) ) ) ) ) ).

% word_mult_less_cancel
tff(fact_5659_smod__word__max,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),signed6721504322012087516modulo(int,aa(word(A),int,ring_1_signed(A,int),Aa2),aa(word(A),int,ring_1_signed(A,int),Ba))),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_5660_le2p__bits__unset,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),Na: nat] :
          ( aa(word(A),$o,aa(word(A),fun(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,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N4)
             => ( aa(nat,$o,aa(nat,fun(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_5661_le__2p__upper__bits,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),Na: nat] :
          ( aa(word(A),$o,aa(word(A),fun(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,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
           => ! [N4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N4)
               => ( aa(nat,$o,aa(nat,fun(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_5662_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,aa(word(A),fun(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,aa(word(A),fun(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,aa(nat,fun(nat,$o),ord_less(nat),Sz),type_len0_len_of(A,type2(A)))
             => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),X),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),Na)))
               => ( ( Sz = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na) )
                 => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),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_5663_div__power__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: nat,Y: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Y)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),type_len0_len_of(A,type2(A)))
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),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_5664_even__mult__exp__div__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),M: nat,Na: nat] :
          ( aa(word(A),$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)),Aa2),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,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
              & ~ aa(word(A),$o,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)),Aa2),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_5665_Suc__2p__unat__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,K: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
         => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,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,aa(nat,fun(nat,nat),plus_plus(nat),K),one_one(nat))) ) ) ) ).

% Suc_2p_unat_mask
tff(fact_5666_sint__of__nat__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Ba: nat,Aa2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ba),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Aa2),Ba)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,ring_1_signed(A,int),aa(nat,word(A),semiring_1_of_nat(word(A)),Aa2))),aa(word(A),int,ring_1_signed(A,int),aa(nat,word(A),semiring_1_of_nat(word(A)),Ba))) ) ) ) ).

% sint_of_nat_le
tff(fact_5667_sint__of__nat__ge__zero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: nat] :
          ( aa(nat,$o,aa(nat,fun(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,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(word(A),int,ring_1_signed(A,int),aa(nat,word(A),semiring_1_of_nat(word(A)),X))) ) ) ).

% sint_of_nat_ge_zero
tff(fact_5668_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_5669_sint__of__int__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: int] :
          ( aa(int,$o,aa(int,fun(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,aa(int,fun(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_5670_word__mult__le__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A),I: word(A),J: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),K)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),J),K))
              <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),J) ) ) ) ) ) ).

% word_mult_le_iff
tff(fact_5671_word__mult__le__mono1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),J: word(A),K: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),J)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),K)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),J),K)) ) ) ) ) ).

% word_mult_le_mono1
tff(fact_5672_sint__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Ba: num] : aa(word(A),int,ring_1_signed(A,int),aa(num,word(A),numeral_numeral(word(A)),Ba)) = aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),Ba)),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_5673_smod__word__min,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] : aa(int,$o,aa(int,fun(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),Aa2),aa(word(A),int,ring_1_signed(A,int),Ba))) ) ).

% smod_word_min
tff(fact_5674_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,aa(int,fun(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_5675_Word_Oword__div__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),Y)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),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_5676_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_5677_word__less__power__trans__ofnat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat,K: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,M),K)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A)))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(nat,word(A),semiring_1_of_nat(word(A)),Na)),aa(nat,word(A),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_5678_word__bit__induct,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(word(A),$o),Aa2: word(A)] :
          ( aa(word(A),$o,P,zero_zero(word(A)))
         => ( ! [A3: word(A)] :
                ( aa(word(A),$o,P,A3)
               => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),A3)
                 => ( aa(word(A),$o,aa(word(A),fun(word(A),$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),$o,P,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),A3)) ) ) )
           => ( ! [A3: word(A)] :
                  ( aa(word(A),$o,P,A3)
                 => ( aa(word(A),$o,aa(word(A),fun(word(A),$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),$o,P,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),A3))) ) )
             => aa(word(A),$o,P,Aa2) ) ) ) ) ).

% word_bit_induct
tff(fact_5679_unat__mult__power__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: nat,Sz: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,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_5680_bit__word__half__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: $o] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Aa2),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa($o,word(A),zero_neq_one_of_bool(word(A)),(Ba))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),Aa2),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))))),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))) = Aa2 ) ) ) ).

% bit_word_half_eq
tff(fact_5681_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,aa(int,fun(int,$o),ord_less(int),aa(int,int,minus_minus(int,i),Shift)),Least)
                            | aa(int,$o,aa(int,fun(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,aa(int,fun(int,$o),ord_less(int),i),Least)
                            | aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Overflow),i) ),
                            aa(int,word(A),Arbitrary2,i),
                            aa(int,word(A),ring_1_of_int(word(A)),i) ) ) ) ) ) ) ) ) ) ) ).

% word_of_int_via_signed
tff(fact_5682_Suc__div__unat__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Sz: nat,Us: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Sz),type_len0_len_of(A,type2(A)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),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_5683_alignUp__div__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: nat,Na: nat,X: word(A),Aa2: word(A)] :
          ( aa(nat,$o,aa(nat,fun(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,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Aa2),X)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
               => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),Aa2),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) != zero_zero(word(A)) )
                 => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Aa2),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_5684_less__eq__decr__length__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A))) ) ) ).

% less_eq_decr_length_iff
tff(fact_5685_decr__length__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Na)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Na) ) ) ).

% decr_length_less_iff
tff(fact_5686_length__not__greater__eq__2__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))
      <=> ( type_len0_len_of(A,type2(A)) = one_one(nat) ) ) ) ).

% length_not_greater_eq_2_iff
tff(fact_5687_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_5688_len__num1,axiom,
    ! [Uu2: itself(numeral_num1)] : type_len0_len_of(numeral_num1,Uu2) = one_one(nat) ).

% len_num1
tff(fact_5689_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_5690_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_5691_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)),aa(word(A),fun(list(word(A)),list(word(A))),cons(word(A)),W),Ws)) = aa(int,int,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_5692_len__bit0,axiom,
    ! [A: $tType] :
      ( type_len0(A)
     => ! [Uu2: itself(numeral_bit0(A))] : type_len0_len_of(numeral_bit0(A),Uu2) = 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_5693_two__less__eq__exp__length,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & linordered_idom(A) )
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,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_5694_len__bit1,axiom,
    ! [A: $tType] :
      ( type_len0(A)
     => ! [Uu2: itself(numeral_bit1(A))] : type_len0_len_of(numeral_bit1(A),Uu2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(nat)) ) ).

% len_bit1
tff(fact_5695_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,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4730199178511100633sh_bit(word(A)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),one_one(word(A)))),Y),
                $ite(aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),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,aa(word(A),fun(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),aa(word(A),fun(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_5696_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)),aa(int,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),K))))) ) ).

% signed_drop_bit_word_minus_numeral
tff(fact_5697_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)),aa(int,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),K))))) ) ).

% signed_drop_bit_word_Suc_minus_numeral
tff(fact_5698_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_5699_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,aa(nat,fun(nat,nat),plus_plus(nat),M),Na),W) ) ).

% signed_drop_bit_signed_drop_bit
tff(fact_5700_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_5701_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_5702_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_5703_sdiv__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: num] : signed7115095781618012415divide(word(A),aa(num,word(A),numeral_numeral(word(A)),Aa2),aa(num,word(A),numeral_numeral(word(A)),Ba)) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(int,aa(int,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),Aa2)),aa(int,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),Ba)))) ) ).

% sdiv_word_numeral_numeral
tff(fact_5704_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)),aa(int,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),K)))) ) ).

% signed_drop_bit_word_Suc_numeral
tff(fact_5705_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)),aa(int,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),K)))) ) ).

% signed_drop_bit_word_numeral
tff(fact_5706_sdiv__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: num] : signed7115095781618012415divide(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Aa2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Ba))) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(int,aa(int,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),Aa2))),aa(int,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),Ba))))) ) ).

% sdiv_word_minus_numeral_minus_numeral
tff(fact_5707_sdiv__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: num] : signed7115095781618012415divide(word(A),aa(num,word(A),numeral_numeral(word(A)),Aa2),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Ba))) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(int,aa(int,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),Aa2)),aa(int,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),Ba))))) ) ).

% sdiv_word_numeral_minus_numeral
tff(fact_5708_sdiv__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: num] : signed7115095781618012415divide(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Aa2)),aa(num,word(A),numeral_numeral(word(A)),Ba)) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(int,aa(int,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),Aa2))),aa(int,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),Ba)))) ) ).

% sdiv_word_minus_numeral_numeral
tff(fact_5709_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_5710_signed__div__arith,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] : aa(word(A),int,ring_1_signed(A,int),signed7115095781618012415divide(word(A),Aa2,Ba)) = aa(int,int,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),Aa2),aa(word(A),int,ring_1_signed(A,int),Ba))) ) ).

% signed_div_arith
tff(fact_5711_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,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),M)),Na)
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A))) ),
                aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),
                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na) )) ) ) ).

% bit_signed_drop_bit_iff
tff(fact_5712_signed__drop__bit__beyond,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] :
          ( aa(nat,$o,aa(nat,fun(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_5713_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,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I4)
                & aa(int,$o,aa(int,fun(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_5714_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,$o,P,word_int_case(A,B,F2,X))
        <=> ~ ? [N2: int] :
                ( ( X = aa(int,word(B),ring_1_of_int(word(B)),N2) )
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),N2)
                & aa(int,$o,aa(int,fun(int,$o),ord_less(int),N2),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(B,type2(B))))
                & ~ aa(A,$o,P,aa(int,A,F2,N2)) ) ) ) ).

% word_int_split_asm
tff(fact_5715_smod__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: num] : signed6721504322012087516modulo(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Aa2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Ba))) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(int,aa(int,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),Aa2))),aa(int,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),Ba))))) ) ).

% smod_word_minus_numeral_minus_numeral
tff(fact_5716_smod__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: num] : signed6721504322012087516modulo(word(A),aa(num,word(A),numeral_numeral(word(A)),Aa2),aa(num,word(A),numeral_numeral(word(A)),Ba)) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(int,aa(int,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),Aa2)),aa(int,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),Ba)))) ) ).

% smod_word_numeral_numeral
tff(fact_5717_smod__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: num] : signed6721504322012087516modulo(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Aa2)),aa(num,word(A),numeral_numeral(word(A)),Ba)) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(int,aa(int,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),Aa2))),aa(int,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),Ba)))) ) ).

% smod_word_minus_numeral_numeral
tff(fact_5718_smod__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: num] : signed6721504322012087516modulo(word(A),aa(num,word(A),numeral_numeral(word(A)),Aa2),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Ba))) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(int,aa(int,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),Aa2)),aa(int,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),Ba))))) ) ).

% smod_word_numeral_minus_numeral
tff(fact_5719_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_5720_sdiv__smod__id,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),signed7115095781618012415divide(word(A),Aa2,Ba)),Ba)),signed6721504322012087516modulo(word(A),Aa2,Ba)) = Aa2 ) ).

% sdiv_smod_id
tff(fact_5721_signed__mod__arith,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] : aa(word(A),int,ring_1_signed(A,int),signed6721504322012087516modulo(word(A),Aa2,Ba)) = aa(int,int,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),Aa2),aa(word(A),int,ring_1_signed(A,int),Ba))) ) ).

% signed_mod_arith
tff(fact_5722_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_5723_uint__word__ariths_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),word_pred(A,Aa2)) = 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),Aa2)),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_5724_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,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(B,type2(B))),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),aa(word(B),word(B),aa(nat,fun(word(B),word(B)),bit_se4197421643247451524op_bit(word(B)),aa(nat,nat,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_5725_uint__word__ariths_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),word_succ(A,Aa2)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Aa2)),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_5726_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_5727_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),aa(word(A),fun(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_5728_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_5729_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),aa(word(A),fun(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_5730_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_5731_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_5732_word__pred__succ,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A)] : word_pred(A,word_succ(A,Aa2)) = Aa2 ) ).

% word_pred_succ
tff(fact_5733_word__succ__pred,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A)] : word_succ(A,word_pred(A,Aa2)) = Aa2 ) ).

% word_succ_pred
tff(fact_5734_Abs__fnat__hom__Suc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: nat] : word_succ(A,aa(nat,word(A),semiring_1_of_nat(word(A)),Aa2)) = aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,suc,Aa2)) ) ).

% Abs_fnat_hom_Suc
tff(fact_5735_word__succ__p1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A)] : word_succ(A,Aa2) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Aa2),one_one(word(A))) ) ).

% word_succ_p1
tff(fact_5736_word__m1__ge,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),word_pred(A,zero_zero(word(A)))) ) ).

% word_m1_ge
tff(fact_5737_mask__eqs_I11_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: 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)),Aa2),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,Aa2)),bit_se2239418461657761734s_mask(word(A),Na)) ) ).

% mask_eqs(11)
tff(fact_5738_word__not__simps_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A)] : ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),word_pred(A,zero_zero(word(A)))),Y) ) ).

% word_not_simps(2)
tff(fact_5739_word__pred__m1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A)] : word_pred(A,Aa2) = aa(word(A),word(A),minus_minus(word(A),Aa2),one_one(word(A))) ) ).

% word_pred_m1
tff(fact_5740_mask__eqs_I12_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: 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)),Aa2),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,Aa2)),bit_se2239418461657761734s_mask(word(A),Na)) ) ).

% mask_eqs(12)
tff(fact_5741_word__mult__succ,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),word_succ(A,Aa2)),Ba) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Ba),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),Aa2),Ba)) ) ).

% word_mult_succ
tff(fact_5742_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,aa(word(B),nat,size_size(word(B)),W)),W) ) ).

% ucast_slice1
tff(fact_5743_wi__hom__succ,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: int] : word_succ(A,aa(int,word(A),ring_1_of_int(word(A)),Aa2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),plus_plus(int),Aa2),one_one(int))) ) ).

% wi_hom_succ
tff(fact_5744_word__arith__nat__Suc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A)] : word_succ(A,Aa2) = aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),Aa2))) ) ).

% word_arith_nat_Suc
tff(fact_5745_wi__hom__pred,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: int] : word_pred(A,aa(int,word(A),ring_1_of_int(word(A)),Aa2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,minus_minus(int,Aa2),one_one(int))) ) ).

% wi_hom_pred
tff(fact_5746_word__succ__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A)] : word_succ(A,Aa2) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Aa2)),one_one(int))) ) ).

% word_succ_alt
tff(fact_5747_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_5748_word__pred__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A)] : word_pred(A,Aa2) = 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),Aa2)),one_one(int))) ) ).

% word_pred_alt
tff(fact_5749_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_5750_uint__word__arith__bintrs_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),word_succ(A,Aa2)) = aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Aa2)),one_one(int))) ) ).

% uint_word_arith_bintrs(5)
tff(fact_5751_uint__word__arith__bintrs_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),word_pred(A,Aa2)) = aa(int,int,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),Aa2)),one_one(int))) ) ).

% uint_word_arith_bintrs(6)
tff(fact_5752_sint__word__ariths_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A)] : aa(word(A),int,ring_1_signed(A,int),word_succ(A,Aa2)) = aa(int,int,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),plus_plus(int),aa(word(A),int,ring_1_signed(A,int),Aa2)),one_one(int))) ) ).

% sint_word_ariths(5)
tff(fact_5753_sint__word__ariths_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A)] : aa(word(A),int,ring_1_signed(A,int),word_pred(A,Aa2)) = aa(int,int,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),Aa2)),one_one(int))) ) ).

% sint_word_ariths(6)
tff(fact_5754_unat__word__ariths_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),word_succ(A,Aa2)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),Aa2))),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_5755_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,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),type_len0_len_of(A,type2(A)))),aa(list(word(A)),nat,size_size(list(word(A))),Ws))
            & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(nat,word(A),nth(word(A),Ws),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),type_len0_len_of(A,type2(A))))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Na),type_len0_len_of(A,type2(A)))) ) ) ) ).

% bit_horner_sum_uint_exp_iff
tff(fact_5756_of__nat__less__two__pow__div__set,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
         => ( collect(word(A),aa(nat,fun(word(A),$o),aTP_Lamp_pa(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_pb(nat,fun(nat,fun(nat,$o)),Na),M))) ) ) ) ).

% of_nat_less_two_pow_div_set
tff(fact_5757_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,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),Na)),K)),aa(nat,int,semiring_1_of_nat(int),type_len0_len_of(A,type2(A)))))) ) ) ) ).

% bit_word_roti_iff
tff(fact_5758_image__ident,axiom,
    ! [A: $tType,Y5: set(A)] : aa(set(A),set(A),image(A,A,aTP_Lamp_pc(A,A)),Y5) = Y5 ).

% image_ident
tff(fact_5759_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_5760_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_5761_image__add__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Sa: set(A)] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A))),Sa) = Sa ) ).

% image_add_0
tff(fact_5762_nth__Cons__Suc,axiom,
    ! [A: $tType,X: A,Xs2: list(A),Na: nat] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs2)),aa(nat,nat,suc,Na)) = aa(nat,A,nth(A,Xs2),Na) ).

% nth_Cons_Suc
tff(fact_5763_length__nth__simps_I4_J,axiom,
    ! [A: $tType,X: A,Xs2: list(A),Na: nat] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs2)),aa(nat,nat,suc,Na)) = aa(nat,A,nth(A,Xs2),Na) ).

% length_nth_simps(4)
tff(fact_5764_length__nth__simps_I3_J,axiom,
    ! [A: $tType,X: A,Xs2: list(A)] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs2)),zero_zero(nat)) = X ).

% length_nth_simps(3)
tff(fact_5765_nth__Cons__0,axiom,
    ! [A: $tType,X: A,Xs2: list(A)] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs2)),zero_zero(nat)) = X ).

% nth_Cons_0
tff(fact_5766_image__add__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I: A,J: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),K)),set_or1337092689740270186AtMost(A,I,J)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)) ) ).

% image_add_atLeastAtMost
tff(fact_5767_image__add__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I: A,J: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),K)),set_or7035219750837199246ssThan(A,I,J)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)) ) ).

% image_add_atLeastLessThan
tff(fact_5768_image__add__atMost,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [C2: A,Aa2: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),C2)),set_ord_atMost(A,Aa2)) = set_ord_atMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),Aa2)) ) ).

% image_add_atMost
tff(fact_5769_bij__betw__of__nat,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N3: set(nat),A2: set(A)] :
          ( bij_betw(nat,A,semiring_1_of_nat(A),N3,A2)
        <=> ( aa(set(nat),set(A),image(nat,A,semiring_1_of_nat(A)),N3) = A2 ) ) ) ).

% bij_betw_of_nat
tff(fact_5770_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_pd(A,fun(A,A),K)),set_or1337092689740270186AtMost(A,I,J)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)) ) ).

% image_add_atLeastAtMost'
tff(fact_5771_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_pd(A,fun(A,A),K)),set_or7035219750837199246ssThan(A,I,J)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)) ) ).

% image_add_atLeastLessThan'
tff(fact_5772_image__minus__const__atLeastAtMost_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [D2: A,Aa2: A,Ba: A] : aa(set(A),set(A),image(A,A,aTP_Lamp_pe(A,fun(A,A),D2)),set_or1337092689740270186AtMost(A,Aa2,Ba)) = set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,Aa2),D2),aa(A,A,minus_minus(A,Ba),D2)) ) ).

% image_minus_const_atLeastAtMost'
tff(fact_5773_image__mult__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D2: A,Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D2)
         => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),times_times(A),D2)),set_or1337092689740270186AtMost(A,Aa2,Ba)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),D2),Aa2),aa(A,A,aa(A,fun(A,A),times_times(A),D2),Ba)) ) ) ) ).

% image_mult_atLeastAtMost
tff(fact_5774_nth__Cons__numeral,axiom,
    ! [A: $tType,X: A,Xs2: list(A),V: num] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs2)),aa(num,nat,numeral_numeral(nat),V)) = aa(nat,A,nth(A,Xs2),aa(nat,nat,minus_minus(nat,aa(num,nat,numeral_numeral(nat),V)),one_one(nat))) ).

% nth_Cons_numeral
tff(fact_5775_image__divide__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D2: A,Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D2)
         => ( aa(set(A),set(A),image(A,A,aTP_Lamp_pf(A,fun(A,A),D2)),set_or1337092689740270186AtMost(A,Aa2,Ba)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),D2),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),D2)) ) ) ) ).

% image_divide_atLeastAtMost
tff(fact_5776_nth__Cons__pos,axiom,
    ! [A: $tType,Na: nat,X: A,Xs2: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs2)),Na) = aa(nat,A,nth(A,Xs2),aa(nat,nat,minus_minus(nat,Na),one_one(nat))) ) ) ).

% nth_Cons_pos
tff(fact_5777_nth__image__indices,axiom,
    ! [A: $tType,L: list(A)] : aa(set(nat),set(A),image(nat,A,nth(A,L)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),L))) = aa(list(A),set(A),set2(A),L) ).

% nth_image_indices
tff(fact_5778_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,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(B),nat,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_5779_set__image__eq__pointwiseI,axiom,
    ! [B: $tType,A: $tType,L: list(A),L3: list(A),F2: fun(A,B)] :
      ( ( aa(list(A),nat,size_size(list(A)),L) = aa(list(A),nat,size_size(list(A)),L3) )
     => ( ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L))
           => ( aa(A,B,F2,aa(nat,A,nth(A,L),I2)) = aa(A,B,F2,aa(nat,A,nth(A,L3),I2)) ) )
       => ( 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),L3)) ) ) ) ).

% set_image_eq_pointwiseI
tff(fact_5780_pigeonhole__infinite,axiom,
    ! [B: $tType,A: $tType,A2: set(A),F2: fun(A,B)] :
      ( ~ aa(set(A),$o,finite_finite2(A),A2)
     => ( aa(set(B),$o,finite_finite2(B),aa(set(A),set(B),image(A,B,F2),A2))
       => ? [X2: A] :
            ( member(A,X2,A2)
            & ~ aa(set(A),$o,finite_finite2(A),collect(A,aa(A,fun(A,$o),aa(fun(A,B),fun(A,fun(A,$o)),aTP_Lamp_pg(set(A),fun(fun(A,B),fun(A,fun(A,$o))),A2),F2),X2))) ) ) ) ).

% pigeonhole_infinite
tff(fact_5781_imageE,axiom,
    ! [A: $tType,B: $tType,Ba: A,F2: fun(B,A),A2: set(B)] :
      ( member(A,Ba,aa(set(B),set(A),image(B,A,F2),A2))
     => ~ ! [X2: B] :
            ( ( Ba = aa(B,A,F2,X2) )
           => ~ member(B,X2,A2) ) ) ).

% imageE
tff(fact_5782_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_ph(fun(B,A),fun(fun(C,B),fun(C,A)),F2),G)),A2) ).

% image_image
tff(fact_5783_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_pi(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_pj(fun(B,A),fun(set(B),fun(fun(A,$o),fun(B,$o))),F2),A2),P))) ).

% Compr_image_eq
tff(fact_5784_word__roti__add,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: int,Na: int,W: word(A)] : word_roti(A,aa(int,int,aa(int,fun(int,int),plus_plus(int),M),Na),W) = word_roti(A,M,word_roti(A,Na,W)) ) ).

% word_roti_add
tff(fact_5785_nth__equalityI,axiom,
    ! [A: $tType,Xs2: list(A),Ysb: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(A),nat,size_size(list(A)),Ysb) )
     => ( ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs2))
           => ( aa(nat,A,nth(A,Xs2),I2) = aa(nat,A,nth(A,Ysb),I2) ) )
       => ( Xs2 = Ysb ) ) ) ).

% nth_equalityI
tff(fact_5786_Skolem__list__nth,axiom,
    ! [A: $tType,K: nat,P: fun(nat,fun(A,$o))] :
      ( ! [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),K)
         => ? [X_1: A] : aa(A,$o,aa(nat,fun(A,$o),P,I4),X_1) )
    <=> ? [Xs3: list(A)] :
          ( ( aa(list(A),nat,size_size(list(A)),Xs3) = K )
          & ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),K)
             => aa(A,$o,aa(nat,fun(A,$o),P,I4),aa(nat,A,nth(A,Xs3),I4)) ) ) ) ).

% Skolem_list_nth
tff(fact_5787_list__eq__iff__nth__eq,axiom,
    ! [A: $tType,Xs2: list(A),Ysb: list(A)] :
      ( ( Xs2 = Ysb )
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(A),nat,size_size(list(A)),Ysb) )
        & ! [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs2))
           => ( aa(nat,A,nth(A,Xs2),I4) = aa(nat,A,nth(A,Ysb),I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_5788_obtain__list__from__elements,axiom,
    ! [A: $tType,Na: nat,P: fun(A,fun(nat,$o))] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Na)
         => ? [Li: A] : aa(nat,$o,aa(A,fun(nat,$o),P,Li),I2) )
     => ~ ! [L2: list(A)] :
            ( ( aa(list(A),nat,size_size(list(A)),L2) = Na )
           => ~ ! [I3: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),Na)
                 => aa(nat,$o,aa(A,fun(nat,$o),P,aa(nat,A,nth(A,L2),I3)),I3) ) ) ) ).

% obtain_list_from_elements
tff(fact_5789_finite__conv__nat__seg__image,axiom,
    ! [A: $tType,A2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A2)
    <=> ? [N2: nat,F5: fun(nat,A)] : A2 = aa(set(nat),set(A),image(nat,A,F5),collect(nat,aTP_Lamp_gx(nat,fun(nat,$o),N2))) ) ).

% finite_conv_nat_seg_image
tff(fact_5790_nat__seg__image__imp__finite,axiom,
    ! [A: $tType,A2: set(A),F2: fun(nat,A),Na: nat] :
      ( ( A2 = aa(set(nat),set(A),image(nat,A,F2),collect(nat,aTP_Lamp_gx(nat,fun(nat,$o),Na))) )
     => aa(set(A),$o,finite_finite2(A),A2) ) ).

% nat_seg_image_imp_finite
tff(fact_5791_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_pk(B,fun(A,B),C2)),A2) = aa(set(B),set(B),insert(B,C2),bot_bot(set(B))) ) ) ).

% image_constant
tff(fact_5792_image__constant__conv,axiom,
    ! [B: $tType,A: $tType,C2: A,A2: set(B)] :
      aa(set(B),set(A),image(B,A,aa(A,fun(B,A),aTP_Lamp_pl(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_5793_sum_Oimage__gen,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Sa: set(A),H: fun(A,B),G: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),Sa)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),Sa) = 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_pn(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),Sa),H),G)),aa(set(A),set(C),image(A,C,G),Sa)) ) ) ) ).

% sum.image_gen
tff(fact_5794_prod_Oimage__gen,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Sa: set(A),H: fun(A,B),G: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),Sa)
         => ( groups7121269368397514597t_prod(A,B,H,Sa) = groups7121269368397514597t_prod(C,B,aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_po(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),Sa),H),G),aa(set(A),set(C),image(A,C,G),Sa)) ) ) ) ).

% prod.image_gen
tff(fact_5795_VEBT__internal_Ointhall,axiom,
    ! [A: $tType,Xs2: list(A),P: fun(A,$o),Na: nat] :
      ( ! [X2: A] :
          ( member(A,X2,aa(list(A),set(A),set2(A),Xs2))
         => aa(A,$o,P,X2) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs2))
       => aa(A,$o,P,aa(nat,A,nth(A,Xs2),Na)) ) ) ).

% VEBT_internal.inthall
tff(fact_5796_all__set__conv__all__nth,axiom,
    ! [A: $tType,Xs2: list(A),P: fun(A,$o)] :
      ( ! [X6: A] :
          ( member(A,X6,aa(list(A),set(A),set2(A),Xs2))
         => aa(A,$o,P,X6) )
    <=> ! [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs2))
         => aa(A,$o,P,aa(nat,A,nth(A,Xs2),I4)) ) ) ).

% all_set_conv_all_nth
tff(fact_5797_all__nth__imp__all__set,axiom,
    ! [A: $tType,Xs2: list(A),P: fun(A,$o),X: A] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs2))
         => aa(A,$o,P,aa(nat,A,nth(A,Xs2),I2)) )
     => ( member(A,X,aa(list(A),set(A),set2(A),Xs2))
       => aa(A,$o,P,X) ) ) ).

% all_nth_imp_all_set
tff(fact_5798_all__set__conv__nth,axiom,
    ! [A: $tType,L: list(A),P: fun(A,$o)] :
      ( ! [X6: A] :
          ( member(A,X6,aa(list(A),set(A),set2(A),L))
         => aa(A,$o,P,X6) )
    <=> ! [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),L))
         => aa(A,$o,P,aa(nat,A,nth(A,L),I4)) ) ) ).

% all_set_conv_nth
tff(fact_5799_in__set__conv__nth,axiom,
    ! [A: $tType,X: A,Xs2: list(A)] :
      ( member(A,X,aa(list(A),set(A),set2(A),Xs2))
    <=> ? [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs2))
          & ( aa(nat,A,nth(A,Xs2),I4) = X ) ) ) ).

% in_set_conv_nth
tff(fact_5800_list__ball__nth,axiom,
    ! [A: $tType,Na: nat,Xs2: list(A),P: fun(A,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs2))
     => ( ! [X2: A] :
            ( member(A,X2,aa(list(A),set(A),set2(A),Xs2))
           => aa(A,$o,P,X2) )
       => aa(A,$o,P,aa(nat,A,nth(A,Xs2),Na)) ) ) ).

% list_ball_nth
tff(fact_5801_nth__mem,axiom,
    ! [A: $tType,Na: nat,Xs2: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs2))
     => member(A,aa(nat,A,nth(A,Xs2),Na),aa(list(A),set(A),set2(A),Xs2)) ) ).

% nth_mem
tff(fact_5802_sum_Ogroup,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [Sa: set(A),T4: set(B),G: fun(A,B),H: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),Sa)
         => ( aa(set(B),$o,finite_finite2(B),T4)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,G),Sa)),T4)
             => ( 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_pq(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),Sa),G),H)),T4) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,H),Sa) ) ) ) ) ) ).

% sum.group
tff(fact_5803_prod_Ogroup,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [Sa: set(A),T4: set(B),G: fun(A,B),H: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),Sa)
         => ( aa(set(B),$o,finite_finite2(B),T4)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,G),Sa)),T4)
             => ( groups7121269368397514597t_prod(B,C,aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_pr(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),Sa),G),H),T4) = groups7121269368397514597t_prod(A,C,H,Sa) ) ) ) ) ) ).

% prod.group
tff(fact_5804_nth__Cons_H,axiom,
    ! [A: $tType,X: A,Xs2: list(A),Na: nat] :
      aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs2)),Na) = $ite(Na = zero_zero(nat),X,aa(nat,A,nth(A,Xs2),aa(nat,nat,minus_minus(nat,Na),one_one(nat)))) ).

% nth_Cons'
tff(fact_5805_nth__non__equal__first__eq,axiom,
    ! [A: $tType,X: A,Y: A,Xs2: list(A),Na: nat] :
      ( ( X != Y )
     => ( ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs2)),Na) = Y )
      <=> ( ( aa(nat,A,nth(A,Xs2),aa(nat,nat,minus_minus(nat,Na),one_one(nat))) = Y )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_5806_nth__equal__first__eq,axiom,
    ! [A: $tType,X: A,Xs2: list(A),Na: nat] :
      ( ~ member(A,X,aa(list(A),set(A),set2(A),Xs2))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(list(A),nat,size_size(list(A)),Xs2))
       => ( ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs2)),Na) = X )
        <=> ( Na = zero_zero(nat) ) ) ) ) ).

% nth_equal_first_eq
tff(fact_5807_nth__rotate1,axiom,
    ! [A: $tType,Na: nat,Xs2: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs2))
     => ( aa(nat,A,nth(A,rotate1(A,Xs2)),Na) = aa(nat,A,nth(A,Xs2),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,Na)),aa(list(A),nat,size_size(list(A)),Xs2))) ) ) ).

% nth_rotate1
tff(fact_5808_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),aa(word(A),nat,size_size(word(A)),W))),W) ) ).

% word_roti_conv_mod'
tff(fact_5809_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,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
            set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),X),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Y)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),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_5810_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_ps(A,fun(A,A),C2)),set_or1337092689740270186AtMost(A,X,Y)) = $ite(
            aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),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_5811_image__affinity__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C2: A,Aa2: A,Ba: A] :
          aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_pt(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,Aa2,Ba)) = $ite(
            set_or1337092689740270186AtMost(A,Aa2,Ba) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),M),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),Aa2)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),Ba)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),Ba)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),Aa2)),C2))) ) ) ).

% image_affinity_atLeastAtMost
tff(fact_5812_image__affinity__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C2: A,Aa2: A,Ba: A] :
          aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_pu(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,Aa2,Ba)) = $ite(
            set_or1337092689740270186AtMost(A,Aa2,Ba) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),M),set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),Aa2)),C2),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),Ba)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),Ba)),C2),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),Aa2)),C2))) ) ) ).

% image_affinity_atLeastAtMost_diff
tff(fact_5813_image__affinity__atLeastAtMost__div,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C2: A,Aa2: A,Ba: A] :
          aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_pv(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,Aa2,Ba)) = $ite(
            set_or1337092689740270186AtMost(A,Aa2,Ba) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),M),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),M)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),M)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),M)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),M)),C2))) ) ) ).

% image_affinity_atLeastAtMost_div
tff(fact_5814_image__affinity__atLeastAtMost__div__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C2: A,Aa2: A,Ba: A] :
          aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_pw(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,Aa2,Ba)) = $ite(
            set_or1337092689740270186AtMost(A,Aa2,Ba) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),M),set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),M)),C2),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),M)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),M)),C2),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Aa2),M)),C2))) ) ) ).

% image_affinity_atLeastAtMost_div_diff
tff(fact_5815_slice__nth,axiom,
    ! [A: $tType,From: nat,To: nat,Xs2: list(A),I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),From),To)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),To),aa(list(A),nat,size_size(list(A)),Xs2))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,minus_minus(nat,To),From))
         => ( aa(nat,A,nth(A,slice(A,From,To,Xs2)),I) = aa(nat,A,nth(A,Xs2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),From),I)) ) ) ) ) ).

% slice_nth
tff(fact_5816_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_5817_horner__sum__eq__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F2: fun(B,A),Aa2: A,Xs2: list(B)] : groups4207007520872428315er_sum(B,A,F2,Aa2,Xs2) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_px(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F2),Aa2),Xs2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs2))) ) ).

% horner_sum_eq_sum
tff(fact_5818_bij__betw__add,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [Aa2: A,A2: set(A),B2: set(A)] :
          ( bij_betw(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),A2,B2)
        <=> ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),Aa2)),A2) = B2 ) ) ) ).

% bij_betw_add
tff(fact_5819_product__nth,axiom,
    ! [A: $tType,B: $tType,Na: nat,Xs2: list(A),Ysb: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(A),nat,size_size(list(A)),Xs2)),aa(list(B),nat,size_size(list(B)),Ysb)))
     => ( aa(nat,product_prod(A,B),nth(product_prod(A,B),product(A,B,Xs2,Ysb)),Na) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xs2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(list(B),nat,size_size(list(B)),Ysb)))),aa(nat,B,nth(B,Ysb),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Na),aa(list(B),nat,size_size(list(B)),Ysb)))) ) ) ).

% product_nth
tff(fact_5820_sless__eq__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Aa2))),aa(num,word(A),numeral_numeral(word(A)),Ba))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,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),Aa2)))),aa(int,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),Ba))) ) ) ).

% sless_eq_word_minus_numeral_numeral
tff(fact_5821_bij__betw__Suc,axiom,
    ! [M2: set(nat),N3: set(nat)] :
      ( bij_betw(nat,nat,suc,M2,N3)
    <=> ( aa(set(nat),set(nat),image(nat,nat,suc),M2) = N3 ) ) ).

% bij_betw_Suc
tff(fact_5822_signed_Oorder__refl,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),X),X) ) ).

% signed.order_refl
tff(fact_5823_signed_Odual__order_Orefl,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Aa2),Aa2) ) ).

% signed.dual_order.refl
tff(fact_5824_pair__imageI,axiom,
    ! [C: $tType,B: $tType,A: $tType,Aa2: A,Ba: 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),Aa2),Ba),A2)
     => member(C,aa(B,C,aa(A,fun(B,C),F2,Aa2),Ba),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_5825_length__product,axiom,
    ! [A: $tType,B: $tType,Xs2: list(A),Ysb: list(B)] : aa(list(product_prod(A,B)),nat,size_size(list(product_prod(A,B))),product(A,B,Xs2,Ysb)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(A),nat,size_size(list(A)),Xs2)),aa(list(B),nat,size_size(list(B)),Ysb)) ).

% length_product
tff(fact_5826_word__sle__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(num,word(A),numeral_numeral(word(A)),Aa2)),aa(num,word(A),numeral_numeral(word(A)),Ba))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,ring_1_signed(A,int),aa(num,word(A),numeral_numeral(word(A)),Aa2))),aa(word(A),int,ring_1_signed(A,int),aa(num,word(A),numeral_numeral(word(A)),Ba))) ) ) ).

% word_sle_no
tff(fact_5827_nth__upto,axiom,
    ! [I: int,K: nat,J: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),I),aa(nat,int,semiring_1_of_nat(int),K))),J)
     => ( aa(nat,int,nth(int,upto(I,J)),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),I),aa(nat,int,semiring_1_of_nat(int),K)) ) ) ).

% nth_upto
tff(fact_5828_extra__sle__sless__unfolds_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(num,word(A),numeral_numeral(word(A)),Na)),one_one(word(A)))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,ring_1_signed(A,int),aa(num,word(A),numeral_numeral(word(A)),Na))),aa(word(A),int,ring_1_signed(A,int),one_one(word(A)))) ) ) ).

% extra_sle_sless_unfolds(6)
tff(fact_5829_extra__sle__sless__unfolds_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),one_one(word(A))),aa(num,word(A),numeral_numeral(word(A)),Na))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,ring_1_signed(A,int),one_one(word(A)))),aa(word(A),int,ring_1_signed(A,int),aa(num,word(A),numeral_numeral(word(A)),Na))) ) ) ).

% extra_sle_sless_unfolds(4)
tff(fact_5830_extra__sle__sless__unfolds_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(num,word(A),numeral_numeral(word(A)),Na)),zero_zero(word(A)))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,ring_1_signed(A,int),aa(num,word(A),numeral_numeral(word(A)),Na))),aa(word(A),int,ring_1_signed(A,int),zero_zero(word(A)))) ) ) ).

% extra_sle_sless_unfolds(5)
tff(fact_5831_extra__sle__sless__unfolds_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),zero_zero(word(A))),aa(num,word(A),numeral_numeral(word(A)),Na))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,ring_1_signed(A,int),zero_zero(word(A)))),aa(word(A),int,ring_1_signed(A,int),aa(num,word(A),numeral_numeral(word(A)),Na))) ) ) ).

% extra_sle_sless_unfolds(2)
tff(fact_5832_extra__sle__sless__unfolds_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),one_one(word(A))),zero_zero(word(A)))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,ring_1_signed(A,int),one_one(word(A)))),aa(word(A),int,ring_1_signed(A,int),zero_zero(word(A)))) ) ) ).

% extra_sle_sless_unfolds(3)
tff(fact_5833_extra__sle__sless__unfolds_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),zero_zero(word(A))),one_one(word(A)))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,ring_1_signed(A,int),zero_zero(word(A)))),aa(word(A),int,ring_1_signed(A,int),one_one(word(A)))) ) ) ).

% extra_sle_sless_unfolds(1)
tff(fact_5834_sless__eq__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(num,word(A),numeral_numeral(word(A)),Aa2)),aa(num,word(A),numeral_numeral(word(A)),Ba))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,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),Aa2))),aa(int,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),Ba))) ) ) ).

% sless_eq_word_numeral_numeral
tff(fact_5835_sless__eq__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Aa2))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Ba)))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,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),Aa2)))),aa(int,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),Ba)))) ) ) ).

% sless_eq_word_minus_numeral_minus_numeral
tff(fact_5836_sless__eq__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(num,word(A),numeral_numeral(word(A)),Aa2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Ba)))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,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),Aa2))),aa(int,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),Ba)))) ) ) ).

% sless_eq_word_numeral_minus_numeral
tff(fact_5837_signed_Olift__Suc__antimono__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F2: fun(nat,word(A)),Na: nat,N6: nat] :
          ( ! [N: nat] : aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(nat,word(A),F2,aa(nat,nat,suc,N))),aa(nat,word(A),F2,N))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N6)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(nat,word(A),F2,N6)),aa(nat,word(A),F2,Na)) ) ) ) ).

% signed.lift_Suc_antimono_le
tff(fact_5838_signed_Olift__Suc__mono__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F2: fun(nat,word(A)),Na: nat,N6: nat] :
          ( ! [N: nat] : aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(nat,word(A),F2,N)),aa(nat,word(A),F2,aa(nat,nat,suc,N)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N6)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(nat,word(A),F2,Na)),aa(nat,word(A),F2,N6)) ) ) ) ).

% signed.lift_Suc_mono_le
tff(fact_5839_product_Osimps_I1_J,axiom,
    ! [B: $tType,A: $tType,Uu2: list(B)] : product(A,B,nil(A),Uu2) = nil(product_prod(A,B)) ).

% product.simps(1)
tff(fact_5840_signed_Olinear,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),X),Y)
          | aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Y),X) ) ) ).

% signed.linear
tff(fact_5841_signed_Onle__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] :
          ( ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Aa2),Ba)
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Ba),Aa2)
            & ( Ba != Aa2 ) ) ) ) ).

% signed.nle_le
tff(fact_5842_signed_Oeq__refl,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ( X = Y )
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),X),Y) ) ) ).

% signed.eq_refl
tff(fact_5843_signed_Ole__cases,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),X),Y)
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Y),X) ) ) ).

% signed.le_cases
tff(fact_5844_signed_Ole__cases3,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A),Z: word(A)] :
          ( ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),X),Y)
           => ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Y),Z) )
         => ( ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Y),X)
             => ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),X),Z) )
           => ( ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),X),Z)
               => ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Z),Y) )
             => ( ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Z),Y)
                 => ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Y),X) )
               => ( ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Y),Z)
                   => ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Z),X) )
                 => ~ ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Z),X)
                     => ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),X),Y) ) ) ) ) ) ) ) ).

% signed.le_cases3
tff(fact_5845_signed_Oorder_Otrans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A),C2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Aa2),Ba)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Ba),C2)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Aa2),C2) ) ) ) ).

% signed.order.trans
tff(fact_5846_signed_Oorder__trans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),X),Y)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Y),Z)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),X),Z) ) ) ) ).

% signed.order_trans
tff(fact_5847_signed_Oantisym__conv,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Y),X)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),X),Y)
          <=> ( X = Y ) ) ) ) ).

% signed.antisym_conv
tff(fact_5848_signed_Oorder_Oeq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] :
          ( ( Aa2 = Ba )
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Aa2),Ba)
            & aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Ba),Aa2) ) ) ) ).

% signed.order.eq_iff
tff(fact_5849_signed_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ( X = Y )
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),X),Y)
            & aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Y),X) ) ) ) ).

% signed.order_eq_iff
tff(fact_5850_signed_Olinorder__wlog,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(word(A),fun(word(A),$o)),Aa2: word(A),Ba: word(A)] :
          ( ! [A3: word(A),B3: word(A)] :
              ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),A3),B3)
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),P,A3),B3) )
         => ( ! [A3: word(A),B3: word(A)] :
                ( aa(word(A),$o,aa(word(A),fun(word(A),$o),P,B3),A3)
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),P,A3),B3) )
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),P,Aa2),Ba) ) ) ) ).

% signed.linorder_wlog
tff(fact_5851_signed_Oorder_Oantisym,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Aa2),Ba)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Ba),Aa2)
           => ( Aa2 = Ba ) ) ) ) ).

% signed.order.antisym
tff(fact_5852_signed_Oorder__antisym,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),X),Y)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Y),X)
           => ( X = Y ) ) ) ) ).

% signed.order_antisym
tff(fact_5853_signed_Oord__eq__le__trans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A),C2: word(A)] :
          ( ( Aa2 = Ba )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Ba),C2)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Aa2),C2) ) ) ) ).

% signed.ord_eq_le_trans
tff(fact_5854_signed_Oord__le__eq__trans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A),C2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Aa2),Ba)
         => ( ( Ba = C2 )
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Aa2),C2) ) ) ) ).

% signed.ord_le_eq_trans
tff(fact_5855_signed_Odual__order_Otrans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Ba: word(A),Aa2: word(A),C2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Ba),Aa2)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),C2),Ba)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),C2),Aa2) ) ) ) ).

% signed.dual_order.trans
tff(fact_5856_signed_Odual__order_Oeq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] :
          ( ( Aa2 = Ba )
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Ba),Aa2)
            & aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Aa2),Ba) ) ) ) ).

% signed.dual_order.eq_iff
tff(fact_5857_signed_Odual__order_Oantisym,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Ba: word(A),Aa2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Ba),Aa2)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Aa2),Ba)
           => ( Aa2 = Ba ) ) ) ) ).

% signed.dual_order.antisym
tff(fact_5858_signed_Ofinite__has__minimal,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: set(word(A))] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A2)
         => ( ( A2 != bot_bot(set(word(A))) )
           => ? [X2: word(A)] :
                ( member(word(A),X2,A2)
                & ! [Xa: word(A)] :
                    ( member(word(A),Xa,A2)
                   => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Xa),X2)
                     => ( X2 = Xa ) ) ) ) ) ) ) ).

% signed.finite_has_minimal
tff(fact_5859_signed_Ofinite__has__maximal,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: set(word(A))] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A2)
         => ( ( A2 != bot_bot(set(word(A))) )
           => ? [X2: word(A)] :
                ( member(word(A),X2,A2)
                & ! [Xa: word(A)] :
                    ( member(word(A),Xa,A2)
                   => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),X2),Xa)
                     => ( X2 = Xa ) ) ) ) ) ) ) ).

% signed.finite_has_maximal
tff(fact_5860_signed_Ofinite__has__minimal2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: set(word(A)),Aa2: word(A)] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A2)
         => ( member(word(A),Aa2,A2)
           => ? [X2: word(A)] :
                ( member(word(A),X2,A2)
                & aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),X2),Aa2)
                & ! [Xa: word(A)] :
                    ( member(word(A),Xa,A2)
                   => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Xa),X2)
                     => ( X2 = Xa ) ) ) ) ) ) ) ).

% signed.finite_has_minimal2
tff(fact_5861_signed_Ofinite__has__maximal2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: set(word(A)),Aa2: word(A)] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A2)
         => ( member(word(A),Aa2,A2)
           => ? [X2: word(A)] :
                ( member(word(A),X2,A2)
                & aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Aa2),X2)
                & ! [Xa: word(A)] :
                    ( member(word(A),Xa,A2)
                   => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),X2),Xa)
                     => ( X2 = Xa ) ) ) ) ) ) ) ).

% signed.finite_has_maximal2
tff(fact_5862_word__sle__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Aa2),Ba)
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,ring_1_signed(A,int),Aa2)),aa(word(A),int,ring_1_signed(A,int),Ba)) ) ) ).

% word_sle_eq
tff(fact_5863_signed_Ofinite__ranking__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [Sa: set(A),P: fun(set(A),$o),F2: fun(A,word(B))] :
          ( aa(set(A),$o,finite_finite2(A),Sa)
         => ( aa(set(A),$o,P,bot_bot(set(A)))
           => ( ! [X2: A,S4: set(A)] :
                  ( aa(set(A),$o,finite_finite2(A),S4)
                 => ( ! [Y3: A] :
                        ( member(A,Y3,S4)
                       => aa(word(B),$o,aa(word(B),fun(word(B),$o),word_sle(B),aa(A,word(B),F2,Y3)),aa(A,word(B),F2,X2)) )
                   => ( aa(set(A),$o,P,S4)
                     => aa(set(A),$o,P,aa(set(A),set(A),insert(A,X2),S4)) ) ) )
             => aa(set(A),$o,P,Sa) ) ) ) ) ).

% signed.finite_ranking_induct
tff(fact_5864_image__int__atLeastAtMost,axiom,
    ! [Aa2: nat,Ba: nat] : aa(set(nat),set(int),image(nat,int,semiring_1_of_nat(int)),set_or1337092689740270186AtMost(nat,Aa2,Ba)) = set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),Aa2),aa(nat,int,semiring_1_of_nat(int),Ba)) ).

% image_int_atLeastAtMost
tff(fact_5865_image__int__atLeastLessThan,axiom,
    ! [Aa2: nat,Ba: nat] : aa(set(nat),set(int),image(nat,int,semiring_1_of_nat(int)),set_or7035219750837199246ssThan(nat,Aa2,Ba)) = set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),Aa2),aa(nat,int,semiring_1_of_nat(int),Ba)) ).

% image_int_atLeastLessThan
tff(fact_5866_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_5867_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_5868_subset__subseqs,axiom,
    ! [A: $tType,X5: set(A),Xs2: list(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X5),aa(list(A),set(A),set2(A),Xs2))
     => member(set(A),X5,aa(set(list(A)),set(set(A)),image(list(A),set(A),set2(A)),aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xs2)))) ) ).

% subset_subseqs
tff(fact_5869_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : aa(set(int),set(int),image(int,int,aTP_Lamp_py(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_5870_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),U)
     => ( set_or7035219750837199246ssThan(int,zero_zero(int),U) = aa(set(nat),set(int),image(nat,int,semiring_1_of_nat(int)),set_ord_lessThan(nat,aa(int,nat,nat2,U))) ) ) ).

% image_atLeastZeroLessThan_int
tff(fact_5871_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C2: nat,X: nat,Y: nat] :
      aa(set(nat),set(nat),image(nat,nat,aTP_Lamp_pz(nat,fun(nat,nat),C2)),set_or7035219750837199246ssThan(nat,X,Y)) = $ite(
        aa(nat,$o,aa(nat,fun(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,aa(nat,fun(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_5872_bit__horner__sum__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Bs2: list($o),Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),groups4207007520872428315er_sum($o,A,zero_neq_one_of_bool(A),aa(num,A,numeral_numeral(A),bit0(one2)),Bs2)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list($o),nat,size_size(list($o)),Bs2))
            & aa(nat,$o,nth($o,Bs2),Na) ) ) ) ).

% bit_horner_sum_bit_iff
tff(fact_5873_word__0__sle__from__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),X),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),zero_zero(word(A))),X) ) ) ).

% word_0_sle_from_less
tff(fact_5874_translation__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Aa2: A,S: set(A),T: set(A)] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),Aa2)),aa(set(A),set(A),minus_minus(set(A),S),T)) = aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),Aa2)),S)),aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),Aa2)),T)) ) ).

% translation_diff
tff(fact_5875_translation__Compl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Aa2: A,T: set(A)] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),Aa2)),aa(set(A),set(A),uminus_uminus(set(A)),T)) = aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),Aa2)),T)) ) ).

% translation_Compl
tff(fact_5876_translation__subtract__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Aa2: A,S: set(A),T: set(A)] : aa(set(A),set(A),image(A,A,aTP_Lamp_qa(A,fun(A,A),Aa2)),aa(set(A),set(A),minus_minus(set(A),S),T)) = aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),image(A,A,aTP_Lamp_qa(A,fun(A,A),Aa2)),S)),aa(set(A),set(A),image(A,A,aTP_Lamp_qa(A,fun(A,A),Aa2)),T)) ) ).

% translation_subtract_diff
tff(fact_5877_translation__subtract__Compl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Aa2: A,T: set(A)] : aa(set(A),set(A),image(A,A,aTP_Lamp_qa(A,fun(A,A),Aa2)),aa(set(A),set(A),uminus_uminus(set(A)),T)) = aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),image(A,A,aTP_Lamp_qa(A,fun(A,A),Aa2)),T)) ) ).

% translation_subtract_Compl
tff(fact_5878_sless__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Aa2))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Ba)))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,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),Aa2)))),aa(int,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),Ba)))) ) ) ).

% sless_word_minus_numeral_minus_numeral
tff(fact_5879_sless__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),aa(num,word(A),numeral_numeral(word(A)),Aa2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Ba)))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,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),Aa2))),aa(int,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),Ba)))) ) ) ).

% sless_word_numeral_minus_numeral
tff(fact_5880_sless__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Aa2))),aa(num,word(A),numeral_numeral(word(A)),Ba))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,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),Aa2)))),aa(int,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),Ba))) ) ) ).

% sless_word_minus_numeral_numeral
tff(fact_5881_word__sless__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),aa(num,word(A),numeral_numeral(word(A)),Aa2)),aa(num,word(A),numeral_numeral(word(A)),Ba))
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(num,word(A),numeral_numeral(word(A)),Aa2)),aa(num,word(A),numeral_numeral(word(A)),Ba))
            & ( aa(num,word(A),numeral_numeral(word(A)),Aa2) != aa(num,word(A),numeral_numeral(word(A)),Ba) ) ) ) ) ).

% word_sless_no
tff(fact_5882_sless__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: num,Ba: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),aa(num,word(A),numeral_numeral(word(A)),Aa2)),aa(num,word(A),numeral_numeral(word(A)),Ba))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,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),Aa2))),aa(int,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),Ba))) ) ) ).

% sless_word_numeral_numeral
tff(fact_5883_signed_OleD,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Y),X)
         => ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),X),Y) ) ) ).

% signed.leD
tff(fact_5884_signed_OleI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),X),Y)
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Y),X) ) ) ).

% signed.leI
tff(fact_5885_signed_Onot__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),X),Y)
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Y),X) ) ) ).

% signed.not_le
tff(fact_5886_signed_Ole__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),X),Y)
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),X),Y)
            | ( X = Y ) ) ) ) ).

% signed.le_less
tff(fact_5887_signed_Onless__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] :
          ( ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Aa2),Ba)
        <=> ( ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Aa2),Ba)
            | ( Aa2 = Ba ) ) ) ) ).

% signed.nless_le
tff(fact_5888_signed_Onot__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),X),Y)
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Y),X) ) ) ).

% signed.not_less
tff(fact_5889_signed_Oless__imp__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),X),Y)
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),X),Y) ) ) ).

% signed.less_imp_le
tff(fact_5890_signed_Ole__neq__trans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Aa2),Ba)
         => ( ( Aa2 != Ba )
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Aa2),Ba) ) ) ) ).

% signed.le_neq_trans
tff(fact_5891_signed_Oantisym__conv1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),X),Y)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),X),Y)
          <=> ( X = Y ) ) ) ) ).

% signed.antisym_conv1
tff(fact_5892_signed_Oantisym__conv2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),X),Y)
         => ( ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),X),Y)
          <=> ( X = Y ) ) ) ) ).

% signed.antisym_conv2
tff(fact_5893_signed_Ole__less__trans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),X),Y)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Y),Z)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),X),Z) ) ) ) ).

% signed.le_less_trans
tff(fact_5894_signed_Oless__le__trans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),X),Y)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Y),Z)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),X),Z) ) ) ) ).

% signed.less_le_trans
tff(fact_5895_signed_Ole__less__linear,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),X),Y)
          | aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Y),X) ) ) ).

% signed.le_less_linear
tff(fact_5896_signed_Oless__le__not__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),X),Y)
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),X),Y)
            & ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Y),X) ) ) ) ).

% signed.less_le_not_le
tff(fact_5897_signed_Onot__le__imp__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A)] :
          ( ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Y),X)
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),X),Y) ) ) ).

% signed.not_le_imp_less
tff(fact_5898_signed_Ole__imp__less__or__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),X),Y)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),X),Y)
            | ( X = Y ) ) ) ) ).

% signed.le_imp_less_or_eq
tff(fact_5899_signed_Oorder_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A),C2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Aa2),Ba)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Ba),C2)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Aa2),C2) ) ) ) ).

% signed.order.strict_trans1
tff(fact_5900_signed_Oorder_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A),C2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Aa2),Ba)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Ba),C2)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Aa2),C2) ) ) ) ).

% signed.order.strict_trans2
tff(fact_5901_signed_Oorder_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Aa2),Ba)
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Aa2),Ba)
            & ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Ba),Aa2) ) ) ) ).

% signed.order.strict_iff_not
tff(fact_5902_signed_Oorder_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Aa2),Ba)
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Aa2),Ba)
            | ( Aa2 = Ba ) ) ) ) ).

% signed.order.order_iff_strict
tff(fact_5903_signed_Oorder_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Aa2),Ba)
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Aa2),Ba)
            & ( Aa2 != Ba ) ) ) ) ).

% signed.order.strict_iff_order
tff(fact_5904_signed_Odual__order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Ba: word(A),Aa2: word(A),C2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Ba),Aa2)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),C2),Ba)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),C2),Aa2) ) ) ) ).

% signed.dual_order.strict_trans1
tff(fact_5905_signed_Odual__order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Ba: word(A),Aa2: word(A),C2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Ba),Aa2)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),C2),Ba)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),C2),Aa2) ) ) ) ).

% signed.dual_order.strict_trans2
tff(fact_5906_signed_Odual__order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Ba: word(A),Aa2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Ba),Aa2)
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Ba),Aa2)
            & ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Aa2),Ba) ) ) ) ).

% signed.dual_order.strict_iff_not
tff(fact_5907_signed_Oorder_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Aa2),Ba)
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Aa2),Ba) ) ) ).

% signed.order.strict_implies_order
tff(fact_5908_signed_Odual__order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Ba: word(A),Aa2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Ba),Aa2)
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Ba),Aa2)
            | ( Aa2 = Ba ) ) ) ) ).

% signed.dual_order.order_iff_strict
tff(fact_5909_signed_Odual__order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Ba: word(A),Aa2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Ba),Aa2)
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Ba),Aa2)
            & ( Aa2 != Ba ) ) ) ) ).

% signed.dual_order.strict_iff_order
tff(fact_5910_signed_Odual__order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Ba: word(A),Aa2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Ba),Aa2)
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Ba),Aa2) ) ) ).

% signed.dual_order.strict_implies_order
tff(fact_5911_signed_Oorder_Onot__eq__order__implies__strict,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] :
          ( ( Aa2 != Ba )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Aa2),Ba)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Aa2),Ba) ) ) ) ).

% signed.order.not_eq_order_implies_strict
tff(fact_5912_signed_Odual__order_Onot__eq__order__implies__strict,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] :
          ( ( Aa2 != Ba )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Ba),Aa2)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Ba),Aa2) ) ) ) ).

% signed.dual_order.not_eq_order_implies_strict
tff(fact_5913_word__sless__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),X),Y)
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),X),Y)
            & ( X != Y ) ) ) ) ).

% word_sless_eq
tff(fact_5914_signed_Ofinite__linorder__max__induct,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: set(word(A)),P: fun(set(word(A)),$o)] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A2)
         => ( aa(set(word(A)),$o,P,bot_bot(set(word(A))))
           => ( ! [B3: word(A),A5: set(word(A))] :
                  ( aa(set(word(A)),$o,finite_finite2(word(A)),A5)
                 => ( ! [X4: word(A)] :
                        ( member(word(A),X4,A5)
                       => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),X4),B3) )
                   => ( aa(set(word(A)),$o,P,A5)
                     => aa(set(word(A)),$o,P,aa(set(word(A)),set(word(A)),insert(word(A),B3),A5)) ) ) )
             => aa(set(word(A)),$o,P,A2) ) ) ) ) ).

% signed.finite_linorder_max_induct
tff(fact_5915_signed_Ofinite__linorder__min__induct,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: set(word(A)),P: fun(set(word(A)),$o)] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A2)
         => ( aa(set(word(A)),$o,P,bot_bot(set(word(A))))
           => ( ! [B3: word(A),A5: set(word(A))] :
                  ( aa(set(word(A)),$o,finite_finite2(word(A)),A5)
                 => ( ! [X4: word(A)] :
                        ( member(word(A),X4,A5)
                       => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),B3),X4) )
                   => ( aa(set(word(A)),$o,P,A5)
                     => aa(set(word(A)),$o,P,aa(set(word(A)),set(word(A)),insert(word(A),B3),A5)) ) ) )
             => aa(set(word(A)),$o,P,A2) ) ) ) ) ).

% signed.finite_linorder_min_induct
tff(fact_5916_signed_Oinfinite__growing,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X5: set(word(A))] :
          ( ( X5 != bot_bot(set(word(A))) )
         => ( ! [X2: word(A)] :
                ( member(word(A),X2,X5)
               => ? [Xa: word(A)] :
                    ( member(word(A),Xa,X5)
                    & aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),X2),Xa) ) )
           => ~ aa(set(word(A)),$o,finite_finite2(word(A)),X5) ) ) ) ).

% signed.infinite_growing
tff(fact_5917_signed_Odual__order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Ba: word(A),Aa2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Ba),Aa2)
         => ( Aa2 != Ba ) ) ) ).

% signed.dual_order.strict_implies_not_eq
tff(fact_5918_signed_Oorder_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Aa2),Ba)
         => ( Aa2 != Ba ) ) ) ).

% signed.order.strict_implies_not_eq
tff(fact_5919_signed_Odual__order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Ba: word(A),Aa2: word(A),C2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Ba),Aa2)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),C2),Ba)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),C2),Aa2) ) ) ) ).

% signed.dual_order.strict_trans
tff(fact_5920_signed_Onot__less__iff__gr__or__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),X),Y)
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Y),X)
            | ( X = Y ) ) ) ) ).

% signed.not_less_iff_gr_or_eq
tff(fact_5921_signed_Oorder_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A),C2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Aa2),Ba)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Ba),C2)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Aa2),C2) ) ) ) ).

% signed.order.strict_trans
tff(fact_5922_signed_Olinorder__less__wlog,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(word(A),fun(word(A),$o)),Aa2: word(A),Ba: word(A)] :
          ( ! [A3: word(A),B3: word(A)] :
              ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),A3),B3)
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),P,A3),B3) )
         => ( ! [A3: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),P,A3),A3)
           => ( ! [A3: word(A),B3: word(A)] :
                  ( aa(word(A),$o,aa(word(A),fun(word(A),$o),P,B3),A3)
                 => aa(word(A),$o,aa(word(A),fun(word(A),$o),P,A3),B3) )
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),P,Aa2),Ba) ) ) ) ) ).

% signed.linorder_less_wlog
tff(fact_5923_signed_Oord__less__eq__trans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A),C2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Aa2),Ba)
         => ( ( Ba = C2 )
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Aa2),C2) ) ) ) ).

% signed.ord_less_eq_trans
tff(fact_5924_signed_Oord__eq__less__trans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A),C2: word(A)] :
          ( ( Aa2 = Ba )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Ba),C2)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Aa2),C2) ) ) ) ).

% signed.ord_eq_less_trans
tff(fact_5925_signed_Oless__imp__not__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),X),Y)
         => ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Y),X) ) ) ).

% signed.less_imp_not_less
tff(fact_5926_signed_Odual__order_Oirrefl,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A)] : ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Aa2),Aa2) ) ).

% signed.dual_order.irrefl
tff(fact_5927_signed_Oless__imp__not__eq2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),X),Y)
         => ( Y != X ) ) ) ).

% signed.less_imp_not_eq2
tff(fact_5928_signed_Oless__imp__not__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),X),Y)
         => ( X != Y ) ) ) ).

% signed.less_imp_not_eq
tff(fact_5929_signed_Odual__order_Oasym,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Ba: word(A),Aa2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Ba),Aa2)
         => ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Aa2),Ba) ) ) ).

% signed.dual_order.asym
tff(fact_5930_signed_Olinorder__cases,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),X),Y)
         => ( ( X != Y )
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Y),X) ) ) ) ).

% signed.linorder_cases
tff(fact_5931_signed_Oless__imp__triv,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A),P: $o] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),X),Y)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Y),X)
           => (P) ) ) ) ).

% signed.less_imp_triv
tff(fact_5932_signed_Oantisym__conv3,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A)] :
          ( ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Y),X)
         => ( ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),X),Y)
          <=> ( X = Y ) ) ) ) ).

% signed.antisym_conv3
tff(fact_5933_signed_Oless__not__sym,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),X),Y)
         => ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Y),X) ) ) ).

% signed.less_not_sym
tff(fact_5934_signed_Oless__imp__neq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),X),Y)
         => ( X != Y ) ) ) ).

% signed.less_imp_neq
tff(fact_5935_signed_Oless__linear,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),X),Y)
          | ( X = Y )
          | aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Y),X) ) ) ).

% signed.less_linear
tff(fact_5936_signed_Oless__irrefl,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),X),X) ) ).

% signed.less_irrefl
tff(fact_5937_signed_Oorder_Oasym,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Aa2),Ba)
         => ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Ba),Aa2) ) ) ).

% signed.order.asym
tff(fact_5938_signed_Oless__trans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),X),Y)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Y),Z)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),X),Z) ) ) ) ).

% signed.less_trans
tff(fact_5939_signed_Oless__asym_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Aa2),Ba)
         => ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Ba),Aa2) ) ) ).

% signed.less_asym'
tff(fact_5940_signed_Oless__asym,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),X),Y)
         => ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Y),X) ) ) ).

% signed.less_asym
tff(fact_5941_signed_Oneq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ( X != Y )
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),X),Y)
            | aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Y),X) ) ) ) ).

% signed.neq_iff
tff(fact_5942_signed_OneqE,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( ( X != Y )
         => ( ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),X),Y)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Y),X) ) ) ) ).

% signed.neqE
tff(fact_5943_signed_Olift__Suc__mono__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F2: fun(nat,word(A)),Na: nat,N6: nat] :
          ( ! [N: nat] : aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),aa(nat,word(A),F2,N)),aa(nat,word(A),F2,aa(nat,nat,suc,N)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),N6)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),aa(nat,word(A),F2,Na)),aa(nat,word(A),F2,N6)) ) ) ) ).

% signed.lift_Suc_mono_less
tff(fact_5944_signed_Olift__Suc__mono__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F2: fun(nat,word(A)),Na: nat,M: nat] :
          ( ! [N: nat] : aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),aa(nat,word(A),F2,N)),aa(nat,word(A),F2,aa(nat,nat,suc,N)))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),aa(nat,word(A),F2,Na)),aa(nat,word(A),F2,M))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M) ) ) ) ).

% signed.lift_Suc_mono_less_iff
tff(fact_5945_word__sless__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Aa2),Ba)
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,ring_1_signed(A,int),Aa2)),aa(word(A),int,ring_1_signed(A,int),Ba)) ) ) ).

% word_sless_alt
tff(fact_5946_word__sless__sint__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),X),Y)
         => aa(int,$o,aa(int,fun(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_5947_take__bit__word__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num,K: num] : aa(word(A),word(A),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)),aa(int,int,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_5948_take__bit__word__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,K: num] : aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),aa(nat,nat,suc,Na)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),K))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,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_5949_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,$o,aa(nat,fun(nat,$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_5950_min_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),ord_min(A,Ba,C2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),C2) ) ) ) ).

% min.bounded_iff
tff(fact_5951_min_Oabsorb2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2)
         => ( ord_min(A,Aa2,Ba) = Ba ) ) ) ).

% min.absorb2
tff(fact_5952_min_Oabsorb1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => ( ord_min(A,Aa2,Ba) = Aa2 ) ) ) ).

% min.absorb1
tff(fact_5953_min__eq__arg_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,Na: A] :
          ( ( ord_min(A,M,Na) = M )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),Na) ) ) ).

% min_eq_arg(1)
tff(fact_5954_min__eq__arg_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,Na: A] :
          ( ( ord_min(A,M,Na) = Na )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Na),M) ) ) ).

% min_eq_arg(2)
tff(fact_5955_min__arg__le_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Na: A,M: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Na),ord_min(A,M,Na))
        <=> ( ord_min(A,M,Na) = Na ) ) ) ).

% min_arg_le(1)
tff(fact_5956_min__arg__le_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [M: A,Na: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),ord_min(A,M,Na))
        <=> ( ord_min(A,M,Na) = M ) ) ) ).

% min_arg_le(2)
tff(fact_5957_min__arg__not__ge_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,Na: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),ord_min(A,M,Na)),M)
        <=> ( ord_min(A,M,Na) = M ) ) ) ).

% min_arg_not_ge(1)
tff(fact_5958_min__arg__not__ge_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,Na: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),ord_min(A,M,Na)),Na)
        <=> ( ord_min(A,M,Na) = Na ) ) ) ).

% min_arg_not_ge(2)
tff(fact_5959_min__less__self__conv_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),ord_min(A,Aa2,Ba)),Aa2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),Aa2) ) ) ).

% min_less_self_conv(1)
tff(fact_5960_min__less__self__conv_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),ord_min(A,Aa2,Ba)),Ba)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba) ) ) ).

% min_less_self_conv(2)
tff(fact_5961_min__simps_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
         => ( ord_min(A,Aa2,Ba) = Aa2 ) ) ) ).

% min_simps(1)
tff(fact_5962_min__simps_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),Aa2)
         => ( ord_min(A,Aa2,Ba) = Ba ) ) ) ).

% min_simps(2)
tff(fact_5963_min__bot2,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [X: A] : ord_min(A,X,bot_bot(A)) = bot_bot(A) ) ).

% min_bot2
tff(fact_5964_min__bot,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [X: A] : ord_min(A,bot_bot(A),X) = bot_bot(A) ) ).

% min_bot
tff(fact_5965_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_5966_min__0R,axiom,
    ! [Na: nat] : ord_min(nat,Na,zero_zero(nat)) = zero_zero(nat) ).

% min_0R
tff(fact_5967_min__0L,axiom,
    ! [Na: nat] : ord_min(nat,zero_zero(nat),Na) = zero_zero(nat) ).

% min_0L
tff(fact_5968_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_5969_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_5970_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_5971_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_5972_take__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: nat,Aa2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),Aa2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,ord_min(nat,M,Na)),Aa2) ) ).

% take_bit_take_bit
tff(fact_5973_signed__take__bit__signed__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,Na: nat,Aa2: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,M),aa(A,A,bit_ri4674362597316999326ke_bit(A,Na),Aa2)) = aa(A,A,bit_ri4674362597316999326ke_bit(A,ord_min(nat,M,Na)),Aa2) ) ).

% signed_take_bit_signed_take_bit
tff(fact_5974_possible__bit__min,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Tyrep: itself(A),I: nat,J: nat] :
          ( bit_se6407376104438227557le_bit(A,Tyrep,ord_min(nat,I,J))
        <=> ( bit_se6407376104438227557le_bit(A,Tyrep,I)
            | bit_se6407376104438227557le_bit(A,Tyrep,J) ) ) ) ).

% possible_bit_min
tff(fact_5975_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_5976_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_5977_min__number__of_I1_J,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          ord_min(A,aa(num,A,numeral_numeral(A),U),aa(num,A,numeral_numeral(A),V)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)),aa(num,A,numeral_numeral(A),U),aa(num,A,numeral_numeral(A),V)) ) ).

% min_number_of(1)
tff(fact_5978_possible__bit__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat] :
          ( bit_se6407376104438227557le_bit(word(A),type2(word(A)),M)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A))) ) ) ).

% possible_bit_word
tff(fact_5979_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_5980_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_5981_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_5982_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_5983_min__Suc__gt_I2_J,axiom,
    ! [Aa2: nat,Ba: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Aa2),Ba)
     => ( ord_min(nat,Ba,aa(nat,nat,suc,Aa2)) = aa(nat,nat,suc,Aa2) ) ) ).

% min_Suc_gt(2)
tff(fact_5984_min__Suc__gt_I1_J,axiom,
    ! [Aa2: nat,Ba: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Aa2),Ba)
     => ( ord_min(nat,aa(nat,nat,suc,Aa2),Ba) = aa(nat,nat,suc,Aa2) ) ) ).

% min_Suc_gt(1)
tff(fact_5985_min__pm,axiom,
    ! [Aa2: nat,Ba: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),ord_min(nat,Aa2,Ba)),aa(nat,nat,minus_minus(nat,Aa2),Ba)) = Aa2 ).

% min_pm
tff(fact_5986_min__pm1,axiom,
    ! [Aa2: nat,Ba: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,Aa2),Ba)),ord_min(nat,Aa2,Ba)) = Aa2 ).

% min_pm1
tff(fact_5987_rev__min__pm,axiom,
    ! [Ba: nat,Aa2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),ord_min(nat,Ba,Aa2)),aa(nat,nat,minus_minus(nat,Aa2),Ba)) = Aa2 ).

% rev_min_pm
tff(fact_5988_rev__min__pm1,axiom,
    ! [Aa2: nat,Ba: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,Aa2),Ba)),ord_min(nat,Ba,Aa2)) = Aa2 ).

% rev_min_pm1
tff(fact_5989_take__bit__of__mask,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,M),bit_se2239418461657761734s_mask(A,Na)) = bit_se2239418461657761734s_mask(A,ord_min(nat,M,Na)) ) ).

% take_bit_of_mask
tff(fact_5990_min__number__of_I4_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: 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),V))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),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),V))) ) ).

% min_number_of(4)
tff(fact_5991_min__number__of_I3_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          ord_min(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)) ) ).

% min_number_of(3)
tff(fact_5992_min__number__of_I2_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          ord_min(A,aa(num,A,numeral_numeral(A),U),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(num,A,numeral_numeral(A),U),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) ) ).

% min_number_of(2)
tff(fact_5993_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_5994_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_5995_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_5996_bit__minus__2__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(one2)))),Na)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Na)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ) ) ).

% bit_minus_2_iff
tff(fact_5997_take__bit__word__Suc__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,K: num] : aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),aa(nat,nat,suc,Na)),aa(num,word(A),numeral_numeral(word(A)),K)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,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_5998_take__bit__word__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num,K: num] : aa(word(A),word(A),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)),aa(int,int,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_5999_bit__signed__take__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,Aa2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,bit_ri4674362597316999326ke_bit(A,M),Aa2)),Na)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Na)
            & aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),ord_min(nat,M,Na)) ) ) ) ).

% bit_signed_take_bit_iff
tff(fact_6000_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_6001_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_6002_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_6003_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_6004_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_6005_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_6006_min__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [X: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),X),ord_min(A,Y,Z)) = ord_min(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z)) ) ).

% min_add_distrib_right
tff(fact_6007_min__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [X: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),ord_min(A,X,Y)),Z) = ord_min(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),Z)) ) ).

% min_add_distrib_left
tff(fact_6008_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_6009_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_6010_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_6011_possible__bit__less__imp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Tyrep: itself(A),I: nat,J: nat] :
          ( bit_se6407376104438227557le_bit(A,Tyrep,I)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
           => bit_se6407376104438227557le_bit(A,Tyrep,J) ) ) ) ).

% possible_bit_less_imp
tff(fact_6012_min__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X4: A,Xa: A] :
          ord_min(A,X4,Xa) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Xa),X4,Xa) ) ).

% min_def_raw
tff(fact_6013_min_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: A,C2: A,Ba: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ord_min(A,Aa2,Ba)),ord_min(A,C2,D2)) ) ) ) ).

% min.mono
tff(fact_6014_min_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => ( Aa2 = ord_min(A,Aa2,Ba) ) ) ) ).

% min.orderE
tff(fact_6015_min_OorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: A,Ba: A] :
          ( ( Aa2 = ord_min(A,Aa2,Ba) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba) ) ) ).

% min.orderI
tff(fact_6016_min_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),ord_min(A,Ba,C2))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),C2) ) ) ) ).

% min.boundedE
tff(fact_6017_min_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),ord_min(A,Ba,C2)) ) ) ) ).

% min.boundedI
tff(fact_6018_min_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
        <=> ( Aa2 = ord_min(A,Aa2,Ba) ) ) ) ).

% min.order_iff
tff(fact_6019_min_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: A,Ba: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ord_min(A,Aa2,Ba)),Aa2) ) ).

% min.cobounded1
tff(fact_6020_min_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: A,Ba: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ord_min(A,Aa2,Ba)),Ba) ) ).

% min.cobounded2
tff(fact_6021_min_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba)
        <=> ( ord_min(A,Aa2,Ba) = Aa2 ) ) ) ).

% min.absorb_iff1
tff(fact_6022_min_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ba: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2)
        <=> ( ord_min(A,Aa2,Ba) = Ba ) ) ) ).

% min.absorb_iff2
tff(fact_6023_min_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: A,C2: A,Ba: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ord_min(A,Aa2,Ba)),C2) ) ) ).

% min.coboundedI1
tff(fact_6024_min_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ba: A,C2: A,Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ord_min(A,Aa2,Ba)),C2) ) ) ).

% min.coboundedI2
tff(fact_6025_min__le__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ord_min(A,X,Y)),Z)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ) ).

% min_le_iff_disj
tff(fact_6026_min__absorb2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X)
         => ( ord_min(A,X,Y) = Y ) ) ) ).

% min_absorb2
tff(fact_6027_min__absorb1,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
         => ( ord_min(A,X,Y) = X ) ) ) ).

% min_absorb1
tff(fact_6028_min__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Aa2: A,Ba: A] :
          ord_min(A,Aa2,Ba) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),Ba),Aa2,Ba) ) ).

% min_def
tff(fact_6029_concat__bit__assoc__sym,axiom,
    ! [M: nat,Na: nat,K: int,L: int,R: int] : aa(int,int,bit_concat_bit(M,aa(int,int,bit_concat_bit(Na,K),L)),R) = aa(int,int,bit_concat_bit(ord_min(nat,M,Na),K),aa(int,int,bit_concat_bit(aa(nat,nat,minus_minus(nat,M),Na),L),R)) ).

% concat_bit_assoc_sym
tff(fact_6030_bit__imp__possible__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Aa2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),Na)
         => bit_se6407376104438227557le_bit(A,type2(A),Na) ) ) ).

% bit_imp_possible_bit
tff(fact_6031_impossible__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Na: nat,Aa2: A] :
          ( ~ bit_se6407376104438227557le_bit(A,type2(A),Na)
         => ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),Na) ) ) ).

% impossible_bit
tff(fact_6032_bit__eq__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Aa2: A,Ba: A] :
          ( ( Aa2 = Ba )
        <=> ! [N2: nat] :
              ( bit_se6407376104438227557le_bit(A,type2(A),N2)
             => ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),N2)
              <=> aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Ba),N2) ) ) ) ) ).

% bit_eq_iff
tff(fact_6033_bit__eqI,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Aa2: A,Ba: A] :
          ( ! [N: nat] :
              ( bit_se6407376104438227557le_bit(A,type2(A),N)
             => ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),N)
              <=> aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Ba),N) ) )
         => ( Aa2 = Ba ) ) ) ).

% bit_eqI
tff(fact_6034_take__bit__concat__bit__eq,axiom,
    ! [M: nat,Na: nat,K: int,L: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,M),aa(int,int,bit_concat_bit(Na,K),L)) = aa(int,int,bit_concat_bit(ord_min(nat,M,Na),K),aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,minus_minus(nat,M),Na)),L)) ).

% take_bit_concat_bit_eq
tff(fact_6035_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,aa(A,fun(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_6036_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,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P2),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),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_6037_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,aa(A,fun(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_6038_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,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P2),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P2),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_6039_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,aa(A,fun(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_6040_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,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P2),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),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_6041_bit__not__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Aa2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Aa2)),Na)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Na)
            & ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),Na) ) ) ) ).

% bit_not_iff
tff(fact_6042_semiring__bit__operations__class_Obit__set__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Aa2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),M),Aa2)),Na)
        <=> ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),Na)
            | ( ( M = Na )
              & bit_se6407376104438227557le_bit(A,type2(A),Na) ) ) ) ) ).

% semiring_bit_operations_class.bit_set_bit_iff
tff(fact_6043_bit__flip__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Aa2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),bit_se8732182000553998342ip_bit(A,M,Aa2)),Na)
        <=> ( ( ( M = Na )
            <=> ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),Na) )
            & bit_se6407376104438227557le_bit(A,type2(A),Na) ) ) ) ).

% bit_flip_bit_iff
tff(fact_6044_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_6045_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_6046_mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Aa2: 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),Aa2),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),Aa2),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),ord_min(nat,M,Na))) ) ).

% mod_exp_eq
tff(fact_6047_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_6048_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,aa(nat,fun(nat,$o),ord_less(nat),Na),M) ) ) ) ).

% bit_mask_iff
tff(fact_6049_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,aa(nat,fun(nat,$o),ord_less(nat),Na),ord_min(nat,type_len0_len_of(A,type2(A)),M)) ) ) ).

% Word.bit_mask_iff
tff(fact_6050_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_6051_bit__of__int__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(int,A,ring_1_of_int(A),K)),Na)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Na)
            & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),Na) ) ) ) ).

% bit_of_int_iff
tff(fact_6052_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_6053_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_6054_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_6055_bit__minus__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Aa2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,uminus_uminus(A),Aa2)),Na)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Na)
            & ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,minus_minus(A,Aa2),one_one(A))),Na) ) ) ) ).

% bit_minus_iff
tff(fact_6056_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)),aa(int,int,bit_se2584673776208193580ke_bit(int,ord_min(nat,type_len0_len_of(A,type2(A)),Na)),I)) ) ).

% and_mask_wi'
tff(fact_6057_bit__push__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Aa2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),M),Aa2)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
            & bit_se6407376104438227557le_bit(A,type2(A),Na)
            & aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),aa(nat,nat,minus_minus(nat,Na),M)) ) ) ) ).

% bit_push_bit_iff
tff(fact_6058_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_6059_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_6060_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_6061_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_6062_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,aa(nat,fun(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,aa(nat,fun(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_6063_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,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,M),type_len0_len_of(B,type2(B)))),Na)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),ord_min(nat,type_len0_len_of(A,type2(A)),M))
            & aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),W),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(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_6064_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_6065_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,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ) ).

% bit_minus_exp_iff
tff(fact_6066_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_6067_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,aa(nat,fun(nat,$o),ord_less(nat),Na),M) ) ) ) ).

% bit_mask_sub_iff
tff(fact_6068_bit__double__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Aa2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Aa2)),Na)
        <=> ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),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_6069_bit__horner__sum__bit__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bs2: list($o),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),groups4207007520872428315er_sum($o,word(A),zero_neq_one_of_bool(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2)),Bs2)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),ord_min(nat,type_len0_len_of(A,type2(A)),aa(list($o),nat,size_size(list($o)),Bs2)))
            & aa(nat,$o,nth($o,Bs2),Na) ) ) ) ).

% bit_horner_sum_bit_word_iff
tff(fact_6070_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_6071_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)) = aa(int,int,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),aa(word(A),word(A),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_6072_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_6073_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_6074_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_6075_min__top,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [X: A] : ord_min(A,top_top(A),X) = X ) ).

% min_top
tff(fact_6076_min__top2,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [X: A] : ord_min(A,X,top_top(A)) = X ) ).

% min_top2
tff(fact_6077_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_6078_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_6079_Collect__const,axiom,
    ! [A: $tType,P: $o] :
      collect(A,aTP_Lamp_qb($o,fun(A,$o),(P))) = $ite((P),top_top(set(A)),bot_bot(set(A))) ).

% Collect_const
tff(fact_6080_finite__Collect__not,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),collect(A,P))
     => ( aa(set(A),$o,finite_finite2(A),collect(A,aTP_Lamp_ds(fun(A,$o),fun(A,$o),P)))
      <=> aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ) ).

% finite_Collect_not
tff(fact_6081_range__add,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [Aa2: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),Aa2)),top_top(set(A))) = top_top(set(A)) ) ).

% range_add
tff(fact_6082_surj__plus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Aa2: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),Aa2)),top_top(set(A))) = top_top(set(A)) ) ).

% surj_plus
tff(fact_6083_surj__diff__right,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Aa2: A] : aa(set(A),set(A),image(A,A,aTP_Lamp_qa(A,fun(A,A),Aa2)),top_top(set(A))) = top_top(set(A)) ) ).

% surj_diff_right
tff(fact_6084_range__constant,axiom,
    ! [B: $tType,A: $tType,X: A] : aa(set(B),set(A),image(B,A,aa(A,fun(B,A),aTP_Lamp_pl(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_6085_top__greatest,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [Aa2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),top_top(A)) ) ).

% top_greatest
tff(fact_6086_top_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),top_top(A)),Aa2)
        <=> ( Aa2 = top_top(A) ) ) ) ).

% top.extremum_unique
tff(fact_6087_top_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [Aa2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),top_top(A)),Aa2)
         => ( Aa2 = top_top(A) ) ) ) ).

% top.extremum_uniqueI
tff(fact_6088_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_6089_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_6090_top_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [Aa2: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),top_top(A)),Aa2) ) ).

% top.extremum_strict
tff(fact_6091_top_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [Aa2: A] :
          ( ( Aa2 != top_top(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),top_top(A)) ) ) ).

% top.not_eq_extremum
tff(fact_6092_UNIV__def,axiom,
    ! [A: $tType] : top_top(set(A)) = collect(A,aTP_Lamp_qc(A,$o)) ).

% UNIV_def
tff(fact_6093_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,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)),W) ) ).

% word_rotr_word_rotr_eq
tff(fact_6094_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_6095_word__rotate_Oword__rot__logs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,V: word(A)] : aa(word(A),word(A),word_rotr(A,Na),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),V)) = aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(word(A),word(A),word_rotr(A,Na),V)) ) ).

% word_rotate.word_rot_logs(2)
tff(fact_6096_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_qd(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_6097_rangeE,axiom,
    ! [A: $tType,B: $tType,Ba: A,F2: fun(B,A)] :
      ( member(A,Ba,aa(set(B),set(A),image(B,A,F2),top_top(set(B))))
     => ~ ! [X2: B] : Ba != aa(B,A,F2,X2) ) ).

% rangeE
tff(fact_6098_finite__range__imageI,axiom,
    ! [A: $tType,C: $tType,B: $tType,G: fun(B,A),F2: fun(A,C)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image(B,A,G),top_top(set(B))))
     => aa(set(C),$o,finite_finite2(C),aa(set(B),set(C),image(B,C,aa(fun(A,C),fun(B,C),aTP_Lamp_qe(fun(B,A),fun(fun(A,C),fun(B,C)),G),F2)),top_top(set(B)))) ) ).

% finite_range_imageI
tff(fact_6099_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,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),X)),Y) ) ) ).

% sup_shunt
tff(fact_6100_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,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M)),type_len0_len_of(A,type2(A)))) ) ) ) ).

% bit_word_rotr_iff
tff(fact_6101_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_6102_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_6103_ucast__range__less,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
       => ( aa(set(word(A)),set(word(B)),image(word(A),word(B),semiring_1_unsigned(A,word(B))),top_top(set(word(A)))) = collect(word(B),aTP_Lamp_qf(word(B),$o)) ) ) ) ).

% ucast_range_less
tff(fact_6104_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,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I),aa(word(A),word(A),word_rotr(A,aa(int,nat,nat2,I)),W),aa(word(A),word(A),word_rotl(A,aa(int,nat,nat2,aa(int,int,uminus_uminus(int),I))),W)) ) ).

% word_roti_eq_word_rotr_word_rotl
tff(fact_6105_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),aa(int,int,bit_concat_bit(aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(int,nat,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),aa(int,nat,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,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),X))),aa(int,int,bit_se2584673776208193580ke_bit(int,aa(int,nat,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_6106_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_6107_word__rot__lr,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: nat,V: word(A)] : aa(word(A),word(A),word_rotr(A,K),aa(word(A),word(A),word_rotl(A,K),V)) = V ) ).

% word_rot_lr
tff(fact_6108_word__rot__rl,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: nat,V: word(A)] : aa(word(A),word(A),word_rotl(A,K),aa(word(A),word(A),word_rotr(A,K),V)) = V ) ).

% word_rot_rl
tff(fact_6109_Collect__const__case__prod,axiom,
    ! [B: $tType,A: $tType,P: $o] :
      collect(product_prod(A,B),product_case_prod(A,B,$o,aTP_Lamp_qg($o,fun(A,fun(B,$o)),(P)))) = $ite((P),top_top(set(product_prod(A,B))),bot_bot(set(product_prod(A,B)))) ).

% Collect_const_case_prod
tff(fact_6110_word__rot__gal_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Na: nat,V: word(A)] :
          ( ( W = aa(word(A),word(A),word_rotr(A,Na),V) )
        <=> ( V = aa(word(A),word(A),word_rotl(A,Na),W) ) ) ) ).

% word_rot_gal'
tff(fact_6111_word__rot__gal,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,V: word(A),W: word(A)] :
          ( ( aa(word(A),word(A),word_rotr(A,Na),V) = W )
        <=> ( aa(word(A),word(A),word_rotl(A,Na),W) = V ) ) ) ).

% word_rot_gal
tff(fact_6112_top__empty__eq2,axiom,
    ! [B: $tType,A: $tType,X4: A,Xa: B] :
      ( aa(B,$o,aa(A,fun(B,$o),top_top(fun(A,fun(B,$o))),X4),Xa)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa),top_top(set(product_prod(A,B)))) ) ).

% top_empty_eq2
tff(fact_6113_infinite__UNIV__listI,axiom,
    ! [A: $tType] : ~ aa(set(list(A)),$o,finite_finite2(list(A)),top_top(set(list(A)))) ).

% infinite_UNIV_listI
tff(fact_6114_infinite__UNIV__int,axiom,
    ~ aa(set(int),$o,finite_finite2(int),top_top(set(int))) ).

% infinite_UNIV_int
tff(fact_6115_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_6116_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_6117_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_6118_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_6119_plus__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)),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,aa(int,fun(int,int),plus_plus(int),Xa2),X)) ) ).

% plus_word.abs_eq
tff(fact_6120_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_6121_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_6122_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_6123_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_6124_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_6125_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_6126_set__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_se5668285175392031749et_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_se5668285175392031749et_bit(int),Xa2),X)) ) ).

% set_bit_word.abs_eq
tff(fact_6127_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_6128_word__rotate_Oword__rot__logs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,V: word(A)] : aa(word(A),word(A),word_rotl(A,Na),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),V)) = aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(word(A),word(A),word_rotl(A,Na),V)) ) ).

% word_rotate.word_rot_logs(1)
tff(fact_6129_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_6130_word_Oabs__induct,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(word(A),$o),X: word(A)] :
          ( ! [Y2: int] : aa(word(A),$o,P,aa(int,word(A),word2(A),Y2))
         => aa(word(A),$o,P,X) ) ) ).

% word.abs_induct
tff(fact_6131_unset__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_se2638667681897837118et_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_se2638667681897837118et_bit(int),Xa2),X)) ) ).

% unset_bit_word.abs_eq
tff(fact_6132_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_6133_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_6134_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_6135_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) )
        <=> ( aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),X) = aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),Y) ) ) ) ).

% word.abs_eq_iff
tff(fact_6136_size__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: int] : aa(word(A),nat,size_size(word(A)),aa(int,word(A),word2(A),X)) = type_len0_len_of(A,type2(A)) ) ).

% size_word.abs_eq
tff(fact_6137_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,aa(int,fun(int,int),plus_plus(int),X),one_one(int))) ) ).

% word_succ.abs_eq
tff(fact_6138_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_6139_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_6140_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_6141_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),aa(int,int,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))))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),X))),aa(int,int,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_6142_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_6143_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,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(B,type2(B))),X)) ) ).

% word_int_case.abs_eq
tff(fact_6144_less__eq__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa2: int,X: int] :
          ( aa(word(A),$o,aa(word(A),fun(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,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),Xa2)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),X)) ) ) ).

% less_eq_word.abs_eq
tff(fact_6145_less__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa2: int,X: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(int,word(A),word2(A),Xa2)),aa(int,word(A),word2(A),X))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),Xa2)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),X)) ) ) ).

% less_word.abs_eq
tff(fact_6146_bit__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: int,X4: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(int,word(A),word2(A),X)),X4)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X4),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),X),X4) ) ) ) ).

% bit_word.abs_eq
tff(fact_6147_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),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),Xa2)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),X))) ) ).

% divide_word.abs_eq
tff(fact_6148_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),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),Xa2)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),X))) ) ).

% modulo_word.abs_eq
tff(fact_6149_take__bit__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa2: nat,X: int] : aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),Xa2),aa(int,word(A),word2(A),X)) = aa(int,word(A),word2(A),aa(int,int,bit_se2584673776208193580ke_bit(int,ord_min(nat,type_len0_len_of(A,type2(A)),Xa2)),X)) ) ).

% take_bit_word.abs_eq
tff(fact_6150_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),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),X))) ) ).

% drop_bit_word.abs_eq
tff(fact_6151_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),aa(int,int,bit_concat_bit(type_len0_len_of(C,type2(C)),X),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(B,type2(B))),Xa2))) ) ).

% word_cat.abs_eq
tff(fact_6152_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),aa(int,nat,nat2,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(B,type2(B))),X))) ) ).

% unsigned.abs_eq
tff(fact_6153_range__mod,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(set(nat),set(nat),image(nat,nat,aTP_Lamp_qh(nat,fun(nat,nat),Na)),top_top(set(nat))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),Na) ) ) ).

% range_mod
tff(fact_6154_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_6155_word__sle_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa2: int,X: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(int,word(A),word2(A),Xa2)),aa(int,word(A),word2(A),X))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,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)),aa(int,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)))),X)) ) ) ).

% word_sle.abs_eq
tff(fact_6156_word__sless_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa2: int,X: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),aa(int,word(A),word2(A),Xa2)),aa(int,word(A),word2(A),X))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,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)),aa(int,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)))),X)) ) ) ).

% word_sless.abs_eq
tff(fact_6157_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),aa(int,int,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_6158_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),aa(int,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)))),X))) ) ).

% signed_drop_bit.abs_eq
tff(fact_6159_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),aa(int,int,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)))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),X))),aa(int,int,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_6160_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,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(nat,nat,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_6161_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),aa(int,int,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_6162_the__signed__int_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: int] : the_signed_int(A,aa(int,word(A),word2(A),X)) = aa(int,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)))),X) ) ).

% the_signed_int.abs_eq
tff(fact_6163_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_6164_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),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,type_len0_len_of(A,type2(A))),X))) ) ).

% of_nat.abs_eq
tff(fact_6165_the__nat_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: int] : the_nat(A,aa(int,word(A),word2(A),X)) = aa(int,nat,nat2,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),X)) ) ).

% the_nat.abs_eq
tff(fact_6166_Word_Oof__int_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: int] : of_int2(A,X) = aa(int,word(A),word2(A),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),X)) ) ).

% Word.of_int.abs_eq
tff(fact_6167_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),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(B,type2(B))),X)) ) ).

% cast.abs_eq
tff(fact_6168_the__int_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: int] : the_int(A,aa(int,word(A),word2(A),X)) = aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),X) ) ).

% the_int.abs_eq
tff(fact_6169_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_qi(nat,fun(real,real),Na),X)) ).

% root_def
tff(fact_6170_the__inv__into__def,axiom,
    ! [B: $tType,A: $tType,A2: set(B),F2: fun(B,A),X4: A] : the_inv_into(B,A,A2,F2,X4) = the(B,aa(A,fun(B,$o),aa(fun(B,A),fun(A,fun(B,$o)),aTP_Lamp_qj(set(B),fun(fun(B,A),fun(A,fun(B,$o))),A2),F2),X4)) ).

% the_inv_into_def
tff(fact_6171_CHAR__eq0__iff,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( ( semiri4206861660011772517g_char(A,type2(A)) = zero_zero(nat) )
      <=> ! [N2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
           => ( aa(nat,A,semiring_1_of_nat(A),N2) != zero_zero(A) ) ) ) ) ).

% CHAR_eq0_iff
tff(fact_6172_CHAR__eq__posI,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [C2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),C2)
         => ( ( aa(nat,A,semiring_1_of_nat(A),C2) = zero_zero(A) )
           => ( ! [X2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),X2)
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X2),C2)
                   => ( aa(nat,A,semiring_1_of_nat(A),X2) != zero_zero(A) ) ) )
             => ( semiri4206861660011772517g_char(A,type2(A)) = C2 ) ) ) ) ) ).

% CHAR_eq_posI
tff(fact_6173_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_6174_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_6175_CHAR__eqI,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [C2: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),C2) = zero_zero(A) )
         => ( ! [X2: nat] :
                ( ( aa(nat,A,semiring_1_of_nat(A),X2) = zero_zero(A) )
               => aa(nat,$o,dvd_dvd(nat,C2),X2) )
           => ( semiri4206861660011772517g_char(A,type2(A)) = C2 ) ) ) ) ).

% CHAR_eqI
tff(fact_6176_CHAR__pos__iff,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),semiri4206861660011772517g_char(A,type2(A)))
      <=> ? [N2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
            & ( aa(nat,A,semiring_1_of_nat(A),N2) = zero_zero(A) ) ) ) ) ).

% CHAR_pos_iff
tff(fact_6177_test__hoare,axiom,
    ! [Xs2: list(nat),Na: nat,Ysb: list(nat)] :
      ( ! [X2: nat] :
          ( member(nat,X2,aa(list(nat),set(nat),set2(nat),Xs2))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X2),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => hoare_hoare_triple(list(nat),one_one(assn),vEBT_test(Na,Xs2,Ysb),aa(list(nat),fun(list(nat),assn),aTP_Lamp_qm(list(nat),fun(list(nat),fun(list(nat),assn)),Xs2),Ysb)) ) ) ).

% test_hoare
tff(fact_6178_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,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),M)),Na)
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A))) ),
                aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),
                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na) )) ) ) ).

% bit_sshiftr_iff
tff(fact_6179_map__ident,axiom,
    ! [A: $tType,X4: list(A)] : aa(list(A),list(A),map(A,A,aTP_Lamp_pc(A,A)),X4) = X4 ).

% map_ident
tff(fact_6180_list_Omap__disc__iff,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Aa2: list(B)] :
      ( ( aa(list(B),list(A),map(B,A,F2),Aa2) = nil(A) )
    <=> ( Aa2 = nil(B) ) ) ).

% list.map_disc_iff
tff(fact_6181_Nil__is__map__conv,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs2: list(B)] :
      ( ( nil(A) = aa(list(B),list(A),map(B,A,F2),Xs2) )
    <=> ( Xs2 = nil(B) ) ) ).

% Nil_is_map_conv
tff(fact_6182_map__is__Nil__conv,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs2: list(B)] :
      ( ( aa(list(B),list(A),map(B,A,F2),Xs2) = nil(A) )
    <=> ( Xs2 = nil(B) ) ) ).

% map_is_Nil_conv
tff(fact_6183_length__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs2: list(B)] : aa(list(A),nat,size_size(list(A)),aa(list(B),list(A),map(B,A,F2),Xs2)) = aa(list(B),nat,size_size(list(B)),Xs2) ).

% length_map
tff(fact_6184_map__eq__conv,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs2: list(B),G: fun(B,A)] :
      ( ( aa(list(B),list(A),map(B,A,F2),Xs2) = aa(list(B),list(A),map(B,A,G),Xs2) )
    <=> ! [X6: B] :
          ( member(B,X6,aa(list(B),set(B),set2(B),Xs2))
         => ( aa(B,A,F2,X6) = aa(B,A,G,X6) ) ) ) ).

% map_eq_conv
tff(fact_6185_list_Oset__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),V: list(B)] : aa(list(A),set(A),set2(A),aa(list(B),list(A),map(B,A,F2),V)) = aa(set(B),set(A),image(B,A,F2),aa(list(B),set(B),set2(B),V)) ).

% list.set_map
tff(fact_6186_nth__map,axiom,
    ! [B: $tType,A: $tType,Na: nat,Xs2: list(A),F2: fun(A,B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs2))
     => ( aa(nat,B,nth(B,aa(list(A),list(B),map(A,B,F2),Xs2)),Na) = aa(A,B,F2,aa(nat,A,nth(A,Xs2),Na)) ) ) ).

% nth_map
tff(fact_6187_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_6188_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_6189_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_6190_image__set,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs2: list(B)] : aa(set(B),set(A),image(B,A,F2),aa(list(B),set(B),set2(B),Xs2)) = aa(list(A),set(A),set2(A),aa(list(B),list(A),map(B,A,F2),Xs2)) ).

% image_set
tff(fact_6191_map__eq__nth__eq,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),L: list(B),L3: list(B),I: nat] :
      ( ( aa(list(B),list(A),map(B,A,F2),L) = aa(list(B),list(A),map(B,A,F2),L3) )
     => ( aa(B,A,F2,aa(nat,B,nth(B,L),I)) = aa(B,A,F2,aa(nat,B,nth(B,L3),I)) ) ) ).

% map_eq_nth_eq
tff(fact_6192_list_Osimps_I8_J,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A)] : aa(list(B),list(A),map(B,A,F2),nil(B)) = nil(A) ).

% list.simps(8)
tff(fact_6193_map__eq__imp__length__eq,axiom,
    ! [A: $tType,B: $tType,C: $tType,F2: fun(B,A),Xs2: list(B),G: fun(C,A),Ysb: list(C)] :
      ( ( aa(list(B),list(A),map(B,A,F2),Xs2) = aa(list(C),list(A),map(C,A,G),Ysb) )
     => ( aa(list(B),nat,size_size(list(B)),Xs2) = aa(list(C),nat,size_size(list(C)),Ysb) ) ) ).

% map_eq_imp_length_eq
tff(fact_6194_GreatestI__ex__nat,axiom,
    ! [P: fun(nat,$o),Ba: nat] :
      ( ? [X_13: nat] : aa(nat,$o,P,X_13)
     => ( ! [Y2: nat] :
            ( aa(nat,$o,P,Y2)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y2),Ba) )
       => aa(nat,$o,P,order_Greatest(nat,P)) ) ) ).

% GreatestI_ex_nat
tff(fact_6195_Greatest__le__nat,axiom,
    ! [P: fun(nat,$o),K: nat,Ba: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [Y2: nat] :
            ( aa(nat,$o,P,Y2)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y2),Ba) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),order_Greatest(nat,P)) ) ) ).

% Greatest_le_nat
tff(fact_6196_GreatestI__nat,axiom,
    ! [P: fun(nat,$o),K: nat,Ba: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [Y2: nat] :
            ( aa(nat,$o,P,Y2)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y2),Ba) )
       => aa(nat,$o,P,order_Greatest(nat,P)) ) ) ).

% GreatestI_nat
tff(fact_6197_rotate1__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs2: list(B)] : rotate1(A,aa(list(B),list(A),map(B,A,F2),Xs2)) = aa(list(B),list(A),map(B,A,F2),rotate1(B,Xs2)) ).

% rotate1_map
tff(fact_6198_list_Omap__ident,axiom,
    ! [A: $tType,T: list(A)] : aa(list(A),list(A),map(A,A,aTP_Lamp_pc(A,A)),T) = T ).

% list.map_ident
tff(fact_6199_list_Omap__cong,axiom,
    ! [B: $tType,A: $tType,X: list(A),Ya: list(A),F2: fun(A,B),G: fun(A,B)] :
      ( ( X = Ya )
     => ( ! [Z3: A] :
            ( member(A,Z3,aa(list(A),set(A),set2(A),Ya))
           => ( aa(A,B,F2,Z3) = aa(A,B,G,Z3) ) )
       => ( aa(list(A),list(B),map(A,B,F2),X) = aa(list(A),list(B),map(A,B,G),Ya) ) ) ) ).

% list.map_cong
tff(fact_6200_list_Omap__cong0,axiom,
    ! [B: $tType,A: $tType,X: list(A),F2: fun(A,B),G: fun(A,B)] :
      ( ! [Z3: A] :
          ( member(A,Z3,aa(list(A),set(A),set2(A),X))
         => ( aa(A,B,F2,Z3) = aa(A,B,G,Z3) ) )
     => ( aa(list(A),list(B),map(A,B,F2),X) = aa(list(A),list(B),map(A,B,G),X) ) ) ).

% list.map_cong0
tff(fact_6201_list_Oinj__map__strong,axiom,
    ! [B: $tType,A: $tType,X: list(A),Xa2: list(A),F2: fun(A,B),Fa: fun(A,B)] :
      ( ! [Z3: A,Za: A] :
          ( member(A,Z3,aa(list(A),set(A),set2(A),X))
         => ( member(A,Za,aa(list(A),set(A),set2(A),Xa2))
           => ( ( aa(A,B,F2,Z3) = aa(A,B,Fa,Za) )
             => ( Z3 = Za ) ) ) )
     => ( ( aa(list(A),list(B),map(A,B,F2),X) = aa(list(A),list(B),map(A,B,Fa),Xa2) )
       => ( X = Xa2 ) ) ) ).

% list.inj_map_strong
tff(fact_6202_map__ext,axiom,
    ! [B: $tType,A: $tType,Xs2: list(A),F2: fun(A,B),G: fun(A,B)] :
      ( ! [X2: A] :
          ( member(A,X2,aa(list(A),set(A),set2(A),Xs2))
         => ( aa(A,B,F2,X2) = aa(A,B,G,X2) ) )
     => ( aa(list(A),list(B),map(A,B,F2),Xs2) = aa(list(A),list(B),map(A,B,G),Xs2) ) ) ).

% map_ext
tff(fact_6203_map__idI,axiom,
    ! [A: $tType,Xs2: list(A),F2: fun(A,A)] :
      ( ! [X2: A] :
          ( member(A,X2,aa(list(A),set(A),set2(A),Xs2))
         => ( aa(A,A,F2,X2) = X2 ) )
     => ( aa(list(A),list(A),map(A,A,F2),Xs2) = Xs2 ) ) ).

% map_idI
tff(fact_6204_map__cong,axiom,
    ! [B: $tType,A: $tType,Xs2: list(A),Ysb: list(A),F2: fun(A,B),G: fun(A,B)] :
      ( ( Xs2 = Ysb )
     => ( ! [X2: A] :
            ( member(A,X2,aa(list(A),set(A),set2(A),Ysb))
           => ( aa(A,B,F2,X2) = aa(A,B,G,X2) ) )
       => ( aa(list(A),list(B),map(A,B,F2),Xs2) = aa(list(A),list(B),map(A,B,G),Ysb) ) ) ) ).

% map_cong
tff(fact_6205_ex__map__conv,axiom,
    ! [B: $tType,A: $tType,Ysb: list(B),F2: fun(A,B)] :
      ( ? [Xs3: list(A)] : Ysb = aa(list(A),list(B),map(A,B,F2),Xs3)
    <=> ! [X6: B] :
          ( member(B,X6,aa(list(B),set(B),set2(B),Ysb))
         => ? [Xa4: A] : X6 = aa(A,B,F2,Xa4) ) ) ).

% ex_map_conv
tff(fact_6206_map__eq__Cons__conv,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),Xs2: list(B),Y: A,Ysb: list(A)] :
      ( ( aa(list(B),list(A),map(B,A,F2),Xs2) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ysb) )
    <=> ? [Z5: B,Zs3: list(B)] :
          ( ( Xs2 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z5),Zs3) )
          & ( aa(B,A,F2,Z5) = Y )
          & ( aa(list(B),list(A),map(B,A,F2),Zs3) = Ysb ) ) ) ).

% map_eq_Cons_conv
tff(fact_6207_Cons__eq__map__conv,axiom,
    ! [A: $tType,B: $tType,X: A,Xs2: list(A),F2: fun(B,A),Ysb: list(B)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs2) = aa(list(B),list(A),map(B,A,F2),Ysb) )
    <=> ? [Z5: B,Zs3: list(B)] :
          ( ( Ysb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z5),Zs3) )
          & ( X = aa(B,A,F2,Z5) )
          & ( Xs2 = aa(list(B),list(A),map(B,A,F2),Zs3) ) ) ) ).

% Cons_eq_map_conv
tff(fact_6208_map__eq__Cons__D,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),Xs2: list(B),Y: A,Ysb: list(A)] :
      ( ( aa(list(B),list(A),map(B,A,F2),Xs2) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ysb) )
     => ? [Z3: B,Zs2: list(B)] :
          ( ( Xs2 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z3),Zs2) )
          & ( aa(B,A,F2,Z3) = Y )
          & ( aa(list(B),list(A),map(B,A,F2),Zs2) = Ysb ) ) ) ).

% map_eq_Cons_D
tff(fact_6209_Cons__eq__map__D,axiom,
    ! [A: $tType,B: $tType,X: A,Xs2: list(A),F2: fun(B,A),Ysb: list(B)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs2) = aa(list(B),list(A),map(B,A,F2),Ysb) )
     => ? [Z3: B,Zs2: list(B)] :
          ( ( Ysb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z3),Zs2) )
          & ( X = aa(B,A,F2,Z3) )
          & ( Xs2 = aa(list(B),list(A),map(B,A,F2),Zs2) ) ) ) ).

% Cons_eq_map_D
tff(fact_6210_map__eq__consE,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),Ls2: list(B),Fa: A,Fl: list(A)] :
      ( ( aa(list(B),list(A),map(B,A,F2),Ls2) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Fa),Fl) )
     => ~ ! [A3: B,L2: list(B)] :
            ( ( Ls2 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A3),L2) )
           => ( ( aa(B,A,F2,A3) = Fa )
             => ( aa(list(B),list(A),map(B,A,F2),L2) != Fl ) ) ) ) ).

% map_eq_consE
tff(fact_6211_list_Osimps_I9_J,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),X21: B,X222: list(B)] : aa(list(B),list(A),map(B,A,F2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X21),X222)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(B,A,F2,X21)),aa(list(B),list(A),map(B,A,F2),X222)) ).

% list.simps(9)
tff(fact_6212_map__consI_I1_J,axiom,
    ! [A: $tType,B: $tType,W: list(A),F2: fun(B,A),Ww: list(B),Aa2: B] :
      ( ( W = aa(list(B),list(A),map(B,A,F2),Ww) )
     => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(B,A,F2,Aa2)),W) = aa(list(B),list(A),map(B,A,F2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Aa2),Ww)) ) ) ).

% map_consI(1)
tff(fact_6213_GreatestI2__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o),X: A,Q: fun(A,$o)] :
          ( aa(A,$o,P,X)
         => ( ! [Y2: A] :
                ( aa(A,$o,P,Y2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y2),X) )
           => ( ! [X2: A] :
                  ( aa(A,$o,P,X2)
                 => ( ! [Y3: A] :
                        ( aa(A,$o,P,Y3)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),X2) )
                   => aa(A,$o,Q,X2) ) )
             => aa(A,$o,Q,order_Greatest(A,P)) ) ) ) ) ).

% GreatestI2_order
tff(fact_6214_Greatest__equality,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o),X: A] :
          ( aa(A,$o,P,X)
         => ( ! [Y2: A] :
                ( aa(A,$o,P,Y2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y2),X) )
           => ( order_Greatest(A,P) = X ) ) ) ) ).

% Greatest_equality
tff(fact_6215_mmap__pure__aux,axiom,
    ! [B: $tType,A: $tType,Xs2: list(A),P: assn,Fi: fun(A,heap_Time_Heap(B)),F2: fun(A,B)] :
      ( ! [X2: A] :
          ( member(A,X2,aa(list(A),set(A),set2(A),Xs2))
         => hoare_hoare_triple(B,P,aa(A,heap_Time_Heap(B),Fi,X2),aa(A,fun(B,assn),aa(fun(A,B),fun(A,fun(B,assn)),aTP_Lamp_qn(assn,fun(fun(A,B),fun(A,fun(B,assn))),P),F2),X2)) )
     => hoare_hoare_triple(list(B),P,vEBT_Example_mmap(A,B,Fi,Xs2),aa(fun(A,B),fun(list(B),assn),aa(assn,fun(fun(A,B),fun(list(B),assn)),aTP_Lamp_qo(list(A),fun(assn,fun(fun(A,B),fun(list(B),assn))),Xs2),P),F2)) ) ).

% mmap_pure_aux
tff(fact_6216_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_6217_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_6218_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_6219_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_6220_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_6221_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_6222_shiftl__of__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A,Na: nat] : bit_Sh4282982442137083160shiftl(A,Aa2,aa(nat,nat,suc,Na)) = bit_Sh4282982442137083160shiftl(A,aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),aa(num,A,numeral_numeral(A),bit0(one2))),Na) ) ).

% shiftl_of_Suc
tff(fact_6223_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_6224_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_6225_bit__shiftl__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A,M: nat,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),bit_Sh4282982442137083160shiftl(A,Aa2,M)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
            & bit_se6407376104438227557le_bit(A,type2(A),Na)
            & aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),aa(nat,nat,minus_minus(nat,Na),M)) ) ) ) ).

% bit_shiftl_iff
tff(fact_6226_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_6227_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_6228_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_6229_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_6230_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_6231_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_6232_listI__assn__extract,axiom,
    ! [A: $tType,B: $tType,I: nat,I5: set(nat),Xs2: list(A),A2: fun(A,fun(B,assn)),Xsi: list(B)] :
      ( member(nat,I,I5)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs2))
       => ( vEBT_List_listI_assn(A,B,I5,A2,Xs2,Xsi) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A2,aa(nat,A,nth(A,Xs2),I)),aa(nat,B,nth(B,Xsi),I))),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),minus_minus(set(nat),I5),aa(set(nat),set(nat),insert(nat,I),bot_bot(set(nat)))),A2,Xs2,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_6233_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,aa(nat,fun(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,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),W),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(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_6234_revcast__slice1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [W: word(B)] : aa(word(B),word(A),slice1(B,A,aa(word(A),nat,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_6235_listI__assn__cong,axiom,
    ! [A: $tType,B: $tType,I5: set(nat),I6: set(nat),Xs2: list(A),Xs4: list(A),Xsi: list(B),Xsi2: list(B),A2: fun(A,fun(B,assn)),A6: fun(A,fun(B,assn))] :
      ( ( I5 = I6 )
     => ( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(A),nat,size_size(list(A)),Xs4) )
       => ( ( aa(list(B),nat,size_size(list(B)),Xsi) = aa(list(B),nat,size_size(list(B)),Xsi2) )
         => ( ! [I2: nat] :
                ( member(nat,I2,I5)
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs2))
                 => ( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(B),nat,size_size(list(B)),Xsi) )
                   => ( ( aa(nat,A,nth(A,Xs2),I2) = aa(nat,A,nth(A,Xs4),I2) )
                      & ( aa(nat,B,nth(B,Xsi),I2) = aa(nat,B,nth(B,Xsi2),I2) )
                      & ( aa(B,assn,aa(A,fun(B,assn),A2,aa(nat,A,nth(A,Xs2),I2)),aa(nat,B,nth(B,Xsi),I2)) = aa(B,assn,aa(A,fun(B,assn),A6,aa(nat,A,nth(A,Xs4),I2)),aa(nat,B,nth(B,Xsi2),I2)) ) ) ) ) )
           => ( vEBT_List_listI_assn(A,B,I5,A2,Xs2,Xsi) = vEBT_List_listI_assn(A,B,I6,A6,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_6236_listI__assn__weak__cong,axiom,
    ! [A: $tType,B: $tType,I5: set(nat),I6: set(nat),A2: fun(A,fun(B,assn)),A6: fun(A,fun(B,assn)),Xs2: list(A),Xs4: list(A),Xsi: list(B),Xsi2: list(B)] :
      ( ( I5 = I6 )
     => ( ( A2 = A6 )
       => ( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(A),nat,size_size(list(A)),Xs4) )
         => ( ( aa(list(B),nat,size_size(list(B)),Xsi) = aa(list(B),nat,size_size(list(B)),Xsi2) )
           => ( ! [I2: nat] :
                  ( member(nat,I2,I5)
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs2))
                   => ( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(B),nat,size_size(list(B)),Xsi) )
                     => ( ( aa(nat,A,nth(A,Xs2),I2) = aa(nat,A,nth(A,Xs4),I2) )
                        & ( aa(nat,B,nth(B,Xsi),I2) = aa(nat,B,nth(B,Xsi2),I2) ) ) ) ) )
             => ( vEBT_List_listI_assn(A,B,I5,A2,Xs2,Xsi) = vEBT_List_listI_assn(A,B,I6,A6,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_6237_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_6238_listI__assn__insert,axiom,
    ! [A: $tType,B: $tType,I: nat,I5: set(nat),Xs2: list(A),A2: fun(A,fun(B,assn)),Xsi: list(B)] :
      ( ~ member(nat,I,I5)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs2))
       => ( vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),insert(nat,I),I5),A2,Xs2,Xsi) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A2,aa(nat,A,nth(A,Xs2),I)),aa(nat,B,nth(B,Xsi),I))),vEBT_List_listI_assn(A,B,I5,A2,Xs2,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_6239_listI__assn__def,axiom,
    ! [A: $tType,B: $tType,I5: set(nat),A2: fun(A,fun(B,assn)),Xs2: list(A),Xsi: list(B)] :
      vEBT_List_listI_assn(A,B,I5,A2,Xs2,Xsi) = aa(assn,assn,
        aa(assn,fun(assn,assn),times_times(assn),
          pure_assn(( ( aa(list(B),nat,size_size(list(B)),Xsi) = aa(list(A),nat,size_size(list(A)),Xs2) )
            & aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),I5),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs2))) ))),
        finite_fold(nat,assn,aa(list(B),fun(nat,fun(assn,assn)),aa(list(A),fun(list(B),fun(nat,fun(assn,assn))),aTP_Lamp_qp(fun(A,fun(B,assn)),fun(list(A),fun(list(B),fun(nat,fun(assn,assn)))),A2),Xs2),Xsi),one_one(assn),I5)) ).

% listI_assn_def
tff(fact_6240_listI__assn__reinsert_H,axiom,
    ! [A: $tType,B: $tType,C: $tType,P: assn,A2: fun(A,fun(B,assn)),Xs2: list(A),I: nat,Xsi: list(B),I5: set(nat),F4: assn,C2: heap_Time_Heap(C),Q: fun(C,assn)] :
      ( entails(P,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A2,aa(nat,A,nth(A,Xs2),I)),aa(nat,B,nth(B,Xsi),I))),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),minus_minus(set(nat),I5),aa(set(nat),set(nat),insert(nat,I),bot_bot(set(nat)))),A2,Xs2,Xsi))),F4))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs2))
       => ( member(nat,I,I5)
         => ( hoare_hoare_triple(C,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),vEBT_List_listI_assn(A,B,I5,A2,Xs2,Xsi)),F4),C2,Q)
           => hoare_hoare_triple(C,P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_6241_fi__rule,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),Ps: assn,F4: assn] :
      ( hoare_hoare_triple(A,P,C2,Q)
     => ( entails(Ps,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),F4))
       => hoare_hoare_triple(A,Ps,C2,aa(assn,fun(A,assn),aTP_Lamp_ec(fun(A,assn),fun(assn,fun(A,assn)),Q),F4)) ) ) ).

% fi_rule
tff(fact_6242_return__cons__rule,axiom,
    ! [A: $tType,P: assn,Q: fun(A,assn),X: A] :
      ( entails(P,aa(A,assn,Q,X))
     => hoare_hoare_triple(A,P,aa(A,heap_Time_Heap(A),heap_Time_return(A),X),Q) ) ).

% return_cons_rule
tff(fact_6243_cond__TBOUND__cons,axiom,
    ! [A: $tType,P: assn,Q: assn,C2: heap_Time_Heap(A),Ba: nat] :
      ( entails(P,Q)
     => ( simple8792101892335181110TBOUND(A,Q,C2,Ba)
       => simple8792101892335181110TBOUND(A,P,C2,Ba) ) ) ).

% cond_TBOUND_cons
tff(fact_6244_cons__pre__rule,axiom,
    ! [A: $tType,P: assn,P3: assn,C2: heap_Time_Heap(A),Q: fun(A,assn)] :
      ( entails(P,P3)
     => ( hoare_hoare_triple(A,P3,C2,Q)
       => hoare_hoare_triple(A,P,C2,Q) ) ) ).

% cons_pre_rule
tff(fact_6245_htt__cons__rule,axiom,
    ! [A: $tType,P3: assn,C2: heap_Time_Heap(A),Q6: fun(A,assn),T3: nat,P: assn,Q: fun(A,assn),T: nat] :
      ( time_htt(A,P3,C2,Q6,T3)
     => ( entails(P,P3)
       => ( ! [X2: A] : entails(aa(A,assn,Q6,X2),aa(A,assn,Q,X2))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),T3),T)
           => time_htt(A,P,C2,Q,T) ) ) ) ) ).

% htt_cons_rule
tff(fact_6246_listI__assn__reinsert,axiom,
    ! [B: $tType,A: $tType,P: assn,A2: fun(A,fun(B,assn)),Xs2: list(A),I: nat,Xsi: list(B),I5: set(nat),F4: assn,Q: assn] :
      ( entails(P,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A2,aa(nat,A,nth(A,Xs2),I)),aa(nat,B,nth(B,Xsi),I))),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),minus_minus(set(nat),I5),aa(set(nat),set(nat),insert(nat,I),bot_bot(set(nat)))),A2,Xs2,Xsi))),F4))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs2))
       => ( member(nat,I,I5)
         => ( entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),vEBT_List_listI_assn(A,B,I5,A2,Xs2,Xsi)),F4),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_6247_aux,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,assn)),Aa2: A,As2: list(A),C2: B,Cs2: list(B)] : finite_fold(nat,assn,aa(list(B),fun(nat,fun(assn,assn)),aa(B,fun(list(B),fun(nat,fun(assn,assn))),aa(list(A),fun(B,fun(list(B),fun(nat,fun(assn,assn)))),aa(A,fun(list(A),fun(B,fun(list(B),fun(nat,fun(assn,assn))))),aTP_Lamp_qq(fun(A,fun(B,assn)),fun(A,fun(list(A),fun(B,fun(list(B),fun(nat,fun(assn,assn)))))),P),Aa2),As2),C2),Cs2),one_one(assn),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),As2)))) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),P,Aa2),C2)),finite_fold(nat,assn,aa(list(B),fun(nat,fun(assn,assn)),aa(list(A),fun(list(B),fun(nat,fun(assn,assn))),aTP_Lamp_qp(fun(A,fun(B,assn)),fun(list(A),fun(list(B),fun(nat,fun(assn,assn)))),P),As2),Cs2),one_one(assn),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),As2)))) ).

% aux
tff(fact_6248_rule__at__index,axiom,
    ! [B: $tType,A: $tType,C: $tType,P: assn,A2: fun(A,fun(B,assn)),Xs2: list(A),Xsi: list(B),F4: assn,I: nat,C2: heap_Time_Heap(C),Q6: fun(C,assn),F6: fun(C,assn)] :
      ( entails(P,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),vEBT_List_list_assn(A,B,A2,Xs2,Xsi)),F4))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs2))
       => ( hoare_hoare_triple(C,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A2,aa(nat,A,nth(A,Xs2),I)),aa(nat,B,nth(B,Xsi),I))),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),minus_minus(set(nat),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs2))),aa(set(nat),set(nat),insert(nat,I),bot_bot(set(nat)))),A2,Xs2,Xsi))),F4),C2,Q6)
         => ( ! [R5: C] : entails(aa(C,assn,Q6,R5),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A2,aa(nat,A,nth(A,Xs2),I)),aa(nat,B,nth(B,Xsi),I))),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),minus_minus(set(nat),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs2))),aa(set(nat),set(nat),insert(nat,I),bot_bot(set(nat)))),A2,Xs2,Xsi))),aa(C,assn,F6,R5)))
           => hoare_hoare_triple(C,P,C2,aa(fun(C,assn),fun(C,assn),aa(list(B),fun(fun(C,assn),fun(C,assn)),aa(list(A),fun(list(B),fun(fun(C,assn),fun(C,assn))),aTP_Lamp_qr(fun(A,fun(B,assn)),fun(list(A),fun(list(B),fun(fun(C,assn),fun(C,assn)))),A2),Xs2),Xsi),F6)) ) ) ) ) ).

% rule_at_index
tff(fact_6249_listI__assn__reinsert__upd_H,axiom,
    ! [A: $tType,B: $tType,C: $tType,P: assn,A2: fun(A,fun(B,assn)),X: A,Xi: B,I5: set(nat),I: nat,Xs2: list(A),Xsi: list(B),F4: assn,C2: heap_Time_Heap(C),Q: fun(C,assn)] :
      ( entails(P,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A2,X),Xi)),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),minus_minus(set(nat),I5),aa(set(nat),set(nat),insert(nat,I),bot_bot(set(nat)))),A2,Xs2,Xsi))),F4))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs2))
       => ( member(nat,I,I5)
         => ( hoare_hoare_triple(C,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),vEBT_List_listI_assn(A,B,I5,A2,list_update(A,Xs2,I,X),list_update(B,Xsi,I,Xi))),F4),C2,Q)
           => hoare_hoare_triple(C,P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_6250_list__update__overwrite,axiom,
    ! [A: $tType,Xs2: list(A),I: nat,X: A,Y: A] : list_update(A,list_update(A,Xs2,I,X),I,Y) = list_update(A,Xs2,I,Y) ).

% list_update_overwrite
tff(fact_6251_list__update__nonempty,axiom,
    ! [A: $tType,Xs2: list(A),K: nat,X: A] :
      ( ( list_update(A,Xs2,K,X) = nil(A) )
    <=> ( Xs2 = nil(A) ) ) ).

% list_update_nonempty
tff(fact_6252_length__list__update,axiom,
    ! [A: $tType,Xs2: list(A),I: nat,X: A] : aa(list(A),nat,size_size(list(A)),list_update(A,Xs2,I,X)) = aa(list(A),nat,size_size(list(A)),Xs2) ).

% length_list_update
tff(fact_6253_list__update__id,axiom,
    ! [A: $tType,Xs2: list(A),I: nat] : list_update(A,Xs2,I,aa(nat,A,nth(A,Xs2),I)) = Xs2 ).

% list_update_id
tff(fact_6254_nth__list__update__neq,axiom,
    ! [A: $tType,I: nat,J: nat,Xs2: list(A),X: A] :
      ( ( I != J )
     => ( aa(nat,A,nth(A,list_update(A,Xs2,I,X)),J) = aa(nat,A,nth(A,Xs2),J) ) ) ).

% nth_list_update_neq
tff(fact_6255_list__update__beyond,axiom,
    ! [A: $tType,Xs2: list(A),I: nat,X: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs2)),I)
     => ( list_update(A,Xs2,I,X) = Xs2 ) ) ).

% list_update_beyond
tff(fact_6256_list__assn__aux__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,assn)),L3: list(B)] : vEBT_List_list_assn(A,B,P,nil(A),L3) = pure_assn(L3 = nil(B)) ).

% list_assn_aux_simps(1)
tff(fact_6257_list__assn__aux__simps_I2_J,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,assn)),L: list(A)] : vEBT_List_list_assn(A,B,P,L,nil(B)) = pure_assn(L = nil(A)) ).

% list_assn_aux_simps(2)
tff(fact_6258_nth__update__invalid,axiom,
    ! [A: $tType,I: nat,L: list(A),J: nat,X: A] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))
     => ( aa(nat,A,nth(A,list_update(A,L,J,X)),I) = aa(nat,A,nth(A,L),I) ) ) ).

% nth_update_invalid
tff(fact_6259_nth__list__update__eq,axiom,
    ! [A: $tType,I: nat,Xs2: list(A),X: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs2))
     => ( aa(nat,A,nth(A,list_update(A,Xs2,I,X)),I) = X ) ) ).

% nth_list_update_eq
tff(fact_6260_set__swap,axiom,
    ! [A: $tType,I: nat,Xs2: list(A),J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs2))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs2))
       => ( aa(list(A),set(A),set2(A),list_update(A,list_update(A,Xs2,I,aa(nat,A,nth(A,Xs2),J)),J,aa(nat,A,nth(A,Xs2),I))) = aa(list(A),set(A),set2(A),Xs2) ) ) ) ).

% set_swap
tff(fact_6261_list__assn__simps_I3_J,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,assn)),Aa2: A,As2: list(A)] : vEBT_List_list_assn(A,B,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Aa2),As2),nil(B)) = bot_bot(assn) ).

% list_assn_simps(3)
tff(fact_6262_list__assn__simps_I4_J,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,assn)),C2: B,Cs2: list(B)] : vEBT_List_list_assn(A,B,P,nil(A),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),C2),Cs2)) = bot_bot(assn) ).

% list_assn_simps(4)
tff(fact_6263_list__assn_Osimps_I3_J,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,fun(B,assn)),V: A,Va: list(A)] : vEBT_List_list_assn(A,B,Uu2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va),nil(B)) = bot_bot(assn) ).

% list_assn.simps(3)
tff(fact_6264_list__assn_Osimps_I4_J,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,fun(B,assn)),V: B,Va: list(B)] : vEBT_List_list_assn(A,B,Uu2,nil(A),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V),Va)) = bot_bot(assn) ).

% list_assn.simps(4)
tff(fact_6265_list__update__code_I3_J,axiom,
    ! [A: $tType,X: A,Xs2: list(A),I: nat,Y: A] : list_update(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs2),aa(nat,nat,suc,I),Y) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),list_update(A,Xs2,I,Y)) ).

% list_update_code(3)
tff(fact_6266_list__update__code_I2_J,axiom,
    ! [A: $tType,X: A,Xs2: list(A),Y: A] : list_update(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs2),zero_zero(nat),Y) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs2) ).

% list_update_code(2)
tff(fact_6267_list__update__swap,axiom,
    ! [A: $tType,I: nat,I7: nat,Xs2: list(A),X: A,X7: A] :
      ( ( I != I7 )
     => ( list_update(A,list_update(A,Xs2,I,X),I7,X7) = list_update(A,list_update(A,Xs2,I7,X7),I,X) ) ) ).

% list_update_swap
tff(fact_6268_set__update__subsetI,axiom,
    ! [A: $tType,Xs2: list(A),A2: set(A),X: A,I: nat] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs2)),A2)
     => ( member(A,X,A2)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),list_update(A,Xs2,I,X))),A2) ) ) ).

% set_update_subsetI
tff(fact_6269_list__assn__aux__ineq__len,axiom,
    ! [B: $tType,A: $tType,L: list(A),Li2: list(B),A2: fun(A,fun(B,assn))] :
      ( ( aa(list(A),nat,size_size(list(A)),L) != aa(list(B),nat,size_size(list(B)),Li2) )
     => ( vEBT_List_list_assn(A,B,A2,L,Li2) = bot_bot(assn) ) ) ).

% list_assn_aux_ineq_len
tff(fact_6270_list__update_Osimps_I1_J,axiom,
    ! [A: $tType,I: nat,V: A] : list_update(A,nil(A),I,V) = nil(A) ).

% list_update.simps(1)
tff(fact_6271_list__update__code_I1_J,axiom,
    ! [A: $tType,I: nat,Y: A] : list_update(A,nil(A),I,Y) = nil(A) ).

% list_update_code(1)
tff(fact_6272_list__assn__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,assn)),Aa2: A,As2: list(A),C2: B,Cs2: list(B)] : vEBT_List_list_assn(A,B,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Aa2),As2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),C2),Cs2)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),P,Aa2),C2)),vEBT_List_list_assn(A,B,P,As2,Cs2)) ).

% list_assn_simps(2)
tff(fact_6273_list__assn_Osimps_I2_J,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,assn)),Aa2: A,As2: list(A),C2: B,Cs2: list(B)] : vEBT_List_list_assn(A,B,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Aa2),As2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),C2),Cs2)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),P,Aa2),C2)),vEBT_List_list_assn(A,B,P,As2,Cs2)) ).

% list_assn.simps(2)
tff(fact_6274_map__update,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs2: list(B),K: nat,Y: B] : aa(list(B),list(A),map(B,A,F2),list_update(B,Xs2,K,Y)) = list_update(A,aa(list(B),list(A),map(B,A,F2),Xs2),K,aa(B,A,F2,Y)) ).

% map_update
tff(fact_6275_list__assn_Osimps_I1_J,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,assn))] : vEBT_List_list_assn(A,B,P,nil(A),nil(B)) = one_one(assn) ).

% list_assn.simps(1)
tff(fact_6276_list__assn__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,assn))] : vEBT_List_list_assn(A,B,P,nil(A),nil(B)) = one_one(assn) ).

% list_assn_simps(1)
tff(fact_6277_in__set__upd__eq__aux,axiom,
    ! [A: $tType,I: nat,L: list(A),X: A,Y: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))
     => ( member(A,X,aa(list(A),set(A),set2(A),list_update(A,L,I,Y)))
      <=> ( ( X = Y )
          | ! [Y4: A] : member(A,X,aa(list(A),set(A),set2(A),list_update(A,L,I,Y4))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_6278_in__set__upd__cases,axiom,
    ! [A: $tType,X: A,L: list(A),I: nat,Y: A] :
      ( member(A,X,aa(list(A),set(A),set2(A),list_update(A,L,I,Y)))
     => ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))
         => ( X != Y ) )
       => member(A,X,aa(list(A),set(A),set2(A),L)) ) ) ).

% in_set_upd_cases
tff(fact_6279_set__update__memI,axiom,
    ! [A: $tType,Na: nat,Xs2: list(A),X: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs2))
     => member(A,X,aa(list(A),set(A),set2(A),list_update(A,Xs2,Na,X))) ) ).

% set_update_memI
tff(fact_6280_in__set__upd__eq,axiom,
    ! [A: $tType,I: nat,L: list(A),X: A,Y: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))
     => ( member(A,X,aa(list(A),set(A),set2(A),list_update(A,L,I,Y)))
      <=> ( ( X = Y )
          | ( member(A,X,aa(list(A),set(A),set2(A),L))
            & ! [Y4: A] : member(A,X,aa(list(A),set(A),set2(A),list_update(A,L,I,Y4))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_6281_list__update__same__conv,axiom,
    ! [A: $tType,I: nat,Xs2: list(A),X: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs2))
     => ( ( list_update(A,Xs2,I,X) = Xs2 )
      <=> ( aa(nat,A,nth(A,Xs2),I) = X ) ) ) ).

% list_update_same_conv
tff(fact_6282_nth__list__update_H,axiom,
    ! [A: $tType,L: list(A),I: nat,X: A,J: nat] :
      aa(nat,A,nth(A,list_update(A,L,I,X)),J) = $ite(
        ( ( I = J )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L)) ),
        X,
        aa(nat,A,nth(A,L),J) ) ).

% nth_list_update'
tff(fact_6283_nth__list__update,axiom,
    ! [A: $tType,I: nat,Xs2: list(A),X: A,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs2))
     => ( aa(nat,A,nth(A,list_update(A,Xs2,I,X)),J) = $ite(I = J,X,aa(nat,A,nth(A,Xs2),J)) ) ) ).

% nth_list_update
tff(fact_6284_set__update__subset__insert,axiom,
    ! [A: $tType,Xs2: list(A),I: nat,X: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),list_update(A,Xs2,I,X))),aa(set(A),set(A),insert(A,X),aa(list(A),set(A),set2(A),Xs2))) ).

% set_update_subset_insert
tff(fact_6285_subst__not__in,axiom,
    ! [A: $tType,B: $tType,I: nat,I5: set(nat),Xs2: list(A),A2: fun(A,fun(B,assn)),X12: A,Xsi: list(B),X23: B] :
      ( ~ member(nat,I,I5)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs2))
       => ( vEBT_List_listI_assn(A,B,I5,A2,list_update(A,Xs2,I,X12),list_update(B,Xsi,I,X23)) = vEBT_List_listI_assn(A,B,I5,A2,Xs2,Xsi) ) ) ) ).

% subst_not_in
tff(fact_6286_map__upd__eq,axiom,
    ! [B: $tType,A: $tType,I: nat,L: list(A),F2: fun(A,B),X: A] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))
       => ( aa(A,B,F2,aa(nat,A,nth(A,L),I)) = aa(A,B,F2,X) ) )
     => ( aa(list(A),list(B),map(A,B,F2),list_update(A,L,I,X)) = aa(list(A),list(B),map(A,B,F2),L) ) ) ).

% map_upd_eq
tff(fact_6287_insert__swap__set__eq,axiom,
    ! [A: $tType,I: nat,L: list(A),X: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))
     => ( aa(set(A),set(A),insert(A,aa(nat,A,nth(A,L),I)),aa(list(A),set(A),set2(A),list_update(A,L,I,X))) = aa(set(A),set(A),insert(A,X),aa(list(A),set(A),set2(A),L)) ) ) ).

% insert_swap_set_eq
tff(fact_6288_list__assn_Oelims,axiom,
    ! [B: $tType,A: $tType,X: fun(A,fun(B,assn)),Xa2: list(A),Xb: list(B),Y: assn] :
      ( ( vEBT_List_list_assn(A,B,X,Xa2,Xb) = Y )
     => ( ( ( Xa2 = nil(A) )
         => ( ( Xb = nil(B) )
           => ( Y != one_one(assn) ) ) )
       => ( ! [A3: A,As: list(A)] :
              ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),As) )
             => ! [C4: B,Cs: list(B)] :
                  ( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),C4),Cs) )
                 => ( Y != aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),X,A3),C4)),vEBT_List_list_assn(A,B,X,As,Cs)) ) ) )
         => ( ( ? [V4: A,Va2: list(A)] : Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V4),Va2)
             => ( ( Xb = nil(B) )
               => ( Y != bot_bot(assn) ) ) )
           => ~ ( ( Xa2 = nil(A) )
               => ( ? [V4: B,Va2: list(B)] : Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V4),Va2)
                 => ( Y != bot_bot(assn) ) ) ) ) ) ) ) ).

% list_assn.elims
tff(fact_6289_list__assn__conv__idx,axiom,
    ! [A: $tType,B: $tType,A2: fun(A,fun(B,assn)),Xs2: list(A),Xsi: list(B)] : vEBT_List_list_assn(A,B,A2,Xs2,Xsi) = vEBT_List_listI_assn(A,B,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs2)),A2,Xs2,Xsi) ).

% list_assn_conv_idx
tff(fact_6290_listI__assn__conv,axiom,
    ! [A: $tType,B: $tType,Na: nat,Xs2: list(A),A2: fun(A,fun(B,assn)),Xsi: list(B)] :
      ( ( Na = aa(list(A),nat,size_size(list(A)),Xs2) )
     => ( vEBT_List_listI_assn(A,B,set_or7035219750837199246ssThan(nat,zero_zero(nat),Na),A2,Xs2,Xsi) = vEBT_List_list_assn(A,B,A2,Xs2,Xsi) ) ) ).

% listI_assn_conv
tff(fact_6291_listI__assn__subst,axiom,
    ! [A: $tType,B: $tType,I: nat,I5: set(nat),Xs2: list(A),A2: fun(A,fun(B,assn)),X12: A,Xsi: list(B),X23: B] :
      ( ~ member(nat,I,I5)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs2))
       => ( vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),insert(nat,I),I5),A2,list_update(A,Xs2,I,X12),list_update(B,Xsi,I,X23)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A2,X12),X23)),vEBT_List_listI_assn(A,B,I5,A2,Xs2,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_6292_listI__assn__conv_H,axiom,
    ! [B: $tType,A: $tType,Na: nat,Xs2: list(A),A2: fun(A,fun(B,assn)),Xsi: list(B),F4: assn] :
      ( ( Na = aa(list(A),nat,size_size(list(A)),Xs2) )
     => ( aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),vEBT_List_listI_assn(A,B,set_or7035219750837199246ssThan(nat,zero_zero(nat),Na),A2,Xs2,Xsi)),F4) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),vEBT_List_list_assn(A,B,A2,Xs2,Xsi)),F4) ) ) ).

% listI_assn_conv'
tff(fact_6293_listI__assn__reinsert__upd,axiom,
    ! [B: $tType,A: $tType,P: assn,A2: fun(A,fun(B,assn)),X: A,Xi: B,I5: set(nat),I: nat,Xs2: list(A),Xsi: list(B),F4: assn,Q: assn] :
      ( entails(P,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A2,X),Xi)),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),minus_minus(set(nat),I5),aa(set(nat),set(nat),insert(nat,I),bot_bot(set(nat)))),A2,Xs2,Xsi))),F4))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs2))
       => ( member(nat,I,I5)
         => ( entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),vEBT_List_listI_assn(A,B,I5,A2,list_update(A,Xs2,I,X),list_update(B,Xsi,I,Xi))),F4),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_6294_set__Cons__sing__Nil,axiom,
    ! [A: $tType,A2: set(A)] : set_Cons(A,A2,aa(set(list(A)),set(list(A)),insert(list(A),nil(A)),bot_bot(set(list(A))))) = aa(set(A),set(list(A)),image(A,list(A),aTP_Lamp_qs(A,list(A))),A2) ).

% set_Cons_sing_Nil
tff(fact_6295_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_6296_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_6297_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_6298_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_6299_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_6300_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_6301_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_6302_not__msb__from__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),V),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),V) ) ) ).

% not_msb_from_less
tff(fact_6303_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))),aa(word(A),nat,size_size(word(A)),X)),zero_zero(int))) ) ).

% word_sint_msb_eq
tff(fact_6304_msb__big,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A)] :
          ( most_s684356279273892711sb_msb(word(A),Aa2)
        <=> aa(word(A),$o,aa(word(A),fun(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))))),Aa2) ) ) ).

% msb_big
tff(fact_6305_listset_Osimps_I1_J,axiom,
    ! [A: $tType] : listset(A,nil(set(A))) = aa(set(list(A)),set(list(A)),insert(list(A),nil(A)),bot_bot(set(list(A)))) ).

% listset.simps(1)
tff(fact_6306_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_6307_inj__map__eq__map,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Xs2: list(A),Ysb: list(A)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( ( aa(list(A),list(B),map(A,B,F2),Xs2) = aa(list(A),list(B),map(A,B,F2),Ysb) )
      <=> ( Xs2 = Ysb ) ) ) ).

% inj_map_eq_map
tff(fact_6308_inj__mult__left,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Aa2: A] :
          ( inj_on(A,A,aa(A,fun(A,A),times_times(A),Aa2),top_top(set(A)))
        <=> ( Aa2 != zero_zero(A) ) ) ) ).

% inj_mult_left
tff(fact_6309_inj__divide__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Aa2: A] :
          ( inj_on(A,A,aTP_Lamp_qt(A,fun(A,A),Aa2),top_top(set(A)))
        <=> ( Aa2 != zero_zero(A) ) ) ) ).

% inj_divide_right
tff(fact_6310_inj__mapI,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => inj_on(list(A),list(B),map(A,B,F2),top_top(set(list(A)))) ) ).

% inj_mapI
tff(fact_6311_inj__map,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] :
      ( inj_on(list(A),list(B),map(A,B,F2),top_top(set(list(A))))
    <=> inj_on(A,B,F2,top_top(set(A))) ) ).

% inj_map
tff(fact_6312_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_qu(B,B),aa(set(A),set(B),image(A,B,G),A2)) = groups7121269368397514597t_prod(A,B,G,A2) ) ) ) ).

% prod.image_eq
tff(fact_6313_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_qv(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_6314_finite__Collect,axiom,
    ! [A: $tType,B: $tType,Sa: set(A),F2: fun(B,A)] :
      ( aa(set(A),$o,finite_finite2(A),Sa)
     => ( inj_on(B,A,F2,top_top(set(B)))
       => aa(set(B),$o,finite_finite2(B),collect(B,aa(fun(B,A),fun(B,$o),aTP_Lamp_qw(set(A),fun(fun(B,A),fun(B,$o)),Sa),F2))) ) ) ).

% finite_Collect
tff(fact_6315_finite__inverse__image,axiom,
    ! [A: $tType,B: $tType,A2: set(A),F2: fun(B,A)] :
      ( aa(set(A),$o,finite_finite2(A),A2)
     => ( inj_on(B,A,F2,top_top(set(B)))
       => aa(set(B),$o,finite_finite2(B),collect(B,aa(fun(B,A),fun(B,$o),aTP_Lamp_qw(set(A),fun(fun(B,A),fun(B,$o)),A2),F2))) ) ) ).

% finite_inverse_image
tff(fact_6316_sorted__list__of__set_Oinj__on,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => inj_on(A,A,aTP_Lamp_qx(A,A),top_top(set(A))) ) ).

% sorted_list_of_set.inj_on
tff(fact_6317_inj__diff__right,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Aa2: A] : inj_on(A,A,aTP_Lamp_qa(A,fun(A,A),Aa2),top_top(set(A))) ) ).

% inj_diff_right
tff(fact_6318_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_qy(fun(A,B),fun(A,fun(C,B)),F2),top_top(set(A))) ) ).

% inj_fun
tff(fact_6319_inj__add__left,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [Aa2: A] : inj_on(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),top_top(set(A))) ) ).

% inj_add_left
tff(fact_6320_inj__on__add_H,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [Aa2: A,A2: set(A)] : inj_on(A,A,aTP_Lamp_qz(A,fun(A,A),Aa2),A2) ) ).

% inj_on_add'
tff(fact_6321_inj__on__add,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [Aa2: A,A2: set(A)] : inj_on(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),A2) ) ).

% inj_on_add
tff(fact_6322_inj__on__id2,axiom,
    ! [A: $tType,A2: set(A)] : inj_on(A,A,aTP_Lamp_pc(A,A),A2) ).

% inj_on_id2
tff(fact_6323_linorder__inj__onI,axiom,
    ! [B: $tType,A: $tType] :
      ( order(A)
     => ! [A2: set(A),F2: fun(A,B)] :
          ( ! [X2: A,Y2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Y2)
             => ( member(A,X2,A2)
               => ( member(A,Y2,A2)
                 => ( aa(A,B,F2,X2) != aa(A,B,F2,Y2) ) ) ) )
         => ( ! [X2: A,Y2: A] :
                ( member(A,X2,A2)
               => ( member(A,Y2,A2)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y2)
                    | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y2),X2) ) ) )
           => inj_on(A,B,F2,A2) ) ) ) ).

% linorder_inj_onI
tff(fact_6324_inj__on__mult,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [Aa2: A,A2: set(A)] :
          ( ( Aa2 != zero_zero(A) )
         => inj_on(A,A,aa(A,fun(A,A),times_times(A),Aa2),A2) ) ) ).

% inj_on_mult
tff(fact_6325_finite__inverse__image__gen,axiom,
    ! [A: $tType,B: $tType,A2: set(A),F2: fun(B,A),D3: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),A2)
     => ( inj_on(B,A,F2,D3)
       => aa(set(B),$o,finite_finite2(B),collect(B,aa(set(B),fun(B,$o),aa(fun(B,A),fun(set(B),fun(B,$o)),aTP_Lamp_ra(set(A),fun(fun(B,A),fun(set(B),fun(B,$o))),A2),F2),D3))) ) ) ).

% finite_inverse_image_gen
tff(fact_6326_map__injective,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs2: list(B),Ysb: list(B)] :
      ( ( aa(list(B),list(A),map(B,A,F2),Xs2) = aa(list(B),list(A),map(B,A,F2),Ysb) )
     => ( inj_on(B,A,F2,top_top(set(B)))
       => ( Xs2 = Ysb ) ) ) ).

% map_injective
tff(fact_6327_inj__mapD,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] :
      ( inj_on(list(A),list(B),map(A,B,F2),top_top(set(list(A))))
     => inj_on(A,B,F2,top_top(set(A))) ) ).

% inj_mapD
tff(fact_6328_map__inj__on,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs2: list(B),Ysb: list(B)] :
      ( ( aa(list(B),list(A),map(B,A,F2),Xs2) = aa(list(B),list(A),map(B,A,F2),Ysb) )
     => ( inj_on(B,A,F2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(list(B),set(B),set2(B),Xs2)),aa(list(B),set(B),set2(B),Ysb)))
       => ( Xs2 = Ysb ) ) ) ).

% map_inj_on
tff(fact_6329_inj__on__map__eq__map,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Xs2: list(A),Ysb: list(A)] :
      ( inj_on(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs2)),aa(list(A),set(A),set2(A),Ysb)))
     => ( ( aa(list(A),list(B),map(A,B,F2),Xs2) = aa(list(A),list(B),map(A,B,F2),Ysb) )
      <=> ( Xs2 = Ysb ) ) ) ).

% inj_on_map_eq_map
tff(fact_6330_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_6331_listset_Osimps_I2_J,axiom,
    ! [A: $tType,A2: set(A),As3: list(set(A))] : listset(A,aa(list(set(A)),list(set(A)),aa(set(A),fun(list(set(A)),list(set(A))),cons(set(A)),A2),As3)) = set_Cons(A,A2,listset(A,As3)) ).

% listset.simps(2)
tff(fact_6332_inj__on__map__inv__f,axiom,
    ! [B: $tType,A: $tType,L: list(A),A2: set(A),F2: fun(A,B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),L)),A2)
     => ( inj_on(A,B,F2,A2)
       => ( aa(list(B),list(A),map(B,A,inv_on(A,B,F2,A2)),aa(list(A),list(B),map(A,B,F2),L)) = L ) ) ) ).

% inj_on_map_inv_f
tff(fact_6333_If__the__inv__into__in__Func,axiom,
    ! [B: $tType,A: $tType,G: fun(A,B),C5: set(A),B2: set(A),X: A] :
      ( inj_on(A,B,G,C5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C5),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_rb(fun(A,B),fun(set(A),fun(A,fun(B,A))),G),C5),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_6334_inv__on__f__f,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A2: set(A),X: A] :
      ( inj_on(A,B,F2,A2)
     => ( member(A,X,A2)
       => ( aa(B,A,inv_on(A,B,F2,A2),aa(A,B,F2,X)) = X ) ) ) ).

% inv_on_f_f
tff(fact_6335_inj__singleton,axiom,
    ! [A: $tType,A2: set(A)] : inj_on(A,set(A),aTP_Lamp_rc(A,set(A)),A2) ).

% inj_singleton
tff(fact_6336_inj__graph,axiom,
    ! [B: $tType,A: $tType] : inj_on(fun(A,B),set(product_prod(A,B)),aTP_Lamp_re(fun(A,B),set(product_prod(A,B))),top_top(set(fun(A,B)))) ).

% inj_graph
tff(fact_6337_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_6338_signed_Osorted__list__of__set_Oinj__on,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => inj_on(word(A),word(A),aTP_Lamp_rf(word(A),word(A)),top_top(set(word(A)))) ) ).

% signed.sorted_list_of_set.inj_on
tff(fact_6339_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_6340_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_6341_inj__on__Cons1,axiom,
    ! [A: $tType,X: A,A2: set(list(A))] : inj_on(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),A2) ).

% inj_on_Cons1
tff(fact_6342_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_rg(list(A),fun(A,list(A)))),X5) ).

% inj_split_Cons
tff(fact_6343_inj__Suc,axiom,
    ! [N3: set(nat)] : inj_on(nat,nat,suc,N3) ).

% inj_Suc
tff(fact_6344_inj__on__of__nat,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N3: set(nat)] : inj_on(nat,A,semiring_1_of_nat(A),N3) ) ).

% inj_on_of_nat
tff(fact_6345_inj__Pair_I2_J,axiom,
    ! [B: $tType,A: $tType,C2: fun(A,B),Sa: set(A)] : inj_on(A,product_prod(B,A),aTP_Lamp_rh(fun(A,B),fun(A,product_prod(B,A)),C2),Sa) ).

% inj_Pair(2)
tff(fact_6346_inj__Pair_I1_J,axiom,
    ! [B: $tType,A: $tType,C2: fun(A,B),Sa: set(A)] : inj_on(A,product_prod(A,B),aTP_Lamp_ri(fun(A,B),fun(A,product_prod(A,B)),C2),Sa) ).

% inj_Pair(1)
tff(fact_6347_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_ri(fun(A,B),fun(A,product_prod(A,B)),F2),X5) ).

% inj_on_convol_ident
tff(fact_6348_inj__on__diff__nat,axiom,
    ! [N3: set(nat),K: nat] :
      ( ! [N: nat] :
          ( member(nat,N,N3)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N) )
     => inj_on(nat,nat,aTP_Lamp_pz(nat,fun(nat,nat),K),N3) ) ).

% inj_on_diff_nat
tff(fact_6349_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_rj(A,fun(B,product_prod(B,A)))),A2) ).

% swap_inj_on
tff(fact_6350_inj__on__set__encode,axiom,
    inj_on(set(nat),nat,nat_set_encode,collect(set(nat),finite_finite2(nat))) ).

% inj_on_set_encode
tff(fact_6351_f__inv__on__f,axiom,
    ! [B: $tType,A: $tType,Y: A,F2: fun(B,A),A2: set(B)] :
      ( member(A,Y,aa(set(B),set(A),image(B,A,F2),A2))
     => ( aa(B,A,F2,aa(A,B,inv_on(B,A,F2,A2),Y)) = Y ) ) ).

% f_inv_on_f
tff(fact_6352_inv__on__f__range,axiom,
    ! [A: $tType,B: $tType,Y: A,F2: fun(B,A),A2: set(B)] :
      ( member(A,Y,aa(set(B),set(A),image(B,A,F2),A2))
     => member(B,aa(A,B,inv_on(B,A,F2,A2),Y),A2) ) ).

% inv_on_f_range
tff(fact_6353_finite__imp__nat__seg__image__inj__on,axiom,
    ! [A: $tType,A2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A2)
     => ? [N: nat,F3: fun(nat,A)] :
          ( ( A2 = aa(set(nat),set(A),image(nat,A,F3),collect(nat,aTP_Lamp_gx(nat,fun(nat,$o),N))) )
          & inj_on(nat,A,F3,collect(nat,aTP_Lamp_gx(nat,fun(nat,$o),N))) ) ) ).

% finite_imp_nat_seg_image_inj_on
tff(fact_6354_finite__imp__inj__to__nat__seg,axiom,
    ! [A: $tType,A2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A2)
     => ? [F3: fun(A,nat),N: nat] :
          ( ( aa(set(A),set(nat),image(A,nat,F3),A2) = collect(nat,aTP_Lamp_gx(nat,fun(nat,$o),N)) )
          & inj_on(A,nat,F3,A2) ) ) ).

% finite_imp_inj_to_nat_seg
tff(fact_6355_finite__imp__inj__to__nat__seg_H,axiom,
    ! [A: $tType,A2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A2)
     => ~ ! [F3: fun(A,nat)] :
            ( ? [N: nat] : aa(set(A),set(nat),image(A,nat,F3),A2) = collect(nat,aTP_Lamp_gx(nat,fun(nat,$o),N))
           => ~ inj_on(A,nat,F3,A2) ) ) ).

% finite_imp_inj_to_nat_seg'
tff(fact_6356_inv__on__def,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A2: set(A),X: B] : aa(B,A,inv_on(A,B,F2,A2),X) = fChoice(A,aa(B,fun(A,$o),aa(set(A),fun(B,fun(A,$o)),aTP_Lamp_rk(fun(A,B),fun(set(A),fun(B,fun(A,$o))),F2),A2),X)) ).

% inv_on_def
tff(fact_6357_inj__sgn__power,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => inj_on(real,real,aTP_Lamp_qi(nat,fun(real,real),Na),top_top(set(real))) ) ).

% inj_sgn_power
tff(fact_6358_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_6359_image__Collect__subsetI,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F2: fun(A,B),B2: set(B)] :
      ( ! [X2: A] :
          ( aa(A,$o,P,X2)
         => member(B,aa(A,B,F2,X2),B2) )
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F2),collect(A,P))),B2) ) ).

% image_Collect_subsetI
tff(fact_6360_DERIV__real__root__generic,axiom,
    ! [Na: nat,X: real,D3: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$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,aa(real,fun(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,$o,aa(real,fun(real,$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_6361_DERIV__at__within__shift,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Y: A,Z: A,X: A,Sa: set(A)] :
          ( has_field_derivative(A,F2,Y,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),X),aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),Z)),Sa)))
        <=> has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_rl(fun(A,A),fun(A,fun(A,A)),F2),Z),Y,topolo174197925503356063within(A,X,Sa)) ) ) ).

% DERIV_at_within_shift
tff(fact_6362_DERIV__cos__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K: A,Xa2: A] : has_field_derivative(A,aTP_Lamp_rm(A,fun(A,A),K),aa(A,A,uminus_uminus(A),sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa2),K))),topolo174197925503356063within(A,Xa2,top_top(set(A)))) ) ).

% DERIV_cos_add
tff(fact_6363_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_rn(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_6364_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_ro(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_6365_DERIV__chain_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,X: A,S: set(A),G: fun(A,A),E4: A] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,X,S))
         => ( 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_rp(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,S)) ) ) ) ).

% DERIV_chain'
tff(fact_6366_DERIV__chain2,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Da: A,G: fun(A,A),X: A,Db: A,S: 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,S))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_rq(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,S)) ) ) ) ).

% DERIV_chain2
tff(fact_6367_DERIV__chain3,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [G: fun(A,A),G2: fun(A,A),F2: fun(A,A),F7: A,X: A] :
          ( ! [X2: A] : has_field_derivative(A,G,aa(A,A,G2,X2),topolo174197925503356063within(A,X2,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_rq(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,G2,aa(A,A,F2,X))),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ) ).

% DERIV_chain3
tff(fact_6368_DERIV__chain__s,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [S: set(A),G: fun(A,A),G2: fun(A,A),F2: fun(A,A),F7: A,X: A] :
          ( ! [X2: A] :
              ( member(A,X2,S)
             => has_field_derivative(A,G,aa(A,A,G2,X2),topolo174197925503356063within(A,X2,top_top(set(A)))) )
         => ( has_field_derivative(A,F2,F7,topolo174197925503356063within(A,X,top_top(set(A))))
           => ( member(A,aa(A,A,F2,X),S)
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_rq(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,G2,aa(A,A,F2,X))),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ) ) ).

% DERIV_chain_s
tff(fact_6369_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_rr(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_6370_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_rs(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_6371_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,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [D4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D4)
            & ! [H4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H4)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H4),D4)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),H4))),aa(real,real,F2,X)) ) ) ) ) ) ).

% DERIV_neg_dec_right
tff(fact_6372_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,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [D4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D4)
            & ! [H4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H4)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H4),D4)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,X)),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),H4))) ) ) ) ) ) ).

% DERIV_pos_inc_right
tff(fact_6373_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,aa(A,fun(A,A),plus_plus(A),X),Z),top_top(set(A))))
        <=> has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_rt(fun(A,A),fun(A,fun(A,A)),F2),Z),Y,topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% DERIV_shift
tff(fact_6374_deriv__nonneg__imp__mono,axiom,
    ! [Aa2: real,Ba: real,G: fun(real,real),G2: fun(real,real)] :
      ( ! [X2: real] :
          ( member(real,X2,set_or1337092689740270186AtMost(real,Aa2,Ba))
         => has_field_derivative(real,G,aa(real,real,G2,X2),topolo174197925503356063within(real,X2,top_top(set(real)))) )
     => ( ! [X2: real] :
            ( member(real,X2,set_or1337092689740270186AtMost(real,Aa2,Ba))
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,G2,X2)) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),Ba)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,G,Aa2)),aa(real,real,G,Ba)) ) ) ) ).

% deriv_nonneg_imp_mono
tff(fact_6375_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [Aa2: real,Ba: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),Ba)
     => ( ! [X2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),X2)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),Ba)
             => ? [Y3: real] :
                  ( has_field_derivative(real,F2,Y3,topolo174197925503356063within(real,X2,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y3) ) ) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,F2,Aa2)),aa(real,real,F2,Ba)) ) ) ).

% DERIV_nonneg_imp_nondecreasing
tff(fact_6376_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [Aa2: real,Ba: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),Ba)
     => ( ! [X2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),X2)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),Ba)
             => ? [Y3: real] :
                  ( has_field_derivative(real,F2,Y3,topolo174197925503356063within(real,X2,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y3),zero_zero(real)) ) ) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,F2,Ba)),aa(real,real,F2,Aa2)) ) ) ).

% DERIV_nonpos_imp_nonincreasing
tff(fact_6377_DERIV__pos__imp__increasing,axiom,
    ! [Aa2: real,Ba: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Aa2),Ba)
     => ( ! [X2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),X2)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),Ba)
             => ? [Y3: real] :
                  ( has_field_derivative(real,F2,Y3,topolo174197925503356063within(real,X2,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y3) ) ) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,Aa2)),aa(real,real,F2,Ba)) ) ) ).

% DERIV_pos_imp_increasing
tff(fact_6378_DERIV__neg__imp__decreasing,axiom,
    ! [Aa2: real,Ba: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Aa2),Ba)
     => ( ! [X2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),X2)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),Ba)
             => ? [Y3: real] :
                  ( has_field_derivative(real,F2,Y3,topolo174197925503356063within(real,X2,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y3),zero_zero(real)) ) ) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,Ba)),aa(real,real,F2,Aa2)) ) ) ).

% DERIV_neg_imp_decreasing
tff(fact_6379_DERIV__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,X: A,S: set(A),G: fun(A,A),E4: A] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,X,S))
         => ( has_field_derivative(A,G,E4,topolo174197925503356063within(A,X,S))
           => ( ( aa(A,A,G,X) != zero_zero(A) )
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_ru(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,S)) ) ) ) ) ).

% DERIV_divide
tff(fact_6380_DERIV__ident,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F4: filter(A)] : has_field_derivative(A,aTP_Lamp_rv(A,A),one_one(A),F4) ) ).

% DERIV_ident
tff(fact_6381_DERIV__diff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,X: A,S: set(A),G: fun(A,A),E4: A] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,X,S))
         => ( has_field_derivative(A,G,E4,topolo174197925503356063within(A,X,S))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_rw(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,minus_minus(A,D3),E4),topolo174197925503356063within(A,X,S)) ) ) ) ).

% DERIV_diff
tff(fact_6382_field__differentiable__diff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F7: A,F4: filter(A),G: fun(A,A),G2: A] :
          ( has_field_derivative(A,F2,F7,F4)
         => ( has_field_derivative(A,G,G2,F4)
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_rw(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,minus_minus(A,F7),G2),F4) ) ) ) ).

% field_differentiable_diff
tff(fact_6383_field__differentiable__add,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F7: A,F4: filter(A),G: fun(A,A),G2: A] :
          ( has_field_derivative(A,F2,F7,F4)
         => ( has_field_derivative(A,G,G2,F4)
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_rx(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),F7),G2),F4) ) ) ) ).

% field_differentiable_add
tff(fact_6384_DERIV__add,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,X: A,S: set(A),G: fun(A,A),E4: A] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,X,S))
         => ( has_field_derivative(A,G,E4,topolo174197925503356063within(A,X,S))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_rx(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),D3),E4),topolo174197925503356063within(A,X,S)) ) ) ) ).

% DERIV_add
tff(fact_6385_has__real__derivative__pos__inc__right,axiom,
    ! [F2: fun(real,real),L: real,X: real,Sa: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,Sa))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [D4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D4)
            & ! [H4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H4)
               => ( member(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),H4),Sa)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H4),D4)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,X)),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),H4))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
tff(fact_6386_has__real__derivative__neg__dec__right,axiom,
    ! [F2: fun(real,real),L: real,X: real,Sa: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,X,Sa))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [D4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D4)
            & ! [H4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H4)
               => ( member(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),H4),Sa)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H4),D4)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),H4))),aa(real,real,F2,X)) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
tff(fact_6387_DERIV__const,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [K: A,F4: filter(A)] : has_field_derivative(A,aTP_Lamp_ry(A,fun(A,A),K),zero_zero(A),F4) ) ).

% DERIV_const
tff(fact_6388_has__field__derivative__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,F4: filter(A),C2: real] :
          ( has_field_derivative(A,F2,D3,F4)
         => has_field_derivative(A,aa(real,fun(A,A),aTP_Lamp_rz(fun(A,A),fun(real,fun(A,A)),F2),C2),aa(A,A,real_V8093663219630862766scaleR(A,C2),D3),F4) ) ) ).

% has_field_derivative_scaleR_right
tff(fact_6389_has__field__derivative__cosh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Db: A,X: A,S: set(A)] :
          ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,X,S))
         => has_field_derivative(A,aTP_Lamp_sa(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,S)) ) ) ).

% has_field_derivative_cosh
tff(fact_6390_has__field__derivative__sinh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Db: A,X: A,S: set(A)] :
          ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,X,S))
         => has_field_derivative(A,aTP_Lamp_sb(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,S)) ) ) ).

% has_field_derivative_sinh
tff(fact_6391_DERIV__mult_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,X: A,S: set(A),G: fun(A,A),E4: A] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,X,S))
         => ( has_field_derivative(A,G,E4,topolo174197925503356063within(A,X,S))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_sc(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,F2,X)),E4)),aa(A,A,aa(A,fun(A,A),times_times(A),D3),aa(A,A,G,X))),topolo174197925503356063within(A,X,S)) ) ) ) ).

% DERIV_mult'
tff(fact_6392_DERIV__mult,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Da: A,X: A,S: set(A),G: fun(A,A),Db: A] :
          ( has_field_derivative(A,F2,Da,topolo174197925503356063within(A,X,S))
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,X,S))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_sc(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(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,S)) ) ) ) ).

% DERIV_mult
tff(fact_6393_DERIV__cmult__Id,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,X: A,S: set(A)] : has_field_derivative(A,aa(A,fun(A,A),times_times(A),C2),C2,topolo174197925503356063within(A,X,S)) ) ).

% DERIV_cmult_Id
tff(fact_6394_DERIV__cmult__right,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,X: A,S: set(A),C2: A] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,X,S))
         => has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_sd(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,S)) ) ) ).

% DERIV_cmult_right
tff(fact_6395_DERIV__cmult,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,X: A,S: set(A),C2: A] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,X,S))
         => has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_se(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,S)) ) ) ).

% DERIV_cmult
tff(fact_6396_DERIV__cdivide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,X: A,S: set(A),C2: A] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,X,S))
         => has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_sf(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,S)) ) ) ).

% DERIV_cdivide
tff(fact_6397_DERIV__inverse_H,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))
         => ( ( aa(A,A,F2,X) != zero_zero(A) )
           => has_field_derivative(A,aTP_Lamp_sg(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,S)) ) ) ) ).

% DERIV_inverse'
tff(fact_6398_DERIV__minus,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))
         => has_field_derivative(A,aTP_Lamp_sh(fun(A,A),fun(A,A),F2),aa(A,A,uminus_uminus(A),D3),topolo174197925503356063within(A,X,S)) ) ) ).

% DERIV_minus
tff(fact_6399_field__differentiable__minus,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F7: A,F4: filter(A)] :
          ( has_field_derivative(A,F2,F7,F4)
         => has_field_derivative(A,aTP_Lamp_sh(fun(A,A),fun(A,A),F2),aa(A,A,uminus_uminus(A),F7),F4) ) ) ).

% field_differentiable_minus
tff(fact_6400_DERIV__sum,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Sa: set(A),F2: fun(B,fun(A,B)),F7: fun(C,fun(A,B)),X: C,F4: filter(B)] :
          ( ! [N: A] :
              ( member(A,N,Sa)
             => has_field_derivative(B,aa(A,fun(B,B),aTP_Lamp_si(fun(B,fun(A,B)),fun(A,fun(B,B)),F2),N),aa(A,B,aa(C,fun(A,B),F7,X),N),F4) )
         => has_field_derivative(B,aa(fun(B,fun(A,B)),fun(B,B),aTP_Lamp_sj(set(A),fun(fun(B,fun(A,B)),fun(B,B)),Sa),F2),aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(C,fun(A,B),F7,X)),Sa),F4) ) ) ).

% DERIV_sum
tff(fact_6401_MVT2,axiom,
    ! [Aa2: real,Ba: real,F2: fun(real,real),F7: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Aa2),Ba)
     => ( ! [X2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),X2)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),Ba)
             => has_field_derivative(real,F2,aa(real,real,F7,X2),topolo174197925503356063within(real,X2,top_top(set(real)))) ) )
       => ? [Z3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Aa2),Z3)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z3),Ba)
            & ( aa(real,real,minus_minus(real,aa(real,real,F2,Ba)),aa(real,real,F2,Aa2)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,Ba),Aa2)),aa(real,real,F7,Z3)) ) ) ) ) ).

% MVT2
tff(fact_6402_DERIV__power__Suc,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,X: A,S: set(A),Na: nat] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,X,S))
         => has_field_derivative(A,aa(nat,fun(A,A),aTP_Lamp_sk(fun(A,A),fun(nat,fun(A,A)),F2),Na),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),Na))),aa(A,A,aa(A,fun(A,A),times_times(A),D3),aa(nat,A,power_power(A,aa(A,A,F2,X)),Na))),topolo174197925503356063within(A,X,S)) ) ) ).

% DERIV_power_Suc
tff(fact_6403_DERIV__const__average,axiom,
    ! [Aa2: real,Ba: real,V: fun(real,real),K: real] :
      ( ( Aa2 != Ba )
     => ( ! [X2: real] : has_field_derivative(real,V,K,topolo174197925503356063within(real,X2,top_top(set(real))))
       => ( aa(real,real,V,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Aa2),Ba)),aa(num,real,numeral_numeral(real),bit0(one2)))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,V,Aa2)),aa(real,real,V,Ba))),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ) ).

% DERIV_const_average
tff(fact_6404_DERIV__power,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,X: A,S: set(A),Na: nat] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,X,S))
         => has_field_derivative(A,aa(nat,fun(A,A),aTP_Lamp_sl(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,S)) ) ) ).

% DERIV_power
tff(fact_6405_DERIV__inverse,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [X: A,S: 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,S)) ) ) ).

% DERIV_inverse
tff(fact_6406_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,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
       => ( ! [Y2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,X),Y2))),D2)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,F2,X)),aa(real,real,F2,Y2)) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_min
tff(fact_6407_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,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
       => ( ! [Y2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,X),Y2))),D2)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,F2,Y2)),aa(real,real,F2,X)) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_max
tff(fact_6408_DERIV__pow,axiom,
    ! [Na: nat,X: real,S: set(real)] : has_field_derivative(real,aTP_Lamp_sm(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,S)) ).

% DERIV_pow
tff(fact_6409_termdiffs__strong__converges__everywhere,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),X: A] :
          ( ! [Y2: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_cv(fun(nat,A),fun(A,fun(nat,A)),C2),Y2))
         => has_field_derivative(A,aTP_Lamp_sn(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_6410_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_so(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_6411_DERIV__quotient,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D2: A,X: A,S: set(A),G: fun(A,A),E2: A] :
          ( has_field_derivative(A,F2,D2,topolo174197925503356063within(A,X,S))
         => ( has_field_derivative(A,G,E2,topolo174197925503356063within(A,X,S))
           => ( ( aa(A,A,G,X) != zero_zero(A) )
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_ru(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,S)) ) ) ) ) ).

% DERIV_quotient
tff(fact_6412_DERIV__inverse__fun,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D2: A,X: A,S: set(A)] :
          ( has_field_derivative(A,F2,D2,topolo174197925503356063within(A,X,S))
         => ( ( aa(A,A,F2,X) != zero_zero(A) )
           => has_field_derivative(A,aTP_Lamp_sg(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,S)) ) ) ) ).

% DERIV_inverse_fun
tff(fact_6413_termdiffs__sums__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K3: real,C2: fun(nat,A),F2: fun(A,A),F7: A,Z: A] :
          ( ! [Z3: A] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z3)),K3)
             => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_cv(fun(nat,A),fun(A,fun(nat,A)),C2),Z3)),aa(A,A,F2,Z3)) )
         => ( has_field_derivative(A,F2,F7,topolo174197925503356063within(A,Z,top_top(set(A))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),K3)
             => 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_6414_has__real__derivative__powr,axiom,
    ! [Z: real,R: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Z)
     => has_field_derivative(real,aTP_Lamp_sp(real,fun(real,real),R),aa(real,real,aa(real,fun(real,real),times_times(real),R),powr(real,Z,aa(real,real,minus_minus(real,R),one_one(real)))),topolo174197925503356063within(real,Z,top_top(set(real)))) ) ).

% has_real_derivative_powr
tff(fact_6415_termdiffs,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K3: A,X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_cv(fun(nat,A),fun(A,fun(nat,A)),C2),K3))
         => ( summable(A,aa(A,fun(nat,A),aTP_Lamp_cw(fun(nat,A),fun(A,fun(nat,A)),C2),K3))
           => ( summable(A,aa(A,fun(nat,A),aTP_Lamp_sq(fun(nat,A),fun(A,fun(nat,A)),C2),K3))
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,K3))
               => has_field_derivative(A,aTP_Lamp_sn(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_6416_termdiffs__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K3: A,X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_cv(fun(nat,A),fun(A,fun(nat,A)),C2),K3))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,K3))
           => has_field_derivative(A,aTP_Lamp_sn(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_6417_termdiffs__strong_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K3: real,C2: fun(nat,A),Z: A] :
          ( ! [Z3: A] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z3)),K3)
             => summable(A,aa(A,fun(nat,A),aTP_Lamp_cv(fun(nat,A),fun(A,fun(nat,A)),C2),Z3)) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),K3)
           => has_field_derivative(A,aTP_Lamp_sn(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_6418_DERIV__fun__powr,axiom,
    ! [G: fun(real,real),M: real,X: real,R: real] :
      ( has_field_derivative(real,G,M,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,G,X))
       => has_field_derivative(real,aa(real,fun(real,real),aTP_Lamp_sr(fun(real,real),fun(real,fun(real,real)),G),R),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),R),powr(real,aa(real,real,G,X),aa(real,real,minus_minus(real,R),aa(nat,real,semiring_1_of_nat(real),one_one(nat)))))),M),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ).

% DERIV_fun_powr
tff(fact_6419_DERIV__log,axiom,
    ! [X: real,Ba: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
     => has_field_derivative(real,log(Ba),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),Ba)),X)),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_log
tff(fact_6420_DERIV__powr,axiom,
    ! [G: fun(real,real),M: real,X: real,F2: fun(real,real),R: real] :
      ( has_field_derivative(real,G,M,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,G,X))
       => ( has_field_derivative(real,F2,R,topolo174197925503356063within(real,X,top_top(set(real))))
         => has_field_derivative(real,aa(fun(real,real),fun(real,real),aTP_Lamp_ss(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,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),R),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_6421_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_6422_artanh__real__has__field__derivative,axiom,
    ! [X: real,A2: set(real)] :
      ( aa(real,$o,aa(real,fun(real,$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_6423_DERIV__real__sqrt,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$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_6424_DERIV__arctan,axiom,
    ! [X: real] : has_field_derivative(real,arctan,aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(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_6425_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,aa(real,fun(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_6426_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_6427_has__field__derivative__tanh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),X: A,Db: A,S: set(A)] :
          ( ( cosh(A,aa(A,A,G,X)) != zero_zero(A) )
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,X,S))
           => has_field_derivative(A,aTP_Lamp_st(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,tanh(A,aa(A,A,G,X))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),Db),topolo174197925503356063within(A,X,S)) ) ) ) ).

% has_field_derivative_tanh
tff(fact_6428_DERIV__real__sqrt__generic,axiom,
    ! [X: real,D3: real] :
      ( ( X != zero_zero(real) )
     => ( ( aa(real,$o,aa(real,fun(real,$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,$o,aa(real,fun(real,$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_6429_arcosh__real__has__field__derivative,axiom,
    ! [X: real,A2: set(real)] :
      ( aa(real,$o,aa(real,fun(real,$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_6430_DERIV__real__root,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),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_6431_DERIV__arccos,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,aa(real,fun(real,$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_6432_DERIV__arcsin,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),X)
     => ( aa(real,$o,aa(real,fun(real,$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_6433_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 )
        & ! [M5: nat,X2: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M5),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M5)),X2),topolo174197925503356063within(real,X2,top_top(set(real)))) )
     => ? [T2: real] :
          ( aa(real,$o,aa(real,fun(real,$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,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aTP_Lamp_su(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_6434_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 )
     => ( ! [M5: nat,X2: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M5),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M5)),X2),topolo174197925503356063within(real,X2,top_top(set(real))))
       => ? [T2: real] :
            ( aa(real,$o,aa(real,fun(real,$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,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aTP_Lamp_su(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_6435_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_6436_Maclaurin__minus,axiom,
    ! [H: real,Na: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H),zero_zero(real))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
         => ( ! [M5: nat,T2: real] :
                ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M5),Na)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),H),T2)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T2),zero_zero(real)) )
               => has_field_derivative(real,aa(nat,fun(real,real),Diff,M5),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M5)),T2),topolo174197925503356063within(real,T2,top_top(set(real)))) )
           => ? [T2: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H),T2)
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T2),zero_zero(real))
                & ( aa(real,real,F2,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_sv(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_6437_Maclaurin2,axiom,
    ! [H: real,Diff: fun(nat,fun(real,real)),F2: fun(real,real),Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M5: nat,T2: real] :
              ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M5),Na)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T2)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T2),H) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M5),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M5)),T2),topolo174197925503356063within(real,T2,top_top(set(real)))) )
         => ? [T2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T2)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T2),H)
              & ( aa(real,real,F2,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_sv(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_6438_Maclaurin,axiom,
    ! [H: real,Na: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
         => ( ! [M5: nat,T2: real] :
                ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M5),Na)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T2)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T2),H) )
               => has_field_derivative(real,aa(nat,fun(real,real),Diff,M5),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M5)),T2),topolo174197925503356063within(real,T2,top_top(set(real)))) )
           => ? [T2: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T2)
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T2),H)
                & ( aa(real,real,F2,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_sv(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_6439_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,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => ( ( X != zero_zero(real) )
         => ( ! [M5: nat,X2: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M5),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M5)),X2),topolo174197925503356063within(real,X2,top_top(set(real))))
           => ? [T2: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,abs_abs(real),T2))
                & aa(real,$o,aa(real,fun(real,$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,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aTP_Lamp_su(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_6440_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 )
     => ( ! [M5: nat,T2: real] :
            ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M5),Na)
              & aa(real,$o,aa(real,fun(real,$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,M5),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M5)),T2),topolo174197925503356063within(real,T2,top_top(set(real)))) )
       => ? [T2: real] :
            ( aa(real,$o,aa(real,fun(real,$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,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aTP_Lamp_su(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_6441_Taylor__down,axiom,
    ! [Na: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real),Aa2: real,Ba: real,C2: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M5: nat,T2: real] :
              ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M5),Na)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),T2)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T2),Ba) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M5),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M5)),T2),topolo174197925503356063within(real,T2,top_top(set(real)))) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Aa2),C2)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),C2),Ba)
             => ? [T2: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Aa2),T2)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T2),C2)
                  & ( aa(real,real,F2,Aa2) = aa(real,real,aa(real,fun(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_sw(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),Aa2),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,Aa2),C2)),Na))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_6442_Taylor__up,axiom,
    ! [Na: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real),Aa2: real,Ba: real,C2: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M5: nat,T2: real] :
              ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M5),Na)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),T2)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T2),Ba) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M5),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M5)),T2),topolo174197925503356063within(real,T2,top_top(set(real)))) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),C2)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),Ba)
             => ? [T2: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),T2)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T2),Ba)
                  & ( aa(real,real,F2,Ba) = aa(real,real,aa(real,fun(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_sw(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),Ba),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,Ba),C2)),Na))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_6443_Taylor,axiom,
    ! [Na: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real),Aa2: real,Ba: real,C2: real,X: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M5: nat,T2: real] :
              ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M5),Na)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),T2)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T2),Ba) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M5),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M5)),T2),topolo174197925503356063within(real,T2,top_top(set(real)))) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),C2)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),C2),Ba)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),X)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Ba)
                 => ( ( X != C2 )
                   => ? [T2: real] :
                        ( $ite(
                            aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),C2),
                            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),T2)
                            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T2),C2) ),
                            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),T2)
                            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T2),X) ) )
                        & ( aa(real,real,F2,X) = aa(real,real,aa(real,fun(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_sx(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_6444_Maclaurin__lemma2,axiom,
    ! [Na: nat,H: real,Diff: fun(nat,fun(real,real)),K: nat,B2: real] :
      ( ! [M5: nat,T2: real] :
          ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M5),Na)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T2)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T2),H) )
         => has_field_derivative(real,aa(nat,fun(real,real),Diff,M5),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M5)),T2),topolo174197925503356063within(real,T2,top_top(set(real)))) )
     => ( ( Na = aa(nat,nat,suc,K) )
       => ! [M3: nat,T6: real] :
            ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),Na)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T6)
              & aa(real,$o,aa(real,fun(real,$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_sz(nat,fun(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real)))),Na),Diff),B2),M3),aa(real,real,minus_minus(real,aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),T6)),aa(real,real,aa(real,fun(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_ta(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Diff),M3),T6)),set_ord_lessThan(nat,aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,M3))))),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,M3)))),semiring_char_0_fact(real,aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,M3))))))),topolo174197925503356063within(real,T6,top_top(set(real)))) ) ) ) ).

% Maclaurin_lemma2
tff(fact_6445_DERIV__arctan__series,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),X)),one_one(real))
     => has_field_derivative(real,aTP_Lamp_tb(real,real),suminf(real,aTP_Lamp_tc(real,fun(nat,real),X)),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_arctan_series
tff(fact_6446_DERIV__even__real__root,axiom,
    ! [Na: nat,X: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$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,aa(real,fun(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_6447_DERIV__power__series_H,axiom,
    ! [R3: real,F2: fun(nat,real),X0: real] :
      ( ! [X2: real] :
          ( member(real,X2,set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),R3),R3))
         => summable(real,aa(real,fun(nat,real),aTP_Lamp_td(fun(nat,real),fun(real,fun(nat,real)),F2),X2)) )
     => ( member(real,X0,set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),R3),R3))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
         => has_field_derivative(real,aTP_Lamp_tf(fun(nat,real),fun(real,real),F2),suminf(real,aa(real,fun(nat,real),aTP_Lamp_td(fun(nat,real),fun(real,fun(nat,real)),F2),X0)),topolo174197925503356063within(real,X0,top_top(set(real)))) ) ) ) ).

% DERIV_power_series'
tff(fact_6448_has__derivative__arcsin,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),X: A,G2: fun(A,real),S: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,G,X))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(A,real,G,X)),one_one(real))
           => ( has_derivative(A,real,G,G2,topolo174197925503356063within(A,X,S))
             => has_derivative(A,real,aTP_Lamp_tg(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_th(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),X),G2),topolo174197925503356063within(A,X,S)) ) ) ) ) ).

% has_derivative_arcsin
tff(fact_6449_greaterThanLessThan__empty__iff2,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Aa2: A,Ba: A] :
          ( ( bot_bot(set(A)) = set_or5935395276787703475ssThan(A,Aa2,Ba) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2) ) ) ).

% greaterThanLessThan_empty_iff2
tff(fact_6450_greaterThanLessThan__empty__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Aa2: A,Ba: A] :
          ( ( set_or5935395276787703475ssThan(A,Aa2,Ba) = bot_bot(set(A)) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2) ) ) ).

% greaterThanLessThan_empty_iff
tff(fact_6451_greaterThanLessThan__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,K: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),K)
         => ( set_or5935395276787703475ssThan(A,K,L) = bot_bot(set(A)) ) ) ) ).

% greaterThanLessThan_empty
tff(fact_6452_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,S: set(A),G: fun(B,C),G2: fun(B,C)] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,S))
         => ( has_derivative(B,C,G,G2,topolo174197925503356063within(B,aa(A,B,F2,X),aa(set(A),set(B),image(A,B,F2),S)))
           => has_derivative(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ti(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),aa(fun(B,C),fun(A,C),aTP_Lamp_ti(fun(A,B),fun(fun(B,C),fun(A,C)),F7),G2),topolo174197925503356063within(A,X,S)) ) ) ) ).

% has_derivative_in_compose
tff(fact_6453_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,S: set(A),G: fun(B,C),G2: fun(B,C)] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,S))
         => ( has_derivative(B,C,G,G2,topolo174197925503356063within(B,aa(A,B,F2,X),top_top(set(B))))
           => has_derivative(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ti(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),aa(fun(B,C),fun(A,C),aTP_Lamp_ti(fun(A,B),fun(fun(B,C),fun(A,C)),F7),G2),topolo174197925503356063within(A,X,S)) ) ) ) ).

% has_derivative_compose
tff(fact_6454_has__derivative__scaleR,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,real),F7: fun(A,real),X: A,S: set(A),G: fun(A,B),G2: fun(A,B)] :
          ( has_derivative(A,real,F2,F7,topolo174197925503356063within(A,X,S))
         => ( has_derivative(A,B,G,G2,topolo174197925503356063within(A,X,S))
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tj(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_tk(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F2),F7),X),G),G2),topolo174197925503356063within(A,X,S)) ) ) ) ).

% has_derivative_scaleR
tff(fact_6455_has__field__derivative__def,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,F4: filter(A)] :
          ( has_field_derivative(A,F2,D3,F4)
        <=> has_derivative(A,A,F2,aa(A,fun(A,A),times_times(A),D3),F4) ) ) ).

% has_field_derivative_def
tff(fact_6456_has__derivative__imp__has__field__derivative,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: fun(A,A),F4: filter(A),D7: A] :
          ( has_derivative(A,A,F2,D3,F4)
         => ( ! [X2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),X2),D7) = aa(A,A,D3,X2)
           => has_field_derivative(A,F2,D7,F4) ) ) ) ).

% has_derivative_imp_has_field_derivative
tff(fact_6457_has__field__derivative__imp__has__derivative,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,F4: filter(A)] :
          ( has_field_derivative(A,F2,D3,F4)
         => has_derivative(A,A,F2,aa(A,fun(A,A),times_times(A),D3),F4) ) ) ).

% has_field_derivative_imp_has_derivative
tff(fact_6458_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)),F4: filter(B)] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => has_derivative(B,C,aa(A,fun(B,C),F2,I2),aa(A,fun(B,C),F7,I2),F4) )
         => has_derivative(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_tm(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2),aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_tm(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F7),F4) ) ) ).

% has_derivative_sum
tff(fact_6459_has__derivative__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),F4: filter(A)] :
          ( has_derivative(A,B,F2,F7,F4)
         => has_derivative(A,B,aTP_Lamp_tn(fun(A,B),fun(A,B),F2),aTP_Lamp_tn(fun(A,B),fun(A,B),F7),F4) ) ) ).

% has_derivative_minus
tff(fact_6460_has__derivative__mult__right,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [G: fun(A,B),G2: fun(A,B),F4: filter(A),X: B] :
          ( has_derivative(A,B,G,G2,F4)
         => has_derivative(A,B,aa(B,fun(A,B),aTP_Lamp_to(fun(A,B),fun(B,fun(A,B)),G),X),aa(B,fun(A,B),aTP_Lamp_to(fun(A,B),fun(B,fun(A,B)),G2),X),F4) ) ) ).

% has_derivative_mult_right
tff(fact_6461_has__derivative__mult__left,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [G: fun(A,B),G2: fun(A,B),F4: filter(A),Y: B] :
          ( has_derivative(A,B,G,G2,F4)
         => has_derivative(A,B,aa(B,fun(A,B),aTP_Lamp_tp(fun(A,B),fun(B,fun(A,B)),G),Y),aa(B,fun(A,B),aTP_Lamp_tp(fun(A,B),fun(B,fun(A,B)),G2),Y),F4) ) ) ).

% has_derivative_mult_left
tff(fact_6462_has__derivative__scaleR__left,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [G: fun(A,real),G2: fun(A,real),F4: filter(A),X: B] :
          ( has_derivative(A,real,G,G2,F4)
         => has_derivative(A,B,aa(B,fun(A,B),aTP_Lamp_tq(fun(A,real),fun(B,fun(A,B)),G),X),aa(B,fun(A,B),aTP_Lamp_tq(fun(A,real),fun(B,fun(A,B)),G2),X),F4) ) ) ).

% has_derivative_scaleR_left
tff(fact_6463_has__derivative__scaleR__right,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [G: fun(A,B),G2: fun(A,B),F4: filter(A),R: real] :
          ( has_derivative(A,B,G,G2,F4)
         => has_derivative(A,B,aa(real,fun(A,B),aTP_Lamp_tr(fun(A,B),fun(real,fun(A,B)),G),R),aa(real,fun(A,B),aTP_Lamp_tr(fun(A,B),fun(real,fun(A,B)),G2),R),F4) ) ) ).

% has_derivative_scaleR_right
tff(fact_6464_has__derivative__const,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [C2: B,F4: filter(A)] : has_derivative(A,B,aTP_Lamp_ts(B,fun(A,B),C2),aTP_Lamp_tt(A,B),F4) ) ).

% has_derivative_const
tff(fact_6465_has__derivative__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),F4: filter(A),G: fun(A,B),G2: fun(A,B)] :
          ( has_derivative(A,B,F2,F7,F4)
         => ( has_derivative(A,B,G,G2,F4)
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tu(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),aa(fun(A,B),fun(A,B),aTP_Lamp_tu(fun(A,B),fun(fun(A,B),fun(A,B)),F7),G2),F4) ) ) ) ).

% has_derivative_add
tff(fact_6466_has__derivative__of__real,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [G: fun(A,real),G2: fun(A,real),F4: filter(A)] :
          ( has_derivative(A,real,G,G2,F4)
         => has_derivative(A,B,aTP_Lamp_tv(fun(A,real),fun(A,B),G),aTP_Lamp_tv(fun(A,real),fun(A,B),G2),F4) ) ) ).

% has_derivative_of_real
tff(fact_6467_has__derivative__ident,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F4: filter(A)] : has_derivative(A,A,aTP_Lamp_tw(A,A),aTP_Lamp_tw(A,A),F4) ) ).

% has_derivative_ident
tff(fact_6468_has__derivative__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),F4: filter(A),G: fun(A,B),G2: fun(A,B)] :
          ( has_derivative(A,B,F2,F7,F4)
         => ( has_derivative(A,B,G,G2,F4)
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tx(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),aa(fun(A,B),fun(A,B),aTP_Lamp_tx(fun(A,B),fun(fun(A,B),fun(A,B)),F7),G2),F4) ) ) ) ).

% has_derivative_diff
tff(fact_6469_has__derivative__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),X: A,S: set(A),G: fun(A,B),G2: fun(A,B)] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,S))
         => ( has_derivative(A,B,G,G2,topolo174197925503356063within(A,X,S))
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ty(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_tz(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F2),F7),X),G),G2),topolo174197925503356063within(A,X,S)) ) ) ) ).

% has_derivative_mult
tff(fact_6470_has__derivative__zero__unique,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [F4: fun(A,B),X: A] :
          ( has_derivative(A,B,aTP_Lamp_tt(A,B),F4,topolo174197925503356063within(A,X,top_top(set(A))))
         => ! [X4: A] : aa(A,B,F4,X4) = zero_zero(B) ) ) ).

% has_derivative_zero_unique
tff(fact_6471_has__derivative__in__compose2,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [T: set(A),G: fun(A,B),G2: fun(A,fun(A,B)),F2: fun(C,A),S: set(C),X: C,F7: fun(C,A)] :
          ( ! [X2: A] :
              ( member(A,X2,T)
             => has_derivative(A,B,G,aa(A,fun(A,B),G2,X2),topolo174197925503356063within(A,X2,T)) )
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(C),set(A),image(C,A,F2),S)),T)
           => ( member(C,X,S)
             => ( has_derivative(C,A,F2,F7,topolo174197925503356063within(C,X,S))
               => has_derivative(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ua(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_ub(fun(A,fun(A,B)),fun(fun(C,A),fun(C,fun(fun(C,A),fun(C,B)))),G2),F2),X),F7),topolo174197925503356063within(C,X,S)) ) ) ) ) ) ).

% has_derivative_in_compose2
tff(fact_6472_has__derivative__exp,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G2: fun(A,real),X: A,S: set(A)] :
          ( has_derivative(A,real,G,G2,topolo174197925503356063within(A,X,S))
         => has_derivative(A,real,aTP_Lamp_uc(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_ud(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G2),X),topolo174197925503356063within(A,X,S)) ) ) ).

% has_derivative_exp
tff(fact_6473_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Aa2: A,Ba: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,Aa2,Ba)),set_or5935395276787703475ssThan(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),Aa2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),D2) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanLessThan
tff(fact_6474_has__derivative__cosh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Db: A,X: A,S: set(A)] :
          ( has_derivative(A,A,G,aa(A,fun(A,A),times_times(A),Db),topolo174197925503356063within(A,X,S))
         => has_derivative(A,A,aTP_Lamp_sa(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,S)) ) ) ).

% has_derivative_cosh
tff(fact_6475_has__derivative__sinh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Db: A,X: A,S: set(A)] :
          ( has_derivative(A,A,G,aa(A,fun(A,A),times_times(A),Db),topolo174197925503356063within(A,X,S))
         => has_derivative(A,A,aTP_Lamp_sb(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,S)) ) ) ).

% has_derivative_sinh
tff(fact_6476_has__derivative__sin,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G2: fun(A,real),X: A,S: set(A)] :
          ( has_derivative(A,real,G,G2,topolo174197925503356063within(A,X,S))
         => has_derivative(A,real,aTP_Lamp_ue(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_uf(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G2),X),topolo174197925503356063within(A,X,S)) ) ) ).

% has_derivative_sin
tff(fact_6477_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Aa2: A,Ba: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,Aa2,Ba)),set_or1337092689740270186AtMost(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),Aa2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),D2) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastAtMost_iff
tff(fact_6478_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Aa2: A,Ba: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,Aa2,Ba)),set_or7035219750837199246ssThan(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),Aa2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),D2) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastLessThan_iff
tff(fact_6479_ivl__disj__un__two_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(1)
tff(fact_6480_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,Sa: set(A),G: fun(A,B),G2: fun(A,B)] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,Sa))
         => ( has_derivative(A,B,G,G2,topolo174197925503356063within(A,X,Sa))
           => ( ( aa(A,B,G,X) != zero_zero(B) )
             => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ug(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_uh(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F2),F7),X),G),G2),topolo174197925503356063within(A,X,Sa)) ) ) ) ) ).

% has_derivative_divide'
tff(fact_6481_has__derivative__inverse,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(B,A),X: B,F7: fun(B,A),Sa: set(B)] :
          ( ( aa(B,A,F2,X) != zero_zero(A) )
         => ( has_derivative(B,A,F2,F7,topolo174197925503356063within(B,X,Sa))
           => has_derivative(B,A,aTP_Lamp_ui(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_uj(fun(B,A),fun(B,fun(fun(B,A),fun(B,A))),F2),X),F7),topolo174197925503356063within(B,X,Sa)) ) ) ) ).

% has_derivative_inverse
tff(fact_6482_has__derivative__inverse_H,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: A,Sa: set(A)] :
          ( ( X != zero_zero(A) )
         => has_derivative(A,A,inverse_inverse(A),aTP_Lamp_uk(A,fun(A,A),X),topolo174197925503356063within(A,X,Sa)) ) ) ).

% has_derivative_inverse'
tff(fact_6483_DERIV__compose__FDERIV,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(real,real),F7: real,G: fun(A,real),X: A,G2: fun(A,real),S: set(A)] :
          ( has_field_derivative(real,F2,F7,topolo174197925503356063within(real,aa(A,real,G,X),top_top(set(real))))
         => ( has_derivative(A,real,G,G2,topolo174197925503356063within(A,X,S))
           => has_derivative(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ul(fun(real,real),fun(fun(A,real),fun(A,real)),F2),G),aa(fun(A,real),fun(A,real),aTP_Lamp_um(real,fun(fun(A,real),fun(A,real)),F7),G2),topolo174197925503356063within(A,X,S)) ) ) ) ).

% DERIV_compose_FDERIV
tff(fact_6484_has__derivative__cos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G2: fun(A,real),X: A,S: set(A)] :
          ( has_derivative(A,real,G,G2,topolo174197925503356063within(A,X,S))
         => has_derivative(A,real,aTP_Lamp_un(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_uo(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G2),X),topolo174197925503356063within(A,X,S)) ) ) ).

% has_derivative_cos
tff(fact_6485_ivl__disj__un__two_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or5935395276787703475ssThan(A,M,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(4)
tff(fact_6486_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,Sa: set(A),Na: nat] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,Sa))
         => has_derivative(A,B,aa(nat,fun(A,B),aTP_Lamp_up(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_uq(fun(A,B),fun(fun(A,B),fun(A,fun(nat,fun(A,B)))),F2),F7),X),Na),topolo174197925503356063within(A,X,Sa)) ) ) ).

% has_derivative_power
tff(fact_6487_has__derivative__ln,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),X: A,G2: fun(A,real),S: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,X))
         => ( has_derivative(A,real,G,G2,topolo174197925503356063within(A,X,S))
           => has_derivative(A,real,aTP_Lamp_ur(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_us(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),X),G2),topolo174197925503356063within(A,X,S)) ) ) ) ).

% has_derivative_ln
tff(fact_6488_has__derivative__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),X: A,Sa: set(A),G: fun(A,B),G2: fun(A,B)] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,Sa))
         => ( has_derivative(A,B,G,G2,topolo174197925503356063within(A,X,Sa))
           => ( ( aa(A,B,G,X) != zero_zero(B) )
             => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ut(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_uu(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F2),F7),X),G),G2),topolo174197925503356063within(A,X,Sa)) ) ) ) ) ).

% has_derivative_divide
tff(fact_6489_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,Sa: set(B)] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => has_derivative(B,C,aa(A,fun(B,C),F2,I2),aa(A,fun(B,C),F7,I2),topolo174197925503356063within(B,X,Sa)) )
         => has_derivative(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_uw(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_uy(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,Sa)) ) ) ).

% has_derivative_prod
tff(fact_6490_has__derivative__powr,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G2: fun(A,real),X: A,X5: set(A),F2: fun(A,real),F7: fun(A,real)] :
          ( has_derivative(A,real,G,G2,topolo174197925503356063within(A,X,X5))
         => ( has_derivative(A,real,F2,F7,topolo174197925503356063within(A,X,X5))
           => ( aa(real,$o,aa(real,fun(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_uz(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_va(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))))),G),G2),X),F2),F7),topolo174197925503356063within(A,X,X5)) ) ) ) ) ) ).

% has_derivative_powr
tff(fact_6491_has__derivative__real__sqrt,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),X: A,G2: fun(A,real),S: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,X))
         => ( has_derivative(A,real,G,G2,topolo174197925503356063within(A,X,S))
           => has_derivative(A,real,aTP_Lamp_vb(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_vc(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),X),G2),topolo174197925503356063within(A,X,S)) ) ) ) ).

% has_derivative_real_sqrt
tff(fact_6492_has__derivative__arctan,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G2: fun(A,real),X: A,S: set(A)] :
          ( has_derivative(A,real,G,G2,topolo174197925503356063within(A,X,S))
         => has_derivative(A,real,aTP_Lamp_vd(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_ve(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G2),X),topolo174197925503356063within(A,X,S)) ) ) ).

% has_derivative_arctan
tff(fact_6493_has__derivative__tan,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),X: A,G2: fun(A,real),S: set(A)] :
          ( ( cos(real,aa(A,real,G,X)) != zero_zero(real) )
         => ( has_derivative(A,real,G,G2,topolo174197925503356063within(A,X,S))
           => has_derivative(A,real,aTP_Lamp_vf(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_vg(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),X),G2),topolo174197925503356063within(A,X,S)) ) ) ) ).

% has_derivative_tan
tff(fact_6494_DERIV__series_H,axiom,
    ! [F2: fun(real,fun(nat,real)),F7: fun(real,fun(nat,real)),X0: real,Aa2: real,Ba: real,L5: fun(nat,real)] :
      ( ! [N: nat] : has_field_derivative(real,aa(nat,fun(real,real),aTP_Lamp_vh(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))))
     => ( ! [X2: real] :
            ( member(real,X2,set_or5935395276787703475ssThan(real,Aa2,Ba))
           => summable(real,aa(real,fun(nat,real),F2,X2)) )
       => ( member(real,X0,set_or5935395276787703475ssThan(real,Aa2,Ba))
         => ( summable(real,aa(real,fun(nat,real),F7,X0))
           => ( summable(real,L5)
             => ( ! [N: nat,X2: real,Y2: real] :
                    ( member(real,X2,set_or5935395276787703475ssThan(real,Aa2,Ba))
                   => ( member(real,Y2,set_or5935395276787703475ssThan(real,Aa2,Ba))
                     => aa(real,$o,aa(real,fun(real,$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,X2),N)),aa(nat,real,aa(real,fun(nat,real),F2,Y2),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,X2),Y2)))) ) )
               => has_field_derivative(real,aTP_Lamp_vi(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_6495_has__derivative__arccos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),X: A,G2: fun(A,real),S: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,G,X))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(A,real,G,X)),one_one(real))
           => ( has_derivative(A,real,G,G2,topolo174197925503356063within(A,X,S))
             => has_derivative(A,real,aTP_Lamp_vj(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_vk(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),X),G2),topolo174197925503356063within(A,X,S)) ) ) ) ) ).

% has_derivative_arccos
tff(fact_6496_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),G2: fun(B,real),S: 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,G2,topolo174197925503356063within(B,X,S))
             => has_derivative(B,real,aa(fun(real,A),fun(B,real),aTP_Lamp_vl(fun(B,real),fun(fun(real,A),fun(B,real)),G),F2),aTP_Lamp_vm(fun(B,real),fun(B,real),G2),topolo174197925503356063within(B,X,S)) ) ) ) ) ).

% has_derivative_floor
tff(fact_6497_termdiffs__aux,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K3: A,X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_sq(fun(nat,A),fun(A,fun(nat,A)),C2),K3))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,K3))
           => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_vo(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_6498_tendsto__const,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [K: B,F4: filter(A)] : filterlim(A,B,aTP_Lamp_vp(B,fun(A,B),K),topolo7230453075368039082e_nhds(B,K),F4) ) ).

% tendsto_const
tff(fact_6499_tendsto__ident__at,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Aa2: A,S: set(A)] : filterlim(A,A,aTP_Lamp_vq(A,A),topolo7230453075368039082e_nhds(A,Aa2),topolo174197925503356063within(A,Aa2,S)) ) ).

% tendsto_ident_at
tff(fact_6500_tendsto__mult__right__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,F2: fun(B,A),L: A,F4: filter(B)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_vr(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)),F4)
          <=> filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F4) ) ) ) ).

% tendsto_mult_right_iff
tff(fact_6501_tendsto__mult__left__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,F2: fun(B,A),L: A,F4: filter(B)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_vs(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)),F4)
          <=> filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F4) ) ) ) ).

% tendsto_mult_left_iff
tff(fact_6502_power__tendsto__0__iff,axiom,
    ! [A: $tType,Na: nat,F2: fun(A,real),F4: filter(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_vt(nat,fun(fun(A,real),fun(A,real)),Na),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
      <=> filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% power_tendsto_0_iff
tff(fact_6503_continuous__within__compose2,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & topolo4958980785337419405_space(C)
        & topological_t2_space(A) )
     => ! [X: A,S: set(A),F2: fun(A,B),G: fun(B,C)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,X,S),F2)
         => ( topolo3448309680560233919inuous(B,C,topolo174197925503356063within(B,aa(A,B,F2,X),aa(set(A),set(B),image(A,B,F2),S)),G)
           => topolo3448309680560233919inuous(A,C,topolo174197925503356063within(A,X,S),aa(fun(B,C),fun(A,C),aTP_Lamp_vu(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G)) ) ) ) ).

% continuous_within_compose2
tff(fact_6504_isCont__Pair,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Aa2: A,F2: fun(A,B),G: fun(A,C)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,C,topolo174197925503356063within(A,Aa2,top_top(set(A))),G)
           => topolo3448309680560233919inuous(A,product_prod(B,C),topolo174197925503356063within(A,Aa2,top_top(set(A))),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_vv(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F2),G)) ) ) ) ).

% isCont_Pair
tff(fact_6505_isCont__LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [Aa2: A,F2: fun(A,B),G: fun(B,C),L: C] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,top_top(set(A))),F2)
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(B,aa(A,B,F2,Aa2),top_top(set(B))))
           => ( ? [D6: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
                  & ! [X2: A] :
                      ( ( ( X2 != Aa2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X2),Aa2))),D6) )
                     => ( aa(A,B,F2,X2) != aa(A,B,F2,Aa2) ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_vw(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(A,Aa2,top_top(set(A)))) ) ) ) ) ).

% isCont_LIM_compose2
tff(fact_6506_isCont__scaleR,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Aa2: A,F2: fun(A,real),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,Aa2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,top_top(set(A))),G)
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_vx(fun(A,real),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% isCont_scaleR
tff(fact_6507_LIM__not__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo8386298272705272623_space(B)
        & zero(A)
        & topological_t2_space(A) )
     => ! [K: A,Aa2: B] :
          ( ( K != zero_zero(A) )
         => ~ filterlim(B,A,aTP_Lamp_vy(A,fun(B,A),K),topolo7230453075368039082e_nhds(A,zero_zero(A)),topolo174197925503356063within(B,Aa2,top_top(set(B)))) ) ) ).

% LIM_not_zero
tff(fact_6508_LIM__offset__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),Aa2: A,L5: B] :
          ( filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_vz(fun(A,B),fun(A,fun(A,B)),F2),Aa2),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,Aa2,top_top(set(A)))) ) ) ).

% LIM_offset_zero_cancel
tff(fact_6509_LIM__offset__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),L5: B,Aa2: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,Aa2,top_top(set(A))))
         => filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_vz(fun(A,B),fun(A,fun(A,B)),F2),Aa2),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% LIM_offset_zero
tff(fact_6510_LIM__isCont__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),Aa2: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,Aa2)),topolo174197925503356063within(A,Aa2,top_top(set(A))))
        <=> filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_vz(fun(A,B),fun(A,fun(A,B)),F2),Aa2),topolo7230453075368039082e_nhds(B,aa(A,B,F2,Aa2)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% LIM_isCont_iff
tff(fact_6511_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_wa(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_6512_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),S: set(C)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,aa(C,A,F2,X),top_top(set(A))),G)
         => ( topolo3448309680560233919inuous(C,A,topolo174197925503356063within(C,X,S),F2)
           => topolo3448309680560233919inuous(C,B,topolo174197925503356063within(C,X,S),aa(fun(A,B),fun(C,B),aTP_Lamp_wb(fun(C,A),fun(fun(A,B),fun(C,B)),F2),G)) ) ) ) ).

% continuous_within_compose3
tff(fact_6513_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),F4: filter(C)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,L,top_top(set(A))),G)
         => ( filterlim(C,A,F2,topolo7230453075368039082e_nhds(A,L),F4)
           => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_wc(fun(A,B),fun(fun(C,A),fun(C,B)),G),F2),topolo7230453075368039082e_nhds(B,aa(A,B,G,L)),F4) ) ) ) ).

% isCont_tendsto_compose
tff(fact_6514_LIM__const__not__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo8386298272705272623_space(B)
        & topological_t2_space(A) )
     => ! [K: A,L5: A,Aa2: B] :
          ( ( K != L5 )
         => ~ filterlim(B,A,aTP_Lamp_wd(A,fun(B,A),K),topolo7230453075368039082e_nhds(A,L5),topolo174197925503356063within(B,Aa2,top_top(set(B)))) ) ) ).

% LIM_const_not_eq
tff(fact_6515_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),F4: 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),F4)
           => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_we(fun(A,B),fun(fun(C,A),fun(C,B)),G),F2),topolo7230453075368039082e_nhds(B,aa(A,B,G,L)),F4) ) ) ) ).

% tendsto_compose
tff(fact_6516_LIM__const__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & topolo8386298272705272623_space(A) )
     => ! [K: B,L5: B,Aa2: A] :
          ( filterlim(A,B,aTP_Lamp_wf(B,fun(A,B),K),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,Aa2,top_top(set(A))))
         => ( K = L5 ) ) ) ).

% LIM_const_eq
tff(fact_6517_isCont__o2,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & topolo4958980785337419405_space(C)
        & topological_t2_space(A) )
     => ! [Aa2: A,F2: fun(A,B),G: fun(B,C)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(B,C,topolo174197925503356063within(B,aa(A,B,F2,Aa2),top_top(set(B))),G)
           => topolo3448309680560233919inuous(A,C,topolo174197925503356063within(A,Aa2,top_top(set(A))),aa(fun(B,C),fun(A,C),aTP_Lamp_vu(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G)) ) ) ) ).

% isCont_o2
tff(fact_6518_filterlim__at__If,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(A,B),G3: filter(B),X: A,P: fun(A,$o),G: fun(A,B)] :
          ( filterlim(A,B,F2,G3,topolo174197925503356063within(A,X,collect(A,P)))
         => ( filterlim(A,B,G,G3,topolo174197925503356063within(A,X,collect(A,aTP_Lamp_wg(fun(A,$o),fun(A,$o),P))))
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,$o),fun(fun(A,B),fun(A,B)),aTP_Lamp_wh(fun(A,B),fun(fun(A,$o),fun(fun(A,B),fun(A,B))),F2),P),G),G3,topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ) ).

% filterlim_at_If
tff(fact_6519_isCont__of__real,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [Aa2: A,G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,Aa2,top_top(set(A))),G)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,top_top(set(A))),aTP_Lamp_wi(fun(A,real),fun(A,B),G)) ) ) ).

% isCont_of_real
tff(fact_6520_isCont__norm,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Aa2: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,Aa2,top_top(set(A))),aTP_Lamp_wj(fun(A,B),fun(A,real),F2)) ) ) ).

% isCont_norm
tff(fact_6521_LIM__offset,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),L5: B,Aa2: A,K: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,Aa2,top_top(set(A))))
         => filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_wk(fun(A,B),fun(A,fun(A,B)),F2),K),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,aa(A,A,minus_minus(A,Aa2),K),top_top(set(A)))) ) ) ).

% LIM_offset
tff(fact_6522_real__LIM__sandwich__zero,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(A,real),Aa2: A,G: fun(A,real)] :
          ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,Aa2,top_top(set(A))))
         => ( ! [X2: A] :
                ( ( X2 != Aa2 )
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(A,real,G,X2)) )
           => ( ! [X2: A] :
                  ( ( X2 != Aa2 )
                 => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(A,real,G,X2)),aa(A,real,F2,X2)) )
             => filterlim(A,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,Aa2,top_top(set(A)))) ) ) ) ) ).

% real_LIM_sandwich_zero
tff(fact_6523_LIM__imp__LIM,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),L: B,Aa2: A,G: fun(A,C),M: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,Aa2,top_top(set(A))))
         => ( ! [X2: A] :
                ( ( X2 != Aa2 )
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(C,aa(C,C,minus_minus(C,aa(A,C,G,X2)),M))),real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(A,B,F2,X2)),L))) )
           => filterlim(A,C,G,topolo7230453075368039082e_nhds(C,M),topolo174197925503356063within(A,Aa2,top_top(set(A)))) ) ) ) ).

% LIM_imp_LIM
tff(fact_6524_continuous__ident,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [X: A,Sa: set(A)] : topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,Sa),aTP_Lamp_wl(A,A)) ) ).

% continuous_ident
tff(fact_6525_has__field__derivative__iff,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))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_wm(fun(A,A),fun(A,fun(A,A)),F2),X),topolo7230453075368039082e_nhds(A,D3),topolo174197925503356063within(A,X,Sa)) ) ) ).

% has_field_derivative_iff
tff(fact_6526_has__field__derivativeD,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))
         => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_wm(fun(A,A),fun(A,fun(A,A)),F2),X),topolo7230453075368039082e_nhds(A,D3),topolo174197925503356063within(A,X,Sa)) ) ) ).

% has_field_derivativeD
tff(fact_6527_tendsto__artanh,axiom,
    ! [A: $tType,F2: fun(A,real),Aa2: real,F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,Aa2),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Aa2)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Aa2),one_one(real))
         => filterlim(A,real,aTP_Lamp_wn(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,artanh(real),Aa2)),F4) ) ) ) ).

% tendsto_artanh
tff(fact_6528_tendsto__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(A,A),Aa2: A,F4: filter(A)] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,Aa2),F4)
         => ( ( sin(A,Aa2) != zero_zero(A) )
           => filterlim(A,A,aTP_Lamp_wo(fun(A,A),fun(A,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,cot(A),Aa2)),F4) ) ) ) ).

% tendsto_cot
tff(fact_6529_tendsto__const__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(B)
     => ! [F4: filter(A),Aa2: B,Ba: B] :
          ( ( F4 != bot_bot(filter(A)) )
         => ( filterlim(A,B,aTP_Lamp_wp(B,fun(A,B),Aa2),topolo7230453075368039082e_nhds(B,Ba),F4)
          <=> ( Aa2 = Ba ) ) ) ) ).

% tendsto_const_iff
tff(fact_6530_continuous__min,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [F4: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( topolo3448309680560233919inuous(A,B,F4,G)
           => topolo3448309680560233919inuous(A,B,F4,aa(fun(A,B),fun(A,B),aTP_Lamp_wq(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_min
tff(fact_6531_tendsto__min,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [X5: fun(A,B),X: B,Net: filter(A),Y5: fun(A,B),Y: B] :
          ( filterlim(A,B,X5,topolo7230453075368039082e_nhds(B,X),Net)
         => ( filterlim(A,B,Y5,topolo7230453075368039082e_nhds(B,Y),Net)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_wr(fun(A,B),fun(fun(A,B),fun(A,B)),X5),Y5),topolo7230453075368039082e_nhds(B,ord_min(B,X,Y)),Net) ) ) ) ).

% tendsto_min
tff(fact_6532_tendsto__rabs__zero__cancel,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,aTP_Lamp_ws(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
     => filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ).

% tendsto_rabs_zero_cancel
tff(fact_6533_tendsto__rabs__zero__iff,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,aTP_Lamp_ws(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
    <=> filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ).

% tendsto_rabs_zero_iff
tff(fact_6534_tendsto__rabs__zero,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
     => filterlim(A,real,aTP_Lamp_ws(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ).

% tendsto_rabs_zero
tff(fact_6535_tendsto__rabs,axiom,
    ! [A: $tType,F2: fun(A,real),L: real,F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,L),F4)
     => filterlim(A,real,aTP_Lamp_ws(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,abs_abs(real),L)),F4) ) ).

% tendsto_rabs
tff(fact_6536_tendsto__arcosh,axiom,
    ! [A: $tType,F2: fun(A,real),Aa2: real,F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,Aa2),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Aa2)
       => filterlim(A,real,aTP_Lamp_wt(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,arcosh(real),Aa2)),F4) ) ) ).

% tendsto_arcosh
tff(fact_6537_tendsto__one__prod_H,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [I5: set(A),F2: fun(B,fun(A,C)),F4: filter(B)] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => filterlim(B,C,aa(A,fun(B,C),aTP_Lamp_wu(fun(B,fun(A,C)),fun(A,fun(B,C)),F2),I2),topolo7230453075368039082e_nhds(C,one_one(C)),F4) )
         => filterlim(B,C,aa(fun(B,fun(A,C)),fun(B,C),aTP_Lamp_wv(set(A),fun(fun(B,fun(A,C)),fun(B,C)),I5),F2),topolo7230453075368039082e_nhds(C,one_one(C)),F4) ) ) ).

% tendsto_one_prod'
tff(fact_6538_tendsto__prod,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(C)
        & comm_ring_1(C) )
     => ! [Sa: set(A),F2: fun(A,fun(B,C)),L5: fun(A,C),F4: filter(B)] :
          ( ! [I2: A] :
              ( member(A,I2,Sa)
             => filterlim(B,C,aa(A,fun(B,C),F2,I2),topolo7230453075368039082e_nhds(C,aa(A,C,L5,I2)),F4) )
         => filterlim(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_wx(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Sa),F2),topolo7230453075368039082e_nhds(C,groups7121269368397514597t_prod(A,C,L5,Sa)),F4) ) ) ).

% tendsto_prod
tff(fact_6539_tendsto__prod_H,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [I5: set(A),F2: fun(A,fun(B,C)),Aa2: fun(A,C),F4: filter(B)] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => filterlim(B,C,aa(A,fun(B,C),F2,I2),topolo7230453075368039082e_nhds(C,aa(A,C,Aa2,I2)),F4) )
         => filterlim(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_wz(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2),topolo7230453075368039082e_nhds(C,groups7121269368397514597t_prod(A,C,Aa2,I5)),F4) ) ) ).

% tendsto_prod'
tff(fact_6540_continuous__prod,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(B)
        & real_V4412858255891104859lgebra(C)
        & comm_ring_1(C) )
     => ! [Sa: set(A),F4: filter(B),F2: fun(A,fun(B,C))] :
          ( ! [I2: A] :
              ( member(A,I2,Sa)
             => topolo3448309680560233919inuous(B,C,F4,aa(A,fun(B,C),F2,I2)) )
         => topolo3448309680560233919inuous(B,C,F4,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_xb(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Sa),F2)) ) ) ).

% continuous_prod
tff(fact_6541_continuous__prod_H,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(B)
        & topolo4987421752381908075d_mult(C) )
     => ! [I5: set(A),F4: filter(B),F2: fun(A,fun(B,C))] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => topolo3448309680560233919inuous(B,C,F4,aa(A,fun(B,C),F2,I2)) )
         => topolo3448309680560233919inuous(B,C,F4,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_xd(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2)) ) ) ).

% continuous_prod'
tff(fact_6542_tendsto__real__root,axiom,
    ! [A: $tType,F2: fun(A,real),X: real,F4: filter(A),Na: nat] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,X),F4)
     => filterlim(A,real,aa(nat,fun(A,real),aTP_Lamp_xe(fun(A,real),fun(nat,fun(A,real)),F2),Na),topolo7230453075368039082e_nhds(real,aa(real,real,root(Na),X)),F4) ) ).

% tendsto_real_root
tff(fact_6543_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,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => ( ~ member(B,L,ring_1_Ints(B))
           => filterlim(A,C,aTP_Lamp_xf(fun(A,B),fun(A,C),F2),topolo7230453075368039082e_nhds(C,aa(int,C,ring_1_of_int(C),archimedean_ceiling(B,L))),F4) ) ) ) ).

% tendsto_of_int_ceiling
tff(fact_6544_tendsto__sinh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),Aa2: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Aa2),F4)
         => filterlim(A,B,aTP_Lamp_xg(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,sinh(B,Aa2)),F4) ) ) ).

% tendsto_sinh
tff(fact_6545_continuous__sinh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F4: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_xh(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_sinh
tff(fact_6546_tendsto__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(A,A),Aa2: A,F4: filter(A)] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,Aa2),F4)
         => ( ( cos(A,Aa2) != zero_zero(A) )
           => filterlim(A,A,aTP_Lamp_xi(fun(A,A),fun(A,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,tan(A),Aa2)),F4) ) ) ) ).

% tendsto_tan
tff(fact_6547_tendsto__power,axiom,
    ! [B: $tType,A: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B) )
     => ! [F2: fun(A,B),Aa2: B,F4: filter(A),Na: nat] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Aa2),F4)
         => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_xj(fun(A,B),fun(nat,fun(A,B)),F2),Na),topolo7230453075368039082e_nhds(B,aa(nat,B,power_power(B,Aa2),Na)),F4) ) ) ).

% tendsto_power
tff(fact_6548_continuous__power_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1898628316856586783d_mult(B) )
     => ! [F4: filter(A),F2: fun(A,B),G: fun(A,nat)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( topolo3448309680560233919inuous(A,nat,F4,G)
           => topolo3448309680560233919inuous(A,B,F4,aa(fun(A,nat),fun(A,B),aTP_Lamp_xk(fun(A,B),fun(fun(A,nat),fun(A,B)),F2),G)) ) ) ) ).

% continuous_power'
tff(fact_6549_tendsto__power__strong,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [F2: fun(A,B),Aa2: B,F4: filter(A),G: fun(A,nat),Ba: nat] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Aa2),F4)
         => ( filterlim(A,nat,G,topolo7230453075368039082e_nhds(nat,Ba),F4)
           => filterlim(A,B,aa(fun(A,nat),fun(A,B),aTP_Lamp_xl(fun(A,B),fun(fun(A,nat),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,aa(nat,B,power_power(B,Aa2),Ba)),F4) ) ) ) ).

% tendsto_power_strong
tff(fact_6550_continuous__power,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [F4: filter(A),F2: fun(A,B),Na: nat] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => topolo3448309680560233919inuous(A,B,F4,aa(nat,fun(A,B),aTP_Lamp_xm(fun(A,B),fun(nat,fun(A,B)),F2),Na)) ) ) ).

% continuous_power
tff(fact_6551_continuous__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [F4: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( topolo3448309680560233919inuous(A,B,F4,G)
           => topolo3448309680560233919inuous(A,B,F4,aa(fun(A,B),fun(A,B),aTP_Lamp_xn(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_diff
tff(fact_6552_tendsto__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [F2: fun(A,B),Aa2: B,F4: filter(A),G: fun(A,B),Ba: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Aa2),F4)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,Ba),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_xo(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,aa(B,B,minus_minus(B,Aa2),Ba)),F4) ) ) ) ).

% tendsto_diff
tff(fact_6553_tendsto__max,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [X5: fun(A,B),X: B,Net: filter(A),Y5: fun(A,B),Y: B] :
          ( filterlim(A,B,X5,topolo7230453075368039082e_nhds(B,X),Net)
         => ( filterlim(A,B,Y5,topolo7230453075368039082e_nhds(B,Y),Net)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_xp(fun(A,B),fun(fun(A,B),fun(A,B)),X5),Y5),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),ord_max(B),X),Y)),Net) ) ) ) ).

% tendsto_max
tff(fact_6554_continuous__max,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [F4: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( topolo3448309680560233919inuous(A,B,F4,G)
           => topolo3448309680560233919inuous(A,B,F4,aa(fun(A,B),fun(A,B),aTP_Lamp_xq(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_max
tff(fact_6555_continuous__cos,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F4: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_xr(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_cos
tff(fact_6556_continuous__sin,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F4: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_xs(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_sin
tff(fact_6557_tendsto__arsinh,axiom,
    ! [A: $tType,F2: fun(A,real),Aa2: real,F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,Aa2),F4)
     => filterlim(A,real,aTP_Lamp_xt(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,arsinh(real),Aa2)),F4) ) ).

% tendsto_arsinh
tff(fact_6558_tendsto__cos,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),Aa2: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Aa2),F4)
         => filterlim(A,B,aTP_Lamp_xr(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,cos(B,Aa2)),F4) ) ) ).

% tendsto_cos
tff(fact_6559_tendsto__sin,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),Aa2: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Aa2),F4)
         => filterlim(A,B,aTP_Lamp_xs(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,sin(B,Aa2)),F4) ) ) ).

% tendsto_sin
tff(fact_6560_tendsto__Complex,axiom,
    ! [A: $tType,F2: fun(A,real),Aa2: real,F4: filter(A),G: fun(A,real),Ba: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,Aa2),F4)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,Ba),F4)
       => filterlim(A,complex,aa(fun(A,real),fun(A,complex),aTP_Lamp_xu(fun(A,real),fun(fun(A,real),fun(A,complex)),F2),G),topolo7230453075368039082e_nhds(complex,complex2(Aa2,Ba)),F4) ) ) ).

% tendsto_Complex
tff(fact_6561_tendsto__arctan,axiom,
    ! [A: $tType,F2: fun(A,real),X: real,F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,X),F4)
     => filterlim(A,real,aTP_Lamp_xv(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,arctan,X)),F4) ) ).

% tendsto_arctan
tff(fact_6562_tendsto__norm,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),Aa2: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Aa2),F4)
         => filterlim(A,real,aTP_Lamp_xw(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,real_V7770717601297561774m_norm(B,Aa2)),F4) ) ) ).

% tendsto_norm
tff(fact_6563_continuous__norm,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [F4: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_wj(fun(A,B),fun(A,real),F2)) ) ) ).

% continuous_norm
tff(fact_6564_tendsto__real__sqrt,axiom,
    ! [A: $tType,F2: fun(A,real),X: real,F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,X),F4)
     => filterlim(A,real,aTP_Lamp_xx(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,sqrt,X)),F4) ) ).

% tendsto_real_sqrt
tff(fact_6565_tendsto__of__real,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [G: fun(A,real),Aa2: real,F4: filter(A)] :
          ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,Aa2),F4)
         => filterlim(A,B,aTP_Lamp_xy(fun(A,real),fun(A,B),G),topolo7230453075368039082e_nhds(B,aa(real,B,real_Vector_of_real(B),Aa2)),F4) ) ) ).

% tendsto_of_real
tff(fact_6566_continuous__of__real,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [F4: filter(A),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,G)
         => topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_wi(fun(A,real),fun(A,B),G)) ) ) ).

% continuous_of_real
tff(fact_6567_tendsto__of__real__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,real),C2: real,F4: filter(A)] :
          ( filterlim(A,B,aTP_Lamp_xz(fun(A,real),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(real,B,real_Vector_of_real(B),C2)),F4)
        <=> filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F4) ) ) ).

% tendsto_of_real_iff
tff(fact_6568_continuous__const,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topological_t2_space(A) )
     => ! [F4: filter(A),C2: B] : topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_ya(B,fun(A,B),C2)) ) ).

% continuous_const
tff(fact_6569_tendsto__add__const__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [C2: B,F2: fun(A,B),D2: B,F4: filter(A)] :
          ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yb(B,fun(fun(A,B),fun(A,B)),C2),F2),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),D2)),F4)
        <=> filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,D2),F4) ) ) ).

% tendsto_add_const_iff
tff(fact_6570_tendsto__add,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [F2: fun(A,B),Aa2: B,F4: filter(A),G: fun(A,B),Ba: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Aa2),F4)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,Ba),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yc(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),Aa2),Ba)),F4) ) ) ) ).

% tendsto_add
tff(fact_6571_continuous__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [F4: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( topolo3448309680560233919inuous(A,B,F4,G)
           => topolo3448309680560233919inuous(A,B,F4,aa(fun(A,B),fun(A,B),aTP_Lamp_yd(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_add
tff(fact_6572_continuous__exp,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F4: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_ye(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_exp
tff(fact_6573_tendsto__exp,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),Aa2: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Aa2),F4)
         => filterlim(A,B,aTP_Lamp_yf(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,exp(B),Aa2)),F4) ) ) ).

% tendsto_exp
tff(fact_6574_tendsto__powr,axiom,
    ! [A: $tType,F2: fun(A,real),Aa2: real,F4: filter(A),G: fun(A,real),Ba: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,Aa2),F4)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,Ba),F4)
       => ( ( Aa2 != zero_zero(real) )
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_yg(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,powr(real,Aa2,Ba)),F4) ) ) ) ).

% tendsto_powr
tff(fact_6575_tendsto__ln,axiom,
    ! [A: $tType,F2: fun(A,real),Aa2: real,F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,Aa2),F4)
     => ( ( Aa2 != zero_zero(real) )
       => filterlim(A,real,aTP_Lamp_oa(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,ln_ln(real),Aa2)),F4) ) ) ).

% tendsto_ln
tff(fact_6576_tendsto__norm__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,real,aTP_Lamp_xw(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_norm_zero_cancel
tff(fact_6577_tendsto__norm__zero__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,real,aTP_Lamp_xw(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
        <=> filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_norm_zero_iff
tff(fact_6578_tendsto__norm__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => filterlim(A,real,aTP_Lamp_xw(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% tendsto_norm_zero
tff(fact_6579_tendsto__add__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yc(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_add_zero
tff(fact_6580_tendsto__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => ( ( L != zero_zero(B) )
           => filterlim(A,B,aTP_Lamp_yh(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,sgn_sgn(B),L)),F4) ) ) ) ).

% tendsto_sgn
tff(fact_6581_continuous__scaleR,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [F4: filter(A),F2: fun(A,real),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => ( topolo3448309680560233919inuous(A,B,F4,G)
           => topolo3448309680560233919inuous(A,B,F4,aa(fun(A,B),fun(A,B),aTP_Lamp_vx(fun(A,real),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_scaleR
tff(fact_6582_tendsto__scaleR,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,real),Aa2: real,F4: filter(A),G: fun(A,B),Ba: B] :
          ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,Aa2),F4)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,Ba),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yi(fun(A,real),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,aa(B,B,real_V8093663219630862766scaleR(B,Aa2),Ba)),F4) ) ) ) ).

% tendsto_scaleR
tff(fact_6583_LIM__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_yj(fun(A,B),fun(B,fun(A,B)),F2),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% LIM_zero
tff(fact_6584_LIM__zero__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_yj(fun(A,B),fun(B,fun(A,B)),F2),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
        <=> filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4) ) ) ).

% LIM_zero_iff
tff(fact_6585_Lim__transform,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [G: fun(A,B),Aa2: B,F4: filter(A),F2: fun(A,B)] :
          ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,Aa2),F4)
         => ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yk(fun(A,B),fun(fun(A,B),fun(A,B)),G),F2),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Aa2),F4) ) ) ) ).

% Lim_transform
tff(fact_6586_Lim__transform2,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),Aa2: B,F4: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Aa2),F4)
         => ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yl(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
           => filterlim(A,B,G,topolo7230453075368039082e_nhds(B,Aa2),F4) ) ) ) ).

% Lim_transform2
tff(fact_6587_LIM__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_yj(fun(A,B),fun(B,fun(A,B)),F2),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4) ) ) ).

% LIM_zero_cancel
tff(fact_6588_Lim__transform__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),G: fun(A,B),F4: filter(A),Aa2: B] :
          ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yl(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Aa2),F4)
          <=> filterlim(A,B,G,topolo7230453075368039082e_nhds(B,Aa2),F4) ) ) ) ).

% Lim_transform_eq
tff(fact_6589_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,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => ( ~ member(B,L,ring_1_Ints(B))
           => filterlim(A,C,aTP_Lamp_ym(fun(A,B),fun(A,C),F2),topolo7230453075368039082e_nhds(C,aa(int,C,ring_1_of_int(C),archim6421214686448440834_floor(B,L))),F4) ) ) ) ).

% tendsto_of_int_floor
tff(fact_6590_tendsto__cosh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),Aa2: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Aa2),F4)
         => filterlim(A,B,aTP_Lamp_yn(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,cosh(B,Aa2)),F4) ) ) ).

% tendsto_cosh
tff(fact_6591_continuous__cosh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F4: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_yo(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_cosh
tff(fact_6592_tendsto__tanh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),Aa2: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Aa2),F4)
         => ( ( cosh(B,Aa2) != zero_zero(B) )
           => filterlim(A,B,aTP_Lamp_yp(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,tanh(B,Aa2)),F4) ) ) ) ).

% tendsto_tanh
tff(fact_6593_tendsto__Pair,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),Aa2: B,F4: filter(A),G: fun(A,C),Ba: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Aa2),F4)
         => ( filterlim(A,C,G,topolo7230453075368039082e_nhds(C,Ba),F4)
           => filterlim(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_yq(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),Aa2),Ba)),F4) ) ) ) ).

% tendsto_Pair
tff(fact_6594_continuous__Pair,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [F4: filter(A),F2: fun(A,B),G: fun(A,C)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( topolo3448309680560233919inuous(A,C,F4,G)
           => topolo3448309680560233919inuous(A,product_prod(B,C),F4,aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_vv(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F2),G)) ) ) ) ).

% continuous_Pair
tff(fact_6595_tendsto__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),Aa2: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Aa2),F4)
         => ( ( Aa2 != zero_zero(B) )
           => filterlim(A,B,aTP_Lamp_yr(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,inverse_inverse(B),Aa2)),F4) ) ) ) ).

% tendsto_inverse
tff(fact_6596_tendsto__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [F2: fun(A,B),Aa2: B,F4: filter(A),G: fun(A,B),Ba: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Aa2),F4)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,Ba),F4)
           => ( ( Ba != zero_zero(B) )
             => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ys(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),Aa2),Ba)),F4) ) ) ) ) ).

% tendsto_divide
tff(fact_6597_tendsto__divide__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [F2: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_yt(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_divide_zero
tff(fact_6598_tendsto__mult__right__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F2: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_yu(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_mult_right_zero
tff(fact_6599_tendsto__mult__left__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F2: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_yv(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_mult_left_zero
tff(fact_6600_tendsto__mult__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yw(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_mult_zero
tff(fact_6601_tendsto__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [F2: fun(A,B),Aa2: B,F4: filter(A),G: fun(A,B),Ba: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Aa2),F4)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,Ba),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yx(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),Aa2),Ba)),F4) ) ) ) ).

% tendsto_mult
tff(fact_6602_tendsto__mult__left,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_yy(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)),F4) ) ) ).

% tendsto_mult_left
tff(fact_6603_tendsto__mult__right,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_yz(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)),F4) ) ) ).

% tendsto_mult_right
tff(fact_6604_continuous__mult__right,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [F4: filter(A),F2: fun(A,B),C2: B] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => topolo3448309680560233919inuous(A,B,F4,aa(B,fun(A,B),aTP_Lamp_za(fun(A,B),fun(B,fun(A,B)),F2),C2)) ) ) ).

% continuous_mult_right
tff(fact_6605_continuous__mult__left,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [F4: filter(A),F2: fun(A,B),C2: B] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => topolo3448309680560233919inuous(A,B,F4,aa(B,fun(A,B),aTP_Lamp_zb(fun(A,B),fun(B,fun(A,B)),F2),C2)) ) ) ).

% continuous_mult_left
tff(fact_6606_continuous__mult_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4211221413907600880p_mult(B) )
     => ! [F4: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( topolo3448309680560233919inuous(A,B,F4,G)
           => topolo3448309680560233919inuous(A,B,F4,aa(fun(A,B),fun(A,B),aTP_Lamp_zc(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_mult'
tff(fact_6607_continuous__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [F4: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( topolo3448309680560233919inuous(A,B,F4,G)
           => topolo3448309680560233919inuous(A,B,F4,aa(fun(A,B),fun(A,B),aTP_Lamp_zd(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_mult
tff(fact_6608_tendsto__mult__one,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,one_one(B)),F4)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,one_one(B)),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ze(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,one_one(B)),F4) ) ) ) ).

% tendsto_mult_one
tff(fact_6609_continuous__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [F4: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_zf(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_minus
tff(fact_6610_tendsto__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [F2: fun(A,B),Aa2: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Aa2),F4)
         => filterlim(A,B,aTP_Lamp_zg(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,uminus_uminus(B),Aa2)),F4) ) ) ).

% tendsto_minus
tff(fact_6611_tendsto__minus__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [F2: fun(A,B),Aa2: B,F4: filter(A)] :
          ( filterlim(A,B,aTP_Lamp_zg(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,uminus_uminus(B),Aa2)),F4)
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Aa2),F4) ) ) ).

% tendsto_minus_cancel
tff(fact_6612_tendsto__minus__cancel__left,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [F2: fun(A,B),Y: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(B,B,uminus_uminus(B),Y)),F4)
        <=> filterlim(A,B,aTP_Lamp_zg(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,Y),F4) ) ) ).

% tendsto_minus_cancel_left
tff(fact_6613_tendsto__sum,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [I5: set(A),F2: fun(A,fun(B,C)),Aa2: fun(A,C),F4: filter(B)] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => filterlim(B,C,aa(A,fun(B,C),F2,I2),topolo7230453075368039082e_nhds(C,aa(A,C,Aa2,I2)),F4) )
         => filterlim(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_zi(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2),topolo7230453075368039082e_nhds(C,aa(set(A),C,groups7311177749621191930dd_sum(A,C,Aa2),I5)),F4) ) ) ).

% tendsto_sum
tff(fact_6614_continuous__sum,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(B)
        & topolo5987344860129210374id_add(C) )
     => ! [I5: set(A),F4: filter(B),F2: fun(A,fun(B,C))] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => topolo3448309680560233919inuous(B,C,F4,aa(A,fun(B,C),F2,I2)) )
         => topolo3448309680560233919inuous(B,C,F4,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_zk(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2)) ) ) ).

% continuous_sum
tff(fact_6615_tendsto__null__sum,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [I5: set(A),F2: fun(B,fun(A,C)),F4: filter(B)] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => filterlim(B,C,aa(A,fun(B,C),aTP_Lamp_zl(fun(B,fun(A,C)),fun(A,fun(B,C)),F2),I2),topolo7230453075368039082e_nhds(C,zero_zero(C)),F4) )
         => filterlim(B,C,aa(fun(B,fun(A,C)),fun(B,C),aTP_Lamp_zm(set(A),fun(fun(B,fun(A,C)),fun(B,C)),I5),F2),topolo7230453075368039082e_nhds(C,zero_zero(C)),F4) ) ) ).

% tendsto_null_sum
tff(fact_6616_tendsto__null__power,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [F2: fun(A,B),F4: filter(A),Na: nat] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
           => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_zn(fun(A,B),fun(nat,fun(A,B)),F2),Na),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_null_power
tff(fact_6617_tendsto__log,axiom,
    ! [A: $tType,F2: fun(A,real),Aa2: real,F4: filter(A),G: fun(A,real),Ba: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,Aa2),F4)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,Ba),F4)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Aa2)
         => ( ( Aa2 != one_one(real) )
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Ba)
             => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_zo(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,aa(real,real,log(Aa2),Ba)),F4) ) ) ) ) ) ).

% tendsto_log
tff(fact_6618_IVT,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo8458572112393995274pology(B) )
     => ! [F2: fun(B,A),Aa2: B,Y: A,Ba: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,Aa2)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(B,A,F2,Ba))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Aa2),Ba)
             => ( ! [X2: B] :
                    ( ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Aa2),X2)
                      & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),Ba) )
                   => topolo3448309680560233919inuous(B,A,topolo174197925503356063within(B,X2,top_top(set(B))),F2) )
               => ? [X2: B] :
                    ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Aa2),X2)
                    & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),Ba)
                    & ( aa(B,A,F2,X2) = Y ) ) ) ) ) ) ) ).

% IVT
tff(fact_6619_IVT2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo8458572112393995274pology(B) )
     => ! [F2: fun(B,A),Ba: B,Y: A,Aa2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,Ba)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(B,A,F2,Aa2))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Aa2),Ba)
             => ( ! [X2: B] :
                    ( ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Aa2),X2)
                      & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),Ba) )
                   => topolo3448309680560233919inuous(B,A,topolo174197925503356063within(B,X2,top_top(set(B))),F2) )
               => ? [X2: B] :
                    ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Aa2),X2)
                    & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),Ba)
                    & ( aa(B,A,F2,X2) = Y ) ) ) ) ) ) ) ).

% IVT2
tff(fact_6620_DERIV__LIM__iff,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [F2: fun(A,A),Aa2: A,D3: A] :
          ( filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_zp(fun(A,A),fun(A,fun(A,A)),F2),Aa2),topolo7230453075368039082e_nhds(A,D3),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_zq(fun(A,A),fun(A,fun(A,A)),F2),Aa2),topolo7230453075368039082e_nhds(A,D3),topolo174197925503356063within(A,Aa2,top_top(set(A)))) ) ) ).

% DERIV_LIM_iff
tff(fact_6621_isCont__Lb__Ub,axiom,
    ! [Aa2: real,Ba: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),Ba)
     => ( ! [X2: real] :
            ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),X2)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),Ba) )
           => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X2,top_top(set(real))),F2) )
       => ? [L6: real,M10: real] :
            ( ! [X4: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),X4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),Ba) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),L6),aa(real,real,F2,X4))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,F2,X4)),M10) ) )
            & ! [Y3: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),L6),Y3)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y3),M10) )
               => ? [X2: real] :
                    ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),X2)
                    & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),Ba)
                    & ( aa(real,real,F2,X2) = Y3 ) ) ) ) ) ) ).

% isCont_Lb_Ub
tff(fact_6622_LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),Ba: B,Aa2: A,G: fun(B,C),C2: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Ba),topolo174197925503356063within(A,Aa2,top_top(set(A))))
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(B,Ba,top_top(set(B))))
           => ( ? [D6: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
                  & ! [X2: A] :
                      ( ( ( X2 != Aa2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X2),Aa2))),D6) )
                     => ( aa(A,B,F2,X2) != Ba ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_vw(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(A,Aa2,top_top(set(A)))) ) ) ) ) ).

% LIM_compose2
tff(fact_6623_continuous__at__within__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [Aa2: A,S: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,S),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,S),G)
           => ( ( aa(A,B,G,Aa2) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,S),aa(fun(A,B),fun(A,B),aTP_Lamp_zr(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% continuous_at_within_divide
tff(fact_6624_isCont__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [Aa2: A,F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,top_top(set(A))),G)
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_yd(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% isCont_add
tff(fact_6625_isCont__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Aa2: A,F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,top_top(set(A))),G)
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_zd(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% isCont_mult
tff(fact_6626_isCont__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Aa2: A,F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,top_top(set(A))),G)
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_zs(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% isCont_diff
tff(fact_6627_isCont__power,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [Aa2: A,F2: fun(A,B),Na: nat] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,top_top(set(A))),aa(nat,fun(A,B),aTP_Lamp_xm(fun(A,B),fun(nat,fun(A,B)),F2),Na)) ) ) ).

% isCont_power
tff(fact_6628_isCont__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Aa2: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,top_top(set(A))),aTP_Lamp_zt(fun(A,B),fun(A,B),F2)) ) ) ).

% isCont_minus
tff(fact_6629_continuous__at__within__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Aa2: A,S: set(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,S),F2)
         => ( ( aa(A,B,F2,Aa2) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,S),aTP_Lamp_zu(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_at_within_inverse
tff(fact_6630_isCont__sum,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(B)
        & topolo5987344860129210374id_add(C) )
     => ! [A2: set(A),Aa2: B,F2: fun(A,fun(B,C))] :
          ( ! [X2: A] :
              ( member(A,X2,A2)
             => topolo3448309680560233919inuous(B,C,topolo174197925503356063within(B,Aa2,top_top(set(B))),aa(A,fun(B,C),F2,X2)) )
         => topolo3448309680560233919inuous(B,C,topolo174197925503356063within(B,Aa2,top_top(set(B))),aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_zk(set(A),fun(fun(A,fun(B,C)),fun(B,C)),A2),F2)) ) ) ).

% isCont_sum
tff(fact_6631_continuous__at__within__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Aa2: A,S: set(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,S),F2)
         => ( ( aa(A,B,F2,Aa2) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,S),aTP_Lamp_zv(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_at_within_sgn
tff(fact_6632_isCont__cos_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Aa2: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,top_top(set(A))),aTP_Lamp_xr(fun(A,B),fun(A,B),F2)) ) ) ).

% isCont_cos'
tff(fact_6633_isCont__sin_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Aa2: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,top_top(set(A))),aTP_Lamp_xs(fun(A,B),fun(A,B),F2)) ) ) ).

% isCont_sin'
tff(fact_6634_isCont__exp_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Aa2: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,top_top(set(A))),aTP_Lamp_ye(fun(A,B),fun(A,B),F2)) ) ) ).

% isCont_exp'
tff(fact_6635_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_zw(nat,fun(A,A),Na)) ) ).

% isCont_pochhammer
tff(fact_6636_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_zx(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_6637_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_zx(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_6638_lim__exp__minus__1,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => filterlim(A,A,aTP_Lamp_zy(A,A),topolo7230453075368039082e_nhds(A,one_one(A)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ).

% lim_exp_minus_1
tff(fact_6639_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : set_or7035219750837199246ssThan(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),L),one_one(int)),U) = set_or5935395276787703475ssThan(int,L,U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_int
tff(fact_6640_lemma__termdiff4,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [K: real,F2: fun(A,B),K3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K)
         => ( ! [H2: A] :
                ( ( H2 != zero_zero(A) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,H2)),K)
                 => aa(real,$o,aa(real,fun(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),K3),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_6641_isCont__bounded,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Aa2: real,Ba: real,F2: fun(real,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),Ba)
         => ( ! [X2: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),X2)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),Ba) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X2,top_top(set(real))),F2) )
           => ? [M10: A] :
              ! [X4: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),X4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),Ba) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(real,A,F2,X4)),M10) ) ) ) ) ).

% isCont_bounded
tff(fact_6642_isCont__eq__Ub,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Aa2: real,Ba: real,F2: fun(real,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),Ba)
         => ( ! [X2: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),X2)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),Ba) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X2,top_top(set(real))),F2) )
           => ? [M10: A] :
                ( ! [X4: real] :
                    ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),X4)
                      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),Ba) )
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(real,A,F2,X4)),M10) )
                & ? [X2: real] :
                    ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),X2)
                    & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),Ba)
                    & ( aa(real,A,F2,X2) = M10 ) ) ) ) ) ) ).

% isCont_eq_Ub
tff(fact_6643_isCont__eq__Lb,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Aa2: real,Ba: real,F2: fun(real,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),Ba)
         => ( ! [X2: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),X2)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),Ba) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X2,top_top(set(real))),F2) )
           => ? [M10: A] :
                ( ! [X4: real] :
                    ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),X4)
                      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),Ba) )
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M10),aa(real,A,F2,X4)) )
                & ? [X2: real] :
                    ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),X2)
                    & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),Ba)
                    & ( aa(real,A,F2,X2) = M10 ) ) ) ) ) ) ).

% isCont_eq_Lb
tff(fact_6644_isCont__inverse__function2,axiom,
    ! [Aa2: real,X: real,Ba: real,G: fun(real,real),F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Aa2),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),Ba)
       => ( ! [Z3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),Z3)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Z3),Ba)
               => ( aa(real,real,G,aa(real,real,F2,Z3)) = Z3 ) ) )
         => ( ! [Z3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),Z3)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Z3),Ba)
                 => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z3,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_6645_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_zx(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_6646_isCont__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [Aa2: A,F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,top_top(set(A))),G)
           => ( ( aa(A,B,G,Aa2) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_zr(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% isCont_divide
tff(fact_6647_isCont__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Aa2: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,top_top(set(A))),F2)
         => ( ( aa(A,B,F2,Aa2) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,top_top(set(A))),aTP_Lamp_zv(fun(A,B),fun(A,B),F2)) ) ) ) ).

% isCont_sgn
tff(fact_6648_filterlim__at__to__0,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(A,B),F4: filter(B),Aa2: A] :
          ( filterlim(A,B,F2,F4,topolo174197925503356063within(A,Aa2,top_top(set(A))))
        <=> filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_zz(fun(A,B),fun(A,fun(A,B)),F2),Aa2),F4,topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% filterlim_at_to_0
tff(fact_6649_continuous__within__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,S: set(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,S),F2)
         => ( ( cos(A,aa(A,A,F2,X)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,S),aTP_Lamp_xi(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_within_tan
tff(fact_6650_continuous__within__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,S: set(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,S),F2)
         => ( ( sin(A,aa(A,A,F2,X)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,S),aTP_Lamp_wo(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_within_cot
tff(fact_6651_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_aaa(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_at_within_tanh
tff(fact_6652_CARAT__DERIV,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),L: A,X: A] :
          ( has_field_derivative(A,F2,L,topolo174197925503356063within(A,X,top_top(set(A))))
        <=> ? [G4: fun(A,A)] :
              ( ! [Z5: A] : aa(A,A,minus_minus(A,aa(A,A,F2,Z5)),aa(A,A,F2,X)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,G4,Z5)),aa(A,A,minus_minus(A,Z5),X))
              & topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),G4)
              & ( aa(A,A,G4,X) = L ) ) ) ) ).

% CARAT_DERIV
tff(fact_6653_isCont__has__Ub,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Aa2: real,Ba: real,F2: fun(real,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),Ba)
         => ( ! [X2: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),X2)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),Ba) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X2,top_top(set(real))),F2) )
           => ? [M10: A] :
                ( ! [X4: real] :
                    ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),X4)
                      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),Ba) )
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(real,A,F2,X4)),M10) )
                & ! [N8: A] :
                    ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),N8),M10)
                   => ? [X2: real] :
                        ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),X2)
                        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),Ba)
                        & aa(A,$o,aa(A,fun(A,$o),ord_less(A),N8),aa(real,A,F2,X2)) ) ) ) ) ) ) ).

% isCont_has_Ub
tff(fact_6654_powser__limit__0__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S: real,Aa2: fun(nat,A),F2: fun(A,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S)
         => ( ! [X2: A] :
                ( ( X2 != zero_zero(A) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X2)),S)
                 => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_cv(fun(nat,A),fun(A,fun(nat,A)),Aa2),X2)),aa(A,A,F2,X2)) ) )
           => filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,aa(nat,A,Aa2,zero_zero(nat))),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% powser_limit_0_strong
tff(fact_6655_powser__limit__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S: real,Aa2: fun(nat,A),F2: fun(A,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S)
         => ( ! [X2: A] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X2)),S)
               => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_cv(fun(nat,A),fun(A,fun(nat,A)),Aa2),X2)),aa(A,A,F2,X2)) )
           => filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,aa(nat,A,Aa2,zero_zero(nat))),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% powser_limit_0
tff(fact_6656_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,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K)
         => ( summable(real,F2)
           => ( ! [H2: A,N: nat] :
                  ( ( H2 != zero_zero(A) )
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,H2)),K)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(nat,B,aa(A,fun(nat,B),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_aab(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_6657_isCont__tan_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Aa2: A,F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Aa2,top_top(set(A))),F2)
         => ( ( cos(A,aa(A,A,F2,Aa2)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Aa2,top_top(set(A))),aTP_Lamp_xi(fun(A,A),fun(A,A),F2)) ) ) ) ).

% isCont_tan'
tff(fact_6658_LIM__cos__div__sin,axiom,
    filterlim(real,real,aTP_Lamp_aac(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_6659_isCont__cot_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Aa2: A,F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Aa2,top_top(set(A))),F2)
         => ( ( sin(A,aa(A,A,F2,Aa2)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Aa2,top_top(set(A))),aTP_Lamp_wo(fun(A,A),fun(A,A),F2)) ) ) ) ).

% isCont_cot'
tff(fact_6660_isCont__polynom,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Aa2: A,C2: fun(nat,A),Na: nat] : topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Aa2,top_top(set(A))),aa(nat,fun(A,A),aTP_Lamp_aad(fun(nat,A),fun(nat,fun(A,A)),C2),Na)) ) ).

% isCont_polynom
tff(fact_6661_isCont__powser__converges__everywhere,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),X: A] :
          ( ! [Y2: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_cv(fun(nat,A),fun(A,fun(nat,A)),C2),Y2))
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),aTP_Lamp_sn(fun(nat,A),fun(A,A),C2)) ) ) ).

% isCont_powser_converges_everywhere
tff(fact_6662_LIM__less__bound,axiom,
    ! [Ba: real,X: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Ba),X)
     => ( ! [X2: real] :
            ( member(real,X2,set_or5935395276787703475ssThan(real,Ba,X))
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,F2,X2)) )
       => ( topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),F2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,F2,X)) ) ) ) ).

% LIM_less_bound
tff(fact_6663_greaterThanLessThan__upto,axiom,
    ! [I: int,J: int] : set_or5935395276787703475ssThan(int,I,J) = aa(list(int),set(int),set2(int),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),aa(int,int,minus_minus(int,J),one_one(int)))) ).

% greaterThanLessThan_upto
tff(fact_6664_isCont__inverse__function,axiom,
    ! [D2: real,X: real,G: fun(real,real),F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
     => ( ! [Z3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,Z3),X))),D2)
           => ( aa(real,real,G,aa(real,real,F2,Z3)) = Z3 ) )
       => ( ! [Z3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,Z3),X))),D2)
             => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z3,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_6665_GMVT_H,axiom,
    ! [Aa2: real,Ba: real,F2: fun(real,real),G: fun(real,real),G2: fun(real,real),F7: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Aa2),Ba)
     => ( ! [Z3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),Z3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Z3),Ba)
             => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z3,top_top(set(real))),F2) ) )
       => ( ! [Z3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),Z3)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Z3),Ba)
               => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z3,top_top(set(real))),G) ) )
         => ( ! [Z3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Aa2),Z3)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z3),Ba)
                 => has_field_derivative(real,G,aa(real,real,G2,Z3),topolo174197925503356063within(real,Z3,top_top(set(real)))) ) )
           => ( ! [Z3: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Aa2),Z3)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z3),Ba)
                   => has_field_derivative(real,F2,aa(real,real,F7,Z3),topolo174197925503356063within(real,Z3,top_top(set(real)))) ) )
             => ? [C4: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Aa2),C4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),C4),Ba)
                  & ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,aa(real,real,F2,Ba)),aa(real,real,F2,Aa2))),aa(real,real,G2,C4)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,aa(real,real,G,Ba)),aa(real,real,G,Aa2))),aa(real,real,F7,C4)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_6666_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_aae(fun(real,A),fun(real,real),F2),zero_zero(real),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ).

% floor_has_real_derivative
tff(fact_6667_isCont__powser,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K3: A,X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_cv(fun(nat,A),fun(A,fun(nat,A)),C2),K3))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,K3))
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),aTP_Lamp_sn(fun(nat,A),fun(A,A),C2)) ) ) ) ).

% isCont_powser
tff(fact_6668_isCont__powser_H,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B)
        & topological_t2_space(A) )
     => ! [Aa2: A,F2: fun(A,B),C2: fun(nat,B),K3: B] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,top_top(set(A))),F2)
         => ( summable(B,aa(B,fun(nat,B),aTP_Lamp_aaf(fun(nat,B),fun(B,fun(nat,B)),C2),K3))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,Aa2))),real_V7770717601297561774m_norm(B,K3))
             => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,top_top(set(A))),aa(fun(nat,B),fun(A,B),aTP_Lamp_aah(fun(A,B),fun(fun(nat,B),fun(A,B)),F2),C2)) ) ) ) ) ).

% isCont_powser'
tff(fact_6669_summable__Leibniz_I2_J,axiom,
    ! [Aa2: fun(nat,real)] :
      ( filterlim(nat,real,Aa2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,Aa2)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(nat,real,Aa2,zero_zero(nat)))
         => ! [N10: nat] : member(real,suminf(real,aTP_Lamp_aai(fun(nat,real),fun(nat,real),Aa2)),set_or1337092689740270186AtMost(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_aai(fun(nat,real),fun(nat,real),Aa2)),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_aai(fun(nat,real),fun(nat,real),Aa2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N10)),one_one(nat)))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_6670_summable__Leibniz_I3_J,axiom,
    ! [Aa2: fun(nat,real)] :
      ( filterlim(nat,real,Aa2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,Aa2)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,Aa2,zero_zero(nat))),zero_zero(real))
         => ! [N10: nat] : member(real,suminf(real,aTP_Lamp_aai(fun(nat,real),fun(nat,real),Aa2)),set_or1337092689740270186AtMost(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_aai(fun(nat,real),fun(nat,real),Aa2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N10)),one_one(nat)))),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_aai(fun(nat,real),fun(nat,real),Aa2)),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_6671_tendsto__zero__mult__left__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,Aa2: fun(nat,A)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_aaj(A,fun(fun(nat,A),fun(nat,A)),C2),Aa2),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
          <=> filterlim(nat,A,Aa2,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ) ).

% tendsto_zero_mult_left_iff
tff(fact_6672_tendsto__zero__mult__right__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,Aa2: fun(nat,A)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_aak(A,fun(fun(nat,A),fun(nat,A)),C2),Aa2),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
          <=> filterlim(nat,A,Aa2,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ) ).

% tendsto_zero_mult_right_iff
tff(fact_6673_tendsto__zero__divide__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,Aa2: fun(nat,A)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_aal(A,fun(fun(nat,A),fun(nat,A)),C2),Aa2),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
          <=> filterlim(nat,A,Aa2,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ) ).

% tendsto_zero_divide_iff
tff(fact_6674_continuous__rabs,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_aam(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_rabs
tff(fact_6675_continuous__real__root,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,real),Na: nat] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => topolo3448309680560233919inuous(A,real,F4,aa(nat,fun(A,real),aTP_Lamp_aan(fun(A,real),fun(nat,fun(A,real)),F2),Na)) ) ) ).

% continuous_real_root
tff(fact_6676_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_aao(fun(nat,A),fun(nat,fun(nat,A)),F2),K),topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% seq_offset_neg
tff(fact_6677_continuous__arctan,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_aap(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_arctan
tff(fact_6678_continuous__real__sqrt,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_aaq(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_real_sqrt
tff(fact_6679_LIMSEQ__imp__Suc,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),L: A] :
          ( filterlim(nat,A,aTP_Lamp_aar(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_6680_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_aar(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% LIMSEQ_Suc
tff(fact_6681_LIMSEQ__const__iff,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [K: A,L: A] :
          ( filterlim(nat,A,aTP_Lamp_aas(A,fun(nat,A),K),topolo7230453075368039082e_nhds(A,L),at_top(nat))
        <=> ( K = L ) ) ) ).

% LIMSEQ_const_iff
tff(fact_6682_LIMSEQ__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),Aa2: A,K: nat] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,Aa2),at_top(nat))
         => filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aat(fun(nat,A),fun(nat,fun(nat,A)),F2),K),topolo7230453075368039082e_nhds(A,Aa2),at_top(nat)) ) ) ).

% LIMSEQ_ignore_initial_segment
tff(fact_6683_LIMSEQ__offset,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),K: nat,Aa2: A] :
          ( filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aat(fun(nat,A),fun(nat,fun(nat,A)),F2),K),topolo7230453075368039082e_nhds(A,Aa2),at_top(nat))
         => filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,Aa2),at_top(nat)) ) ) ).

% LIMSEQ_offset
tff(fact_6684_continuous__arsinh,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_aau(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_arsinh
tff(fact_6685_lim__mono,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [N3: nat,X5: fun(nat,A),Y5: fun(nat,A),X: A,Y: A] :
          ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N3),N)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N)),aa(nat,A,Y5,N)) )
         => ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,X),at_top(nat))
           => ( filterlim(nat,A,Y5,topolo7230453075368039082e_nhds(A,Y),at_top(nat))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ) ) ).

% lim_mono
tff(fact_6686_LIMSEQ__le,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X5: fun(nat,A),X: A,Y5: fun(nat,A),Y: A] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,X),at_top(nat))
         => ( filterlim(nat,A,Y5,topolo7230453075368039082e_nhds(A,Y),at_top(nat))
           => ( ? [N8: nat] :
                ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N8),N)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N)),aa(nat,A,Y5,N)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) ) ) ) ) ).

% LIMSEQ_le
tff(fact_6687_Lim__bounded,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F2: fun(nat,A),L: A,M2: nat,C5: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => ( ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,N)),C5) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),C5) ) ) ) ).

% Lim_bounded
tff(fact_6688_Lim__bounded2,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F2: fun(nat,A),L: A,N3: nat,C5: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => ( ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N3),N)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C5),aa(nat,A,F2,N)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C5),L) ) ) ) ).

% Lim_bounded2
tff(fact_6689_LIMSEQ__le__const,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X5: fun(nat,A),X: A,Aa2: A] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,X),at_top(nat))
         => ( ? [N8: nat] :
              ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N8),N)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),aa(nat,A,X5,N)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),X) ) ) ) ).

% LIMSEQ_le_const
tff(fact_6690_LIMSEQ__le__const2,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X5: fun(nat,A),X: A,Aa2: A] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,X),at_top(nat))
         => ( ? [N8: nat] :
              ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N8),N)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N)),Aa2) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Aa2) ) ) ) ).

% LIMSEQ_le_const2
tff(fact_6691_isCont__rabs,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Aa2: A,F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,Aa2,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,Aa2,top_top(set(A))),aTP_Lamp_aam(fun(A,real),fun(A,real),F2)) ) ) ).

% isCont_rabs
tff(fact_6692_continuous__at__within__powr,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Aa2: A,S: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,Aa2,S),F2)
         => ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,Aa2,S),G)
           => ( ( aa(A,real,F2,Aa2) != zero_zero(real) )
             => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,Aa2,S),aa(fun(A,real),fun(A,real),aTP_Lamp_aav(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ).

% continuous_at_within_powr
tff(fact_6693_continuous__within__ln,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [X: A,S: set(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,X,S),F2)
         => ( ( aa(A,real,F2,X) != zero_zero(real) )
           => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,X,S),aTP_Lamp_aaw(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_within_ln
tff(fact_6694_mult__nat__left__at__top,axiom,
    ! [C2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_6695_mult__nat__right__at__top,axiom,
    ! [C2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),C2)
     => filterlim(nat,nat,aTP_Lamp_aax(nat,fun(nat,nat),C2),at_top(nat),at_top(nat)) ) ).

% mult_nat_right_at_top
tff(fact_6696_monoseq__convergent,axiom,
    ! [X5: fun(nat,real),B2: real] :
      ( topological_monoseq(real,X5)
     => ( ! [I2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(nat,real,X5,I2))),B2)
       => ~ ! [L6: real] : ~ filterlim(nat,real,X5,topolo7230453075368039082e_nhds(real,L6),at_top(nat)) ) ) ).

% monoseq_convergent
tff(fact_6697_LIMSEQ__lessThan__iff__atMost,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(set(nat),A),X: A] :
          ( filterlim(nat,A,aTP_Lamp_aay(fun(set(nat),A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,X),at_top(nat))
        <=> filterlim(nat,A,aTP_Lamp_aaz(fun(set(nat),A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,X),at_top(nat)) ) ) ).

% LIMSEQ_lessThan_iff_atMost
tff(fact_6698_LIMSEQ__root,axiom,
    filterlim(nat,real,aTP_Lamp_aba(nat,real),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat)) ).

% LIMSEQ_root
tff(fact_6699_isCont__powr,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Aa2: A,F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,Aa2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,Aa2,top_top(set(A))),G)
           => ( ( aa(A,real,F2,Aa2) != zero_zero(real) )
             => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,Aa2,top_top(set(A))),aa(fun(A,real),fun(A,real),aTP_Lamp_aav(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ).

% isCont_powr
tff(fact_6700_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_aaw(fun(A,real),fun(A,real),F2)) ) ) ) ).

% isCont_ln'
tff(fact_6701_monoseq__le,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Aa2: fun(nat,A),X: A] :
          ( topological_monoseq(A,Aa2)
         => ( filterlim(nat,A,Aa2,topolo7230453075368039082e_nhds(A,X),at_top(nat))
           => ( ( ! [N10: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,Aa2,N10)),X)
                & ! [M3: nat,N10: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M3),N10)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,Aa2,M3)),aa(nat,A,Aa2,N10)) ) )
              | ( ! [N10: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(nat,A,Aa2,N10))
                & ! [M3: nat,N10: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M3),N10)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,Aa2,N10)),aa(nat,A,Aa2,M3)) ) ) ) ) ) ) ).

% monoseq_le
tff(fact_6702_lim__const__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Aa2: A] : filterlim(nat,A,aTP_Lamp_abb(A,fun(nat,A),Aa2),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_const_over_n
tff(fact_6703_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,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),L)
           => filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_abc(fun(nat,A),fun(nat,fun(nat,A)),X5),L),topolo7230453075368039082e_nhds(A,X),at_top(nat)) ) ) ) ).

% LIMSEQ_linear
tff(fact_6704_lim__inverse__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_abd(nat,A),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_inverse_n
tff(fact_6705_LIMSEQ__SEQ__conv,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [Aa2: A,X5: fun(A,B),L5: B] :
          ( ! [S6: fun(nat,A)] :
              ( ( ! [N2: nat] : aa(nat,A,S6,N2) != Aa2
                & filterlim(nat,A,S6,topolo7230453075368039082e_nhds(A,Aa2),at_top(nat)) )
             => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_abe(fun(A,B),fun(fun(nat,A),fun(nat,B)),X5),S6),topolo7230453075368039082e_nhds(B,L5),at_top(nat)) )
        <=> filterlim(A,B,X5,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,Aa2,top_top(set(A)))) ) ) ).

% LIMSEQ_SEQ_conv
tff(fact_6706_LIMSEQ__SEQ__conv1,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),L: B,Aa2: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,Aa2,top_top(set(A))))
         => ! [S7: fun(nat,A)] :
              ( ( ! [N: nat] : aa(nat,A,S7,N) != Aa2
                & filterlim(nat,A,S7,topolo7230453075368039082e_nhds(A,Aa2),at_top(nat)) )
             => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_abf(fun(A,B),fun(fun(nat,A),fun(nat,B)),F2),S7),topolo7230453075368039082e_nhds(B,L),at_top(nat)) ) ) ) ).

% LIMSEQ_SEQ_conv1
tff(fact_6707_LIMSEQ__SEQ__conv2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [Aa2: A,F2: fun(A,B),L: B] :
          ( ! [S4: fun(nat,A)] :
              ( ( ! [N10: nat] : aa(nat,A,S4,N10) != Aa2
                & filterlim(nat,A,S4,topolo7230453075368039082e_nhds(A,Aa2),at_top(nat)) )
             => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_abe(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,Aa2,top_top(set(A)))) ) ) ).

% LIMSEQ_SEQ_conv2
tff(fact_6708_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_abg(fun(nat,A),fun(nat,A),F2)) ) ) ).

% telescope_summable'
tff(fact_6709_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_abh(fun(nat,A),fun(nat,A),F2)) ) ) ).

% telescope_summable
tff(fact_6710_nested__sequence__unique,axiom,
    ! [F2: fun(nat,real),G: fun(nat,real)] :
      ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F2,N)),aa(nat,real,F2,aa(nat,nat,suc,N)))
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,G,aa(nat,nat,suc,N))),aa(nat,real,G,N))
       => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$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_abi(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,$o,aa(real,fun(real,$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,$o,aa(real,fun(real,$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_6711_LIMSEQ__inverse__zero,axiom,
    ! [X5: fun(nat,real)] :
      ( ! [R5: real] :
        ? [N8: nat] :
        ! [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N8),N)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),R5),aa(nat,real,X5,N)) )
     => filterlim(nat,real,aTP_Lamp_abj(fun(nat,real),fun(nat,real),X5),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_inverse_zero
tff(fact_6712_lim__inverse__n_H,axiom,
    filterlim(nat,real,aTP_Lamp_abk(nat,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ).

% lim_inverse_n'
tff(fact_6713_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim(nat,real,aTP_Lamp_abl(nat,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_6714_LIMSEQ__root__const,axiom,
    ! [C2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
     => filterlim(nat,real,aTP_Lamp_abm(real,fun(nat,real),C2),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat)) ) ).

% LIMSEQ_root_const
tff(fact_6715_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R: real] : filterlim(nat,real,aTP_Lamp_abn(real,fun(nat,real),R),topolo7230453075368039082e_nhds(real,R),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_6716_sums__def,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A),S: A] :
          ( aa(A,$o,sums(A,F2),S)
        <=> filterlim(nat,A,aTP_Lamp_abo(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,S),at_top(nat)) ) ) ).

% sums_def
tff(fact_6717_sums__def__le,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A),S: A] :
          ( aa(A,$o,sums(A,F2),S)
        <=> filterlim(nat,A,aTP_Lamp_abp(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,S),at_top(nat)) ) ) ).

% sums_def_le
tff(fact_6718_continuous__at__within__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Aa2: A,S: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,Aa2,S),F2)
         => ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,Aa2,S),G)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F2,Aa2))
             => ( ( aa(A,real,F2,Aa2) != one_one(real) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,Aa2))
                 => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,Aa2,S),aa(fun(A,real),fun(A,real),aTP_Lamp_abq(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% continuous_at_within_log
tff(fact_6719_increasing__LIMSEQ,axiom,
    ! [F2: fun(nat,real),L: real] :
      ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F2,N)),aa(nat,real,F2,aa(nat,nat,suc,N)))
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F2,N)),L)
       => ( ! [E: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
             => ? [N10: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),L),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,F2,N10)),E)) )
         => filterlim(nat,real,F2,topolo7230453075368039082e_nhds(real,L),at_top(nat)) ) ) ) ).

% increasing_LIMSEQ
tff(fact_6720_lim__1__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_abr(nat,A),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_1_over_n
tff(fact_6721_LIMSEQ__n__over__Suc__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_abs(nat,A),topolo7230453075368039082e_nhds(A,one_one(A)),at_top(nat)) ) ).

% LIMSEQ_n_over_Suc_n
tff(fact_6722_LIMSEQ__Suc__n__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_abt(nat,A),topolo7230453075368039082e_nhds(A,one_one(A)),at_top(nat)) ) ).

% LIMSEQ_Suc_n_over_n
tff(fact_6723_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_abh(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_6724_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_abg(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_6725_LIMSEQ__realpow__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(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_6726_LIMSEQ__divide__realpow__zero,axiom,
    ! [X: real,Aa2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),X)
     => filterlim(nat,real,aa(real,fun(nat,real),aTP_Lamp_abu(real,fun(real,fun(nat,real)),X),Aa2),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_6727_LIMSEQ__abs__realpow__zero,axiom,
    ! [C2: real] :
      ( aa(real,$o,aa(real,fun(real,$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_6728_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C2: real] :
      ( aa(real,$o,aa(real,fun(real,$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_6729_LIMSEQ__inverse__realpow__zero,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),X)
     => filterlim(nat,real,aTP_Lamp_abv(real,fun(nat,real),X),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_6730_sums__def_H,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A),S: A] :
          ( aa(A,$o,sums(A,F2),S)
        <=> filterlim(nat,A,aTP_Lamp_abw(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,S),at_top(nat)) ) ) ).

% sums_def'
tff(fact_6731_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R: real] : filterlim(nat,real,aTP_Lamp_abx(real,fun(nat,real),R),topolo7230453075368039082e_nhds(real,R),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_6732_root__test__convergence,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),X: real] :
          ( filterlim(nat,real,aTP_Lamp_aby(fun(nat,A),fun(nat,real),F2),topolo7230453075368039082e_nhds(real,X),at_top(nat))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),one_one(real))
           => summable(A,F2) ) ) ) ).

% root_test_convergence
tff(fact_6733_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_abz(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,suminf(A,F2)),at_top(nat)) ) ) ).

% summable_LIMSEQ
tff(fact_6734_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_aca(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,suminf(A,F2)),at_top(nat)) ) ) ).

% summable_LIMSEQ'
tff(fact_6735_isCont__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Aa2: A,F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,Aa2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,Aa2,top_top(set(A))),G)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F2,Aa2))
             => ( ( aa(A,real,F2,Aa2) != one_one(real) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,Aa2))
                 => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,Aa2,top_top(set(A))),aa(fun(A,real),fun(A,real),aTP_Lamp_abq(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% isCont_log
tff(fact_6736_LIMSEQ__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),L5: A] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
        <=> ! [R7: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R7)
             => ? [No2: nat] :
                ! [N2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No2),N2)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X5,N2)),L5))),R7) ) ) ) ) ).

% LIMSEQ_iff
tff(fact_6737_LIMSEQ__I,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),L5: A] :
          ( ! [R5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R5)
             => ? [No3: nat] :
                ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No3),N)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X5,N)),L5))),R5) ) )
         => filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat)) ) ) ).

% LIMSEQ_I
tff(fact_6738_LIMSEQ__D,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),L5: A,R: real] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
           => ? [No: nat] :
              ! [N10: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No),N10)
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X5,N10)),L5))),R) ) ) ) ) ).

% LIMSEQ_D
tff(fact_6739_LIMSEQ__power__zero,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [X: A] :
          ( aa(real,$o,aa(real,fun(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_6740_tendsto__exp__limit__sequentially,axiom,
    ! [X: real] : filterlim(nat,real,aTP_Lamp_acb(real,fun(nat,real),X),topolo7230453075368039082e_nhds(real,aa(real,real,exp(real),X)),at_top(nat)) ).

% tendsto_exp_limit_sequentially
tff(fact_6741_tendsto__power__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [F2: fun(A,nat),F4: filter(A),X: B] :
          ( filterlim(A,nat,F2,at_top(nat),F4)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,X)),one_one(real))
           => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_acc(fun(A,nat),fun(B,fun(A,B)),F2),X),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_power_zero
tff(fact_6742_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R: real] : filterlim(nat,real,aTP_Lamp_acd(real,fun(nat,real),R),topolo7230453075368039082e_nhds(real,R),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_6743_LIMSEQ__norm__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( ! [N: nat] : aa(real,$o,aa(real,fun(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_6744_summable__Leibniz_I1_J,axiom,
    ! [Aa2: fun(nat,real)] :
      ( filterlim(nat,real,Aa2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,Aa2)
       => summable(real,aTP_Lamp_aai(fun(nat,real),fun(nat,real),Aa2)) ) ) ).

% summable_Leibniz(1)
tff(fact_6745_field__derivative__lim__unique,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Df: A,Z: A,S: fun(nat,A),Aa2: A] :
          ( has_field_derivative(A,F2,Df,topolo174197925503356063within(A,Z,top_top(set(A))))
         => ( filterlim(nat,A,S,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
           => ( ! [N: nat] : aa(nat,A,S,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_ace(fun(A,A),fun(A,fun(fun(nat,A),fun(nat,A))),F2),Z),S),topolo7230453075368039082e_nhds(A,Aa2),at_top(nat))
               => ( Df = Aa2 ) ) ) ) ) ) ).

% field_derivative_lim_unique
tff(fact_6746_powser__times__n__limit__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [X: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X)),one_one(real))
         => filterlim(nat,A,aTP_Lamp_acf(A,fun(nat,A),X),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% powser_times_n_limit_0
tff(fact_6747_lim__n__over__pown,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),real_V7770717601297561774m_norm(A,X))
         => filterlim(nat,A,aTP_Lamp_acg(A,fun(nat,A),X),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% lim_n_over_pown
tff(fact_6748_summable,axiom,
    ! [Aa2: fun(nat,real)] :
      ( filterlim(nat,real,Aa2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,Aa2,N))
       => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,Aa2,aa(nat,nat,suc,N))),aa(nat,real,Aa2,N))
         => summable(real,aTP_Lamp_aai(fun(nat,real),fun(nat,real),Aa2)) ) ) ) ).

% summable
tff(fact_6749_cos__diff__limit__1,axiom,
    ! [Theta: fun(nat,real),Theta2: real] :
      ( filterlim(nat,real,aa(real,fun(nat,real),aTP_Lamp_ach(fun(nat,real),fun(real,fun(nat,real)),Theta),Theta2),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat))
     => ~ ! [K5: fun(nat,int)] : ~ filterlim(nat,real,aa(fun(nat,int),fun(nat,real),aTP_Lamp_aci(fun(nat,real),fun(fun(nat,int),fun(nat,real)),Theta),K5),topolo7230453075368039082e_nhds(real,Theta2),at_top(nat)) ) ).

% cos_diff_limit_1
tff(fact_6750_cos__limit__1,axiom,
    ! [Theta: fun(nat,real)] :
      ( filterlim(nat,real,aTP_Lamp_acj(fun(nat,real),fun(nat,real),Theta),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat))
     => ? [K5: fun(nat,int)] : filterlim(nat,real,aa(fun(nat,int),fun(nat,real),aTP_Lamp_aci(fun(nat,real),fun(fun(nat,int),fun(nat,real)),Theta),K5),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% cos_limit_1
tff(fact_6751_summable__Leibniz_I4_J,axiom,
    ! [Aa2: fun(nat,real)] :
      ( filterlim(nat,real,Aa2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,Aa2)
       => filterlim(nat,real,aTP_Lamp_ack(fun(nat,real),fun(nat,real),Aa2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_aai(fun(nat,real),fun(nat,real),Aa2))),at_top(nat)) ) ) ).

% summable_Leibniz(4)
tff(fact_6752_zeroseq__arctan__series,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real))
     => filterlim(nat,real,aTP_Lamp_ax(real,fun(nat,real),X),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% zeroseq_arctan_series
tff(fact_6753_summable__Leibniz_H_I2_J,axiom,
    ! [Aa2: fun(nat,real),Na: nat] :
      ( filterlim(nat,real,Aa2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,Aa2,N))
       => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,Aa2,aa(nat,nat,suc,N))),aa(nat,real,Aa2,N))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_aai(fun(nat,real),fun(nat,real),Aa2)),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_aai(fun(nat,real),fun(nat,real),Aa2))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_6754_summable__Leibniz_H_I3_J,axiom,
    ! [Aa2: fun(nat,real)] :
      ( filterlim(nat,real,Aa2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,Aa2,N))
       => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,Aa2,aa(nat,nat,suc,N))),aa(nat,real,Aa2,N))
         => filterlim(nat,real,aTP_Lamp_ack(fun(nat,real),fun(nat,real),Aa2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_aai(fun(nat,real),fun(nat,real),Aa2))),at_top(nat)) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_6755_sums__alternating__upper__lower,axiom,
    ! [Aa2: fun(nat,real)] :
      ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,Aa2,aa(nat,nat,suc,N))),aa(nat,real,Aa2,N))
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,Aa2,N))
       => ( filterlim(nat,real,Aa2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
         => ? [L2: real] :
              ( ! [N10: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_aai(fun(nat,real),fun(nat,real),Aa2)),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_ack(fun(nat,real),fun(nat,real),Aa2),topolo7230453075368039082e_nhds(real,L2),at_top(nat))
              & ! [N10: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),L2),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_aai(fun(nat,real),fun(nat,real),Aa2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N10)),one_one(nat)))))
              & filterlim(nat,real,aTP_Lamp_acl(fun(nat,real),fun(nat,real),Aa2),topolo7230453075368039082e_nhds(real,L2),at_top(nat)) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_6756_summable__Leibniz_I5_J,axiom,
    ! [Aa2: fun(nat,real)] :
      ( filterlim(nat,real,Aa2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,Aa2)
       => filterlim(nat,real,aTP_Lamp_acl(fun(nat,real),fun(nat,real),Aa2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_aai(fun(nat,real),fun(nat,real),Aa2))),at_top(nat)) ) ) ).

% summable_Leibniz(5)
tff(fact_6757_summable__Leibniz_H_I4_J,axiom,
    ! [Aa2: fun(nat,real),Na: nat] :
      ( filterlim(nat,real,Aa2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,Aa2,N))
       => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,Aa2,aa(nat,nat,suc,N))),aa(nat,real,Aa2,N))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),suminf(real,aTP_Lamp_aai(fun(nat,real),fun(nat,real),Aa2))),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_aai(fun(nat,real),fun(nat,real),Aa2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),one_one(nat))))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_6758_summable__Leibniz_H_I5_J,axiom,
    ! [Aa2: fun(nat,real)] :
      ( filterlim(nat,real,Aa2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,Aa2,N))
       => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,Aa2,aa(nat,nat,suc,N))),aa(nat,real,Aa2,N))
         => filterlim(nat,real,aTP_Lamp_acl(fun(nat,real),fun(nat,real),Aa2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_aai(fun(nat,real),fun(nat,real),Aa2))),at_top(nat)) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_6759_filterlim__sequentially__Suc,axiom,
    ! [A: $tType,F2: fun(nat,A),F4: filter(A)] :
      ( filterlim(nat,A,aTP_Lamp_acm(fun(nat,A),fun(nat,A),F2),F4,at_top(nat))
    <=> filterlim(nat,A,F2,F4,at_top(nat)) ) ).

% filterlim_sequentially_Suc
tff(fact_6760_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_acn(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_6761_bounded__linear_Otendsto,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [F2: fun(A,B),G: fun(C,A),Aa2: A,F4: filter(C)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( filterlim(C,A,G,topolo7230453075368039082e_nhds(A,Aa2),F4)
           => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_aco(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G),topolo7230453075368039082e_nhds(B,aa(A,B,F2,Aa2)),F4) ) ) ) ).

% bounded_linear.tendsto
tff(fact_6762_bounded__linear_Ocontinuous,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F4: filter(C),G: fun(C,A)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( topolo3448309680560233919inuous(C,A,F4,G)
           => topolo3448309680560233919inuous(C,B,F4,aa(fun(C,A),fun(C,B),aTP_Lamp_acp(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G)) ) ) ) ).

% bounded_linear.continuous
tff(fact_6763_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_acq(fun(A,B),fun(fun(nat,A),fun(nat,B)),F2),X5)) ) ) ) ) ).

% bounded_linear.suminf
tff(fact_6764_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_acq(fun(A,B),fun(fun(nat,A),fun(nat,B)),F2),X5)) ) ) ) ).

% bounded_linear.Cauchy
tff(fact_6765_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_tx(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% bounded_linear_sub
tff(fact_6766_bounded__linear_Osums,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),X5: fun(nat,A),Aa2: A] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( aa(A,$o,sums(A,X5),Aa2)
           => aa(B,$o,sums(B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_acq(fun(A,B),fun(fun(nat,A),fun(nat,B)),F2),X5)),aa(A,B,F2,Aa2)) ) ) ) ).

% bounded_linear.sums
tff(fact_6767_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_6768_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_acq(fun(A,B),fun(fun(nat,A),fun(nat,B)),F2),X5)) ) ) ) ).

% bounded_linear.summable
tff(fact_6769_bounded__linear__ident,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => real_V3181309239436604168linear(A,A,aTP_Lamp_tw(A,A)) ) ).

% bounded_linear_ident
tff(fact_6770_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_ua(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G)) ) ) ) ).

% bounded_linear_compose
tff(fact_6771_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_tu(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% bounded_linear_add
tff(fact_6772_bounded__linear__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => real_V3181309239436604168linear(A,B,aTP_Lamp_tt(A,B)) ) ).

% bounded_linear_zero
tff(fact_6773_bounded__linear__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R: real] : real_V3181309239436604168linear(A,A,real_V8093663219630862766scaleR(A,R)) ) ).

% bounded_linear_scaleR_right
tff(fact_6774_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_tq(fun(A,real),fun(B,fun(A,B)),G),X)) ) ) ).

% bounded_linear_scaleR_const
tff(fact_6775_bounded__linear__const__scaleR,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [G: fun(A,B),R: real] :
          ( real_V3181309239436604168linear(A,B,G)
         => real_V3181309239436604168linear(A,B,aa(real,fun(A,B),aTP_Lamp_tr(fun(A,B),fun(real,fun(A,B)),G),R)) ) ) ).

% bounded_linear_const_scaleR
tff(fact_6776_bounded__linear__scaleR__left,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] : real_V3181309239436604168linear(real,A,aTP_Lamp_acr(A,fun(real,A),X)) ) ).

% bounded_linear_scaleR_left
tff(fact_6777_bounded__linear__mult__left,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Y: A] : real_V3181309239436604168linear(A,A,aTP_Lamp_acs(A,fun(A,A),Y)) ) ).

% bounded_linear_mult_left
tff(fact_6778_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_to(fun(A,B),fun(B,fun(A,B)),G),X)) ) ) ).

% bounded_linear_const_mult
tff(fact_6779_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_tp(fun(A,B),fun(B,fun(A,B)),G),Y)) ) ) ).

% bounded_linear_mult_const
tff(fact_6780_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_6781_bounded__linear__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Y: A] : real_V3181309239436604168linear(A,A,aTP_Lamp_act(A,fun(A,A),Y)) ) ).

% bounded_linear_divide
tff(fact_6782_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_tn(fun(A,B),fun(A,B),F2)) ) ) ).

% bounded_linear_minus
tff(fact_6783_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))] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => real_V3181309239436604168linear(B,C,aa(A,fun(B,C),F2,I2)) )
         => real_V3181309239436604168linear(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_tm(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2)) ) ) ).

% bounded_linear_sum
tff(fact_6784_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),G2: fun(C,A),F4: filter(C)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( has_derivative(C,A,G,G2,F4)
           => has_derivative(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ua(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G),aa(fun(C,A),fun(C,B),aTP_Lamp_ua(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G2),F4) ) ) ) ).

% bounded_linear.has_derivative
tff(fact_6785_bounded__linear_Obounded,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ? [K8: real] :
            ! [X4: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,X4))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X4)),K8)) ) ) ).

% bounded_linear.bounded
tff(fact_6786_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),F4: filter(C)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( filterlim(C,A,G,topolo7230453075368039082e_nhds(A,zero_zero(A)),F4)
           => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_aco(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% bounded_linear.tendsto_zero
tff(fact_6787_bounded__linear_OisCont,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),Aa2: C,G: fun(C,A)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( topolo3448309680560233919inuous(C,A,topolo174197925503356063within(C,Aa2,top_top(set(C))),G)
           => topolo3448309680560233919inuous(C,B,topolo174197925503356063within(C,Aa2,top_top(set(C))),aa(fun(C,A),fun(C,B),aTP_Lamp_acp(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G)) ) ) ) ).

% bounded_linear.isCont
tff(fact_6788_bounded__linear_Ononneg__bounded,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ? [K8: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),K8)
              & ! [X4: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,X4))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X4)),K8)) ) ) ) ).

% bounded_linear.nonneg_bounded
tff(fact_6789_filterlim__compose,axiom,
    ! [B: $tType,A: $tType,C: $tType,G: fun(A,B),F32: filter(B),F22: filter(A),F2: fun(C,A),F1: filter(C)] :
      ( filterlim(A,B,G,F32,F22)
     => ( filterlim(C,A,F2,F22,F1)
       => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_acu(fun(A,B),fun(fun(C,A),fun(C,B)),G),F2),F32,F1) ) ) ).

% filterlim_compose
tff(fact_6790_filterlim__ident,axiom,
    ! [A: $tType,F4: filter(A)] : filterlim(A,A,aTP_Lamp_pc(A,A),F4,F4) ).

% filterlim_ident
tff(fact_6791_bounded__linear_Opos__bounded,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ? [K8: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K8)
              & ! [X4: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,X4))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X4)),K8)) ) ) ) ).

% bounded_linear.pos_bounded
tff(fact_6792_bounded__linear__intro,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),K3: real] :
          ( ! [X2: A,Y2: A] : aa(A,B,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),Y2)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F2,X2)),aa(A,B,F2,Y2))
         => ( ! [R5: real,X2: A] : aa(A,B,F2,aa(A,A,real_V8093663219630862766scaleR(A,R5),X2)) = aa(B,B,real_V8093663219630862766scaleR(B,R5),aa(A,B,F2,X2))
           => ( ! [X2: A] : aa(real,$o,aa(real,fun(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)),K3))
             => real_V3181309239436604168linear(A,B,F2) ) ) ) ) ).

% bounded_linear_intro
tff(fact_6793_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,S: set(A)] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,S))
        <=> ( real_V3181309239436604168linear(A,B,F7)
            & filterlim(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_acv(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),F2),F7),X),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,X,S)) ) ) ) ).

% has_derivative_iff_norm
tff(fact_6794_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,S: set(A)] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,S))
        <=> ( real_V3181309239436604168linear(A,B,F7)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_acw(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F2),F7),X),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,X,S)) ) ) ) ).

% has_derivative_at_within
tff(fact_6795_has__derivativeI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F7: fun(A,B),X: A,F2: fun(A,B),S: 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_acx(fun(A,B),fun(A,fun(fun(A,B),fun(A,B))),F7),X),F2),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,X,S))
           => has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,S)) ) ) ) ).

% has_derivativeI
tff(fact_6796_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)] :
                ( ! [H5: A] : aa(A,B,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),H5)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F2,X)),aa(A,B,F7,H5))),aa(A,B,E3,H5))
                & filterlim(A,real,aTP_Lamp_acy(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_6797_has__derivative__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),X: A,S: set(A)] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,S))
        <=> ( real_V3181309239436604168linear(A,B,F7)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_acn(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F2),F7),X),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,X,S)) ) ) ) ).

% has_derivative_within
tff(fact_6798_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_acz(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_6799_has__derivative__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),F4: filter(A)] :
          ( has_derivative(A,B,F2,F7,F4)
        <=> ( 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_ada(fun(A,B),fun(fun(A,B),fun(filter(A),fun(A,B))),F2),F7),F4),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% has_derivative_def
tff(fact_6800_has__derivative__at__within__iff__Ex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [X: A,Sa: set(A),F2: fun(A,B),F7: fun(A,B)] :
          ( member(A,X,Sa)
         => ( topolo1002775350975398744n_open(A,Sa)
           => ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,Sa))
            <=> ( real_V3181309239436604168linear(A,B,F7)
                & ? [E3: fun(A,B)] :
                    ( ! [H5: A] :
                        ( member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),H5),Sa)
                       => ( aa(A,B,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),H5)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F2,X)),aa(A,B,F7,H5))),aa(A,B,E3,H5)) ) )
                    & filterlim(A,real,aTP_Lamp_acy(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_6801_lim__const,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Aa2: A] : topolo3827282254853284352ce_Lim(nat,A,at_top(nat),aTP_Lamp_aas(A,fun(nat,A),Aa2)) = Aa2 ) ).

% lim_const
tff(fact_6802_open__Collect__const,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [P: $o] : topolo1002775350975398744n_open(A,collect(A,aTP_Lamp_adb($o,fun(A,$o),(P)))) ) ).

% open_Collect_const
tff(fact_6803_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,$o),aTP_Lamp_adc(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q))) ) ) ) ).

% open_Collect_disj
tff(fact_6804_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,$o),aTP_Lamp_add(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q))) ) ) ) ).

% open_Collect_conj
tff(fact_6805_Lim__ident__at,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [X: A,S: set(A)] :
          ( ( topolo174197925503356063within(A,X,S) != bot_bot(filter(A)) )
         => ( topolo3827282254853284352ce_Lim(A,A,topolo174197925503356063within(A,X,S),aTP_Lamp_wl(A,A)) = X ) ) ) ).

% Lim_ident_at
tff(fact_6806_lim__explicit,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),F0: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,F0),at_top(nat))
        <=> ! [S6: set(A)] :
              ( topolo1002775350975398744n_open(A,S6)
             => ( member(A,F0,S6)
               => ? [N7: nat] :
                  ! [N2: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N2)
                   => member(A,aa(nat,A,F2,N2),S6) ) ) ) ) ) ).

% lim_explicit
tff(fact_6807_continuous__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [F4: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( topolo3448309680560233919inuous(A,B,F4,G)
           => ( ( aa(A,B,G,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_wl(A,A))) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,F4,aa(fun(A,B),fun(A,B),aTP_Lamp_zr(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% continuous_divide
tff(fact_6808_continuous__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [F4: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( ( aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_wl(A,A))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_zu(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_inverse
tff(fact_6809_continuous__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [F4: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
        <=> filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_wl(A,A)))),F4) ) ) ).

% continuous_def
tff(fact_6810_continuous__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [F4: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( ( aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_wl(A,A))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_zv(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_sgn
tff(fact_6811_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_ade(filter(B),fun(fun(B,A),fun(A,$o)),A2),F2)) ) ).

% t2_space_class.Lim_def
tff(fact_6812_continuous__powr,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => ( topolo3448309680560233919inuous(A,real,F4,G)
           => ( ( aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_wl(A,A))) != zero_zero(real) )
             => topolo3448309680560233919inuous(A,real,F4,aa(fun(A,real),fun(A,real),aTP_Lamp_aav(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ).

% continuous_powr
tff(fact_6813_continuous__ln,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => ( ( aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_wl(A,A))) != zero_zero(real) )
           => topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_aaw(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_ln
tff(fact_6814_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_abz(fun(nat,A),fun(nat,A),F2)) ) ).

% suminf_eq_lim
tff(fact_6815_Topological__Spaces_Olim__def,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [X5: fun(nat,A)] : topolo3827282254853284352ce_Lim(nat,A,at_top(nat),X5) = the(A,aTP_Lamp_adf(fun(nat,A),fun(A,$o),X5)) ) ).

% Topological_Spaces.lim_def
tff(fact_6816_continuous__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F4: filter(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,F4,F2)
         => ( ( cos(A,aa(A,A,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_adg(A,A)))) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,F4,aTP_Lamp_xi(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_tan
tff(fact_6817_continuous__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F4: filter(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,F4,F2)
         => ( ( sin(A,aa(A,A,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_adg(A,A)))) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,F4,aTP_Lamp_wo(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_cot
tff(fact_6818_continuous__tanh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F4: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( ( cosh(B,aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_wl(A,A)))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_aaa(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_tanh
tff(fact_6819_continuous__arcosh,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_wl(A,A))))
           => topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_adh(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_arcosh
tff(fact_6820_continuous__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => ( topolo3448309680560233919inuous(A,real,F4,G)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_wl(A,A))))
             => ( ( aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_wl(A,A))) != one_one(real) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_wl(A,A))))
                 => topolo3448309680560233919inuous(A,real,F4,aa(fun(A,real),fun(A,real),aTP_Lamp_abq(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% continuous_log
tff(fact_6821_continuous__artanh,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => ( member(real,aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_wl(A,A))),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real)))
           => topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_adi(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_artanh
tff(fact_6822_tendsto__offset__zero__iff,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & topolo4958980785337419405_space(C)
        & zero(A) )
     => ! [Aa2: B,Sa: set(B),F2: fun(B,C),L5: C] :
          ( nO_MATCH(A,B,zero_zero(A),Aa2)
         => ( member(B,Aa2,Sa)
           => ( topolo1002775350975398744n_open(B,Sa)
             => ( filterlim(B,C,F2,topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,Aa2,Sa))
              <=> filterlim(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_adj(B,fun(fun(B,C),fun(B,C)),Aa2),F2),topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,zero_zero(B),top_top(set(B)))) ) ) ) ) ) ).

% tendsto_offset_zero_iff
tff(fact_6823_has__derivativeI__sandwich,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [E2: real,F7: fun(A,B),S: set(A),X: A,F2: fun(A,B),H6: fun(A,real)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
         => ( real_V3181309239436604168linear(A,B,F7)
           => ( ! [Y2: A] :
                  ( member(A,Y2,S)
                 => ( ( Y2 != X )
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Y2,X)),E2)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(B,B,minus_minus(B,aa(A,B,F2,Y2)),aa(A,B,F2,X))),aa(A,B,F7,aa(A,A,minus_minus(A,Y2),X))))),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Y2),X)))),aa(A,real,H6,Y2)) ) ) )
             => ( filterlim(A,real,H6,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,X,S))
               => has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,S)) ) ) ) ) ) ).

% has_derivativeI_sandwich
tff(fact_6824_dist__add__cancel2,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Ba: A,Aa2: A,C2: A] : real_V557655796197034286t_dist(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Ba),Aa2),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),Aa2)) = real_V557655796197034286t_dist(A,Ba,C2) ) ).

% dist_add_cancel2
tff(fact_6825_dist__add__cancel,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Aa2: A,Ba: A,C2: A] : real_V557655796197034286t_dist(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba),aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),C2)) = real_V557655796197034286t_dist(A,Ba,C2) ) ).

% dist_add_cancel
tff(fact_6826_dist__le__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(A,X,Y)),zero_zero(real))
        <=> ( X = Y ) ) ) ).

% dist_le_zero_iff
tff(fact_6827_div__add__self1__no__field,axiom,
    ! [A: $tType,B: $tType] :
      ( ( euclid4440199948858584721cancel(B)
        & field(A) )
     => ! [X: A,Ba: B,Aa2: B] :
          ( nO_MATCH(A,B,X,Ba)
         => ( ( Ba != zero_zero(B) )
           => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),Ba),Aa2)),Ba) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),Aa2),Ba)),one_one(B)) ) ) ) ) ).

% div_add_self1_no_field
tff(fact_6828_div__add__self2__no__field,axiom,
    ! [A: $tType,B: $tType] :
      ( ( euclid4440199948858584721cancel(B)
        & field(A) )
     => ! [X: A,Ba: B,Aa2: B] :
          ( nO_MATCH(A,B,X,Ba)
         => ( ( Ba != zero_zero(B) )
           => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),Aa2),Ba)),Ba) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),Aa2),Ba)),one_one(B)) ) ) ) ) ).

% div_add_self2_no_field
tff(fact_6829_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_adk(A,fun(real,fun(A,$o)),X),D2))) ) ).

% open_ball
tff(fact_6830_continuous__dist,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [F4: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( topolo3448309680560233919inuous(A,B,F4,G)
           => topolo3448309680560233919inuous(A,real,F4,aa(fun(A,B),fun(A,real),aTP_Lamp_adl(fun(A,B),fun(fun(A,B),fun(A,real)),F2),G)) ) ) ) ).

% continuous_dist
tff(fact_6831_zero__le__dist,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),real_V557655796197034286t_dist(A,X,Y)) ) ).

% zero_le_dist
tff(fact_6832_dist__triangle,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Z: A,Y: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(A,X,Z)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,X,Y)),real_V557655796197034286t_dist(A,Y,Z))) ) ).

% dist_triangle
tff(fact_6833_dist__triangle2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y: A,Z: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(A,X,Y)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,X,Z)),real_V557655796197034286t_dist(A,Y,Z))) ) ).

% dist_triangle2
tff(fact_6834_dist__triangle3,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y: A,Aa2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(A,X,Y)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,Aa2,X)),real_V557655796197034286t_dist(A,Aa2,Y))) ) ).

% dist_triangle3
tff(fact_6835_dist__triangle__le,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Z: A,Y: A,E2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,X,Z)),real_V557655796197034286t_dist(A,Y,Z))),E2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(A,X,Y)),E2) ) ) ).

% dist_triangle_le
tff(fact_6836_dist__triangle__lt,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Z: A,Y: A,E2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,X,Z)),real_V557655796197034286t_dist(A,Y,Z))),E2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X,Y)),E2) ) ) ).

% dist_triangle_lt
tff(fact_6837_dist__triangle__less__add,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X12: A,Y: A,E1: real,X23: A,E22: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X12,Y)),E1)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X23,Y)),E22)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X12,X23)),aa(real,real,aa(real,fun(real,real),plus_plus(real),E1),E22)) ) ) ) ).

% dist_triangle_less_add
tff(fact_6838_abs__dist__diff__le,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Aa2: A,Ba: A,C2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,real_V557655796197034286t_dist(A,Aa2,Ba)),real_V557655796197034286t_dist(A,Ba,C2)))),real_V557655796197034286t_dist(A,Aa2,C2)) ) ).

% abs_dist_diff_le
tff(fact_6839_metric__CauchyI,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A)] :
          ( ! [E: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
             => ? [M12: nat] :
                ! [M5: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M12),M5)
                 => ! [N: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M12),N)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X5,M5),aa(nat,A,X5,N))),E) ) ) )
         => topolo3814608138187158403Cauchy(A,X5) ) ) ).

% metric_CauchyI
tff(fact_6840_metric__CauchyD,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A),E2: real] :
          ( topolo3814608138187158403Cauchy(A,X5)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
           => ? [M10: nat] :
              ! [M3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M10),M3)
               => ! [N10: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M10),N10)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X5,M3),aa(nat,A,X5,N10))),E2) ) ) ) ) ) ).

% metric_CauchyD
tff(fact_6841_Cauchy__altdef2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [S: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,S)
        <=> ! [E3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
             => ? [N7: nat] :
                ! [N2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N2)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,S,N2),aa(nat,A,S,N7))),E3) ) ) ) ) ).

% Cauchy_altdef2
tff(fact_6842_Cauchy__def,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X5)
        <=> ! [E3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
             => ? [M11: nat] :
                ! [M4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M11),M4)
                 => ! [N2: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M11),N2)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X5,M4),aa(nat,A,X5,N2))),E3) ) ) ) ) ) ).

% Cauchy_def
tff(fact_6843_tendsto__dist,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A),G: fun(A,B),M: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,M),F4)
           => filterlim(A,real,aa(fun(A,B),fun(A,real),aTP_Lamp_adm(fun(A,B),fun(fun(A,B),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,real_V557655796197034286t_dist(B,L,M)),F4) ) ) ) ).

% tendsto_dist
tff(fact_6844_scale__right__distrib__NO__MATCH,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,Y: A,Aa2: real] :
          ( nO_MATCH(A,real,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y),Aa2)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,Aa2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,Aa2),X)),aa(A,A,real_V8093663219630862766scaleR(A,Aa2),Y)) ) ) ) ).

% scale_right_distrib_NO_MATCH
tff(fact_6845_distrib__right__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring(B)
     => ! [X: A,Y: A,C2: B,Aa2: B,Ba: 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,aa(B,fun(B,B),plus_plus(B),Aa2),Ba)),C2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),Aa2),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),Ba),C2)) ) ) ) ).

% distrib_right_NO_MATCH
tff(fact_6846_distrib__left__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring(B)
     => ! [X: A,Y: A,Aa2: B,Ba: B,C2: B] :
          ( nO_MATCH(A,B,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y),Aa2)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),Aa2),aa(B,B,aa(B,fun(B,B),plus_plus(B),Ba),C2)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),Aa2),Ba)),aa(B,B,aa(B,fun(B,B),times_times(B),Aa2),C2)) ) ) ) ).

% distrib_left_NO_MATCH
tff(fact_6847_left__diff__distrib__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( ring(B)
     => ! [X: A,Y: A,C2: B,Aa2: B,Ba: 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,Aa2),Ba)),C2) = aa(B,B,minus_minus(B,aa(B,B,aa(B,fun(B,B),times_times(B),Aa2),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),Ba),C2)) ) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_6848_right__diff__distrib__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( ring(B)
     => ! [X: A,Y: A,Aa2: B,Ba: B,C2: B] :
          ( nO_MATCH(A,B,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y),Aa2)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),Aa2),aa(B,B,minus_minus(B,Ba),C2)) = aa(B,B,minus_minus(B,aa(B,B,aa(B,fun(B,B),times_times(B),Aa2),Ba)),aa(B,B,aa(B,fun(B,B),times_times(B),Aa2),C2)) ) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_6849_dist__triangle__half__r,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Y: A,X12: A,E2: real,X23: A] :
          ( aa(real,$o,aa(real,fun(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,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Y,X23)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E2),aa(num,real,numeral_numeral(real),bit0(one2))))
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X12,X23)),E2) ) ) ) ).

% dist_triangle_half_r
tff(fact_6850_dist__triangle__half__l,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X12: A,Y: A,E2: real,X23: A] :
          ( aa(real,$o,aa(real,fun(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,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X23,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E2),aa(num,real,numeral_numeral(real),bit0(one2))))
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X12,X23)),E2) ) ) ) ).

% dist_triangle_half_l
tff(fact_6851_dist__triangle__third,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X12: A,X23: A,E2: real,X32: A,X42: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X12,X23)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E2),aa(num,real,numeral_numeral(real),bit1(one2))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X23,X32)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E2),aa(num,real,numeral_numeral(real),bit1(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X32,X42)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E2),aa(num,real,numeral_numeral(real),bit1(one2))))
             => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X12,X42)),E2) ) ) ) ) ).

% dist_triangle_third
tff(fact_6852_metric__LIM__imp__LIM,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V7819770556892013058_space(C)
        & real_V7819770556892013058_space(B) )
     => ! [F2: fun(A,B),L: B,Aa2: A,G: fun(A,C),M: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,Aa2,top_top(set(A))))
         => ( ! [X2: A] :
                ( ( X2 != Aa2 )
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(C,aa(A,C,G,X2),M)),real_V557655796197034286t_dist(B,aa(A,B,F2,X2),L)) )
           => filterlim(A,C,G,topolo7230453075368039082e_nhds(C,M),topolo174197925503356063within(A,Aa2,top_top(set(A)))) ) ) ) ).

% metric_LIM_imp_LIM
tff(fact_6853_Cauchy__altdef,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [F2: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,F2)
        <=> ! [E3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
             => ? [M11: nat] :
                ! [M4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M11),M4)
                 => ! [N2: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),N2)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,F2,M4),aa(nat,A,F2,N2))),E3) ) ) ) ) ) ).

% Cauchy_altdef
tff(fact_6854_CauchyI_H,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A)] :
          ( ! [E: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
             => ? [M12: nat] :
                ! [M5: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M12),M5)
                 => ! [N: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M5),N)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X5,M5),aa(nat,A,X5,N))),E) ) ) )
         => topolo3814608138187158403Cauchy(A,X5) ) ) ).

% CauchyI'
tff(fact_6855_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_6856_tendsto__dist__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
        <=> filterlim(A,real,aa(B,fun(A,real),aTP_Lamp_adn(fun(A,B),fun(B,fun(A,real)),F2),L),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% tendsto_dist_iff
tff(fact_6857_lim__sequentially,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A),L5: A] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
        <=> ! [R7: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R7)
             => ? [No2: nat] :
                ! [N2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No2),N2)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X5,N2),L5)),R7) ) ) ) ) ).

% lim_sequentially
tff(fact_6858_metric__LIMSEQ__I,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A),L5: A] :
          ( ! [R5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R5)
             => ? [No3: nat] :
                ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No3),N)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X5,N),L5)),R5) ) )
         => filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat)) ) ) ).

% metric_LIMSEQ_I
tff(fact_6859_metric__LIMSEQ__D,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A),L5: A,R: real] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
           => ? [No: nat] :
              ! [N10: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No),N10)
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X5,N10),L5)),R) ) ) ) ) ).

% metric_LIMSEQ_D
tff(fact_6860_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_6861_metric__Cauchy__iff2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X5)
        <=> ! [J3: nat] :
            ? [M11: nat] :
            ! [M4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M11),M4)
             => ! [N2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M11),N2)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X5,M4),aa(nat,A,X5,N2))),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,J3)))) ) ) ) ) ).

% metric_Cauchy_iff2
tff(fact_6862_scale__left__distrib__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,Y: A,C2: B,Aa2: real,Ba: real] :
          ( nO_MATCH(A,B,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y),C2)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),plus_plus(real),Aa2),Ba)),X) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,Aa2),X)),aa(A,A,real_V8093663219630862766scaleR(A,Ba),X)) ) ) ) ).

% scale_left_distrib_NO_MATCH
tff(fact_6863_metric__LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),Ba: B,Aa2: A,G: fun(B,C),C2: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Ba),topolo174197925503356063within(A,Aa2,top_top(set(A))))
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(B,Ba,top_top(set(B))))
           => ( ? [D6: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
                  & ! [X2: A] :
                      ( ( ( X2 != Aa2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X2,Aa2)),D6) )
                     => ( aa(A,B,F2,X2) != Ba ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ado(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(A,Aa2,top_top(set(A)))) ) ) ) ) ).

% metric_LIM_compose2
tff(fact_6864_metric__isCont__LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [Aa2: A,F2: fun(A,B),G: fun(B,C),L: C] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,top_top(set(A))),F2)
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(B,aa(A,B,F2,Aa2),top_top(set(B))))
           => ( ? [D6: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
                  & ! [X2: A] :
                      ( ( ( X2 != Aa2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X2,Aa2)),D6) )
                     => ( aa(A,B,F2,X2) != aa(A,B,F2,Aa2) ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ado(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(A,Aa2,top_top(set(A)))) ) ) ) ) ).

% metric_isCont_LIM_compose2
tff(fact_6865_LIMSEQ__iff__nz,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A),L5: A] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
        <=> ! [R7: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R7)
             => ? [No2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),No2)
                  & ! [N2: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No2),N2)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X5,N2),L5)),R7) ) ) ) ) ) ).

% LIMSEQ_iff_nz
tff(fact_6866_LIM__offset__zero__iff,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & topolo4958980785337419405_space(C)
        & zero(A) )
     => ! [Aa2: B,F2: fun(B,C),L5: C] :
          ( nO_MATCH(A,B,zero_zero(A),Aa2)
         => ( filterlim(B,C,F2,topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,Aa2,top_top(set(B))))
          <=> filterlim(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_adj(B,fun(fun(B,C),fun(B,C)),Aa2),F2),topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,zero_zero(B),top_top(set(B)))) ) ) ) ).

% LIM_offset_zero_iff
tff(fact_6867_tendsto__exp__limit__at__right,axiom,
    ! [X: real] : filterlim(real,real,aTP_Lamp_adp(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_6868_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_6869_greaterThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_ord_greaterThan(A,X)),set_ord_greaterThan(A,Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) ) ) ).

% greaterThan_subset_iff
tff(fact_6870_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_6871_greaterThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A] : set_ord_greaterThan(A,L) = collect(A,aa(A,fun(A,$o),ord_less(A),L)) ) ).

% greaterThan_def
tff(fact_6872_filterlim__at__left__to__right,axiom,
    ! [A: $tType,F2: fun(real,A),F4: filter(A),Aa2: real] :
      ( filterlim(real,A,F2,F4,topolo174197925503356063within(real,Aa2,set_ord_lessThan(real,Aa2)))
    <=> filterlim(real,A,aTP_Lamp_adq(fun(real,A),fun(real,A),F2),F4,topolo174197925503356063within(real,aa(real,real,uminus_uminus(real),Aa2),set_ord_greaterThan(real,aa(real,real,uminus_uminus(real),Aa2)))) ) ).

% filterlim_at_left_to_right
tff(fact_6873_filterlim__at__right__to__0,axiom,
    ! [A: $tType,F2: fun(real,A),F4: filter(A),Aa2: real] :
      ( filterlim(real,A,F2,F4,topolo174197925503356063within(real,Aa2,set_ord_greaterThan(real,Aa2)))
    <=> filterlim(real,A,aa(real,fun(real,A),aTP_Lamp_adr(fun(real,A),fun(real,fun(real,A)),F2),Aa2),F4,topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ) ).

% filterlim_at_right_to_0
tff(fact_6874_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_6875_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_6876_filterlim__times__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linordered_field(B)
        & topolo1944317154257567458pology(B) )
     => ! [F2: fun(A,B),P2: B,F1: filter(A),C2: B,L: B] :
          ( filterlim(A,B,F2,topolo174197925503356063within(B,P2,set_ord_greaterThan(B,P2)),F1)
         => ( aa(B,$o,aa(B,fun(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_ads(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo174197925503356063within(B,L,set_ord_greaterThan(B,L)),F1) ) ) ) ) ).

% filterlim_times_pos
tff(fact_6877_log__inj,axiom,
    ! [Ba: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Ba)
     => inj_on(real,real,log(Ba),set_ord_greaterThan(real,zero_zero(real))) ) ).

% log_inj
tff(fact_6878_filterlim__tendsto__pos__mult__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F4: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
       => ( filterlim(A,real,G,at_bot(real),F4)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_adt(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_bot(real),F4) ) ) ) ).

% filterlim_tendsto_pos_mult_at_bot
tff(fact_6879_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_adu(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_6880_isCont__If__ge,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [Aa2: A,G: fun(A,B),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,set_ord_lessThan(A,Aa2)),G)
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,G,Aa2)),topolo174197925503356063within(A,Aa2,set_ord_greaterThan(A,Aa2)))
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,top_top(set(A))),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_adv(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Aa2),G),F2)) ) ) ) ).

% isCont_If_ge
tff(fact_6881_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [Ba: real,F2: fun(real,real),Flim: real] :
      ( ! [X2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),Ba)
         => ? [Y3: real] :
              ( has_field_derivative(real,F2,Y3,topolo174197925503356063within(real,X2,top_top(set(real))))
              & aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y3) ) )
     => ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,Flim),at_bot(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),Flim),aa(real,real,F2,Ba)) ) ) ).

% DERIV_pos_imp_increasing_at_bot
tff(fact_6882_filterlim__pow__at__bot__odd,axiom,
    ! [Na: nat,F2: fun(real,real),F4: filter(real)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( filterlim(real,real,F2,at_bot(real),F4)
       => ( ~ 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_adw(nat,fun(fun(real,real),fun(real,real)),Na),F2),at_bot(real),F4) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_6883_filterlim__pow__at__bot__even,axiom,
    ! [Na: nat,F2: fun(real,real),F4: filter(real)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( filterlim(real,real,F2,at_bot(real),F4)
       => ( 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_adw(nat,fun(fun(real,real),fun(real,real)),Na),F2),at_top(real),F4) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_6884_lim__zero__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),L: A] :
          ( filterlim(A,A,aTP_Lamp_adx(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_6885_exp__at__top,axiom,
    filterlim(real,real,exp(real),at_top(real),at_top(real)) ).

% exp_at_top
tff(fact_6886_filterlim__at__top__mult__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,at_top(real),F4)
     => ( filterlim(A,real,G,at_top(real),F4)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_adt(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F4) ) ) ).

% filterlim_at_top_mult_at_top
tff(fact_6887_filterlim__at__top__add__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,at_top(real),F4)
     => ( filterlim(A,real,G,at_top(real),F4)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ady(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F4) ) ) ).

% filterlim_at_top_add_at_top
tff(fact_6888_sqrt__at__top,axiom,
    filterlim(real,real,sqrt,at_top(real),at_top(real)) ).

% sqrt_at_top
tff(fact_6889_ln__at__top,axiom,
    filterlim(real,real,ln_ln(real),at_top(real),at_top(real)) ).

% ln_at_top
tff(fact_6890_filterlim__at__infinity__conv__norm__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),G3: filter(A)] :
          ( filterlim(A,B,F2,at_infinity(B),G3)
        <=> filterlim(A,real,aTP_Lamp_xw(fun(A,B),fun(A,real),F2),at_top(real),G3) ) ) ).

% filterlim_at_infinity_conv_norm_at_top
tff(fact_6891_filterlim__norm__at__top__imp__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,real,aTP_Lamp_xw(fun(A,B),fun(A,real),F2),at_top(real),F4)
         => filterlim(A,B,F2,at_infinity(B),F4) ) ) ).

% filterlim_norm_at_top_imp_at_infinity
tff(fact_6892_filterlim__at__infinity__imp__norm__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F2,at_infinity(B),F4)
         => filterlim(A,real,aTP_Lamp_xw(fun(A,B),fun(A,real),F2),at_top(real),F4) ) ) ).

% filterlim_at_infinity_imp_norm_at_top
tff(fact_6893_filterlim__tendsto__add__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F4: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F4)
     => ( filterlim(A,real,G,at_top(real),F4)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ady(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F4) ) ) ).

% filterlim_tendsto_add_at_top
tff(fact_6894_filterlim__real__sequentially,axiom,
    filterlim(nat,real,semiring_1_of_nat(real),at_top(real),at_top(nat)) ).

% filterlim_real_sequentially
tff(fact_6895_filterlim__uminus__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F2,at_top(real),F4)
    <=> filterlim(A,real,aTP_Lamp_adz(fun(A,real),fun(A,real),F2),at_bot(real),F4) ) ).

% filterlim_uminus_at_top
tff(fact_6896_filterlim__uminus__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F2,at_bot(real),F4)
    <=> filterlim(A,real,aTP_Lamp_adz(fun(A,real),fun(A,real),F2),at_top(real),F4) ) ).

% filterlim_uminus_at_bot
tff(fact_6897_filterlim__at__top__mirror,axiom,
    ! [A: $tType,F2: fun(real,A),F4: filter(A)] :
      ( filterlim(real,A,F2,F4,at_top(real))
    <=> filterlim(real,A,aTP_Lamp_adq(fun(real,A),fun(real,A),F2),F4,at_bot(real)) ) ).

% filterlim_at_top_mirror
tff(fact_6898_filterlim__at__bot__mirror,axiom,
    ! [A: $tType,F2: fun(real,A),F4: filter(A)] :
      ( filterlim(real,A,F2,F4,at_bot(real))
    <=> filterlim(real,A,aTP_Lamp_adq(fun(real,A),fun(real,A),F2),F4,at_top(real)) ) ).

% filterlim_at_bot_mirror
tff(fact_6899_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_6900_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_6901_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_6902_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_6903_filterlim__pow__at__top,axiom,
    ! [A: $tType,Na: nat,F2: fun(A,real),F4: filter(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( filterlim(A,real,F2,at_top(real),F4)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_vt(nat,fun(fun(A,real),fun(A,real)),Na),F2),at_top(real),F4) ) ) ).

% filterlim_pow_at_top
tff(fact_6904_tendsto__add__filterlim__at__infinity_H,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,B),C2: B] :
          ( filterlim(A,B,F2,at_infinity(B),F4)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,C2),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aea(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),at_infinity(B),F4) ) ) ) ).

% tendsto_add_filterlim_at_infinity'
tff(fact_6905_tendsto__add__filterlim__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),C2: B,F4: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F4)
         => ( filterlim(A,B,G,at_infinity(B),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aea(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),at_infinity(B),F4) ) ) ) ).

% tendsto_add_filterlim_at_infinity
tff(fact_6906_real__tendsto__divide__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F4: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F4)
     => ( filterlim(A,real,G,at_top(real),F4)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_aeb(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% real_tendsto_divide_at_top
tff(fact_6907_tendsto__inverse__0__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F2,at_top(real),F4)
     => filterlim(A,real,aTP_Lamp_aec(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ).

% tendsto_inverse_0_at_top
tff(fact_6908_filterlim__sequentially__iff__filterlim__real,axiom,
    ! [A: $tType,F2: fun(A,nat),F4: filter(A)] :
      ( filterlim(A,nat,F2,at_top(nat),F4)
    <=> filterlim(A,real,aTP_Lamp_aed(fun(A,nat),fun(A,real),F2),at_top(real),F4) ) ).

% filterlim_sequentially_iff_filterlim_real
tff(fact_6909_filterlim__at__top__mult__tendsto__pos,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F4: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
       => ( filterlim(A,real,G,at_top(real),F4)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_aee(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F4) ) ) ) ).

% filterlim_at_top_mult_tendsto_pos
tff(fact_6910_filterlim__tendsto__pos__mult__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F4: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
       => ( filterlim(A,real,G,at_top(real),F4)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_adt(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F4) ) ) ) ).

% filterlim_tendsto_pos_mult_at_top
tff(fact_6911_tendsto__neg__powr,axiom,
    ! [A: $tType,S: real,F2: fun(A,real),F4: filter(A)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),S),zero_zero(real))
     => ( filterlim(A,real,F2,at_top(real),F4)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_aef(real,fun(fun(A,real),fun(A,real)),S),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% tendsto_neg_powr
tff(fact_6912_tendsto__mult__filterlim__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [F2: fun(A,B),C2: B,F4: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F4)
         => ( ( C2 != zero_zero(B) )
           => ( filterlim(A,B,G,at_infinity(B),F4)
             => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aeg(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),at_infinity(B),F4) ) ) ) ) ).

% tendsto_mult_filterlim_at_infinity
tff(fact_6913_filterlim__power__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),F4: filter(A),Na: nat] :
          ( filterlim(A,B,F2,at_infinity(B),F4)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
           => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_aeh(fun(A,B),fun(nat,fun(A,B)),F2),Na),at_infinity(B),F4) ) ) ) ).

% filterlim_power_at_infinity
tff(fact_6914_tendsto__divide__0,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),C2: B,F4: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F4)
         => ( filterlim(A,B,G,at_infinity(B),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aei(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_divide_0
tff(fact_6915_ln__x__over__x__tendsto__0,axiom,
    filterlim(real,real,aTP_Lamp_aej(real,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(real)) ).

% ln_x_over_x_tendsto_0
tff(fact_6916_filterlim__at__top__to__right,axiom,
    ! [A: $tType,F2: fun(real,A),F4: filter(A)] :
      ( filterlim(real,A,F2,F4,at_top(real))
    <=> filterlim(real,A,aTP_Lamp_aek(fun(real,A),fun(real,A),F2),F4,topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ) ).

% filterlim_at_top_to_right
tff(fact_6917_filterlim__at__right__to__top,axiom,
    ! [A: $tType,F2: fun(real,A),F4: filter(A)] :
      ( filterlim(real,A,F2,F4,topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
    <=> filterlim(real,A,aTP_Lamp_aek(fun(real,A),fun(real,A),F2),F4,at_top(real)) ) ).

% filterlim_at_right_to_top
tff(fact_6918_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_6919_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_6920_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_adu(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_6921_filterlim__tendsto__neg__mult__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F4: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
       => ( filterlim(A,real,G,at_top(real),F4)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_adt(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_bot(real),F4) ) ) ) ).

% filterlim_tendsto_neg_mult_at_bot
tff(fact_6922_tendsto__power__div__exp__0,axiom,
    ! [K: nat] : filterlim(real,real,aTP_Lamp_ael(nat,fun(real,real),K),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(real)) ).

% tendsto_power_div_exp_0
tff(fact_6923_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_aem(fun(real,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% lim_infinity_imp_sequentially
tff(fact_6924_filterlim__inverse__at__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [G: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,aTP_Lamp_yr(fun(A,B),fun(A,B),G),topolo174197925503356063within(B,zero_zero(B),top_top(set(B))),F4)
        <=> filterlim(A,B,G,at_infinity(B),F4) ) ) ).

% filterlim_inverse_at_iff
tff(fact_6925_tendsto__exp__limit__at__top,axiom,
    ! [X: real] : filterlim(real,real,aTP_Lamp_aen(real,fun(real,real),X),topolo7230453075368039082e_nhds(real,aa(real,real,exp(real),X)),at_top(real)) ).

% tendsto_exp_limit_at_top
tff(fact_6926_filterlim__divide__at__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),C2: A,F4: filter(A),G: fun(A,A)] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,C2),F4)
         => ( filterlim(A,A,G,topolo174197925503356063within(A,zero_zero(A),top_top(set(A))),F4)
           => ( ( C2 != zero_zero(A) )
             => filterlim(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_ru(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),at_infinity(A),F4) ) ) ) ) ).

% filterlim_divide_at_infinity
tff(fact_6927_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_6928_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_6929_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [Ba: real,F2: fun(real,real),Flim: real] :
      ( ! [X2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ba),X2)
         => ? [Y3: real] :
              ( has_field_derivative(real,F2,Y3,topolo174197925503356063within(real,X2,top_top(set(real))))
              & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y3),zero_zero(real)) ) )
     => ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,Flim),at_top(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),Flim),aa(real,real,F2,Ba)) ) ) ).

% DERIV_neg_imp_decreasing_at_top
tff(fact_6930_filterlim__realpow__sequentially__gt1,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: A] :
          ( aa(real,$o,aa(real,fun(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_6931_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,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),K)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
             => eventually(A,aa(real,fun(A,$o),aa(nat,fun(real,fun(A,$o)),aTP_Lamp_aeo(fun(nat,A),fun(nat,fun(real,fun(A,$o))),C2),Na),B2),at_infinity(A)) ) ) ) ) ).

% polyfun_extremal
tff(fact_6932_lhopital__left__at__top,axiom,
    ! [G: fun(real,real),X: real,G2: 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_aep(fun(real,real),fun(real,$o),G2),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_aeq(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_aeq(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G2),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aer(fun(real,real),fun(fun(real,real),fun(real,real)),G2),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_aer(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_6933_eventually__sequentially__Suc,axiom,
    ! [P: fun(nat,$o)] :
      ( eventually(nat,aTP_Lamp_aes(fun(nat,$o),fun(nat,$o),P),at_top(nat))
    <=> eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentially_Suc
tff(fact_6934_eventually__sequentially__seg,axiom,
    ! [P: fun(nat,$o),K: nat] :
      ( eventually(nat,aa(nat,fun(nat,$o),aTP_Lamp_aet(fun(nat,$o),fun(nat,fun(nat,$o)),P),K),at_top(nat))
    <=> eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentially_seg
tff(fact_6935_eventually__const,axiom,
    ! [A: $tType,F4: filter(A),P: $o] :
      ( ( F4 != bot_bot(filter(A)) )
     => ( eventually(A,aTP_Lamp_qb($o,fun(A,$o),(P)),F4)
      <=> (P) ) ) ).

% eventually_const
tff(fact_6936_sequentially__offset,axiom,
    ! [P: fun(nat,$o),K: nat] :
      ( eventually(nat,P,at_top(nat))
     => eventually(nat,aa(nat,fun(nat,$o),aTP_Lamp_aet(fun(nat,$o),fun(nat,fun(nat,$o)),P),K),at_top(nat)) ) ).

% sequentially_offset
tff(fact_6937_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_aeu(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_6938_eventually__not__equal__at__infinity,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Aa2: A] : eventually(A,aTP_Lamp_aev(A,fun(A,$o),Aa2),at_infinity(A)) ) ).

% eventually_not_equal_at_infinity
tff(fact_6939_eventually__sequentially,axiom,
    ! [P: fun(nat,$o)] :
      ( eventually(nat,P,at_top(nat))
    <=> ? [N7: nat] :
        ! [N2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N2)
         => aa(nat,$o,P,N2) ) ) ).

% eventually_sequentially
tff(fact_6940_eventually__sequentiallyI,axiom,
    ! [C2: nat,P: fun(nat,$o)] :
      ( ! [X2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),C2),X2)
         => aa(nat,$o,P,X2) )
     => eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentiallyI
tff(fact_6941_eventually__at__top__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o)] :
          ( eventually(A,P,at_top(A))
        <=> ? [N7: A] :
            ! [N2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),N7),N2)
             => aa(A,$o,P,N2) ) ) ) ).

% eventually_at_top_linorder
tff(fact_6942_eventually__at__top__linorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,P: fun(A,$o)] :
          ( ! [X2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),X2)
             => aa(A,$o,P,X2) )
         => eventually(A,P,at_top(A)) ) ) ).

% eventually_at_top_linorderI
tff(fact_6943_le__sequentially,axiom,
    ! [F4: filter(nat)] :
      ( aa(filter(nat),$o,aa(filter(nat),fun(filter(nat),$o),ord_less_eq(filter(nat)),F4),at_top(nat))
    <=> ! [N7: nat] : eventually(nat,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),F4) ) ).

% le_sequentially
tff(fact_6944_eventually__ge__at__top,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A] : eventually(A,aa(A,fun(A,$o),ord_less_eq(A),C2),at_top(A)) ) ).

% eventually_ge_at_top
tff(fact_6945_eventually__gt__at__top,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [C2: A] : eventually(A,aa(A,fun(A,$o),ord_less(A),C2),at_top(A)) ) ).

% eventually_gt_at_top
tff(fact_6946_eventually__False__sequentially,axiom,
    ~ eventually(nat,aTP_Lamp_aew(nat,$o),at_top(nat)) ).

% eventually_False_sequentially
tff(fact_6947_eventually__at__top__not__equal,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [C2: A] : eventually(A,aTP_Lamp_aex(A,fun(A,$o),C2),at_top(A)) ) ).

% eventually_at_top_not_equal
tff(fact_6948_eventually__nhds__iff__sequentially,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [P: fun(A,$o),Aa2: A] :
          ( eventually(A,P,topolo7230453075368039082e_nhds(A,Aa2))
        <=> ! [F5: fun(nat,A)] :
              ( filterlim(nat,A,F5,topolo7230453075368039082e_nhds(A,Aa2),at_top(nat))
             => eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_aey(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),P),F5),at_top(nat)) ) ) ) ).

% eventually_nhds_iff_sequentially
tff(fact_6949_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_aez(fun(A,B),fun(B,fun(A,$o)),F2),L),Net)
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),Net) ) ) ).

% tendsto_eventually
tff(fact_6950_tendsto__imp__eventually__ne,axiom,
    ! [B: $tType,A: $tType] :
      ( topological_t1_space(B)
     => ! [F2: fun(A,B),C2: B,F4: filter(A),C7: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F4)
         => ( ( C2 != C7 )
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_afa(fun(A,B),fun(B,fun(A,$o)),F2),C7),F4) ) ) ) ).

% tendsto_imp_eventually_ne
tff(fact_6951_tendsto__discrete,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo8865339358273720382pology(B)
     => ! [F2: fun(A,B),Y: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Y),F4)
        <=> eventually(A,aa(B,fun(A,$o),aTP_Lamp_afb(fun(A,B),fun(B,fun(A,$o)),F2),Y),F4) ) ) ).

% tendsto_discrete
tff(fact_6952_tendsto__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),G: fun(A,B),F4: filter(A),C2: B] :
          ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_afc(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G),F4)
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F4)
          <=> filterlim(A,B,G,topolo7230453075368039082e_nhds(B,C2),F4) ) ) ) ).

% tendsto_cong
tff(fact_6953_filterlim__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),F22: filter(B),F1: filter(A)] :
      ( filterlim(A,B,F2,F22,F1)
    <=> ! [P6: fun(B,$o)] :
          ( eventually(B,P6,F22)
         => eventually(A,aa(fun(B,$o),fun(A,$o),aTP_Lamp_afd(fun(A,B),fun(fun(B,$o),fun(A,$o)),F2),P6),F1) ) ) ).

% filterlim_iff
tff(fact_6954_filterlim__cong,axiom,
    ! [A: $tType,B: $tType,F1: filter(A),F12: filter(A),F22: filter(B),F23: filter(B),F2: fun(B,A),G: fun(B,A)] :
      ( ( F1 = F12 )
     => ( ( F22 = F23 )
       => ( eventually(B,aa(fun(B,A),fun(B,$o),aTP_Lamp_afe(fun(B,A),fun(fun(B,A),fun(B,$o)),F2),G),F22)
         => ( filterlim(B,A,F2,F1,F22)
          <=> filterlim(B,A,G,F12,F23) ) ) ) ) ).

% filterlim_cong
tff(fact_6955_eventually__compose__filterlim,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F4: filter(A),F2: fun(B,A),G3: filter(B)] :
      ( eventually(A,P,F4)
     => ( filterlim(B,A,F2,F4,G3)
       => eventually(B,aa(fun(B,A),fun(B,$o),aTP_Lamp_aff(fun(A,$o),fun(fun(B,A),fun(B,$o)),P),F2),G3) ) ) ).

% eventually_compose_filterlim
tff(fact_6956_filterlim__at__infinity__imp__eventually__ne,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_infinity(B),F4)
         => eventually(A,aa(B,fun(A,$o),aTP_Lamp_afg(fun(A,B),fun(B,fun(A,$o)),F2),C2),F4) ) ) ).

% filterlim_at_infinity_imp_eventually_ne
tff(fact_6957_Lim__transform__eventually,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_afc(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G),F4)
           => filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),F4) ) ) ) ).

% Lim_transform_eventually
tff(fact_6958_filterlim__mono__eventually,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),F4: filter(B),G3: filter(A),F6: filter(B),G5: filter(A),F7: fun(A,B)] :
      ( filterlim(A,B,F2,F4,G3)
     => ( aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),F4),F6)
       => ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),G5),G3)
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_afh(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),F7),G5)
           => filterlim(A,B,F7,F6,G5) ) ) ) ) ).

% filterlim_mono_eventually
tff(fact_6959_filterlim__at__within__not__equal,axiom,
    ! [B: $tType,A: $tType] :
      ( topological_t2_space(B)
     => ! [F2: fun(A,B),Aa2: B,S: set(B),F4: filter(A),Ba: B] :
          ( filterlim(A,B,F2,topolo174197925503356063within(B,Aa2,S),F4)
         => eventually(A,aa(B,fun(A,$o),aa(set(B),fun(B,fun(A,$o)),aTP_Lamp_afi(fun(A,B),fun(set(B),fun(B,fun(A,$o))),F2),S),Ba),F4) ) ) ).

% filterlim_at_within_not_equal
tff(fact_6960_sequentially__imp__eventually__within,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [S: set(A),Aa2: A,P: fun(A,$o)] :
          ( ! [F3: fun(nat,A)] :
              ( ( ! [N10: nat] :
                    ( member(A,aa(nat,A,F3,N10),S)
                    & ( aa(nat,A,F3,N10) != Aa2 ) )
                & filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,Aa2),at_top(nat)) )
             => eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_aey(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),P),F3),at_top(nat)) )
         => eventually(A,P,topolo174197925503356063within(A,Aa2,S)) ) ) ).

% sequentially_imp_eventually_within
tff(fact_6961_sequentially__imp__eventually__at,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [Aa2: A,P: fun(A,$o)] :
          ( ! [F3: fun(nat,A)] :
              ( ( ! [N10: nat] : aa(nat,A,F3,N10) != Aa2
                & filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,Aa2),at_top(nat)) )
             => eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_aey(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),P),F3),at_top(nat)) )
         => eventually(A,P,topolo174197925503356063within(A,Aa2,top_top(set(A)))) ) ) ).

% sequentially_imp_eventually_at
tff(fact_6962_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,$o),P,topolo3827282254853284352ce_Lim(A,A,topolo174197925503356063within(A,X,X5),aTP_Lamp_wl(A,A))),topolo174197925503356063within(A,X,X5))
        <=> eventually(A,aa(A,fun(A,$o),P,X),topolo174197925503356063within(A,X,X5)) ) ) ).

% eventually_Lim_ident_at
tff(fact_6963_trivial__limit__def,axiom,
    ! [A: $tType,F4: filter(A)] :
      ( ( F4 = bot_bot(filter(A)) )
    <=> eventually(A,aTP_Lamp_dd(A,$o),F4) ) ).

% trivial_limit_def
tff(fact_6964_eventually__const__iff,axiom,
    ! [A: $tType,P: $o,F4: filter(A)] :
      ( eventually(A,aTP_Lamp_qb($o,fun(A,$o),(P)),F4)
    <=> ( (P)
        | ( F4 = bot_bot(filter(A)) ) ) ) ).

% eventually_const_iff
tff(fact_6965_False__imp__not__eventually,axiom,
    ! [A: $tType,P: fun(A,$o),Net: filter(A)] :
      ( ! [X2: A] : ~ aa(A,$o,P,X2)
     => ( ( Net != bot_bot(filter(A)) )
       => ~ eventually(A,P,Net) ) ) ).

% False_imp_not_eventually
tff(fact_6966_not__eventually__impI,axiom,
    ! [A: $tType,P: fun(A,$o),F4: filter(A),Q: fun(A,$o)] :
      ( eventually(A,P,F4)
     => ( ~ eventually(A,Q,F4)
       => ~ eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_dk(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F4) ) ) ).

% not_eventually_impI
tff(fact_6967_eventually__conj__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),F4: filter(A)] :
      ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_oe(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F4)
    <=> ( eventually(A,P,F4)
        & eventually(A,Q,F4) ) ) ).

% eventually_conj_iff
tff(fact_6968_eventually__rev__mp,axiom,
    ! [A: $tType,P: fun(A,$o),F4: filter(A),Q: fun(A,$o)] :
      ( eventually(A,P,F4)
     => ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_dk(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F4)
       => eventually(A,Q,F4) ) ) ).

% eventually_rev_mp
tff(fact_6969_eventually__subst,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),F4: filter(A)] :
      ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_afj(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F4)
     => ( eventually(A,P,F4)
      <=> eventually(A,Q,F4) ) ) ).

% eventually_subst
tff(fact_6970_eventually__elim2,axiom,
    ! [A: $tType,P: fun(A,$o),F4: filter(A),Q: fun(A,$o),R3: fun(A,$o)] :
      ( eventually(A,P,F4)
     => ( eventually(A,Q,F4)
       => ( ! [I2: A] :
              ( aa(A,$o,P,I2)
             => ( aa(A,$o,Q,I2)
               => aa(A,$o,R3,I2) ) )
         => eventually(A,R3,F4) ) ) ) ).

% eventually_elim2
tff(fact_6971_eventually__conj,axiom,
    ! [A: $tType,P: fun(A,$o),F4: filter(A),Q: fun(A,$o)] :
      ( eventually(A,P,F4)
     => ( eventually(A,Q,F4)
       => eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_oe(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F4) ) ) ).

% eventually_conj
tff(fact_6972_eventually__True,axiom,
    ! [A: $tType,F4: filter(A)] : eventually(A,aTP_Lamp_qc(A,$o),F4) ).

% eventually_True
tff(fact_6973_eventually__mp,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),F4: filter(A)] :
      ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_dk(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F4)
     => ( eventually(A,P,F4)
       => eventually(A,Q,F4) ) ) ).

% eventually_mp
tff(fact_6974_eventually__frequently__const__simps_I3_J,axiom,
    ! [A: $tType,P: fun(A,$o),C5: $o,F4: filter(A)] :
      ( eventually(A,aa($o,fun(A,$o),aTP_Lamp_afk(fun(A,$o),fun($o,fun(A,$o)),P),(C5)),F4)
    <=> ( eventually(A,P,F4)
        | (C5) ) ) ).

% eventually_frequently_const_simps(3)
tff(fact_6975_eventually__frequently__const__simps_I4_J,axiom,
    ! [A: $tType,C5: $o,P: fun(A,$o),F4: filter(A)] :
      ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_afl($o,fun(fun(A,$o),fun(A,$o)),(C5)),P),F4)
    <=> ( (C5)
        | eventually(A,P,F4) ) ) ).

% eventually_frequently_const_simps(4)
tff(fact_6976_eventually__frequently__const__simps_I6_J,axiom,
    ! [A: $tType,C5: $o,P: fun(A,$o),F4: filter(A)] :
      ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_afm($o,fun(fun(A,$o),fun(A,$o)),(C5)),P),F4)
    <=> ( (C5)
       => eventually(A,P,F4) ) ) ).

% eventually_frequently_const_simps(6)
tff(fact_6977_has__derivative__transform__eventually,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F7: fun(A,B),X: A,S: set(A),G: fun(A,B)] :
          ( has_derivative(A,B,F2,F7,topolo174197925503356063within(A,X,S))
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_afn(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G),topolo174197925503356063within(A,X,S))
           => ( ( aa(A,B,F2,X) = aa(A,B,G,X) )
             => ( member(A,X,S)
               => has_derivative(A,B,G,F7,topolo174197925503356063within(A,X,S)) ) ) ) ) ) ).

% has_derivative_transform_eventually
tff(fact_6978_has__field__derivative__cong__eventually,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),G: fun(A,A),X: A,Sa: set(A),U: A] :
          ( eventually(A,aa(fun(A,A),fun(A,$o),aTP_Lamp_afo(fun(A,A),fun(fun(A,A),fun(A,$o)),F2),G),topolo174197925503356063within(A,X,Sa))
         => ( ( aa(A,A,F2,X) = aa(A,A,G,X) )
           => ( has_field_derivative(A,F2,U,topolo174197925503356063within(A,X,Sa))
            <=> has_field_derivative(A,G,U,topolo174197925503356063within(A,X,Sa)) ) ) ) ) ).

% has_field_derivative_cong_eventually
tff(fact_6979_eventually__eventually,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [P: fun(A,$o),X: A] :
          ( eventually(A,aTP_Lamp_afp(fun(A,$o),fun(A,$o),P),topolo7230453075368039082e_nhds(A,X))
        <=> eventually(A,P,topolo7230453075368039082e_nhds(A,X)) ) ) ).

% eventually_eventually
tff(fact_6980_t1__space__nhds,axiom,
    ! [A: $tType] :
      ( topological_t1_space(A)
     => ! [X: A,Y: A] :
          ( ( X != Y )
         => eventually(A,aTP_Lamp_afq(A,fun(A,$o),Y),topolo7230453075368039082e_nhds(A,X)) ) ) ).

% t1_space_nhds
tff(fact_6981_eventually__at__filter,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [P: fun(A,$o),Aa2: A,S: set(A)] :
          ( eventually(A,P,topolo174197925503356063within(A,Aa2,S))
        <=> eventually(A,aa(set(A),fun(A,$o),aa(A,fun(set(A),fun(A,$o)),aTP_Lamp_afr(fun(A,$o),fun(A,fun(set(A),fun(A,$o))),P),Aa2),S),topolo7230453075368039082e_nhds(A,Aa2)) ) ) ).

% eventually_at_filter
tff(fact_6982_eventually__nhds__in__open,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),X: A] :
          ( topolo1002775350975398744n_open(A,S)
         => ( member(A,X,S)
           => eventually(A,aTP_Lamp_afs(set(A),fun(A,$o),S),topolo7230453075368039082e_nhds(A,X)) ) ) ) ).

% eventually_nhds_in_open
tff(fact_6983_eventually__at__bot__not__equal,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [C2: A] : eventually(A,aTP_Lamp_aft(A,fun(A,$o),C2),at_bot(A)) ) ).

% eventually_at_bot_not_equal
tff(fact_6984_eventually__gt__at__bot,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [C2: A] : eventually(A,aTP_Lamp_afu(A,fun(A,$o),C2),at_bot(A)) ) ).

% eventually_gt_at_bot
tff(fact_6985_eventually__le__at__bot,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A] : eventually(A,aa(A,fun(A,$o),aTP_Lamp_afv(A,fun(A,$o)),C2),at_bot(A)) ) ).

% eventually_le_at_bot
tff(fact_6986_eventually__at__bot__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o)] :
          ( eventually(A,P,at_bot(A))
        <=> ? [N7: A] :
            ! [N2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),N2),N7)
             => aa(A,$o,P,N2) ) ) ) ).

% eventually_at_bot_linorder
tff(fact_6987_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,$o),G: fun(B,A)] :
          ( ! [X2: A,Y2: A] :
              ( aa(A,$o,Q,X2)
             => ( aa(A,$o,Q,Y2)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y2)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X2)),aa(A,B,F2,Y2)) ) ) )
         => ( ! [X2: B] :
                ( aa(B,$o,P,X2)
               => ( aa(A,B,F2,aa(B,A,G,X2)) = X2 ) )
           => ( ! [X2: B] :
                  ( aa(B,$o,P,X2)
                 => aa(A,$o,Q,aa(B,A,G,X2)) )
             => ( 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_6988_eventually__at__infinity,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [P: fun(A,$o)] :
          ( eventually(A,P,at_infinity(A))
        <=> ? [B7: real] :
            ! [X6: A] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B7),real_V7770717601297561774m_norm(A,X6))
             => aa(A,$o,P,X6) ) ) ) ).

% eventually_at_infinity
tff(fact_6989_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_afw(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G),Net)
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_afw(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_6990_order__tendstoD_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),Y: B,F4: filter(A),Aa2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Y),F4)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y),Aa2)
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_afx(fun(A,B),fun(B,fun(A,$o)),F2),Aa2),F4) ) ) ) ).

% order_tendstoD(2)
tff(fact_6991_order__tendstoD_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),Y: B,F4: filter(A),Aa2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Y),F4)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Aa2),Y)
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_afy(fun(A,B),fun(B,fun(A,$o)),F2),Aa2),F4) ) ) ) ).

% order_tendstoD(1)
tff(fact_6992_order__tendstoI,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Y: A,F2: fun(B,A),F4: filter(B)] :
          ( ! [A3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),Y)
             => eventually(B,aa(A,fun(B,$o),aTP_Lamp_afz(fun(B,A),fun(A,fun(B,$o)),F2),A3),F4) )
         => ( ! [A3: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),A3)
               => eventually(B,aa(A,fun(B,$o),aTP_Lamp_aga(fun(B,A),fun(A,fun(B,$o)),F2),A3),F4) )
           => filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,Y),F4) ) ) ) ).

% order_tendstoI
tff(fact_6993_order__tendsto__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),X: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,X),F4)
        <=> ( ! [L4: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),L4),X)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_afy(fun(A,B),fun(B,fun(A,$o)),F2),L4),F4) )
            & ! [U2: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),U2)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_afx(fun(A,B),fun(B,fun(A,$o)),F2),U2),F4) ) ) ) ) ).

% order_tendsto_iff
tff(fact_6994_filterlim__at__top__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,at_top(B),F4)
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_agb(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G),F4)
           => filterlim(A,B,G,at_top(B),F4) ) ) ) ).

% filterlim_at_top_mono
tff(fact_6995_filterlim__at__top__ge,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_top(B),F4)
        <=> ! [Z6: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),Z6)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_agc(fun(A,B),fun(B,fun(A,$o)),F2),Z6),F4) ) ) ) ).

% filterlim_at_top_ge
tff(fact_6996_filterlim__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F2,at_top(B),F4)
        <=> ! [Z6: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_agc(fun(A,B),fun(B,fun(A,$o)),F2),Z6),F4) ) ) ).

% filterlim_at_top
tff(fact_6997_filterlim__at__top__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F2,at_top(B),F4)
        <=> ! [Z6: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_agd(fun(A,B),fun(B,fun(A,$o)),F2),Z6),F4) ) ) ).

% filterlim_at_top_dense
tff(fact_6998_filterlim__at,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),Ba: B,S: set(B),F4: filter(A)] :
          ( filterlim(A,B,F2,topolo174197925503356063within(B,Ba,S),F4)
        <=> ( eventually(A,aa(set(B),fun(A,$o),aa(B,fun(set(B),fun(A,$o)),aTP_Lamp_age(fun(A,B),fun(B,fun(set(B),fun(A,$o))),F2),Ba),S),F4)
            & filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Ba),F4) ) ) ) ).

% filterlim_at
tff(fact_6999_eventually__at__right__less,axiom,
    ! [A: $tType] :
      ( ( no_top(A)
        & topolo1944317154257567458pology(A) )
     => ! [X: A] : eventually(A,aa(A,fun(A,$o),ord_less(A),X),topolo174197925503356063within(A,X,set_ord_greaterThan(A,X))) ) ).

% eventually_at_right_less
tff(fact_7000_has__field__derivative__cong__ev,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [X: A,Y: A,Sa: set(A),F2: fun(A,A),G: fun(A,A),U: A,V: A,T: set(A)] :
          ( ( X = Y )
         => ( eventually(A,aa(fun(A,A),fun(A,$o),aa(fun(A,A),fun(fun(A,A),fun(A,$o)),aTP_Lamp_agf(set(A),fun(fun(A,A),fun(fun(A,A),fun(A,$o))),Sa),F2),G),topolo7230453075368039082e_nhds(A,X))
           => ( ( U = V )
             => ( ( Sa = T )
               => ( member(A,X,Sa)
                 => ( has_field_derivative(A,F2,U,topolo174197925503356063within(A,X,Sa))
                  <=> has_field_derivative(A,G,V,topolo174197925503356063within(A,Y,T)) ) ) ) ) ) ) ) ).

% has_field_derivative_cong_ev
tff(fact_7001_tendsto__def,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
        <=> ! [S6: set(B)] :
              ( topolo1002775350975398744n_open(B,S6)
             => ( member(B,L,S6)
               => eventually(A,aa(set(B),fun(A,$o),aTP_Lamp_agg(fun(A,B),fun(set(B),fun(A,$o)),F2),S6),F4) ) ) ) ) ).

% tendsto_def
tff(fact_7002_topological__tendstoD,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A),Sa: set(B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => ( topolo1002775350975398744n_open(B,Sa)
           => ( member(B,L,Sa)
             => eventually(A,aa(set(B),fun(A,$o),aTP_Lamp_agg(fun(A,B),fun(set(B),fun(A,$o)),F2),Sa),F4) ) ) ) ) ).

% topological_tendstoD
tff(fact_7003_topological__tendstoI,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [L: A,F2: fun(B,A),F4: filter(B)] :
          ( ! [S4: set(A)] :
              ( topolo1002775350975398744n_open(A,S4)
             => ( member(A,L,S4)
               => eventually(B,aa(set(A),fun(B,$o),aTP_Lamp_agh(fun(B,A),fun(set(A),fun(B,$o)),F2),S4),F4) ) )
         => filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F4) ) ) ).

% topological_tendstoI
tff(fact_7004_filterlim__at__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F2,at_bot(B),F4)
        <=> ! [Z6: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_agi(fun(A,B),fun(B,fun(A,$o)),F2),Z6),F4) ) ) ).

% filterlim_at_bot
tff(fact_7005_filterlim__at__bot__le,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_bot(B),F4)
        <=> ! [Z6: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Z6),C2)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_agi(fun(A,B),fun(B,fun(A,$o)),F2),Z6),F4) ) ) ) ).

% filterlim_at_bot_le
tff(fact_7006_filterlim__at__bot__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( ( dense_linorder(B)
        & no_bot(B) )
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F2,at_bot(B),F4)
        <=> ! [Z6: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_agj(fun(A,B),fun(B,fun(A,$o)),F2),Z6),F4) ) ) ).

% filterlim_at_bot_dense
tff(fact_7007_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_agk(fun(A,real),fun(fun(A,real),fun(A,$o)),F2),G),Net)
     => ( eventually(A,aa(fun(A,real),fun(A,$o),aTP_Lamp_agk(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_7008_countable__basis__at__decseq,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [X: A] :
          ~ ! [A5: fun(nat,set(A))] :
              ( ! [I3: nat] : topolo1002775350975398744n_open(A,aa(nat,set(A),A5,I3))
             => ( ! [I3: nat] : member(A,X,aa(nat,set(A),A5,I3))
               => ~ ! [S7: set(A)] :
                      ( topolo1002775350975398744n_open(A,S7)
                     => ( member(A,X,S7)
                       => eventually(nat,aa(set(A),fun(nat,$o),aTP_Lamp_agl(fun(nat,set(A)),fun(set(A),fun(nat,$o)),A5),S7),at_top(nat)) ) ) ) ) ) ).

% countable_basis_at_decseq
tff(fact_7009_eventually__at__to__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [P: fun(A,$o),Aa2: A] :
          ( eventually(A,P,topolo174197925503356063within(A,Aa2,top_top(set(A))))
        <=> eventually(A,aa(A,fun(A,$o),aTP_Lamp_agm(fun(A,$o),fun(A,fun(A,$o)),P),Aa2),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% eventually_at_to_0
tff(fact_7010_increasing__tendsto,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_agn(fun(A,B),fun(B,fun(A,$o)),F2),L),F4)
         => ( ! [X2: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X2),L)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_afy(fun(A,B),fun(B,fun(A,$o)),F2),X2),F4) )
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4) ) ) ) ).

% increasing_tendsto
tff(fact_7011_decreasing__tendsto,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [L: B,F2: fun(A,B),F4: filter(A)] :
          ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_ago(B,fun(fun(A,B),fun(A,$o)),L),F2),F4)
         => ( ! [X2: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),L),X2)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_afx(fun(A,B),fun(B,fun(A,$o)),F2),X2),F4) )
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4) ) ) ) ).

% decreasing_tendsto
tff(fact_7012_filterlim__at__top__gt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F2: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_top(B),F4)
        <=> ! [Z6: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),Z6)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_agp(fun(A,B),fun(B,fun(A,$o)),F2),Z6),F4) ) ) ) ).

% filterlim_at_top_gt
tff(fact_7013_filterlim__atI,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),C2: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F4)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_agq(fun(A,B),fun(B,fun(A,$o)),F2),C2),F4)
           => filterlim(A,B,F2,topolo174197925503356063within(B,C2,top_top(set(B))),F4) ) ) ) ).

% filterlim_atI
tff(fact_7014_LIM__compose__eventually,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),Ba: B,Aa2: A,G: fun(B,C),C2: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Ba),topolo174197925503356063within(A,Aa2,top_top(set(A))))
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(B,Ba,top_top(set(B))))
           => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_agr(fun(A,B),fun(B,fun(A,$o)),F2),Ba),topolo174197925503356063within(A,Aa2,top_top(set(A))))
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ags(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(A,Aa2,top_top(set(A)))) ) ) ) ) ).

% LIM_compose_eventually
tff(fact_7015_tendsto__compose__eventually,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [G: fun(A,B),M: B,L: A,F2: fun(C,A),F4: filter(C)] :
          ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,M),topolo174197925503356063within(A,L,top_top(set(A))))
         => ( filterlim(C,A,F2,topolo7230453075368039082e_nhds(A,L),F4)
           => ( eventually(C,aa(fun(C,A),fun(C,$o),aTP_Lamp_agt(A,fun(fun(C,A),fun(C,$o)),L),F2),F4)
             => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_we(fun(A,B),fun(fun(C,A),fun(C,B)),G),F2),topolo7230453075368039082e_nhds(B,M),F4) ) ) ) ) ).

% tendsto_compose_eventually
tff(fact_7016_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_agu(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_7017_DERIV__cong__ev,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [X: A,Y: A,F2: fun(A,A),G: fun(A,A),U: A,V: A] :
          ( ( X = Y )
         => ( eventually(A,aa(fun(A,A),fun(A,$o),aTP_Lamp_afo(fun(A,A),fun(fun(A,A),fun(A,$o)),F2),G),topolo7230453075368039082e_nhds(A,X))
           => ( ( U = V )
             => ( has_field_derivative(A,F2,U,topolo174197925503356063within(A,X,top_top(set(A))))
              <=> has_field_derivative(A,G,V,topolo174197925503356063within(A,Y,top_top(set(A)))) ) ) ) ) ) ).

% DERIV_cong_ev
tff(fact_7018_filterlim__at__bot__lt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F2: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_bot(B),F4)
        <=> ! [Z6: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Z6),C2)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_agv(fun(A,B),fun(B,fun(A,$o)),F2),Z6),F4) ) ) ) ).

% filterlim_at_bot_lt
tff(fact_7019_tendsto__le,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [F4: filter(A),F2: fun(A,B),X: B,G: fun(A,B),Y: B] :
          ( ( F4 != bot_bot(filter(A)) )
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,X),F4)
           => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,Y),F4)
             => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_agw(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G),F4)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),X) ) ) ) ) ) ).

% tendsto_le
tff(fact_7020_tendsto__lowerbound,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [F2: fun(A,B),X: B,F4: filter(A),Aa2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,X),F4)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_agx(fun(A,B),fun(B,fun(A,$o)),F2),Aa2),F4)
           => ( ( F4 != bot_bot(filter(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Aa2),X) ) ) ) ) ).

% tendsto_lowerbound
tff(fact_7021_tendsto__upperbound,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [F2: fun(A,B),X: B,F4: filter(A),Aa2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,X),F4)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_agy(fun(A,B),fun(B,fun(A,$o)),F2),Aa2),F4)
           => ( ( F4 != bot_bot(filter(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Aa2) ) ) ) ) ).

% tendsto_upperbound
tff(fact_7022_metric__tendsto__imp__tendsto,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(C)
        & real_V7819770556892013058_space(B) )
     => ! [F2: fun(A,B),Aa2: B,F4: filter(A),G: fun(A,C),Ba: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Aa2),F4)
         => ( eventually(A,aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aa(B,fun(fun(A,C),fun(C,fun(A,$o))),aTP_Lamp_agz(fun(A,B),fun(B,fun(fun(A,C),fun(C,fun(A,$o)))),F2),Aa2),G),Ba),F4)
           => filterlim(A,C,G,topolo7230453075368039082e_nhds(C,Ba),F4) ) ) ) ).

% metric_tendsto_imp_tendsto
tff(fact_7023_filterlim__at__infinity__imp__filterlim__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F2,at_infinity(real),F4)
     => ( eventually(A,aTP_Lamp_aha(fun(A,real),fun(A,$o),F2),F4)
       => filterlim(A,real,F2,at_top(real),F4) ) ) ).

% filterlim_at_infinity_imp_filterlim_at_top
tff(fact_7024_filterlim__at__infinity__imp__filterlim__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F2,at_infinity(real),F4)
     => ( eventually(A,aTP_Lamp_ahb(fun(A,real),fun(A,$o),F2),F4)
       => filterlim(A,real,F2,at_bot(real),F4) ) ) ).

% filterlim_at_infinity_imp_filterlim_at_bot
tff(fact_7025_eventually__floor__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => ( ~ member(B,L,ring_1_Ints(B))
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_ahc(fun(A,B),fun(B,fun(A,$o)),F2),L),F4) ) ) ) ).

% eventually_floor_eq
tff(fact_7026_eventually__ceiling__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => ( ~ member(B,L,ring_1_Ints(B))
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_ahd(fun(A,B),fun(B,fun(A,$o)),F2),L),F4) ) ) ) ).

% eventually_ceiling_eq
tff(fact_7027_eventually__at__right__to__0,axiom,
    ! [P: fun(real,$o),Aa2: real] :
      ( eventually(real,P,topolo174197925503356063within(real,Aa2,set_ord_greaterThan(real,Aa2)))
    <=> eventually(real,aa(real,fun(real,$o),aTP_Lamp_ahe(fun(real,$o),fun(real,fun(real,$o)),P),Aa2),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ) ).

% eventually_at_right_to_0
tff(fact_7028_eventually__at__left__to__right,axiom,
    ! [P: fun(real,$o),Aa2: real] :
      ( eventually(real,P,topolo174197925503356063within(real,Aa2,set_ord_lessThan(real,Aa2)))
    <=> eventually(real,aTP_Lamp_ahf(fun(real,$o),fun(real,$o),P),topolo174197925503356063within(real,aa(real,real,uminus_uminus(real),Aa2),set_ord_greaterThan(real,aa(real,real,uminus_uminus(real),Aa2)))) ) ).

% eventually_at_left_to_right
tff(fact_7029_continuous__arcosh__strong,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => ( eventually(A,aTP_Lamp_ahg(fun(A,real),fun(A,$o),F2),F4)
           => topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_adh(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_arcosh_strong
tff(fact_7030_eventually__at__right__real,axiom,
    ! [Aa2: real,Ba: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Aa2),Ba)
     => eventually(real,aa(real,fun(real,$o),aTP_Lamp_ahh(real,fun(real,fun(real,$o)),Aa2),Ba),topolo174197925503356063within(real,Aa2,set_ord_greaterThan(real,Aa2))) ) ).

% eventually_at_right_real
tff(fact_7031_eventually__at__left__real,axiom,
    ! [Ba: real,Aa2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Ba),Aa2)
     => eventually(real,aa(real,fun(real,$o),aTP_Lamp_ahh(real,fun(real,fun(real,$o)),Ba),Aa2),topolo174197925503356063within(real,Aa2,set_ord_lessThan(real,Aa2))) ) ).

% eventually_at_left_real
tff(fact_7032_eventually__at__le,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [P: fun(A,$o),Aa2: A,Sa: set(A)] :
          ( eventually(A,P,topolo174197925503356063within(A,Aa2,Sa))
        <=> ? [D5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D5)
              & ! [X6: A] :
                  ( member(A,X6,Sa)
                 => ( ( ( X6 != Aa2 )
                      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(A,X6,Aa2)),D5) )
                   => aa(A,$o,P,X6) ) ) ) ) ) ).

% eventually_at_le
tff(fact_7033_eventually__at__infinity__pos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [P2: fun(A,$o)] :
          ( eventually(A,P2,at_infinity(A))
        <=> ? [B7: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B7)
              & ! [X6: A] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B7),real_V7770717601297561774m_norm(A,X6))
                 => aa(A,$o,P2,X6) ) ) ) ) ).

% eventually_at_infinity_pos
tff(fact_7034_tendsto__imp__filterlim__at__left,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),L5: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),F4)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_afx(fun(A,B),fun(B,fun(A,$o)),F2),L5),F4)
           => filterlim(A,B,F2,topolo174197925503356063within(B,L5,set_ord_lessThan(B,L5)),F4) ) ) ) ).

% tendsto_imp_filterlim_at_left
tff(fact_7035_tendsto__imp__filterlim__at__right,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),L5: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),F4)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_afy(fun(A,B),fun(B,fun(A,$o)),F2),L5),F4)
           => filterlim(A,B,F2,topolo174197925503356063within(B,L5,set_ord_greaterThan(B,L5)),F4) ) ) ) ).

% tendsto_imp_filterlim_at_right
tff(fact_7036_tendstoD,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A),E2: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
           => eventually(A,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_ahi(fun(A,B),fun(B,fun(real,fun(A,$o))),F2),L),E2),F4) ) ) ) ).

% tendstoD
tff(fact_7037_tendstoI,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( ! [E: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
             => eventually(A,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_ahi(fun(A,B),fun(B,fun(real,fun(A,$o))),F2),L),E),F4) )
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4) ) ) ).

% tendstoI
tff(fact_7038_tendsto__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
        <=> ! [E3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
             => eventually(A,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_ahi(fun(A,B),fun(B,fun(real,fun(A,$o))),F2),L),E3),F4) ) ) ) ).

% tendsto_iff
tff(fact_7039_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_ahj(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_7040_eventually__at__top__to__right,axiom,
    ! [P: fun(real,$o)] :
      ( eventually(real,P,at_top(real))
    <=> eventually(real,aTP_Lamp_ahk(fun(real,$o),fun(real,$o),P),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ) ).

% eventually_at_top_to_right
tff(fact_7041_eventually__at__right__to__top,axiom,
    ! [P: fun(real,$o)] :
      ( eventually(real,P,topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
    <=> eventually(real,aTP_Lamp_ahk(fun(real,$o),fun(real,$o),P),at_top(real)) ) ).

% eventually_at_right_to_top
tff(fact_7042_tendsto__arcosh__strong,axiom,
    ! [A: $tType,F2: fun(A,real),Aa2: real,F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,Aa2),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Aa2)
       => ( eventually(A,aTP_Lamp_ahl(fun(A,real),fun(A,$o),F2),F4)
         => filterlim(A,real,aTP_Lamp_wt(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,arcosh(real),Aa2)),F4) ) ) ) ).

% tendsto_arcosh_strong
tff(fact_7043_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,$o),G: fun(B,A),Aa2: A] :
          ( ! [X2: A,Y2: A] :
              ( aa(A,$o,Q,X2)
             => ( aa(A,$o,Q,Y2)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y2)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X2)),aa(A,B,F2,Y2)) ) ) )
         => ( ! [X2: B] :
                ( aa(B,$o,P,X2)
               => ( aa(A,B,F2,aa(B,A,G,X2)) = X2 ) )
           => ( ! [X2: B] :
                  ( aa(B,$o,P,X2)
                 => aa(A,$o,Q,aa(B,A,G,X2)) )
             => ( eventually(A,Q,topolo174197925503356063within(A,Aa2,set_ord_lessThan(A,Aa2)))
               => ( ! [B3: A] :
                      ( aa(A,$o,Q,B3)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),Aa2) )
                 => ( eventually(B,P,at_top(B))
                   => filterlim(A,B,F2,at_top(B),topolo174197925503356063within(A,Aa2,set_ord_lessThan(A,Aa2))) ) ) ) ) ) ) ) ).

% filterlim_at_top_at_left
tff(fact_7044_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,$o),G: fun(B,A),Aa2: A] :
          ( ! [X2: A,Y2: A] :
              ( aa(A,$o,Q,X2)
             => ( aa(A,$o,Q,Y2)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y2)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X2)),aa(A,B,F2,Y2)) ) ) )
         => ( ! [X2: B] :
                ( aa(B,$o,P,X2)
               => ( aa(A,B,F2,aa(B,A,G,X2)) = X2 ) )
           => ( ! [X2: B] :
                  ( aa(B,$o,P,X2)
                 => aa(A,$o,Q,aa(B,A,G,X2)) )
             => ( eventually(A,Q,topolo174197925503356063within(A,Aa2,set_ord_greaterThan(A,Aa2)))
               => ( ! [B3: A] :
                      ( aa(A,$o,Q,B3)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),B3) )
                 => ( eventually(B,P,at_bot(B))
                   => filterlim(A,B,F2,at_bot(B),topolo174197925503356063within(A,Aa2,set_ord_greaterThan(A,Aa2))) ) ) ) ) ) ) ) ).

% filterlim_at_bot_at_right
tff(fact_7045_filterlim__at__withinI,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),C2: B,F4: filter(A),A2: set(B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F4)
         => ( eventually(A,aa(set(B),fun(A,$o),aa(B,fun(set(B),fun(A,$o)),aTP_Lamp_ahm(fun(A,B),fun(B,fun(set(B),fun(A,$o))),F2),C2),A2),F4)
           => filterlim(A,B,F2,topolo174197925503356063within(B,C2,A2),F4) ) ) ) ).

% filterlim_at_withinI
tff(fact_7046_tendsto__0__le,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,C),K3: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( eventually(A,aa(real,fun(A,$o),aa(fun(A,C),fun(real,fun(A,$o)),aTP_Lamp_ahn(fun(A,B),fun(fun(A,C),fun(real,fun(A,$o))),F2),G),K3),F4)
           => filterlim(A,C,G,topolo7230453075368039082e_nhds(C,zero_zero(C)),F4) ) ) ) ).

% tendsto_0_le
tff(fact_7047_eventually__floor__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => ( ~ member(B,L,ring_1_Ints(B))
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_aho(fun(A,B),fun(B,fun(A,$o)),F2),L),F4) ) ) ) ).

% eventually_floor_less
tff(fact_7048_eventually__less__ceiling,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => ( ~ member(B,L,ring_1_Ints(B))
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_ahp(fun(A,B),fun(B,fun(A,$o)),F2),L),F4) ) ) ) ).

% eventually_less_ceiling
tff(fact_7049_filterlim__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [C2: real,F2: fun(A,B),F4: filter(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),C2)
         => ( filterlim(A,B,F2,at_infinity(B),F4)
          <=> ! [R7: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),R7)
               => eventually(A,aa(real,fun(A,$o),aTP_Lamp_ahq(fun(A,B),fun(real,fun(A,$o)),F2),R7),F4) ) ) ) ) ).

% filterlim_at_infinity
tff(fact_7050_tendsto__powr_H,axiom,
    ! [A: $tType,F2: fun(A,real),Aa2: real,F4: filter(A),G: fun(A,real),Ba: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,Aa2),F4)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,Ba),F4)
       => ( ( ( Aa2 != zero_zero(real) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Ba)
              & eventually(A,aTP_Lamp_ahr(fun(A,real),fun(A,$o),F2),F4) ) )
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_yg(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,powr(real,Aa2,Ba)),F4) ) ) ) ).

% tendsto_powr'
tff(fact_7051_tendsto__powr2,axiom,
    ! [A: $tType,F2: fun(A,real),Aa2: real,F4: filter(A),G: fun(A,real),Ba: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,Aa2),F4)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,Ba),F4)
       => ( eventually(A,aTP_Lamp_ahr(fun(A,real),fun(A,$o),F2),F4)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Ba)
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_yg(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,powr(real,Aa2,Ba)),F4) ) ) ) ) ).

% tendsto_powr2
tff(fact_7052_tendsto__zero__powrI,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A),G: fun(A,real),Ba: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,Ba),F4)
       => ( eventually(A,aTP_Lamp_ahr(fun(A,real),fun(A,$o),F2),F4)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Ba)
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_yg(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ) ) ).

% tendsto_zero_powrI
tff(fact_7053_LIM__at__top__divide,axiom,
    ! [A: $tType,F2: fun(A,real),Aa2: real,F4: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,Aa2),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Aa2)
       => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
         => ( eventually(A,aTP_Lamp_aha(fun(A,real),fun(A,$o),G),F4)
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_aeb(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F4) ) ) ) ) ).

% LIM_at_top_divide
tff(fact_7054_filterlim__inverse__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
     => ( eventually(A,aTP_Lamp_aha(fun(A,real),fun(A,$o),F2),F4)
       => filterlim(A,real,aTP_Lamp_aec(fun(A,real),fun(A,real),F2),at_top(real),F4) ) ) ).

% filterlim_inverse_at_top
tff(fact_7055_filterlim__inverse__at__top__iff,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( eventually(A,aTP_Lamp_aha(fun(A,real),fun(A,$o),F2),F4)
     => ( filterlim(A,real,aTP_Lamp_aec(fun(A,real),fun(A,real),F2),at_top(real),F4)
      <=> filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% filterlim_inverse_at_top_iff
tff(fact_7056_filterlim__inverse__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
     => ( eventually(A,aTP_Lamp_ahb(fun(A,real),fun(A,$o),F2),F4)
       => filterlim(A,real,aTP_Lamp_aec(fun(A,real),fun(A,real),F2),at_bot(real),F4) ) ) ).

% filterlim_inverse_at_bot
tff(fact_7057_lhopital__at__top__at__top,axiom,
    ! [F2: fun(real,real),Aa2: real,G: fun(real,real),F7: fun(real,real),G2: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,Aa2,top_top(set(real))))
     => ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,Aa2,top_top(set(real))))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_aeq(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F7),topolo174197925503356063within(real,Aa2,top_top(set(real))))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_aeq(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G2),topolo174197925503356063within(real,Aa2,top_top(set(real))))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ahs(fun(real,real),fun(fun(real,real),fun(real,real)),F7),G2),at_top(real),topolo174197925503356063within(real,Aa2,top_top(set(real))))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ahs(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_top(real),topolo174197925503356063within(real,Aa2,top_top(set(real)))) ) ) ) ) ) ).

% lhopital_at_top_at_top
tff(fact_7058_lhopital,axiom,
    ! [F2: fun(real,real),X: real,G: fun(real,real),G2: fun(real,real),F7: fun(real,real),F4: 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_aep(fun(real,real),fun(real,$o),G),topolo174197925503356063within(real,X,top_top(set(real))))
         => ( eventually(real,aTP_Lamp_aep(fun(real,real),fun(real,$o),G2),topolo174197925503356063within(real,X,top_top(set(real))))
           => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_aeq(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F7),topolo174197925503356063within(real,X,top_top(set(real))))
             => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_aeq(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G2),topolo174197925503356063within(real,X,top_top(set(real))))
               => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aer(fun(real,real),fun(fun(real,real),fun(real,real)),G2),F7),F4,topolo174197925503356063within(real,X,top_top(set(real))))
                 => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ahs(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),F4,topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ) ) ) ) ).

% lhopital
tff(fact_7059_lhopital__right__at__top__at__top,axiom,
    ! [F2: fun(real,real),Aa2: real,G: fun(real,real),F7: fun(real,real),G2: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,Aa2,set_ord_greaterThan(real,Aa2)))
     => ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,Aa2,set_ord_greaterThan(real,Aa2)))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_aeq(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F7),topolo174197925503356063within(real,Aa2,set_ord_greaterThan(real,Aa2)))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_aeq(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G2),topolo174197925503356063within(real,Aa2,set_ord_greaterThan(real,Aa2)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ahs(fun(real,real),fun(fun(real,real),fun(real,real)),F7),G2),at_top(real),topolo174197925503356063within(real,Aa2,set_ord_greaterThan(real,Aa2)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ahs(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_top(real),topolo174197925503356063within(real,Aa2,set_ord_greaterThan(real,Aa2))) ) ) ) ) ) ).

% lhopital_right_at_top_at_top
tff(fact_7060_lhopital__at__top__at__bot,axiom,
    ! [F2: fun(real,real),Aa2: real,G: fun(real,real),F7: fun(real,real),G2: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,Aa2,top_top(set(real))))
     => ( filterlim(real,real,G,at_bot(real),topolo174197925503356063within(real,Aa2,top_top(set(real))))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_aeq(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F7),topolo174197925503356063within(real,Aa2,top_top(set(real))))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_aeq(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G2),topolo174197925503356063within(real,Aa2,top_top(set(real))))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ahs(fun(real,real),fun(fun(real,real),fun(real,real)),F7),G2),at_bot(real),topolo174197925503356063within(real,Aa2,top_top(set(real))))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ahs(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_bot(real),topolo174197925503356063within(real,Aa2,top_top(set(real)))) ) ) ) ) ) ).

% lhopital_at_top_at_bot
tff(fact_7061_lhopital__left__at__top__at__top,axiom,
    ! [F2: fun(real,real),Aa2: real,G: fun(real,real),F7: fun(real,real),G2: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,Aa2,set_ord_lessThan(real,Aa2)))
     => ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,Aa2,set_ord_lessThan(real,Aa2)))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_aeq(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F7),topolo174197925503356063within(real,Aa2,set_ord_lessThan(real,Aa2)))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_aeq(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G2),topolo174197925503356063within(real,Aa2,set_ord_lessThan(real,Aa2)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ahs(fun(real,real),fun(fun(real,real),fun(real,real)),F7),G2),at_top(real),topolo174197925503356063within(real,Aa2,set_ord_lessThan(real,Aa2)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ahs(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_top(real),topolo174197925503356063within(real,Aa2,set_ord_lessThan(real,Aa2))) ) ) ) ) ) ).

% lhopital_left_at_top_at_top
tff(fact_7062_lhospital__at__top__at__top,axiom,
    ! [G: fun(real,real),G2: 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_aep(fun(real,real),fun(real,$o),G2),at_top(real))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_aeq(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_aeq(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G2),at_top(real))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aer(fun(real,real),fun(fun(real,real),fun(real,real)),G2),F7),topolo7230453075368039082e_nhds(real,X),at_top(real))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aer(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_7063_lhopital__at__top,axiom,
    ! [G: fun(real,real),X: real,G2: 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_aep(fun(real,real),fun(real,$o),G2),topolo174197925503356063within(real,X,top_top(set(real))))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_aeq(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_aeq(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G2),topolo174197925503356063within(real,X,top_top(set(real))))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aer(fun(real,real),fun(fun(real,real),fun(real,real)),G2),F7),topolo7230453075368039082e_nhds(real,Y),topolo174197925503356063within(real,X,top_top(set(real))))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aer(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_7064_lhopital__right,axiom,
    ! [F2: fun(real,real),X: real,G: fun(real,real),G2: fun(real,real),F7: fun(real,real),F4: 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_aep(fun(real,real),fun(real,$o),G),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
         => ( eventually(real,aTP_Lamp_aep(fun(real,real),fun(real,$o),G2),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
           => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_aeq(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F7),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
             => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_aeq(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G2),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
               => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aer(fun(real,real),fun(fun(real,real),fun(real,real)),G2),F7),F4,topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
                 => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ahs(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),F4,topolo174197925503356063within(real,X,set_ord_greaterThan(real,X))) ) ) ) ) ) ) ) ).

% lhopital_right
tff(fact_7065_lhopital__right__0,axiom,
    ! [F0: fun(real,real),G0: fun(real,real),G2: fun(real,real),F7: fun(real,real),F4: 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_aep(fun(real,real),fun(real,$o),G0),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
         => ( eventually(real,aTP_Lamp_aep(fun(real,real),fun(real,$o),G2),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
           => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_aeq(fun(real,real),fun(fun(real,real),fun(real,$o)),F0),F7),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
             => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_aeq(fun(real,real),fun(fun(real,real),fun(real,$o)),G0),G2),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
               => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aer(fun(real,real),fun(fun(real,real),fun(real,real)),G2),F7),F4,topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
                 => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ahs(fun(real,real),fun(fun(real,real),fun(real,real)),F0),G0),F4,topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ) ) ) ) ) ) ) ).

% lhopital_right_0
tff(fact_7066_lhopital__left,axiom,
    ! [F2: fun(real,real),X: real,G: fun(real,real),G2: fun(real,real),F7: fun(real,real),F4: 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_aep(fun(real,real),fun(real,$o),G),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
         => ( eventually(real,aTP_Lamp_aep(fun(real,real),fun(real,$o),G2),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
           => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_aeq(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F7),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
             => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_aeq(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G2),topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
               => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aer(fun(real,real),fun(fun(real,real),fun(real,real)),G2),F7),F4,topolo174197925503356063within(real,X,set_ord_lessThan(real,X)))
                 => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ahs(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),F4,topolo174197925503356063within(real,X,set_ord_lessThan(real,X))) ) ) ) ) ) ) ) ).

% lhopital_left
tff(fact_7067_lhopital__right__at__top__at__bot,axiom,
    ! [F2: fun(real,real),Aa2: real,G: fun(real,real),F7: fun(real,real),G2: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,Aa2,set_ord_greaterThan(real,Aa2)))
     => ( filterlim(real,real,G,at_bot(real),topolo174197925503356063within(real,Aa2,set_ord_greaterThan(real,Aa2)))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_aeq(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F7),topolo174197925503356063within(real,Aa2,set_ord_greaterThan(real,Aa2)))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_aeq(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G2),topolo174197925503356063within(real,Aa2,set_ord_greaterThan(real,Aa2)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ahs(fun(real,real),fun(fun(real,real),fun(real,real)),F7),G2),at_bot(real),topolo174197925503356063within(real,Aa2,set_ord_greaterThan(real,Aa2)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ahs(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_bot(real),topolo174197925503356063within(real,Aa2,set_ord_greaterThan(real,Aa2))) ) ) ) ) ) ).

% lhopital_right_at_top_at_bot
tff(fact_7068_lhopital__left__at__top__at__bot,axiom,
    ! [F2: fun(real,real),Aa2: real,G: fun(real,real),F7: fun(real,real),G2: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,Aa2,set_ord_lessThan(real,Aa2)))
     => ( filterlim(real,real,G,at_bot(real),topolo174197925503356063within(real,Aa2,set_ord_lessThan(real,Aa2)))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_aeq(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F7),topolo174197925503356063within(real,Aa2,set_ord_lessThan(real,Aa2)))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_aeq(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G2),topolo174197925503356063within(real,Aa2,set_ord_lessThan(real,Aa2)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ahs(fun(real,real),fun(fun(real,real),fun(real,real)),F7),G2),at_bot(real),topolo174197925503356063within(real,Aa2,set_ord_lessThan(real,Aa2)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ahs(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_bot(real),topolo174197925503356063within(real,Aa2,set_ord_lessThan(real,Aa2))) ) ) ) ) ) ).

% lhopital_left_at_top_at_bot
tff(fact_7069_lhopital__right__at__top,axiom,
    ! [G: fun(real,real),X: real,G2: 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_aep(fun(real,real),fun(real,$o),G2),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_aeq(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_aeq(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G2),topolo174197925503356063within(real,X,set_ord_greaterThan(real,X)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aer(fun(real,real),fun(fun(real,real),fun(real,real)),G2),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_aer(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_7070_lhopital__right__0__at__top,axiom,
    ! [G: fun(real,real),G2: 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_aep(fun(real,real),fun(real,$o),G2),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_aeq(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_aeq(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G2),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real))))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aer(fun(real,real),fun(fun(real,real),fun(real,real)),G2),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_aer(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_7071_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_aht(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_7072_Bfun__metric__def,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( bfun(A,B,F2,F4)
        <=> ? [Y4: B,K7: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K7)
              & eventually(A,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_ahu(fun(A,B),fun(B,fun(real,fun(A,$o))),F2),Y4),K7),F4) ) ) ) ).

% Bfun_metric_def
tff(fact_7073_eventually__all__finite,axiom,
    ! [A: $tType,B: $tType] :
      ( finite_finite(A)
     => ! [P: fun(B,fun(A,$o)),Net: filter(B)] :
          ( ! [Y2: A] : eventually(B,aa(A,fun(B,$o),aTP_Lamp_ahv(fun(B,fun(A,$o)),fun(A,fun(B,$o)),P),Y2),Net)
         => eventually(B,aTP_Lamp_ahw(fun(B,fun(A,$o)),fun(B,$o),P),Net) ) ) ).

% eventually_all_finite
tff(fact_7074_filterlim__int__sequentially,axiom,
    filterlim(nat,int,semiring_1_of_nat(int),at_top(int),at_top(nat)) ).

% filterlim_int_sequentially
tff(fact_7075_Bfun__const,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [C2: B,F4: filter(A)] : bfun(A,B,aTP_Lamp_ahx(B,fun(A,B),C2),F4) ) ).

% Bfun_const
tff(fact_7076_finite__set__of__finite__funs,axiom,
    ! [A: $tType,B: $tType,A2: set(A),B2: set(B),D2: B] :
      ( aa(set(A),$o,finite_finite2(A),A2)
     => ( aa(set(B),$o,finite_finite2(B),B2)
       => aa(set(fun(A,B)),$o,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_ahy(set(A),fun(set(B),fun(B,fun(fun(A,B),$o))),A2),B2),D2))) ) ) ).

% finite_set_of_finite_funs
tff(fact_7077_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_ahz(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),F2),G),at_top(nat)) ) ) ).

% Bseq_subseq
tff(fact_7078_Bseq__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( bfun(nat,A,aTP_Lamp_bl(fun(nat,A),fun(nat,A),F2),at_top(nat))
        <=> bfun(nat,A,F2,at_top(nat)) ) ) ).

% Bseq_Suc_iff
tff(fact_7079_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_aia(fun(nat,A),fun(nat,fun(nat,A)),X5),K),at_top(nat)) ) ) ).

% Bseq_ignore_initial_segment
tff(fact_7080_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_aia(fun(nat,A),fun(nat,fun(nat,A)),X5),K),at_top(nat))
         => bfun(nat,A,X5,at_top(nat)) ) ) ).

% Bseq_offset
tff(fact_7081_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_aib(fun(nat,A),fun(A,fun(nat,A)),F2),C2),at_top(nat)) ) ) ).

% Bseq_add
tff(fact_7082_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_aib(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_7083_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_aic(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G),at_top(nat)) ) ) ) ).

% Bseq_mult
tff(fact_7084_Bseq__minus__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( bfun(nat,A,aTP_Lamp_bo(fun(nat,A),fun(nat,A),X5),at_top(nat))
        <=> bfun(nat,A,X5,at_top(nat)) ) ) ).

% Bseq_minus_iff
tff(fact_7085_filterlim__nat__sequentially,axiom,
    filterlim(int,nat,nat2,at_top(nat),at_top(int)) ).

% filterlim_nat_sequentially
tff(fact_7086_BseqI_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),K3: real] :
          ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X5,N))),K3)
         => bfun(nat,A,X5,at_top(nat)) ) ) ).

% BseqI'
tff(fact_7087_eventually__all__ge__at__top,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o)] :
          ( eventually(A,P,at_top(A))
         => eventually(A,aTP_Lamp_aid(fun(A,$o),fun(A,$o),P),at_top(A)) ) ) ).

% eventually_all_ge_at_top
tff(fact_7088_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_bi(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_7089_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_aie(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_7090_Greatest__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o)] : order_Greatest(A,P) = the(A,aTP_Lamp_aif(fun(A,$o),fun(A,$o),P)) ) ).

% Greatest_def
tff(fact_7091_filterlim__int__of__nat__at__topD,axiom,
    ! [A: $tType,F2: fun(int,A),F4: filter(A)] :
      ( filterlim(nat,A,aTP_Lamp_aig(fun(int,A),fun(nat,A),F2),F4,at_top(nat))
     => filterlim(int,A,F2,F4,at_top(int)) ) ).

% filterlim_int_of_nat_at_topD
tff(fact_7092_BseqD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( bfun(nat,A,X5,at_top(nat))
         => ? [K8: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K8)
              & ! [N10: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X5,N10))),K8) ) ) ) ).

% BseqD
tff(fact_7093_BseqE,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( bfun(nat,A,X5,at_top(nat))
         => ~ ! [K8: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K8)
               => ~ ! [N10: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X5,N10))),K8) ) ) ) ).

% BseqE
tff(fact_7094_BseqI,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [K3: real,X5: fun(nat,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K3)
         => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X5,N))),K3)
           => bfun(nat,A,X5,at_top(nat)) ) ) ) ).

% BseqI
tff(fact_7095_Bseq__def,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( bfun(nat,A,X5,at_top(nat))
        <=> ? [K7: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K7)
              & ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X5,N2))),K7) ) ) ) ).

% Bseq_def
tff(fact_7096_Bseq__iff1a,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( bfun(nat,A,X5,at_top(nat))
        <=> ? [N7: nat] :
            ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(nat,A,X5,N2))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N7))) ) ) ).

% Bseq_iff1a
tff(fact_7097_Bseq__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( bfun(nat,A,X5,at_top(nat))
        <=> ? [N7: nat] :
            ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X5,N2))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N7))) ) ) ).

% Bseq_iff
tff(fact_7098_Bseq__realpow,axiom,
    ! [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),one_one(real))
       => bfun(nat,real,power_power(real,X),at_top(nat)) ) ) ).

% Bseq_realpow
tff(fact_7099_BfunI,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),K3: real,F4: filter(A)] :
          ( eventually(A,aa(real,fun(A,$o),aTP_Lamp_aih(fun(A,B),fun(real,fun(A,$o)),F2),K3),F4)
         => bfun(A,B,F2,F4) ) ) ).

% BfunI
tff(fact_7100_Bseq__iff2,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( bfun(nat,A,X5,at_top(nat))
        <=> ? [K2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K2)
              & ? [X6: A] :
                ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,X5,N2)),aa(A,A,uminus_uminus(A),X6)))),K2) ) ) ) ).

% Bseq_iff2
tff(fact_7101_Bseq__iff3,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( bfun(nat,A,X5,at_top(nat))
        <=> ? [K2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K2)
              & ? [N7: nat] :
                ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,X5,N2)),aa(A,A,uminus_uminus(A),aa(nat,A,X5,N7))))),K2) ) ) ) ).

% Bseq_iff3
tff(fact_7102_Bfun__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),Aa2: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Aa2),F4)
         => ( ( Aa2 != zero_zero(B) )
           => bfun(A,B,aTP_Lamp_yr(fun(A,B),fun(A,B),F2),F4) ) ) ) ).

% Bfun_inverse
tff(fact_7103_Bfun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( bfun(A,B,F2,F4)
        <=> ? [K7: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K7)
              & eventually(A,aa(real,fun(A,$o),aTP_Lamp_aih(fun(A,B),fun(real,fun(A,$o)),F2),K7),F4) ) ) ) ).

% Bfun_def
tff(fact_7104_BfunE,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( bfun(A,B,F2,F4)
         => ~ ! [B6: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B6)
               => ~ eventually(A,aa(real,fun(A,$o),aTP_Lamp_aih(fun(A,B),fun(real,fun(A,$o)),F2),B6),F4) ) ) ) ).

% BfunE
tff(fact_7105_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_aii(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_7106_sequentially__imp__eventually__at__right,axiom,
    ! [A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [Aa2: A,Ba: A,P: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
         => ( ! [F3: fun(nat,A)] :
                ( ! [N10: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),aa(nat,A,F3,N10))
               => ( ! [N10: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F3,N10)),Ba)
                 => ( order_antimono(nat,A,F3)
                   => ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,Aa2),at_top(nat))
                     => eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_aij(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),P),F3),at_top(nat)) ) ) ) )
           => eventually(A,P,topolo174197925503356063within(A,Aa2,set_ord_greaterThan(A,Aa2))) ) ) ) ).

% sequentially_imp_eventually_at_right
tff(fact_7107_decseq__const,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [K: A] : order_antimono(nat,A,aTP_Lamp_aik(A,fun(nat,A),K)) ) ).

% decseq_const
tff(fact_7108_greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( member(A,I,set_or3652927894154168847AtMost(A,L,U))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),I)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),U) ) ) ) ).

% greaterThanAtMost_iff
tff(fact_7109_greaterThanAtMost__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,K: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),K)
         => ( set_or3652927894154168847AtMost(A,K,L) = bot_bot(set(A)) ) ) ) ).

% greaterThanAtMost_empty
tff(fact_7110_image__add__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [C2: A,Aa2: A,Ba: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),C2)),set_or3652927894154168847AtMost(A,Aa2,Ba)) = set_or3652927894154168847AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),Aa2),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),Ba)) ) ).

% image_add_greaterThanAtMost
tff(fact_7111_decseqD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),I: nat,J: nat] :
          ( order_antimono(nat,A,F2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,J)),aa(nat,A,F2,I)) ) ) ) ).

% decseqD
tff(fact_7112_decseq__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( order_antimono(nat,A,X5)
        <=> ! [M4: nat,N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M4),N2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N2)),aa(nat,A,X5,M4)) ) ) ) ).

% decseq_def
tff(fact_7113_ivl__disj__un__two_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or3652927894154168847AtMost(A,M,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(6)
tff(fact_7114_Ioc__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: A,Ba: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,Aa2,Ba)),set_or3652927894154168847AtMost(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2)
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),Aa2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),D2) ) ) ) ) ).

% Ioc_subset_iff
tff(fact_7115_Ioc__inj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Aa2: A,Ba: A,C2: A,D2: A] :
          ( ( set_or3652927894154168847AtMost(A,Aa2,Ba) = set_or3652927894154168847AtMost(A,C2,D2) )
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),Aa2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D2),C2) )
            | ( ( Aa2 = C2 )
              & ( Ba = D2 ) ) ) ) ) ).

% Ioc_inj
tff(fact_7116_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,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Y)),aa(A,B,F2,X)) ) ) ) ).

% antimonoD
tff(fact_7117_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,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Y)),aa(A,B,F2,X)) ) ) ) ).

% antimonoE
tff(fact_7118_antimonoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( ! [X2: A,Y2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Y2)),aa(A,B,F2,X2)) )
         => order_antimono(A,B,F2) ) ) ).

% antimonoI
tff(fact_7119_antimono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( order_antimono(A,B,F2)
        <=> ! [X6: A,Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X6),Y4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Y4)),aa(A,B,F2,X6)) ) ) ) ).

% antimono_def
tff(fact_7120_decseq__Suc__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A)] :
          ( order_antimono(nat,A,F2)
        <=> ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,aa(nat,nat,suc,N2))),aa(nat,A,F2,N2)) ) ) ).

% decseq_Suc_iff
tff(fact_7121_decseq__SucI,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,aa(nat,nat,suc,N))),aa(nat,A,X5,N))
         => order_antimono(nat,A,X5) ) ) ).

% decseq_SucI
tff(fact_7122_decseq__SucD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: fun(nat,A),I: nat] :
          ( order_antimono(nat,A,A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A2,aa(nat,nat,suc,I))),aa(nat,A,A2,I)) ) ) ).

% decseq_SucD
tff(fact_7123_decseq__bounded,axiom,
    ! [X5: fun(nat,real),B2: real] :
      ( order_antimono(nat,real,X5)
     => ( ! [I2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B2),aa(nat,real,X5,I2))
       => bfun(nat,real,X5,at_top(nat)) ) ) ).

% decseq_bounded
tff(fact_7124_ivl__disj__un__two_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or3652927894154168847AtMost(A,M,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(8)
tff(fact_7125_ivl__disj__un__one_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_ord_atMost(A,L)),set_or3652927894154168847AtMost(A,L,U)) = set_ord_atMost(A,U) ) ) ) ).

% ivl_disj_un_one(3)
tff(fact_7126_ivl__disj__un__one_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,U)),set_ord_greaterThan(A,U)) = set_ord_greaterThan(A,L) ) ) ) ).

% ivl_disj_un_one(5)
tff(fact_7127_sum_Ohead,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,Na)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,M)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or3652927894154168847AtMost(nat,M,Na))) ) ) ) ).

% sum.head
tff(fact_7128_prod_Ohead,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(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_or3652927894154168847AtMost(nat,M,Na))) ) ) ) ).

% prod.head
tff(fact_7129_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Aa2: A,Ba: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,Aa2,Ba)),set_or1337092689740270186AtMost(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),Aa2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),D2) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastAtMost_iff
tff(fact_7130_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Aa2: A,Ba: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,Aa2,Ba)),set_or7035219750837199246ssThan(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),Aa2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ba),D2) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastLessThan_iff
tff(fact_7131_ivl__disj__un__two__touch_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(3)
tff(fact_7132_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Aa2: A,Ba: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,Aa2,Ba)),set_or3652927894154168847AtMost(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),Aa2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ba),D2) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanAtMost_iff
tff(fact_7133_ivl__disj__un__two_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or5935395276787703475ssThan(A,M,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(2)
tff(fact_7134_decseq__ge,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X5: fun(nat,A),L5: A,Na: nat] :
          ( order_antimono(nat,A,X5)
         => ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L5),aa(nat,A,X5,Na)) ) ) ) ).

% decseq_ge
tff(fact_7135_decseq__convergent,axiom,
    ! [X5: fun(nat,real),B2: real] :
      ( order_antimono(nat,real,X5)
     => ( ! [I2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B2),aa(nat,real,X5,I2))
       => ~ ! [L6: real] :
              ( filterlim(nat,real,X5,topolo7230453075368039082e_nhds(real,L6),at_top(nat))
             => ~ ! [I3: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),L6),aa(nat,real,X5,I3)) ) ) ) ).

% decseq_convergent
tff(fact_7136_ivl__disj__un__singleton_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),insert(A,L),bot_bot(set(A)))),set_or3652927894154168847AtMost(A,L,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(5)
tff(fact_7137_ivl__disj__un__two_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(5)
tff(fact_7138_tendsto__at__right__sequentially,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [Aa2: A,Ba: A,X5: fun(A,B),L5: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
         => ( ! [S4: fun(nat,A)] :
                ( ! [N10: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),aa(nat,A,S4,N10))
               => ( ! [N10: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,S4,N10)),Ba)
                 => ( order_antimono(nat,A,S4)
                   => ( filterlim(nat,A,S4,topolo7230453075368039082e_nhds(A,Aa2),at_top(nat))
                     => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_ail(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,Aa2,set_ord_greaterThan(A,Aa2))) ) ) ) ).

% tendsto_at_right_sequentially
tff(fact_7139_GMVT,axiom,
    ! [Aa2: real,Ba: real,F2: fun(real,real),G: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Aa2),Ba)
     => ( ! [X2: real] :
            ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),X2)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),Ba) )
           => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X2,top_top(set(real))),F2) )
       => ( ! [X2: real] :
              ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Aa2),X2)
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X2),Ba) )
             => differentiable(real,real,F2,topolo174197925503356063within(real,X2,top_top(set(real)))) )
         => ( ! [X2: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Aa2),X2)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),Ba) )
               => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X2,top_top(set(real))),G) )
           => ( ! [X2: real] :
                  ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Aa2),X2)
                    & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X2),Ba) )
                 => differentiable(real,real,G,topolo174197925503356063within(real,X2,top_top(set(real)))) )
             => ? [G_c: real,F_c: real,C4: real] :
                  ( has_field_derivative(real,G,G_c,topolo174197925503356063within(real,C4,top_top(set(real))))
                  & has_field_derivative(real,F2,F_c,topolo174197925503356063within(real,C4,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Aa2),C4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),C4),Ba)
                  & ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,aa(real,real,F2,Ba)),aa(real,real,F2,Aa2))),G_c) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,aa(real,real,G,Ba)),aa(real,real,G,Aa2))),F_c) ) ) ) ) ) ) ) ).

% GMVT
tff(fact_7140_interval__cases,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [Sa: set(A)] :
          ( ! [A3: A,B3: A,X2: A] :
              ( member(A,A3,Sa)
             => ( member(A,B3,Sa)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),X2)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),B3)
                   => member(A,X2,Sa) ) ) ) )
         => ? [A3: A,B3: A] :
              ( ( Sa = bot_bot(set(A)) )
              | ( Sa = top_top(set(A)) )
              | ( Sa = set_ord_lessThan(A,B3) )
              | ( Sa = set_ord_atMost(A,B3) )
              | ( Sa = set_ord_greaterThan(A,A3) )
              | ( Sa = set_ord_atLeast(A,A3) )
              | ( Sa = set_or5935395276787703475ssThan(A,A3,B3) )
              | ( Sa = set_or3652927894154168847AtMost(A,A3,B3) )
              | ( Sa = set_or7035219750837199246ssThan(A,A3,B3) )
              | ( Sa = set_or1337092689740270186AtMost(A,A3,B3) ) ) ) ) ).

% interval_cases
tff(fact_7141_atLeast__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,K: A] :
          ( member(A,I,set_ord_atLeast(A,K))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K),I) ) ) ).

% atLeast_iff
tff(fact_7142_image__add__atLeast,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),K)),set_ord_atLeast(A,I)) = set_ord_atLeast(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),K),I)) ) ).

% image_add_atLeast
tff(fact_7143_atLeast__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_ord_atLeast(A,X)),set_ord_atLeast(A,Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) ) ) ).

% atLeast_subset_iff
tff(fact_7144_Icc__subset__Ici__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [L: A,H: A,L3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,L,H)),set_ord_atLeast(A,L3))
        <=> ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),H)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L3),L) ) ) ) ).

% Icc_subset_Ici_iff
tff(fact_7145_differentiable__cmult__left__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [C2: B,Q3: fun(A,B),T: A] :
          ( differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aim(B,fun(fun(A,B),fun(A,B)),C2),Q3),topolo174197925503356063within(A,T,top_top(set(A))))
        <=> ( ( C2 = zero_zero(B) )
            | differentiable(A,B,Q3,topolo174197925503356063within(A,T,top_top(set(A)))) ) ) ) ).

% differentiable_cmult_left_iff
tff(fact_7146_differentiable__cmult__right__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Q3: fun(A,B),C2: B,T: A] :
          ( differentiable(A,B,aa(B,fun(A,B),aTP_Lamp_ain(fun(A,B),fun(B,fun(A,B)),Q3),C2),topolo174197925503356063within(A,T,top_top(set(A))))
        <=> ( ( C2 = zero_zero(B) )
            | differentiable(A,B,Q3,topolo174197925503356063within(A,T,top_top(set(A)))) ) ) ) ).

% differentiable_cmult_right_iff
tff(fact_7147_differentiable__sum,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [S: set(A),F2: fun(A,fun(B,C)),Net: filter(B)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( ! [X2: A] :
                ( member(A,X2,S)
               => differentiable(B,C,aa(A,fun(B,C),F2,X2),Net) )
           => differentiable(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_tm(set(A),fun(fun(A,fun(B,C)),fun(B,C)),S),F2),Net) ) ) ) ).

% differentiable_sum
tff(fact_7148_differentiable__scaleR,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,real),X: A,S: set(A),G: fun(A,B)] :
          ( differentiable(A,real,F2,topolo174197925503356063within(A,X,S))
         => ( differentiable(A,B,G,topolo174197925503356063within(A,X,S))
           => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tj(fun(A,real),fun(fun(A,B),fun(A,B)),F2),G),topolo174197925503356063within(A,X,S)) ) ) ) ).

% differentiable_scaleR
tff(fact_7149_differentiable__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,B)] :
          ( differentiable(A,B,F2,F4)
         => ( differentiable(A,B,G,F4)
           => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tu(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),F4) ) ) ) ).

% differentiable_add
tff(fact_7150_differentiable__const,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Aa2: B,F4: filter(A)] : differentiable(A,B,aTP_Lamp_ts(B,fun(A,B),Aa2),F4) ) ).

% differentiable_const
tff(fact_7151_differentiable__ident,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F4: filter(A)] : differentiable(A,A,aTP_Lamp_tw(A,A),F4) ) ).

% differentiable_ident
tff(fact_7152_differentiable__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,B)] :
          ( differentiable(A,B,F2,F4)
         => ( differentiable(A,B,G,F4)
           => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tx(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),F4) ) ) ) ).

% differentiable_diff
tff(fact_7153_differentiable__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( differentiable(A,B,F2,F4)
         => differentiable(A,B,aTP_Lamp_tn(fun(A,B),fun(A,B),F2),F4) ) ) ).

% differentiable_minus
tff(fact_7154_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,S: set(C)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,aa(C,A,G,X),top_top(set(A))))
         => ( differentiable(C,A,G,topolo174197925503356063within(C,X,S))
           => differentiable(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ua(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G),topolo174197925503356063within(C,X,S)) ) ) ) ).

% differentiable_compose
tff(fact_7155_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,S: set(C)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,aa(C,A,G,X),aa(set(C),set(A),image(C,A,G),S)))
         => ( differentiable(C,A,G,topolo174197925503356063within(C,X,S))
           => differentiable(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ua(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G),topolo174197925503356063within(C,X,S)) ) ) ) ).

% differentiable_in_compose
tff(fact_7156_differentiable__power,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),X: A,S: set(A),Na: nat] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,X,S))
         => differentiable(A,B,aa(nat,fun(A,B),aTP_Lamp_up(fun(A,B),fun(nat,fun(A,B)),F2),Na),topolo174197925503356063within(A,X,S)) ) ) ).

% differentiable_power
tff(fact_7157_differentiable__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [F2: fun(A,B),X: A,S: set(A),G: fun(A,B)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,X,S))
         => ( differentiable(A,B,G,topolo174197925503356063within(A,X,S))
           => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ty(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo174197925503356063within(A,X,S)) ) ) ) ).

% differentiable_mult
tff(fact_7158_atLeast__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A] : set_ord_atLeast(A,L) = collect(A,aa(A,fun(A,$o),ord_less_eq(A),L)) ) ).

% atLeast_def
tff(fact_7159_ivl__disj__un__one_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,U)),set_ord_atLeast(A,U)) = set_ord_atLeast(A,L) ) ) ) ).

% ivl_disj_un_one(8)
tff(fact_7160_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : set_or1337092689740270186AtMost(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),L),one_one(int)),U) = set_or3652927894154168847AtMost(int,L,U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_int
tff(fact_7161_differentiable__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),X: A,S: set(A),G: fun(A,B)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,X,S))
         => ( differentiable(A,B,G,topolo174197925503356063within(A,X,S))
           => ( ( aa(A,B,G,X) != zero_zero(B) )
             => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ug(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo174197925503356063within(A,X,S)) ) ) ) ) ).

% differentiable_divide
tff(fact_7162_differentiable__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),X: A,S: set(A)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,X,S))
         => ( ( aa(A,B,F2,X) != zero_zero(B) )
           => differentiable(A,B,aTP_Lamp_aio(fun(A,B),fun(A,B),F2),topolo174197925503356063within(A,X,S)) ) ) ) ).

% differentiable_inverse
tff(fact_7163_ivl__disj__un__one_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,U)),set_ord_greaterThan(A,U)) = set_ord_atLeast(A,L) ) ) ) ).

% ivl_disj_un_one(7)
tff(fact_7164_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),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),M),one_one(word(A))),U) ) ) ) ).

% word_atLeastAtMost_Suc_greaterThanAtMost
tff(fact_7165_greaterThanAtMost__upto,axiom,
    ! [I: int,J: int] : set_or3652927894154168847AtMost(int,I,J) = aa(list(int),set(int),set2(int),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)) ).

% greaterThanAtMost_upto
tff(fact_7166_complete__interval,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [Aa2: A,Ba: A,P: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Aa2),Ba)
         => ( aa(A,$o,P,Aa2)
           => ( ~ aa(A,$o,P,Ba)
             => ? [C4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),C4)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C4),Ba)
                  & ! [X4: A] :
                      ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),X4)
                        & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),C4) )
                     => aa(A,$o,P,X4) )
                  & ! [D6: A] :
                      ( ! [X2: A] :
                          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Aa2),X2)
                            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),D6) )
                         => aa(A,$o,P,X2) )
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D6),C4) ) ) ) ) ) ) ).

% complete_interval
tff(fact_7167_VEBT__internal_Oreplicatei_Opelims,axiom,
    ! [A: $tType,X: nat,Xa2: heap_Time_Heap(A),Y: heap_Time_Heap(list(A))] :
      ( ( vEBT_VEBT_replicatei(A,X,Xa2) = Y )
     => ( accp(product_prod(nat,heap_Time_Heap(A)),vEBT_V8535202610096940344ei_rel(A),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)),X),Xa2))
       => ( ( ( X = zero_zero(nat) )
           => ( ( Y = aa(list(A),heap_Time_Heap(list(A)),heap_Time_return(list(A)),nil(A)) )
             => ~ accp(product_prod(nat,heap_Time_Heap(A)),vEBT_V8535202610096940344ei_rel(A),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)),Xa2)) ) )
         => ~ ! [N: nat] :
                ( ( X = aa(nat,nat,suc,N) )
               => ( ( Y = heap_Time_bind(A,list(A),Xa2,aa(nat,fun(A,heap_Time_Heap(list(A))),aTP_Lamp_az(heap_Time_Heap(A),fun(nat,fun(A,heap_Time_Heap(list(A)))),Xa2),N)) )
                 => ~ accp(product_prod(nat,heap_Time_Heap(A)),vEBT_V8535202610096940344ei_rel(A),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)),Xa2)) ) ) ) ) ) ).

% VEBT_internal.replicatei.pelims
tff(fact_7168_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,aa(real,fun(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_7169_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_7170_complex__Re__numeral,axiom,
    ! [V: num] : re(aa(num,complex,numeral_numeral(complex),V)) = aa(num,real,numeral_numeral(real),V) ).

% complex_Re_numeral
tff(fact_7171_Re__sum,axiom,
    ! [A: $tType,F2: fun(A,complex),S: set(A)] : re(aa(set(A),complex,groups7311177749621191930dd_sum(A,complex,F2),S)) = aa(set(A),real,groups7311177749621191930dd_sum(A,real,aTP_Lamp_aip(fun(A,complex),fun(A,real),F2)),S) ).

% Re_sum
tff(fact_7172_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_7173_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_7174_complex__Re__le__cmod,axiom,
    ! [X: complex] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),re(X)),real_V7770717601297561774m_norm(complex,X)) ).

% complex_Re_le_cmod
tff(fact_7175_has__derivative__Re,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,complex),G2: fun(A,complex),F4: filter(A)] :
          ( has_derivative(A,complex,G,G2,F4)
         => has_derivative(A,real,aTP_Lamp_aiq(fun(A,complex),fun(A,real),G),aTP_Lamp_aiq(fun(A,complex),fun(A,real),G2),F4) ) ) ).

% has_derivative_Re
tff(fact_7176_plus__complex_Osimps_I1_J,axiom,
    ! [X: complex,Y: complex] : re(aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),X),Y)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),re(X)),re(Y)) ).

% plus_complex.simps(1)
tff(fact_7177_Cauchy__Re,axiom,
    ! [X5: fun(nat,complex)] :
      ( topolo3814608138187158403Cauchy(complex,X5)
     => topolo3814608138187158403Cauchy(real,aTP_Lamp_air(fun(nat,complex),fun(nat,real),X5)) ) ).

% Cauchy_Re
tff(fact_7178_sums__Re,axiom,
    ! [X5: fun(nat,complex),Aa2: complex] :
      ( aa(complex,$o,sums(complex,X5),Aa2)
     => aa(real,$o,sums(real,aTP_Lamp_air(fun(nat,complex),fun(nat,real),X5)),re(Aa2)) ) ).

% sums_Re
tff(fact_7179_continuous__Re,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),G: fun(A,complex)] :
          ( topolo3448309680560233919inuous(A,complex,F4,G)
         => topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_ais(fun(A,complex),fun(A,real),G)) ) ) ).

% continuous_Re
tff(fact_7180_summable__Re,axiom,
    ! [F2: fun(nat,complex)] :
      ( summable(complex,F2)
     => summable(real,aTP_Lamp_air(fun(nat,complex),fun(nat,real),F2)) ) ).

% summable_Re
tff(fact_7181_abs__Re__le__cmod,axiom,
    ! [X: complex] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),re(X))),real_V7770717601297561774m_norm(complex,X)) ).

% abs_Re_le_cmod
tff(fact_7182_Re__csqrt,axiom,
    ! [Z: complex] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),re(csqrt(Z))) ).

% Re_csqrt
tff(fact_7183_isCont__Re,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Aa2: A,G: fun(A,complex)] :
          ( topolo3448309680560233919inuous(A,complex,topolo174197925503356063within(A,Aa2,top_top(set(A))),G)
         => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,Aa2,top_top(set(A))),aTP_Lamp_ais(fun(A,complex),fun(A,real),G)) ) ) ).

% isCont_Re
tff(fact_7184_cmod__plus__Re__le__0__iff,axiom,
    ! [Z: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(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_7185_tendsto__Re,axiom,
    ! [A: $tType,G: fun(A,complex),Aa2: complex,F4: filter(A)] :
      ( filterlim(A,complex,G,topolo7230453075368039082e_nhds(complex,Aa2),F4)
     => filterlim(A,real,aTP_Lamp_aip(fun(A,complex),fun(A,real),G),topolo7230453075368039082e_nhds(real,re(Aa2)),F4) ) ).

% tendsto_Re
tff(fact_7186_cos__n__Re__cis__pow__n,axiom,
    ! [Na: nat,Aa2: real] : cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),Aa2)) = re(aa(nat,complex,power_power(complex,cis(Aa2)),Na)) ).

% cos_n_Re_cis_pow_n
tff(fact_7187_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,aa(real,fun(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_7188_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_7189_Im__sum,axiom,
    ! [A: $tType,F2: fun(A,complex),S: set(A)] : im(aa(set(A),complex,groups7311177749621191930dd_sum(A,complex,F2),S)) = aa(set(A),real,groups7311177749621191930dd_sum(A,real,aTP_Lamp_ait(fun(A,complex),fun(A,real),F2)),S) ).

% Im_sum
tff(fact_7190_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_7191_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_7192_csqrt__of__real__nonneg,axiom,
    ! [X: complex] :
      ( ( im(X) = zero_zero(real) )
     => ( aa(real,$o,aa(real,fun(real,$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_7193_csqrt__minus,axiom,
    ! [X: complex] :
      ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),im(X)),zero_zero(real))
        | ( ( im(X) = zero_zero(real) )
          & aa(real,$o,aa(real,fun(real,$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_7194_csqrt__of__real__nonpos,axiom,
    ! [X: complex] :
      ( ( im(X) = zero_zero(real) )
     => ( aa(real,$o,aa(real,fun(real,$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_7195_sums__complex__iff,axiom,
    ! [F2: fun(nat,complex),X: complex] :
      ( aa(complex,$o,sums(complex,F2),X)
    <=> ( aa(real,$o,sums(real,aTP_Lamp_air(fun(nat,complex),fun(nat,real),F2)),re(X))
        & aa(real,$o,sums(real,aTP_Lamp_aiu(fun(nat,complex),fun(nat,real),F2)),im(X)) ) ) ).

% sums_complex_iff
tff(fact_7196_continuous__complex__iff,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,complex)] :
          ( topolo3448309680560233919inuous(A,complex,F4,F2)
        <=> ( topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_ais(fun(A,complex),fun(A,real),F2))
            & topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_aiv(fun(A,complex),fun(A,real),F2)) ) ) ) ).

% continuous_complex_iff
tff(fact_7197_continuous__Im,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),G: fun(A,complex)] :
          ( topolo3448309680560233919inuous(A,complex,F4,G)
         => topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_aiv(fun(A,complex),fun(A,real),G)) ) ) ).

% continuous_Im
tff(fact_7198_sums__Im,axiom,
    ! [X5: fun(nat,complex),Aa2: complex] :
      ( aa(complex,$o,sums(complex,X5),Aa2)
     => aa(real,$o,sums(real,aTP_Lamp_aiu(fun(nat,complex),fun(nat,real),X5)),im(Aa2)) ) ).

% sums_Im
tff(fact_7199_Cauchy__Im,axiom,
    ! [X5: fun(nat,complex)] :
      ( topolo3814608138187158403Cauchy(complex,X5)
     => topolo3814608138187158403Cauchy(real,aTP_Lamp_aiu(fun(nat,complex),fun(nat,real),X5)) ) ).

% Cauchy_Im
tff(fact_7200_plus__complex_Osimps_I2_J,axiom,
    ! [X: complex,Y: complex] : im(aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),X),Y)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),im(X)),im(Y)) ).

% plus_complex.simps(2)
tff(fact_7201_has__derivative__Im,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,complex),G2: fun(A,complex),F4: filter(A)] :
          ( has_derivative(A,complex,G,G2,F4)
         => has_derivative(A,real,aTP_Lamp_aiw(fun(A,complex),fun(A,real),G),aTP_Lamp_aiw(fun(A,complex),fun(A,real),G2),F4) ) ) ).

% has_derivative_Im
tff(fact_7202_summable__Im,axiom,
    ! [F2: fun(nat,complex)] :
      ( summable(complex,F2)
     => summable(real,aTP_Lamp_aiu(fun(nat,complex),fun(nat,real),F2)) ) ).

% summable_Im
tff(fact_7203_abs__Im__le__cmod,axiom,
    ! [X: complex] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),im(X))),real_V7770717601297561774m_norm(complex,X)) ).

% abs_Im_le_cmod
tff(fact_7204_isCont__Im,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Aa2: A,G: fun(A,complex)] :
          ( topolo3448309680560233919inuous(A,complex,topolo174197925503356063within(A,Aa2,top_top(set(A))),G)
         => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,Aa2,top_top(set(A))),aTP_Lamp_aiv(fun(A,complex),fun(A,real),G)) ) ) ).

% isCont_Im
tff(fact_7205_summable__complex__iff,axiom,
    ! [F2: fun(nat,complex)] :
      ( summable(complex,F2)
    <=> ( summable(real,aTP_Lamp_air(fun(nat,complex),fun(nat,real),F2))
        & summable(real,aTP_Lamp_aiu(fun(nat,complex),fun(nat,real),F2)) ) ) ).

% summable_complex_iff
tff(fact_7206_times__complex_Osimps_I2_J,axiom,
    ! [X: complex,Y: complex] : im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),X),Y)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),re(X)),im(Y))),aa(real,real,aa(real,fun(real,real),times_times(real),im(X)),re(Y))) ).

% times_complex.simps(2)
tff(fact_7207_cmod__Re__le__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( im(X) = im(Y) )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(complex,X)),real_V7770717601297561774m_norm(complex,Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),re(X))),aa(real,real,abs_abs(real),re(Y))) ) ) ).

% cmod_Re_le_iff
tff(fact_7208_cmod__Im__le__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( re(X) = re(Y) )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(complex,X)),real_V7770717601297561774m_norm(complex,Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),im(X))),aa(real,real,abs_abs(real),im(Y))) ) ) ).

% cmod_Im_le_iff
tff(fact_7209_plus__complex_Ocode,axiom,
    ! [X: complex,Y: complex] : aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),X),Y) = complex2(aa(real,real,aa(real,fun(real,real),plus_plus(real),re(X)),re(Y)),aa(real,real,aa(real,fun(real,real),plus_plus(real),im(X)),im(Y))) ).

% plus_complex.code
tff(fact_7210_csqrt__principal,axiom,
    ! [Z: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),re(csqrt(Z)))
      | ( ( re(csqrt(Z)) = zero_zero(real) )
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),im(csqrt(Z))) ) ) ).

% csqrt_principal
tff(fact_7211_cmod__le,axiom,
    ! [Z: complex] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(complex,Z)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,abs_abs(real),re(Z))),aa(real,real,abs_abs(real),im(Z)))) ).

% cmod_le
tff(fact_7212_tendsto__Im,axiom,
    ! [A: $tType,G: fun(A,complex),Aa2: complex,F4: filter(A)] :
      ( filterlim(A,complex,G,topolo7230453075368039082e_nhds(complex,Aa2),F4)
     => filterlim(A,real,aTP_Lamp_ait(fun(A,complex),fun(A,real),G),topolo7230453075368039082e_nhds(real,im(Aa2)),F4) ) ).

% tendsto_Im
tff(fact_7213_sin__n__Im__cis__pow__n,axiom,
    ! [Na: nat,Aa2: real] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),Aa2)) = im(aa(nat,complex,power_power(complex,cis(Aa2)),Na)) ).

% sin_n_Im_cis_pow_n
tff(fact_7214_fun__complex__eq,axiom,
    ! [A: $tType,F2: fun(A,complex),X4: A] : aa(A,complex,F2,X4) = aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),aa(real,complex,real_Vector_of_real(complex),re(aa(A,complex,F2,X4)))),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),aa(real,complex,real_Vector_of_real(complex),im(aa(A,complex,F2,X4))))) ).

% fun_complex_eq
tff(fact_7215_complex__eq,axiom,
    ! [Aa2: complex] : Aa2 = aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),aa(real,complex,real_Vector_of_real(complex),re(Aa2))),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),aa(real,complex,real_Vector_of_real(complex),im(Aa2)))) ).

% complex_eq
tff(fact_7216_times__complex_Ocode,axiom,
    ! [X: complex,Y: complex] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),X),Y) = complex2(aa(real,real,minus_minus(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,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),re(X)),im(Y))),aa(real,real,aa(real,fun(real,real),times_times(real),im(X)),re(Y)))) ).

% times_complex.code
tff(fact_7217_tendsto__complex__iff,axiom,
    ! [A: $tType,F2: fun(A,complex),X: complex,F4: filter(A)] :
      ( filterlim(A,complex,F2,topolo7230453075368039082e_nhds(complex,X),F4)
    <=> ( filterlim(A,real,aTP_Lamp_aip(fun(A,complex),fun(A,real),F2),topolo7230453075368039082e_nhds(real,re(X)),F4)
        & filterlim(A,real,aTP_Lamp_ait(fun(A,complex),fun(A,real),F2),topolo7230453075368039082e_nhds(real,im(X)),F4) ) ) ).

% tendsto_complex_iff
tff(fact_7218_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,aa(real,fun(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_7219_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_7220_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_7221_complex__eq__0,axiom,
    ! [Z: complex] :
      ( ( Z = zero_zero(complex) )
    <=> ( aa(real,real,aa(real,fun(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_7222_norm__complex__def,axiom,
    ! [Z: complex] : real_V7770717601297561774m_norm(complex,Z) = aa(real,real,sqrt,aa(real,real,aa(real,fun(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_7223_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,aa(real,fun(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_7224_complex__neq__0,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero(complex) )
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,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_7225_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,aa(real,fun(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,aa(real,fun(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_7226_csqrt__square,axiom,
    ! [Ba: complex] :
      ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),re(Ba))
        | ( ( re(Ba) = zero_zero(real) )
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),im(Ba)) ) )
     => ( csqrt(aa(nat,complex,power_power(complex,Ba),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = Ba ) ) ).

% csqrt_square
tff(fact_7227_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,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),re(W))
          | ( ( re(W) = zero_zero(real) )
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),im(W)) ) )
       => ( csqrt(Z) = W ) ) ) ).

% csqrt_unique
tff(fact_7228_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,aa(real,fun(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_7229_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,aa(real,fun(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_7230_complex__abs__le__norm,axiom,
    ! [Z: complex] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,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_7231_complex__unit__circle,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero(complex) )
     => ( aa(real,real,aa(real,fun(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_7232_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,aa(real,fun(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,aa(real,fun(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_7233_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,aa(real,fun(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,aa(real,fun(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,aa(real,fun(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_7234_Im__Reals__divide,axiom,
    ! [R: complex,Z: complex] :
      ( member(complex,R,real_Vector_Reals(complex))
     => ( im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),R),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(R))),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_7235_Re__Reals__divide,axiom,
    ! [R: complex,Z: complex] :
      ( member(complex,R,real_Vector_Reals(complex))
     => ( re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),R),Z)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),re(R)),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_7236_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_7237_Reals__mult,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Aa2: A,Ba: A] :
          ( member(A,Aa2,real_Vector_Reals(A))
         => ( member(A,Ba,real_Vector_Reals(A))
           => member(A,aa(A,A,aa(A,fun(A,A),times_times(A),Aa2),Ba),real_Vector_Reals(A)) ) ) ) ).

% Reals_mult
tff(fact_7238_Reals__add,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Aa2: A,Ba: A] :
          ( member(A,Aa2,real_Vector_Reals(A))
         => ( member(A,Ba,real_Vector_Reals(A))
           => member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa2),Ba),real_Vector_Reals(A)) ) ) ) ).

% Reals_add
tff(fact_7239_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_7240_Reals__1,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => member(A,one_one(A),real_Vector_Reals(A)) ) ).

% Reals_1
tff(fact_7241_Re__prod__Reals,axiom,
    ! [A: $tType,A2: set(A),F2: fun(A,complex)] :
      ( ! [X2: A] :
          ( member(A,X2,A2)
         => member(complex,aa(A,complex,F2,X2),real_Vector_Reals(complex)) )
     => ( re(groups7121269368397514597t_prod(A,complex,F2,A2)) = groups7121269368397514597t_prod(A,real,aTP_Lamp_aip(fun(A,complex),fun(A,real),F2),A2) ) ) ).

% Re_prod_Reals
tff(fact_7242_series__comparison__complex,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [G: fun(nat,complex),N3: nat,F2: fun(nat,A)] :
          ( summable(complex,G)
         => ( ! [N: nat] : member(complex,aa(nat,complex,G,N),real_Vector_Reals(complex))
           => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),re(aa(nat,complex,G,N)))
             => ( ! [N: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N3),N)
                   => aa(real,$o,aa(real,fun(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_7243_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,aa(real,fun(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_7244_even__sum__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_parity(B)
     => ! [A2: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,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,$o),aTP_Lamp_aix(set(A),fun(fun(A,B),fun(A,$o)),A2),F2)))) ) ) ) ).

% even_sum_iff
tff(fact_7245_card__UNIV__unit,axiom,
    finite_card(product_unit,top_top(set(product_unit))) = one_one(nat) ).

% card_UNIV_unit
tff(fact_7246_card__Collect__less__nat,axiom,
    ! [Na: nat] : finite_card(nat,collect(nat,aTP_Lamp_gx(nat,fun(nat,$o),Na))) = Na ).

% card_Collect_less_nat
tff(fact_7247_card__eq__UNIV2,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Sa: set(A)] :
          ( ( finite_card(A,top_top(set(A))) = finite_card(A,Sa) )
        <=> ( Sa = top_top(set(A)) ) ) ) ).

% card_eq_UNIV2
tff(fact_7248_card__eq__UNIV,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Sa: set(A)] :
          ( ( finite_card(A,Sa) = finite_card(A,top_top(set(A))) )
        <=> ( Sa = top_top(set(A)) ) ) ) ).

% card_eq_UNIV
tff(fact_7249_card__Collect__le__nat,axiom,
    ! [Na: nat] : finite_card(nat,collect(nat,aTP_Lamp_oc(nat,fun(nat,$o),Na))) = aa(nat,nat,suc,Na) ).

% card_Collect_le_nat
tff(fact_7250_complex__cnj__add,axiom,
    ! [X: complex,Y: complex] : cnj(aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),X),Y)) = aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),cnj(X)),cnj(Y)) ).

% complex_cnj_add
tff(fact_7251_card__UNIV__bool,axiom,
    finite_card($o,top_top(set($o))) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% card_UNIV_bool
tff(fact_7252_card__ge__UNIV,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Sa: set(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),finite_card(A,top_top(set(A)))),finite_card(A,Sa))
        <=> ( Sa = top_top(set(A)) ) ) ) ).

% card_ge_UNIV
tff(fact_7253_cnj__sum,axiom,
    ! [A: $tType,F2: fun(A,complex),S: set(A)] : cnj(aa(set(A),complex,groups7311177749621191930dd_sum(A,complex,F2),S)) = aa(set(A),complex,groups7311177749621191930dd_sum(A,complex,aTP_Lamp_aiy(fun(A,complex),fun(A,complex),F2)),S) ).

% cnj_sum
tff(fact_7254_card__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : finite_card(int,set_or7035219750837199246ssThan(int,L,U)) = aa(int,nat,nat2,aa(int,int,minus_minus(int,U),L)) ).

% card_atLeastLessThan_int
tff(fact_7255_cnj__prod,axiom,
    ! [A: $tType,F2: fun(A,complex),S: set(A)] : cnj(groups7121269368397514597t_prod(A,complex,F2,S)) = groups7121269368397514597t_prod(A,complex,aTP_Lamp_aiy(fun(A,complex),fun(A,complex),F2),S) ).

% cnj_prod
tff(fact_7256_prod__constant,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Y: A,A2: set(B)] : groups7121269368397514597t_prod(B,A,aTP_Lamp_aiz(A,fun(B,A),Y),A2) = aa(nat,A,power_power(A,Y),finite_card(B,A2)) ) ).

% prod_constant
tff(fact_7257_card__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : finite_card(int,set_or3652927894154168847AtMost(int,L,U)) = aa(int,nat,nat2,aa(int,int,minus_minus(int,U),L)) ).

% card_greaterThanAtMost_int
tff(fact_7258_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_aja(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_7259_card__Diff__insert,axiom,
    ! [A: $tType,Aa2: A,A2: set(A),B2: set(A)] :
      ( member(A,Aa2,A2)
     => ( ~ member(A,Aa2,B2)
       => ( finite_card(A,aa(set(A),set(A),minus_minus(set(A),A2),aa(set(A),set(A),insert(A,Aa2),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_7260_card__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : finite_card(int,set_or1337092689740270186AtMost(int,L,U)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,minus_minus(int,U),L)),one_one(int))) ).

% card_atLeastAtMost_int
tff(fact_7261_card__doubleton__eq__2__iff,axiom,
    ! [A: $tType,Aa2: A,Ba: A] :
      ( ( finite_card(A,aa(set(A),set(A),insert(A,Aa2),aa(set(A),set(A),insert(A,Ba),bot_bot(set(A))))) = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
    <=> ( Aa2 != Ba ) ) ).

% card_doubleton_eq_2_iff
tff(fact_7262_card__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : finite_card(int,set_or5935395276787703475ssThan(int,L,U)) = aa(int,nat,nat2,aa(int,int,minus_minus(int,U),aa(int,int,aa(int,fun(int,int),plus_plus(int),L),one_one(int)))) ).

% card_greaterThanLessThan_int
tff(fact_7263_card__image__le,axiom,
    ! [B: $tType,A: $tType,A2: set(A),F2: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),A2)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),finite_card(B,aa(set(A),set(B),image(A,B,F2),A2))),finite_card(A,A2)) ) ).

% card_image_le
tff(fact_7264_obtain__subset__with__card__n,axiom,
    ! [A: $tType,Na: nat,Sa: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),finite_card(A,Sa))
     => ~ ! [T7: set(A)] :
            ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T7),Sa)
           => ( ( finite_card(A,T7) = Na )
             => ~ aa(set(A),$o,finite_finite2(A),T7) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_7265_finite__if__finite__subsets__card__bdd,axiom,
    ! [A: $tType,F4: set(A),C5: nat] :
      ( ! [G6: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),G6),F4)
         => ( aa(set(A),$o,finite_finite2(A),G6)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),finite_card(A,G6)),C5) ) )
     => ( aa(set(A),$o,finite_finite2(A),F4)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),finite_card(A,F4)),C5) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_7266_card__seteq,axiom,
    ! [A: $tType,B2: set(A),A2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A2),B2)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),finite_card(A,B2)),finite_card(A,A2))
         => ( A2 = B2 ) ) ) ) ).

% card_seteq
tff(fact_7267_card__mono,axiom,
    ! [A: $tType,B2: set(A),A2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A2),B2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),finite_card(A,A2)),finite_card(A,B2)) ) ) ).

% card_mono
tff(fact_7268_differentiable__cnj__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(A,complex),X: A,A2: set(A)] :
          ( differentiable(A,complex,aTP_Lamp_ajb(fun(A,complex),fun(A,complex),F2),topolo174197925503356063within(A,X,A2))
        <=> differentiable(A,complex,F2,topolo174197925503356063within(A,X,A2)) ) ) ).

% differentiable_cnj_iff
tff(fact_7269_card__lists__length__eq,axiom,
    ! [A: $tType,A2: set(A),Na: nat] :
      ( aa(set(A),$o,finite_finite2(A),A2)
     => ( finite_card(list(A),collect(list(A),aa(nat,fun(list(A),$o),aTP_Lamp_nd(set(A),fun(nat,fun(list(A),$o)),A2),Na))) = aa(nat,nat,power_power(nat,finite_card(A,A2)),Na) ) ) ).

% card_lists_length_eq
tff(fact_7270_card__le__if__inj__on__rel,axiom,
    ! [B: $tType,A: $tType,B2: set(A),A2: set(B),R: fun(B,fun(A,$o))] :
      ( aa(set(A),$o,finite_finite2(A),B2)
     => ( ! [A3: B] :
            ( member(B,A3,A2)
           => ? [B8: A] :
                ( member(A,B8,B2)
                & aa(A,$o,aa(B,fun(A,$o),R,A3),B8) ) )
       => ( ! [A12: B,A23: B,B3: A] :
              ( member(B,A12,A2)
             => ( member(B,A23,A2)
               => ( member(A,B3,B2)
                 => ( aa(A,$o,aa(B,fun(A,$o),R,A12),B3)
                   => ( aa(A,$o,aa(B,fun(A,$o),R,A23),B3)
                     => ( A12 = A23 ) ) ) ) ) )
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),finite_card(B,A2)),finite_card(A,B2)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_7271_n__subsets,axiom,
    ! [A: $tType,A2: set(A),K: nat] :
      ( aa(set(A),$o,finite_finite2(A),A2)
     => ( finite_card(set(A),collect(set(A),aa(nat,fun(set(A),$o),aTP_Lamp_ajc(set(A),fun(nat,fun(set(A),$o)),A2),K))) = aa(nat,nat,binomial(finite_card(A,A2)),K) ) ) ).

% n_subsets
tff(fact_7272_sum__multicount__gen,axiom,
    ! [A: $tType,B: $tType,S: set(A),T: set(B),R3: fun(A,fun(B,$o)),K: fun(B,nat)] :
      ( aa(set(A),$o,finite_finite2(A),S)
     => ( aa(set(B),$o,finite_finite2(B),T)
       => ( ! [X2: B] :
              ( member(B,X2,T)
             => ( finite_card(A,collect(A,aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_my(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),S),R3),X2))) = aa(B,nat,K,X2) ) )
         => ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aa(fun(A,fun(B,$o)),fun(A,nat),aTP_Lamp_ajd(set(B),fun(fun(A,fun(B,$o)),fun(A,nat)),T),R3)),S) = aa(set(B),nat,groups7311177749621191930dd_sum(B,nat,K),T) ) ) ) ) ).

% sum_multicount_gen
tff(fact_7273_card__2__iff_H,axiom,
    ! [A: $tType,Sa: set(A)] :
      ( ( finite_card(A,Sa) = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
    <=> ? [X6: A] :
          ( member(A,X6,Sa)
          & ? [Xa4: A] :
              ( member(A,Xa4,Sa)
              & ( X6 != Xa4 )
              & ! [Xb4: A] :
                  ( member(A,Xb4,Sa)
                 => ( ( Xb4 = X6 )
                    | ( Xb4 = Xa4 ) ) ) ) ) ) ).

% card_2_iff'
tff(fact_7274_card__eq__sum,axiom,
    ! [A: $tType,A2: set(A)] : finite_card(A,A2) = aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aTP_Lamp_aje(A,nat)),A2) ).

% card_eq_sum
tff(fact_7275_card__insert__le,axiom,
    ! [A: $tType,A2: set(A),X: A] : aa(nat,$o,aa(nat,fun(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_7276_has__derivative__cnj,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,complex),G2: fun(A,complex),F4: filter(A)] :
          ( has_derivative(A,complex,G,G2,F4)
         => has_derivative(A,complex,aTP_Lamp_ajb(fun(A,complex),fun(A,complex),G),aTP_Lamp_ajb(fun(A,complex),fun(A,complex),G2),F4) ) ) ).

% has_derivative_cnj
tff(fact_7277_sums__cnj,axiom,
    ! [F2: fun(nat,complex),L: complex] :
      ( aa(complex,$o,sums(complex,aTP_Lamp_ajf(fun(nat,complex),fun(nat,complex),F2)),cnj(L))
    <=> aa(complex,$o,sums(complex,F2),L) ) ).

% sums_cnj
tff(fact_7278_continuous__cnj,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),G: fun(A,complex)] :
          ( topolo3448309680560233919inuous(A,complex,F4,G)
         => topolo3448309680560233919inuous(A,complex,F4,aTP_Lamp_ajg(fun(A,complex),fun(A,complex),G)) ) ) ).

% continuous_cnj
tff(fact_7279_card__1__singletonE,axiom,
    ! [A: $tType,A2: set(A)] :
      ( ( finite_card(A,A2) = one_one(nat) )
     => ~ ! [X2: A] : A2 != aa(set(A),set(A),insert(A,X2),bot_bot(set(A))) ) ).

% card_1_singletonE
tff(fact_7280_card__length,axiom,
    ! [A: $tType,Xs2: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),finite_card(A,aa(list(A),set(A),set2(A),Xs2))),aa(list(A),nat,size_size(list(A)),Xs2)) ).

% card_length
tff(fact_7281_card__le__sym__Diff,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A2)
     => ( aa(set(A),$o,finite_finite2(A),B2)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),finite_card(A,A2)),finite_card(A,B2))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(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_le_sym_Diff
tff(fact_7282_card__Un__le,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A2),B2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),finite_card(A,A2)),finite_card(A,B2))) ).

% card_Un_le
tff(fact_7283_tendsto__cnj,axiom,
    ! [A: $tType,G: fun(A,complex),Aa2: complex,F4: filter(A)] :
      ( filterlim(A,complex,G,topolo7230453075368039082e_nhds(complex,Aa2),F4)
     => filterlim(A,complex,aTP_Lamp_aiy(fun(A,complex),fun(A,complex),G),topolo7230453075368039082e_nhds(complex,cnj(Aa2)),F4) ) ).

% tendsto_cnj
tff(fact_7284_lim__cnj,axiom,
    ! [A: $tType,F2: fun(A,complex),L: complex,F4: filter(A)] :
      ( filterlim(A,complex,aTP_Lamp_aiy(fun(A,complex),fun(A,complex),F2),topolo7230453075368039082e_nhds(complex,cnj(L)),F4)
    <=> filterlim(A,complex,F2,topolo7230453075368039082e_nhds(complex,L),F4) ) ).

% lim_cnj
tff(fact_7285_card__atLeastZeroLessThan__int,axiom,
    ! [U: int] : finite_card(int,set_or7035219750837199246ssThan(int,zero_zero(int),U)) = aa(int,nat,nat2,U) ).

% card_atLeastZeroLessThan_int
tff(fact_7286_card__less__Suc2,axiom,
    ! [M2: set(nat),I: nat] :
      ( ~ member(nat,zero_zero(nat),M2)
     => ( finite_card(nat,collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_ajh(set(nat),fun(nat,fun(nat,$o)),M2),I))) = finite_card(nat,collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_aji(set(nat),fun(nat,fun(nat,$o)),M2),I))) ) ) ).

% card_less_Suc2
tff(fact_7287_card__less__Suc,axiom,
    ! [M2: set(nat),I: nat] :
      ( member(nat,zero_zero(nat),M2)
     => ( aa(nat,nat,suc,finite_card(nat,collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_ajh(set(nat),fun(nat,fun(nat,$o)),M2),I)))) = finite_card(nat,collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_aji(set(nat),fun(nat,fun(nat,$o)),M2),I))) ) ) ).

% card_less_Suc
tff(fact_7288_card__less,axiom,
    ! [M2: set(nat),I: nat] :
      ( member(nat,zero_zero(nat),M2)
     => ( finite_card(nat,collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_aji(set(nat),fun(nat,fun(nat,$o)),M2),I))) != zero_zero(nat) ) ) ).

% card_less
tff(fact_7289_card__map__elide,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(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_7290_subset__card__intvl__is__intvl,axiom,
    ! [A2: set(nat),K: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),A2),set_or7035219750837199246ssThan(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),finite_card(nat,A2))))
     => ( A2 = set_or7035219750837199246ssThan(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),finite_card(nat,A2))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_7291_sum__Suc,axiom,
    ! [A: $tType,F2: fun(A,nat),A2: set(A)] : aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aTP_Lamp_ajj(fun(A,nat),fun(A,nat),F2)),A2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A2)),finite_card(A,A2)) ).

% sum_Suc
tff(fact_7292_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_ajk(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_7293_sum__multicount,axiom,
    ! [A: $tType,B: $tType,Sa: set(A),T4: set(B),R3: fun(A,fun(B,$o)),K: nat] :
      ( aa(set(A),$o,finite_finite2(A),Sa)
     => ( aa(set(B),$o,finite_finite2(B),T4)
       => ( ! [X2: B] :
              ( member(B,X2,T4)
             => ( finite_card(A,collect(A,aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_my(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Sa),R3),X2))) = K ) )
         => ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aa(fun(A,fun(B,$o)),fun(A,nat),aTP_Lamp_ajd(set(B),fun(fun(A,fun(B,$o)),fun(A,nat)),T4),R3)),Sa) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),finite_card(B,T4)) ) ) ) ) ).

% sum_multicount
tff(fact_7294_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_ajl(A,real)),A2) ).

% real_of_card
tff(fact_7295_isCont__cnj,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Aa2: A,G: fun(A,complex)] :
          ( topolo3448309680560233919inuous(A,complex,topolo174197925503356063within(A,Aa2,top_top(set(A))),G)
         => topolo3448309680560233919inuous(A,complex,topolo174197925503356063within(A,Aa2,top_top(set(A))),aTP_Lamp_ajg(fun(A,complex),fun(A,complex),G)) ) ) ).

% isCont_cnj
tff(fact_7296_sum__bounded__below,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [A2: set(A),K3: B,F2: fun(A,B)] :
          ( ! [I2: A] :
              ( member(A,I2,A2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),K3),aa(A,B,F2,I2)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),finite_card(A,A2))),K3)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A2)) ) ) ).

% sum_bounded_below
tff(fact_7297_sum__bounded__above,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [A2: set(A),F2: fun(A,B),K3: B] :
          ( ! [I2: A] :
              ( member(A,I2,A2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),K3) )
         => aa(B,$o,aa(B,fun(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))),K3)) ) ) ).

% sum_bounded_above
tff(fact_7298_card__le__Suc0__iff__eq,axiom,
    ! [A: $tType,A2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),finite_card(A,A2)),aa(nat,nat,suc,zero_zero(nat)))
      <=> ! [X6: A] :
            ( member(A,X6,A2)
           => ! [Xa4: A] :
                ( member(A,Xa4,A2)
               => ( X6 = Xa4 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_7299_card__le__Suc__iff,axiom,
    ! [A: $tType,Na: nat,A2: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Na)),finite_card(A,A2))
    <=> ? [A7: A,B9: set(A)] :
          ( ( A2 = aa(set(A),set(A),insert(A,A7),B9) )
          & ~ member(A,A7,B9)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),finite_card(A,B9))
          & aa(set(A),$o,finite_finite2(A),B9) ) ) ).

% card_le_Suc_iff
tff(fact_7300_surj__card__le,axiom,
    ! [B: $tType,A: $tType,A2: set(A),B2: set(B),F2: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),A2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B2),aa(set(A),set(B),image(A,B,F2),A2))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),finite_card(B,B2)),finite_card(A,A2)) ) ) ).

% surj_card_le
tff(fact_7301_card__1__singletonI,axiom,
    ! [A: $tType,Sa: set(A),X: A] :
      ( aa(set(A),$o,finite_finite2(A),Sa)
     => ( ( finite_card(A,Sa) = one_one(nat) )
       => ( member(A,X,Sa)
         => ( Sa = aa(set(A),set(A),insert(A,X),bot_bot(set(A))) ) ) ) ) ).

% card_1_singletonI
tff(fact_7302_card__Diff1__le,axiom,
    ! [A: $tType,A2: set(A),X: A] : aa(nat,$o,aa(nat,fun(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_7303_diff__card__le__card__Diff,axiom,
    ! [A: $tType,B2: set(A),A2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B2)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,finite_card(A,A2)),finite_card(A,B2))),finite_card(A,aa(set(A),set(A),minus_minus(set(A),A2),B2))) ) ).

% diff_card_le_card_Diff
tff(fact_7304_card__lists__length__le,axiom,
    ! [A: $tType,A2: set(A),Na: nat] :
      ( aa(set(A),$o,finite_finite2(A),A2)
     => ( finite_card(list(A),collect(list(A),aa(nat,fun(list(A),$o),aTP_Lamp_ne(set(A),fun(nat,fun(list(A),$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_7305_ex__bij__betw__nat__finite__1,axiom,
    ! [A: $tType,M2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),M2)
     => ? [H2: fun(nat,A)] : bij_betw(nat,A,H2,set_or1337092689740270186AtMost(nat,one_one(nat),finite_card(A,M2)),M2) ) ).

% ex_bij_betw_nat_finite_1
tff(fact_7306_card__roots__unity,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Na)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),finite_card(A,collect(A,aTP_Lamp_nv(nat,fun(A,$o),Na)))),Na) ) ) ).

% card_roots_unity
tff(fact_7307_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N3: set(nat),Na: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),N3),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),finite_card(nat,N3)),Na) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_7308_card__sum__le__nat__sum,axiom,
    ! [Sa: set(nat)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_je(nat,nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),finite_card(nat,Sa)))),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_je(nat,nat)),Sa)) ).

% card_sum_le_nat_sum
tff(fact_7309_card__nth__roots,axiom,
    ! [C2: complex,Na: nat] :
      ( ( C2 != zero_zero(complex) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => ( finite_card(complex,collect(complex,aa(nat,fun(complex,$o),aTP_Lamp_kq(complex,fun(nat,fun(complex,$o)),C2),Na))) = Na ) ) ) ).

% card_nth_roots
tff(fact_7310_card__roots__unity__eq,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( finite_card(complex,collect(complex,aTP_Lamp_hl(nat,fun(complex,$o),Na))) = Na ) ) ).

% card_roots_unity_eq
tff(fact_7311_card__2__iff,axiom,
    ! [A: $tType,Sa: set(A)] :
      ( ( finite_card(A,Sa) = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
    <=> ? [X6: A,Y4: A] :
          ( ( Sa = aa(set(A),set(A),insert(A,X6),aa(set(A),set(A),insert(A,Y4),bot_bot(set(A)))) )
          & ( X6 != Y4 ) ) ) ).

% card_2_iff
tff(fact_7312_card__3__iff,axiom,
    ! [A: $tType,Sa: set(A)] :
      ( ( finite_card(A,Sa) = aa(num,nat,numeral_numeral(nat),bit1(one2)) )
    <=> ? [X6: A,Y4: A,Z5: A] :
          ( ( Sa = aa(set(A),set(A),insert(A,X6),aa(set(A),set(A),insert(A,Y4),aa(set(A),set(A),insert(A,Z5),bot_bot(set(A))))) )
          & ( X6 != Y4 )
          & ( Y4 != Z5 )
          & ( X6 != Z5 ) ) ) ).

% card_3_iff
tff(fact_7313_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_7314_card__insert__disjoint_H,axiom,
    ! [A: $tType,A2: set(A),X: A] :
      ( aa(set(A),$o,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_7315_inj__on__iff__card__le,axiom,
    ! [A: $tType,B: $tType,A2: set(A),B2: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),A2)
     => ( aa(set(B),$o,finite_finite2(B),B2)
       => ( ? [F5: fun(A,B)] :
              ( inj_on(A,B,F5,A2)
              & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F5),A2)),B2) )
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),finite_card(A,A2)),finite_card(B,B2)) ) ) ) ).

% inj_on_iff_card_le
tff(fact_7316_card__inj__on__le,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A2: set(A),B2: set(B)] :
      ( inj_on(A,B,F2,A2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F2),A2)),B2)
       => ( aa(set(B),$o,finite_finite2(B),B2)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),finite_card(A,A2)),finite_card(B,B2)) ) ) ) ).

% card_inj_on_le
tff(fact_7317_card__le__inj,axiom,
    ! [B: $tType,A: $tType,A2: set(A),B2: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),A2)
     => ( aa(set(B),$o,finite_finite2(B),B2)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),finite_card(A,A2)),finite_card(B,B2))
         => ? [F3: fun(A,B)] :
              ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F3),A2)),B2)
              & inj_on(A,B,F3,A2) ) ) ) ) ).

% card_le_inj
tff(fact_7318_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_7319_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_7320_sum__norm__bound,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Sa: set(A),F2: fun(A,B),K3: real] :
          ( ! [X2: A] :
              ( member(A,X2,Sa)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,X2))),K3) )
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),Sa))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),finite_card(A,Sa))),K3)) ) ) ).

% sum_norm_bound
tff(fact_7321_Re__complex__div__ge__0,axiom,
    ! [Aa2: complex,Ba: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Aa2),Ba)))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Aa2),cnj(Ba)))) ) ).

% Re_complex_div_ge_0
tff(fact_7322_Re__complex__div__le__0,axiom,
    ! [Aa2: complex,Ba: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Aa2),Ba))),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Aa2),cnj(Ba)))),zero_zero(real)) ) ).

% Re_complex_div_le_0
tff(fact_7323_Im__complex__div__ge__0,axiom,
    ! [Aa2: complex,Ba: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Aa2),Ba)))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Aa2),cnj(Ba)))) ) ).

% Im_complex_div_ge_0
tff(fact_7324_Im__complex__div__le__0,axiom,
    ! [Aa2: complex,Ba: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Aa2),Ba))),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Aa2),cnj(Ba)))),zero_zero(real)) ) ).

% Im_complex_div_le_0
tff(fact_7325_prod__le__power,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(B)
     => ! [A2: set(A),F2: fun(A,B),Na: B,K: nat] :
          ( ! [I2: A] :
              ( member(A,I2,A2)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I2))
                & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),Na) ) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),finite_card(A,A2)),K)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),Na)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),groups7121269368397514597t_prod(A,B,F2,A2)),aa(nat,B,power_power(B,Na),K)) ) ) ) ) ).

% prod_le_power
tff(fact_7326_sum__bounded__above__strict,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere8940638589300402666id_add(B)
        & semiring_1(B) )
     => ! [A2: set(A),F2: fun(A,B),K3: B] :
          ( ! [I2: A] :
              ( member(A,I2,A2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,I2)),K3) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),finite_card(A,A2))
           => aa(B,$o,aa(B,fun(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))),K3)) ) ) ) ).

% sum_bounded_above_strict
tff(fact_7327_sum__bounded__above__divide,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_field(B)
     => ! [A2: set(A),F2: fun(A,B),K3: B] :
          ( ! [I2: A] :
              ( member(A,I2,A2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),aa(B,B,aa(B,fun(B,B),divide_divide(B),K3),aa(nat,B,semiring_1_of_nat(B),finite_card(A,A2)))) )
         => ( aa(set(A),$o,finite_finite2(A),A2)
           => ( ( A2 != bot_bot(set(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A2)),K3) ) ) ) ) ).

% sum_bounded_above_divide
tff(fact_7328_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_7329_card__insert__le__m1,axiom,
    ! [A: $tType,Na: nat,Y: set(A),X: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),finite_card(A,Y)),aa(nat,nat,minus_minus(nat,Na),one_one(nat)))
       => aa(nat,$o,aa(nat,fun(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_7330_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))),aa(word(A),nat,size_size(word(A)),X)) ) ).

% card_word_size
tff(fact_7331_sum__fun__comp,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [Sa: set(A),R3: set(B),G: fun(A,B),F2: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),Sa)
         => ( aa(set(B),$o,finite_finite2(B),R3)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,G),Sa)),R3)
             => ( aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ajm(fun(A,B),fun(fun(B,C),fun(A,C)),G),F2)),Sa) = 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_ajn(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),Sa),G),F2)),R3) ) ) ) ) ) ).

% sum_fun_comp
tff(fact_7332_prod__gen__delta,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Sa: set(A),Aa2: A,Ba: fun(A,B),C2: B] :
          ( aa(set(A),$o,finite_finite2(A),Sa)
         => ( groups7121269368397514597t_prod(A,B,aa(B,fun(A,B),aa(fun(A,B),fun(B,fun(A,B)),aTP_Lamp_ajo(A,fun(fun(A,B),fun(B,fun(A,B))),Aa2),Ba),C2),Sa) = $ite(member(A,Aa2,Sa),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Ba,Aa2)),aa(nat,B,power_power(B,C2),aa(nat,nat,minus_minus(nat,finite_card(A,Sa)),one_one(nat)))),aa(nat,B,power_power(B,C2),finite_card(A,Sa))) ) ) ) ).

% prod_gen_delta
tff(fact_7333_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,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),finite_card(A,collect(A,aa(nat,fun(A,$o),aTP_Lamp_ob(fun(nat,A),fun(nat,fun(A,$o)),C2),Na)))),Na) ) ) ) ).

% polyfun_roots_card
tff(fact_7334_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_7335_card__map__elide2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(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_7336_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_7337_complex__add__cnj,axiom,
    ! [Z: complex] : aa(complex,complex,aa(complex,fun(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_7338_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,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
           => ( aa(set(A),$o,finite_finite2(A),collect(A,aa(nat,fun(A,$o),aTP_Lamp_ob(fun(nat,A),fun(nat,fun(A,$o)),C2),Na)))
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),finite_card(A,collect(A,aa(nat,fun(A,$o),aTP_Lamp_ob(fun(nat,A),fun(nat,fun(A,$o)),C2),Na)))),Na) ) ) ) ) ).

% polyfun_rootbound
tff(fact_7339_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_7340_complex__div__cnj,axiom,
    ! [Aa2: complex,Ba: complex] : aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Aa2),Ba) = aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Aa2),cnj(Ba))),aa(real,complex,real_Vector_of_real(complex),aa(nat,real,power_power(real,real_V7770717601297561774m_norm(complex,Ba)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% complex_div_cnj
tff(fact_7341_cnj__add__mult__eq__Re,axiom,
    ! [Z: complex,W: complex] : aa(complex,complex,aa(complex,fun(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_7342_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_7343_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_7344_card__num1,axiom,
    finite_card(numeral_num1,top_top(set(numeral_num1))) = one_one(nat) ).

% card_num1
tff(fact_7345_card__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( ( finite_finite(B)
        & finite_finite(A) )
     => ( finite_card(sum_sum(A,B),top_top(set(sum_sum(A,B)))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),finite_card(A,top_top(set(A)))),finite_card(B,top_top(set(B)))) ) ) ).

% card_sum
tff(fact_7346_card__prod,axiom,
    ! [A: $tType,B: $tType] : finite_card(product_prod(A,B),top_top(set(product_prod(A,B)))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),finite_card(A,top_top(set(A)))),finite_card(B,top_top(set(B)))) ).

% card_prod
tff(fact_7347_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_7348_bit0_OCARD__eq,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( finite_card(numeral_bit0(A),top_top(set(numeral_bit0(A)))) = aa(int,nat,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_7349_bit1_OCARD__eq,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A)))) = aa(int,nat,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_7350_CARD__1,axiom,
    ! [A: $tType] :
      ( cARD_1(A)
     => ( finite_card(A,top_top(set(A))) = one_one(nat) ) ) ).

% CARD_1
tff(fact_7351_bit0_Osize0,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit0(A),top_top(set(numeral_bit0(A)))))) ) ).

% bit0.size0
tff(fact_7352_bit1_Osize0,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A)))))) ) ).

% bit1.size0
tff(fact_7353_bit0_Osize1,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit0(A),top_top(set(numeral_bit0(A)))))) ) ).

% bit0.size1
tff(fact_7354_bit1_Osize1,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A)))))) ) ).

% bit1.size1
tff(fact_7355_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,aa(nat,fun(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_7356_bit0__cases,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [X: numeral_bit0(A)] :
          ~ ! [Z3: int] :
              ( ( X = aa(int,numeral_bit0(A),ring_1_of_int(numeral_bit0(A)),Z3) )
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z3)
               => ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z3),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit0(A),top_top(set(numeral_bit0(A)))))) ) ) ) ).

% bit0_cases
tff(fact_7357_bit1__cases,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [X: numeral_bit1(A)] :
          ~ ! [Z3: int] :
              ( ( X = aa(int,numeral_bit1(A),ring_1_of_int(numeral_bit1(A)),Z3) )
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z3)
               => ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z3),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A)))))) ) ) ) ).

% bit1_cases
tff(fact_7358_bit0__induct,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [P: fun(numeral_bit0(A),$o),X: numeral_bit0(A)] :
          ( ! [Z3: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z3)
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z3),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit0(A),top_top(set(numeral_bit0(A))))))
               => aa(numeral_bit0(A),$o,P,aa(int,numeral_bit0(A),ring_1_of_int(numeral_bit0(A)),Z3)) ) )
         => aa(numeral_bit0(A),$o,P,X) ) ) ).

% bit0_induct
tff(fact_7359_bit1__induct,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [P: fun(numeral_bit1(A),$o),X: numeral_bit1(A)] :
          ( ! [Z3: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z3)
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z3),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A))))))
               => aa(numeral_bit1(A),$o,P,aa(int,numeral_bit1(A),ring_1_of_int(numeral_bit1(A)),Z3)) ) )
         => aa(numeral_bit1(A),$o,P,X) ) ) ).

% bit1_induct
tff(fact_7360_one__le__card__finite,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),finite_card(A,top_top(set(A)))) ) ).

% one_le_card_finite
tff(fact_7361_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,aa(nat,fun(nat,nat),plus_plus(nat),finite_card(A,top_top(set(A)))),one_one(nat))) ).

% card_UNIV_option
tff(fact_7362_one__less__int__card,axiom,
    ! [A: $tType] :
      ( card2(A)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),aa(nat,int,semiring_1_of_nat(int),finite_card(A,top_top(set(A))))) ) ).

% one_less_int_card
tff(fact_7363_two__le__card,axiom,
    ! [A: $tType] :
      ( card2(A)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),finite_card(A,top_top(set(A)))) ) ).

% two_le_card
tff(fact_7364_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_7365_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_7366_finite__UNIV__fun,axiom,
    ! [A: $tType,B: $tType] :
      ( aa(set(fun(A,B)),$o,finite_finite2(fun(A,B)),top_top(set(fun(A,B))))
    <=> ( ( aa(set(A),$o,finite_finite2(A),top_top(set(A)))
          & aa(set(B),$o,finite_finite2(B),top_top(set(B))) )
        | ( finite_card(B,top_top(set(B))) = one_one(nat) ) ) ) ).

% finite_UNIV_fun
tff(fact_7367_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,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,int,semiring_1_of_nat(int),finite_card(A,top_top(set(A)))))))) ) ).

% inj_on_Abs_bit1
tff(fact_7368_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,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(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,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(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_7369_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_7370_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_7371_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_7372_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_7373_Abs__bit1__cases,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [X: numeral_bit1(A)] :
          ~ ! [Y2: int] :
              ( ( X = aa(int,numeral_bit1(A),numeral_Abs_bit12(A),Y2) )
             => ~ member(int,Y2,set_or7035219750837199246ssThan(int,zero_zero(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,int,semiring_1_of_nat(int),finite_card(A,top_top(set(A)))))))) ) ) ).

% Abs_bit1_cases
tff(fact_7374_Abs__bit1__induct,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [P: fun(numeral_bit1(A),$o),X: numeral_bit1(A)] :
          ( ! [Y2: int] :
              ( member(int,Y2,set_or7035219750837199246ssThan(int,zero_zero(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,int,semiring_1_of_nat(int),finite_card(A,top_top(set(A))))))))
             => aa(numeral_bit1(A),$o,P,aa(int,numeral_bit1(A),numeral_Abs_bit12(A),Y2)) )
         => aa(numeral_bit1(A),$o,P,X) ) ) ).

% Abs_bit1_induct
tff(fact_7375_Abs__bit1__inverse,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Y: int] :
          ( member(int,Y,set_or7035219750837199246ssThan(int,zero_zero(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),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_7376_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_7377_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_7378_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_7379_plus__bit1__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)),plus_plus(numeral_bit1(A)),X),Y) = numeral_Abs_bit1(A,aa(int,int,aa(int,fun(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))) ) ).

% plus_bit1_def
tff(fact_7380_times__bit1__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) = numeral_Abs_bit1(A,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))) ) ).

% times_bit1_def
tff(fact_7381_less__eq__bit1__def,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Aa2: numeral_bit1(A),Ba: numeral_bit1(A)] :
          ( aa(numeral_bit1(A),$o,aa(numeral_bit1(A),fun(numeral_bit1(A),$o),ord_less_eq(numeral_bit1(A)),Aa2),Ba)
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(numeral_bit1(A),int,numeral_Rep_bit1(A),Aa2)),aa(numeral_bit1(A),int,numeral_Rep_bit1(A),Ba)) ) ) ).

% less_eq_bit1_def
tff(fact_7382_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_7383_bit1_ORep__le__n,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [X: numeral_bit1(A)] : aa(int,$o,aa(int,fun(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_7384_bit1_ORep__less__n,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [X: numeral_bit1(A)] : aa(int,$o,aa(int,fun(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_7385_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_7386_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_7387_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_7388_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_7389_bit1_Oadd__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)),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,aa(int,fun(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_7390_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_7391_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_7392_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_7393_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,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,int,semiring_1_of_nat(int),finite_card(A,top_top(set(A))))))))
         => ( ! [X2: numeral_bit1(A)] : aa(int,$o,P,aa(numeral_bit1(A),int,numeral_Rep_bit1(A),X2))
           => aa(int,$o,P,Y) ) ) ) ).

% Rep_bit1_induct
tff(fact_7394_Rep__bit1__cases,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Y: int] :
          ( member(int,Y,set_or7035219750837199246ssThan(int,zero_zero(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,int,semiring_1_of_nat(int),finite_card(A,top_top(set(A))))))))
         => ~ ! [X2: numeral_bit1(A)] : Y != aa(numeral_bit1(A),int,numeral_Rep_bit1(A),X2) ) ) ).

% Rep_bit1_cases
tff(fact_7395_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,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,int,semiring_1_of_nat(int),finite_card(A,top_top(set(A)))))))) ) ).

% Rep_bit1
tff(fact_7396_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,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,int,semiring_1_of_nat(int),finite_card(A,top_top(set(A)))))))) ) ).

% type_definition_bit1
tff(fact_7397_card__lists__distinct__length__eq,axiom,
    ! [A: $tType,A2: set(A),K: nat] :
      ( aa(set(A),$o,finite_finite2(A),A2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),finite_card(A,A2))
       => ( finite_card(list(A),collect(list(A),aa(nat,fun(list(A),$o),aTP_Lamp_ajp(set(A),fun(nat,fun(list(A),$o)),A2),K))) = groups7121269368397514597t_prod(nat,nat,aTP_Lamp_je(nat,nat),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(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_7398_distinct1__rotate,axiom,
    ! [A: $tType,Xs2: list(A)] :
      ( distinct(A,rotate1(A,Xs2))
    <=> distinct(A,Xs2) ) ).

% distinct1_rotate
tff(fact_7399_distinct__union,axiom,
    ! [A: $tType,Xs2: list(A),Ysb: list(A)] :
      ( distinct(A,union(A,Xs2,Ysb))
    <=> distinct(A,Ysb) ) ).

% distinct_union
tff(fact_7400_distinct__swap,axiom,
    ! [A: $tType,I: nat,Xs2: list(A),J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs2))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs2))
       => ( distinct(A,list_update(A,list_update(A,Xs2,I,aa(nat,A,nth(A,Xs2),J)),J,aa(nat,A,nth(A,Xs2),I)))
        <=> distinct(A,Xs2) ) ) ) ).

% distinct_swap
tff(fact_7401_finite__lists__distinct__length__eq,axiom,
    ! [A: $tType,A2: set(A),Na: nat] :
      ( aa(set(A),$o,finite_finite2(A),A2)
     => aa(set(list(A)),$o,finite_finite2(list(A)),collect(list(A),aa(nat,fun(list(A),$o),aTP_Lamp_ajp(set(A),fun(nat,fun(list(A),$o)),A2),Na))) ) ).

% finite_lists_distinct_length_eq
tff(fact_7402_distinct__finite__set,axiom,
    ! [A: $tType,X: set(A)] : aa(set(list(A)),$o,finite_finite2(list(A)),collect(list(A),aTP_Lamp_ajq(set(A),fun(list(A),$o),X))) ).

% distinct_finite_set
tff(fact_7403_finite__distinct__list,axiom,
    ! [A: $tType,A2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A2)
     => ? [Xs: list(A)] :
          ( ( aa(list(A),set(A),set2(A),Xs) = A2 )
          & distinct(A,Xs) ) ) ).

% finite_distinct_list
tff(fact_7404_distinct__conv__nth,axiom,
    ! [A: $tType,Xs2: list(A)] :
      ( distinct(A,Xs2)
    <=> ! [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs2))
         => ! [J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs2))
             => ( ( I4 != J3 )
               => ( aa(nat,A,nth(A,Xs2),I4) != aa(nat,A,nth(A,Xs2),J3) ) ) ) ) ) ).

% distinct_conv_nth
tff(fact_7405_nth__eq__iff__index__eq,axiom,
    ! [A: $tType,Xs2: list(A),I: nat,J: nat] :
      ( distinct(A,Xs2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs2))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs2))
         => ( ( aa(nat,A,nth(A,Xs2),I) = aa(nat,A,nth(A,Xs2),J) )
          <=> ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
tff(fact_7406_distinct__length__le,axiom,
    ! [A: $tType,Ysb: list(A),Xs2: list(A)] :
      ( distinct(A,Ysb)
     => ( ( aa(list(A),set(A),set2(A),Ysb) = aa(list(A),set(A),set2(A),Xs2) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ysb)),aa(list(A),nat,size_size(list(A)),Xs2)) ) ) ).

% distinct_length_le
tff(fact_7407_subseqs__distinctD,axiom,
    ! [A: $tType,Ysb: list(A),Xs2: list(A)] :
      ( member(list(A),Ysb,aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xs2)))
     => ( distinct(A,Xs2)
       => distinct(A,Ysb) ) ) ).

% subseqs_distinctD
tff(fact_7408_distinct__singleton,axiom,
    ! [A: $tType,X: A] : distinct(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) ).

% distinct_singleton
tff(fact_7409_distinct__length__2__or__more,axiom,
    ! [A: $tType,Aa2: A,Ba: A,Xs2: list(A)] :
      ( distinct(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Aa2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Ba),Xs2)))
    <=> ( ( Aa2 != Ba )
        & distinct(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Aa2),Xs2))
        & distinct(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Ba),Xs2)) ) ) ).

% distinct_length_2_or_more
tff(fact_7410_distinct_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Xs2: list(A)] :
      ( distinct(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs2))
    <=> ( ~ member(A,X,aa(list(A),set(A),set2(A),Xs2))
        & distinct(A,Xs2) ) ) ).

% distinct.simps(2)
tff(fact_7411_distinct__product,axiom,
    ! [A: $tType,B: $tType,Xs2: list(A),Ysb: list(B)] :
      ( distinct(A,Xs2)
     => ( distinct(B,Ysb)
       => distinct(product_prod(A,B),product(A,B,Xs2,Ysb)) ) ) ).

% distinct_product
tff(fact_7412_distinct__product__lists,axiom,
    ! [A: $tType,Xss2: list(list(A))] :
      ( ! [X2: list(A)] :
          ( member(list(A),X2,aa(list(list(A)),set(list(A)),set2(list(A)),Xss2))
         => distinct(A,X2) )
     => distinct(list(A),product_lists(A,Xss2)) ) ).

% distinct_product_lists
tff(fact_7413_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs2: list(A)] :
          ( distinct(A,Xs2)
         => distinct(A,Xs2) ) ) ).

% sorted_list_of_set.distinct_if_distinct_map
tff(fact_7414_distinct__upto,axiom,
    ! [I: int,J: int] : distinct(int,upto(I,J)) ).

% distinct_upto
tff(fact_7415_distinct__set__subseqs,axiom,
    ! [A: $tType,Xs2: list(A)] :
      ( distinct(A,Xs2)
     => distinct(set(A),aa(list(list(A)),list(set(A)),map(list(A),set(A),set2(A)),subseqs(A,Xs2))) ) ).

% distinct_set_subseqs
tff(fact_7416_distinct__map__eq,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),L: list(B),X: B,Y: B] :
      ( distinct(A,aa(list(B),list(A),map(B,A,F2),L))
     => ( ( aa(B,A,F2,X) = aa(B,A,F2,Y) )
       => ( member(B,X,aa(list(B),set(B),set2(B),L))
         => ( member(B,Y,aa(list(B),set(B),set2(B),L))
           => ( X = Y ) ) ) ) ) ).

% distinct_map_eq
tff(fact_7417_distinct__mapI,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),L: list(B)] :
      ( distinct(A,aa(list(B),list(A),map(B,A,F2),L))
     => distinct(B,L) ) ).

% distinct_mapI
tff(fact_7418_distinct_Osimps_I1_J,axiom,
    ! [A: $tType] : distinct(A,nil(A)) ).

% distinct.simps(1)
tff(fact_7419_distinct__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs2: list(B)] :
      ( distinct(A,aa(list(B),list(A),map(B,A,F2),Xs2))
    <=> ( distinct(B,Xs2)
        & inj_on(B,A,F2,aa(list(B),set(B),set2(B),Xs2)) ) ) ).

% distinct_map
tff(fact_7420_card__distinct,axiom,
    ! [A: $tType,Xs2: list(A)] :
      ( ( finite_card(A,aa(list(A),set(A),set2(A),Xs2)) = aa(list(A),nat,size_size(list(A)),Xs2) )
     => distinct(A,Xs2) ) ).

% card_distinct
tff(fact_7421_distinct__card,axiom,
    ! [A: $tType,Xs2: list(A)] :
      ( distinct(A,Xs2)
     => ( finite_card(A,aa(list(A),set(A),set2(A),Xs2)) = aa(list(A),nat,size_size(list(A)),Xs2) ) ) ).

% distinct_card
tff(fact_7422_distinct__Ex1,axiom,
    ! [A: $tType,Xs2: list(A),X: A] :
      ( distinct(A,Xs2)
     => ( member(A,X,aa(list(A),set(A),set2(A),Xs2))
       => ? [X2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X2),aa(list(A),nat,size_size(list(A)),Xs2))
            & ( aa(nat,A,nth(A,Xs2),X2) = X )
            & ! [Y3: nat] :
                ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y3),aa(list(A),nat,size_size(list(A)),Xs2))
                  & ( aa(nat,A,nth(A,Xs2),Y3) = X ) )
               => ( Y3 = X2 ) ) ) ) ) ).

% distinct_Ex1
tff(fact_7423_distinct__idx,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),L: list(B),I: nat,J: nat] :
      ( distinct(A,aa(list(B),list(A),map(B,A,F2),L))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),L))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(B),nat,size_size(list(B)),L))
         => ( ( aa(B,A,F2,aa(nat,B,nth(B,L),I)) = aa(B,A,F2,aa(nat,B,nth(B,L),J)) )
           => ( I = J ) ) ) ) ) ).

% distinct_idx
tff(fact_7424_inj__on__nth,axiom,
    ! [A: $tType,Xs2: list(A),I5: set(nat)] :
      ( distinct(A,Xs2)
     => ( ! [X2: nat] :
            ( member(nat,X2,I5)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X2),aa(list(A),nat,size_size(list(A)),Xs2)) )
       => inj_on(nat,A,nth(A,Xs2),I5) ) ) ).

% inj_on_nth
tff(fact_7425_distinct__finite__subset,axiom,
    ! [A: $tType,X: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),X)
     => aa(set(list(A)),$o,finite_finite2(list(A)),collect(list(A),aTP_Lamp_ajr(set(A),fun(list(A),$o),X))) ) ).

% distinct_finite_subset
tff(fact_7426_finite__set__image,axiom,
    ! [A: $tType,A2: set(list(A))] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),aa(set(list(A)),set(set(A)),image(list(A),set(A),set2(A)),A2))
     => ( ! [Xs: list(A)] :
            ( member(list(A),Xs,A2)
           => distinct(A,Xs) )
       => aa(set(list(A)),$o,finite_finite2(list(A)),A2) ) ) ).

% finite_set_image
tff(fact_7427_bij__betw__nth,axiom,
    ! [A: $tType,Xs2: list(A),A2: set(nat),B2: set(A)] :
      ( distinct(A,Xs2)
     => ( ( A2 = set_ord_lessThan(nat,aa(list(A),nat,size_size(list(A)),Xs2)) )
       => ( ( B2 = aa(list(A),set(A),set2(A),Xs2) )
         => bij_betw(nat,A,nth(A,Xs2),A2,B2) ) ) ) ).

% bij_betw_nth
tff(fact_7428_distinct__list__update,axiom,
    ! [A: $tType,Xs2: list(A),Aa2: A,I: nat] :
      ( distinct(A,Xs2)
     => ( ~ member(A,Aa2,aa(set(A),set(A),minus_minus(set(A),aa(list(A),set(A),set2(A),Xs2)),aa(set(A),set(A),insert(A,aa(nat,A,nth(A,Xs2),I)),bot_bot(set(A)))))
       => distinct(A,list_update(A,Xs2,I,Aa2)) ) ) ).

% distinct_list_update
tff(fact_7429_set__update__distinct,axiom,
    ! [A: $tType,Xs2: list(A),Na: nat,X: A] :
      ( distinct(A,Xs2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs2))
       => ( aa(list(A),set(A),set2(A),list_update(A,Xs2,Na,X)) = aa(set(A),set(A),insert(A,X),aa(set(A),set(A),minus_minus(set(A),aa(list(A),set(A),set2(A),Xs2)),aa(set(A),set(A),insert(A,aa(nat,A,nth(A,Xs2),Na)),bot_bot(set(A))))) ) ) ) ).

% set_update_distinct
tff(fact_7430_bit1_Otype,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(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A))))))) ) ).

% bit1.type
tff(fact_7431_card__lists__distinct__length__eq_H,axiom,
    ! [A: $tType,K: nat,A2: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),finite_card(A,A2))
     => ( finite_card(list(A),collect(list(A),aa(set(A),fun(list(A),$o),aTP_Lamp_ajs(nat,fun(set(A),fun(list(A),$o)),K),A2))) = groups7121269368397514597t_prod(nat,nat,aTP_Lamp_je(nat,nat),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(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_7432_mod__type_Ointro,axiom,
    ! [A: $tType] :
      ( ( minus(A)
        & one(A)
        & plus(A)
        & times(A)
        & uminus(A)
        & zero(A) )
     => ! [Rep: fun(A,int),Abs: fun(int,A),Na: int] :
          ( type_definition(A,int,Rep,Abs,set_or7035219750837199246ssThan(int,zero_zero(int),Na))
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),Na)
           => ( ( zero_zero(A) = aa(int,A,Abs,zero_zero(int)) )
             => ( ( one_one(A) = aa(int,A,Abs,one_one(int)) )
               => ( ! [X2: A,Y2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),Y2) = aa(int,A,Abs,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(A,int,Rep,X2)),aa(A,int,Rep,Y2))),Na))
                 => ( ! [X2: A,Y2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),X2),Y2) = aa(int,A,Abs,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(A,int,Rep,X2)),aa(A,int,Rep,Y2))),Na))
                   => ( ! [X2: A,Y2: A] : aa(A,A,minus_minus(A,X2),Y2) = aa(int,A,Abs,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,minus_minus(int,aa(A,int,Rep,X2)),aa(A,int,Rep,Y2))),Na))
                     => ( ! [X2: A] : aa(A,A,uminus_uminus(A),X2) = aa(int,A,Abs,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),aa(A,int,Rep,X2))),Na))
                       => numeral_mod_type(A,Na,Rep,Abs) ) ) ) ) ) ) ) ) ) ).

% mod_type.intro
tff(fact_7433_mod__type__def,axiom,
    ! [A: $tType] :
      ( ( minus(A)
        & one(A)
        & plus(A)
        & times(A)
        & uminus(A)
        & zero(A) )
     => ! [Na: int,Rep: fun(A,int),Abs: fun(int,A)] :
          ( numeral_mod_type(A,Na,Rep,Abs)
        <=> ( type_definition(A,int,Rep,Abs,set_or7035219750837199246ssThan(int,zero_zero(int),Na))
            & aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),Na)
            & ( zero_zero(A) = aa(int,A,Abs,zero_zero(int)) )
            & ( one_one(A) = aa(int,A,Abs,one_one(int)) )
            & ! [X6: A,Y4: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),X6),Y4) = aa(int,A,Abs,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(A,int,Rep,X6)),aa(A,int,Rep,Y4))),Na))
            & ! [X6: A,Y4: A] : aa(A,A,aa(A,fun(A,A),times_times(A),X6),Y4) = aa(int,A,Abs,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(A,int,Rep,X6)),aa(A,int,Rep,Y4))),Na))
            & ! [X6: A,Y4: A] : aa(A,A,minus_minus(A,X6),Y4) = aa(int,A,Abs,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,minus_minus(int,aa(A,int,Rep,X6)),aa(A,int,Rep,Y4))),Na))
            & ! [X6: A] : aa(A,A,uminus_uminus(A),X6) = aa(int,A,Abs,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),aa(A,int,Rep,X6))),Na)) ) ) ) ).

% mod_type_def
tff(fact_7434_mod__type_ORep__le__n,axiom,
    ! [A: $tType] :
      ( ( minus(A)
        & one(A)
        & plus(A)
        & times(A)
        & uminus(A)
        & zero(A) )
     => ! [Na: int,Rep: fun(A,int),Abs: fun(int,A),X: A] :
          ( numeral_mod_type(A,Na,Rep,Abs)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(A,int,Rep,X)),Na) ) ) ).

% mod_type.Rep_le_n
tff(fact_7435_mod__type_Osize1,axiom,
    ! [A: $tType] :
      ( ( minus(A)
        & one(A)
        & plus(A)
        & times(A)
        & uminus(A)
        & zero(A) )
     => ! [Na: int,Rep: fun(A,int),Abs: fun(int,A)] :
          ( numeral_mod_type(A,Na,Rep,Abs)
         => aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),Na) ) ) ).

% mod_type.size1
tff(fact_7436_mod__type_Oone__def,axiom,
    ! [A: $tType] :
      ( ( minus(A)
        & one(A)
        & plus(A)
        & times(A)
        & uminus(A)
        & zero(A) )
     => ! [Na: int,Rep: fun(A,int),Abs: fun(int,A)] :
          ( numeral_mod_type(A,Na,Rep,Abs)
         => ( one_one(A) = aa(int,A,Abs,one_one(int)) ) ) ) ).

% mod_type.one_def
tff(fact_7437_mod__type_ORep__1,axiom,
    ! [A: $tType] :
      ( ( minus(A)
        & one(A)
        & plus(A)
        & times(A)
        & uminus(A)
        & zero(A) )
     => ! [Na: int,Rep: fun(A,int),Abs: fun(int,A)] :
          ( numeral_mod_type(A,Na,Rep,Abs)
         => ( aa(A,int,Rep,one_one(A)) = one_one(int) ) ) ) ).

% mod_type.Rep_1
tff(fact_7438_mod__type_ORep__Abs__1,axiom,
    ! [A: $tType] :
      ( ( minus(A)
        & one(A)
        & plus(A)
        & times(A)
        & uminus(A)
        & zero(A) )
     => ! [Na: int,Rep: fun(A,int),Abs: fun(int,A)] :
          ( numeral_mod_type(A,Na,Rep,Abs)
         => ( aa(A,int,Rep,aa(int,A,Abs,one_one(int))) = one_one(int) ) ) ) ).

% mod_type.Rep_Abs_1
tff(fact_7439_mod__type_Oadd__def,axiom,
    ! [A: $tType] :
      ( ( minus(A)
        & one(A)
        & plus(A)
        & times(A)
        & uminus(A)
        & zero(A) )
     => ! [Na: int,Rep: fun(A,int),Abs: fun(int,A),X: A,Y: A] :
          ( numeral_mod_type(A,Na,Rep,Abs)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y) = aa(int,A,Abs,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(A,int,Rep,X)),aa(A,int,Rep,Y))),Na)) ) ) ) ).

% mod_type.add_def
tff(fact_7440_mod__type_Omult__def,axiom,
    ! [A: $tType] :
      ( ( minus(A)
        & one(A)
        & plus(A)
        & times(A)
        & uminus(A)
        & zero(A) )
     => ! [Na: int,Rep: fun(A,int),Abs: fun(int,A),X: A,Y: A] :
          ( numeral_mod_type(A,Na,Rep,Abs)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),X),Y) = aa(int,A,Abs,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(A,int,Rep,X)),aa(A,int,Rep,Y))),Na)) ) ) ) ).

% mod_type.mult_def
tff(fact_7441_bit1_Omod__type__axioms,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => numeral_mod_type(numeral_bit1(A),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A))))),numeral_Rep_bit1(A),numeral_Abs_bit12(A)) ) ).

% bit1.mod_type_axioms
tff(fact_7442_bit1__iszero__numeral,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [W: num] :
          ( ring_1_iszero(numeral_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)))))) = zero_zero(int) ) ) ) ).

% bit1_iszero_numeral
tff(fact_7443_inj__on__Abs__bit0,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => inj_on(int,numeral_bit0(A),numeral_Abs_bit02(A),set_or7035219750837199246ssThan(int,zero_zero(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_bit0
tff(fact_7444_iszero__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [W: num] :
          ( ring_1_iszero(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),W)) ) ) ).

% iszero_neg_numeral
tff(fact_7445_not__iszero__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ~ ring_1_iszero(A,one_one(A)) ) ).

% not_iszero_1
tff(fact_7446_one__bit0__def,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( one_one(numeral_bit0(A)) = aa(int,numeral_bit0(A),numeral_Abs_bit02(A),one_one(int)) ) ) ).

% one_bit0_def
tff(fact_7447_iszero__0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ring_1_iszero(A,zero_zero(A)) ) ).

% iszero_0
tff(fact_7448_iszero__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: A] :
          ( ring_1_iszero(A,Z)
        <=> ( Z = zero_zero(A) ) ) ) ).

% iszero_def
tff(fact_7449_eq__numeral__iff__iszero_I10_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Y: num] :
          ( ( zero_zero(A) = aa(num,A,numeral_numeral(A),Y) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),Y)) ) ) ).

% eq_numeral_iff_iszero(10)
tff(fact_7450_eq__numeral__iff__iszero_I9_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: num] :
          ( ( aa(num,A,numeral_numeral(A),X) = zero_zero(A) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),X)) ) ) ).

% eq_numeral_iff_iszero(9)
tff(fact_7451_not__iszero__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [W: num] : ~ ring_1_iszero(A,aa(num,A,numeral_numeral(A),W)) ) ).

% not_iszero_numeral
tff(fact_7452_not__iszero__Numeral1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ~ ring_1_iszero(A,aa(num,A,numeral_numeral(A),one2)) ) ).

% not_iszero_Numeral1
tff(fact_7453_eq__iff__iszero__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: A,Y: A] :
          ( ( X = Y )
        <=> ring_1_iszero(A,aa(A,A,minus_minus(A,X),Y)) ) ) ).

% eq_iff_iszero_diff
tff(fact_7454_not__iszero__neg__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ~ ring_1_iszero(A,aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% not_iszero_neg_1
tff(fact_7455_eq__numeral__iff__iszero_I11_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X)) = zero_zero(A) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),X)) ) ) ).

% eq_numeral_iff_iszero(11)
tff(fact_7456_eq__numeral__iff__iszero_I12_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Y: num] :
          ( ( zero_zero(A) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Y)) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),Y)) ) ) ).

% eq_numeral_iff_iszero(12)
tff(fact_7457_not__iszero__neg__Numeral1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ~ ring_1_iszero(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),one2))) ) ).

% not_iszero_neg_Numeral1
tff(fact_7458_eq__numeral__iff__iszero_I3_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: num,Y: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X)) = aa(num,A,numeral_numeral(A),Y) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),X),Y))) ) ) ).

% eq_numeral_iff_iszero(3)
tff(fact_7459_eq__numeral__iff__iszero_I2_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: num,Y: num] :
          ( ( aa(num,A,numeral_numeral(A),X) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Y)) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),X),Y))) ) ) ).

% eq_numeral_iff_iszero(2)
tff(fact_7460_bit0_Oof__nat__eq,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [K: nat] : aa(nat,numeral_bit0(A),semiring_1_of_nat(numeral_bit0(A)),K) = aa(int,numeral_bit0(A),numeral_Abs_bit02(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_bit0(A),top_top(set(numeral_bit0(A))))))) ) ).

% bit0.of_nat_eq
tff(fact_7461_eq__numeral__iff__iszero_I8_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Y: num] :
          ( ( one_one(A) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Y)) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),Y))) ) ) ).

% eq_numeral_iff_iszero(8)
tff(fact_7462_eq__numeral__iff__iszero_I7_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X)) = one_one(A) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),X),one2))) ) ) ).

% eq_numeral_iff_iszero(7)
tff(fact_7463_bit0_OUNIV__eq,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( top_top(set(numeral_bit0(A))) = aa(set(int),set(numeral_bit0(A)),image(int,numeral_bit0(A),numeral_Abs_bit02(A)),set_or7035219750837199246ssThan(int,zero_zero(int),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit0(A),top_top(set(numeral_bit0(A))))))) ) ) ).

% bit0.UNIV_eq
tff(fact_7464_bit0_Oof__int__eq,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Z: int] : aa(int,numeral_bit0(A),ring_1_of_int(numeral_bit0(A)),Z) = aa(int,numeral_bit0(A),numeral_Abs_bit02(A),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Z),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit0(A),top_top(set(numeral_bit0(A))))))) ) ).

% bit0.of_int_eq
tff(fact_7465_Abs__bit0__inject,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [X: int,Y: int] :
          ( member(int,X,set_or7035219750837199246ssThan(int,zero_zero(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,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_bit0(A),numeral_Abs_bit02(A),X) = aa(int,numeral_bit0(A),numeral_Abs_bit02(A),Y) )
            <=> ( X = Y ) ) ) ) ) ).

% Abs_bit0_inject
tff(fact_7466_Abs__bit0__induct,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [P: fun(numeral_bit0(A),$o),X: numeral_bit0(A)] :
          ( ! [Y2: int] :
              ( member(int,Y2,set_or7035219750837199246ssThan(int,zero_zero(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_bit0(A),$o,P,aa(int,numeral_bit0(A),numeral_Abs_bit02(A),Y2)) )
         => aa(numeral_bit0(A),$o,P,X) ) ) ).

% Abs_bit0_induct
tff(fact_7467_Abs__bit0__cases,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [X: numeral_bit0(A)] :
          ~ ! [Y2: int] :
              ( ( X = aa(int,numeral_bit0(A),numeral_Abs_bit02(A),Y2) )
             => ~ member(int,Y2,set_or7035219750837199246ssThan(int,zero_zero(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_bit0_cases
tff(fact_7468_Abs__bit0_H__def,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [X: int] : numeral_Abs_bit0(A,X) = aa(int,numeral_bit0(A),numeral_Abs_bit02(A),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),X),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit0(A),top_top(set(numeral_bit0(A))))))) ) ).

% Abs_bit0'_def
tff(fact_7469_Abs__bit0__inverse,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Y: int] :
          ( member(int,Y,set_or7035219750837199246ssThan(int,zero_zero(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_bit0(A),int,numeral_Rep_bit0(A),aa(int,numeral_bit0(A),numeral_Abs_bit02(A),Y)) = Y ) ) ) ).

% Abs_bit0_inverse
tff(fact_7470_word__eq__numeral__iff__iszero_I9_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: num] :
          ( ( aa(num,word(A),numeral_numeral(word(A)),X) = zero_zero(word(A)) )
        <=> ring_1_iszero(word(A),aa(num,word(A),numeral_numeral(word(A)),X)) ) ) ).

% word_eq_numeral_iff_iszero(9)
tff(fact_7471_word__eq__numeral__iff__iszero_I10_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: num] :
          ( ( zero_zero(word(A)) = aa(num,word(A),numeral_numeral(word(A)),Y) )
        <=> ring_1_iszero(word(A),aa(num,word(A),numeral_numeral(word(A)),Y)) ) ) ).

% word_eq_numeral_iff_iszero(10)
tff(fact_7472_word__eq__numeral__iff__iszero_I11_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: num] :
          ( ( aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),X)) = zero_zero(word(A)) )
        <=> ring_1_iszero(word(A),aa(num,word(A),numeral_numeral(word(A)),X)) ) ) ).

% word_eq_numeral_iff_iszero(11)
tff(fact_7473_word__eq__numeral__iff__iszero_I12_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: num] :
          ( ( zero_zero(word(A)) = aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Y)) )
        <=> ring_1_iszero(word(A),aa(num,word(A),numeral_numeral(word(A)),Y)) ) ) ).

% word_eq_numeral_iff_iszero(12)
tff(fact_7474_word__eq__numeral__iff__iszero_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: num,Y: num] :
          ( ( aa(num,word(A),numeral_numeral(word(A)),X) = aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Y)) )
        <=> ring_1_iszero(word(A),aa(num,word(A),numeral_numeral(word(A)),aa(num,num,aa(num,fun(num,num),plus_plus(num),X),Y))) ) ) ).

% word_eq_numeral_iff_iszero(2)
tff(fact_7475_word__eq__numeral__iff__iszero_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: num,Y: num] :
          ( ( aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),X)) = aa(num,word(A),numeral_numeral(word(A)),Y) )
        <=> ring_1_iszero(word(A),aa(num,word(A),numeral_numeral(word(A)),aa(num,num,aa(num,fun(num,num),plus_plus(num),X),Y))) ) ) ).

% word_eq_numeral_iff_iszero(3)
tff(fact_7476_iszero__word__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: num] :
          ( ring_1_iszero(word(A),aa(num,word(A),numeral_numeral(word(A)),Bin))
        <=> ring_1_iszero(int,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),Bin))) ) ) ).

% iszero_word_no
tff(fact_7477_word__eq__numeral__iff__iszero_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: num] :
          ( ( one_one(word(A)) = aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Y)) )
        <=> ring_1_iszero(word(A),aa(num,word(A),numeral_numeral(word(A)),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),Y))) ) ) ).

% word_eq_numeral_iff_iszero(8)
tff(fact_7478_word__eq__numeral__iff__iszero_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: num] :
          ( ( aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),X)) = one_one(word(A)) )
        <=> ring_1_iszero(word(A),aa(num,word(A),numeral_numeral(word(A)),aa(num,num,aa(num,fun(num,num),plus_plus(num),X),one2))) ) ) ).

% word_eq_numeral_iff_iszero(7)
tff(fact_7479_bit0__iszero__numeral,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [W: num] :
          ( ring_1_iszero(numeral_bit0(A),aa(num,numeral_bit0(A),numeral_numeral(numeral_bit0(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_bit0(A),top_top(set(numeral_bit0(A)))))) = zero_zero(int) ) ) ) ).

% bit0_iszero_numeral
tff(fact_7480_bit0_ORep__Abs__1,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( aa(numeral_bit0(A),int,numeral_Rep_bit0(A),aa(int,numeral_bit0(A),numeral_Abs_bit02(A),one_one(int))) = one_one(int) ) ) ).

% bit0.Rep_Abs_1
tff(fact_7481_less__eq__bit0__def,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Aa2: numeral_bit0(A),Ba: numeral_bit0(A)] :
          ( aa(numeral_bit0(A),$o,aa(numeral_bit0(A),fun(numeral_bit0(A),$o),ord_less_eq(numeral_bit0(A)),Aa2),Ba)
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(numeral_bit0(A),int,numeral_Rep_bit0(A),Aa2)),aa(numeral_bit0(A),int,numeral_Rep_bit0(A),Ba)) ) ) ).

% less_eq_bit0_def
tff(fact_7482_times__bit0__def,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [X: numeral_bit0(A),Y: numeral_bit0(A)] : aa(numeral_bit0(A),numeral_bit0(A),aa(numeral_bit0(A),fun(numeral_bit0(A),numeral_bit0(A)),times_times(numeral_bit0(A)),X),Y) = numeral_Abs_bit0(A,aa(int,int,aa(int,fun(int,int),times_times(int),aa(numeral_bit0(A),int,numeral_Rep_bit0(A),X)),aa(numeral_bit0(A),int,numeral_Rep_bit0(A),Y))) ) ).

% times_bit0_def
tff(fact_7483_plus__bit0__def,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [X: numeral_bit0(A),Y: numeral_bit0(A)] : aa(numeral_bit0(A),numeral_bit0(A),aa(numeral_bit0(A),fun(numeral_bit0(A),numeral_bit0(A)),plus_plus(numeral_bit0(A)),X),Y) = numeral_Abs_bit0(A,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(numeral_bit0(A),int,numeral_Rep_bit0(A),X)),aa(numeral_bit0(A),int,numeral_Rep_bit0(A),Y))) ) ).

% plus_bit0_def
tff(fact_7484_bit0_ORep__1,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( aa(numeral_bit0(A),int,numeral_Rep_bit0(A),one_one(numeral_bit0(A))) = one_one(int) ) ) ).

% bit0.Rep_1
tff(fact_7485_Abs__bit0_H__code,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [X: int] : aa(numeral_bit0(A),int,numeral_Rep_bit0(A),numeral_Abs_bit0(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_bit0(A),top_top(set(numeral_bit0(A)))))) ) ).

% Abs_bit0'_code
tff(fact_7486_bit0_ORep__le__n,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [X: numeral_bit0(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(numeral_bit0(A),int,numeral_Rep_bit0(A),X)),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit0(A),top_top(set(numeral_bit0(A)))))) ) ).

% bit0.Rep_le_n
tff(fact_7487_bit0_ORep__less__n,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [X: numeral_bit0(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(numeral_bit0(A),int,numeral_Rep_bit0(A),X)),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit0(A),top_top(set(numeral_bit0(A)))))) ) ).

% bit0.Rep_less_n
tff(fact_7488_bit0_ORep__mod,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [X: numeral_bit0(A)] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(numeral_bit0(A),int,numeral_Rep_bit0(A),X)),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit0(A),top_top(set(numeral_bit0(A)))))) = aa(numeral_bit0(A),int,numeral_Rep_bit0(A),X) ) ).

% bit0.Rep_mod
tff(fact_7489_bit0_Omod__type__axioms,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => numeral_mod_type(numeral_bit0(A),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit0(A),top_top(set(numeral_bit0(A))))),numeral_Rep_bit0(A),numeral_Abs_bit02(A)) ) ).

% bit0.mod_type_axioms
tff(fact_7490_bit0_ORep__numeral,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [W: num] : aa(numeral_bit0(A),int,numeral_Rep_bit0(A),aa(num,numeral_bit0(A),numeral_numeral(numeral_bit0(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_bit0(A),top_top(set(numeral_bit0(A)))))) ) ).

% bit0.Rep_numeral
tff(fact_7491_bit0_ORep__Abs__mod,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [M: int] : aa(numeral_bit0(A),int,numeral_Rep_bit0(A),aa(int,numeral_bit0(A),numeral_Abs_bit02(A),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),M),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit0(A),top_top(set(numeral_bit0(A)))))))) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),M),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit0(A),top_top(set(numeral_bit0(A)))))) ) ).

% bit0.Rep_Abs_mod
tff(fact_7492_bit0_Ominus__def,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [X: numeral_bit0(A)] : aa(numeral_bit0(A),numeral_bit0(A),uminus_uminus(numeral_bit0(A)),X) = aa(int,numeral_bit0(A),numeral_Abs_bit02(A),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),aa(numeral_bit0(A),int,numeral_Rep_bit0(A),X))),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit0(A),top_top(set(numeral_bit0(A))))))) ) ).

% bit0.minus_def
tff(fact_7493_bit0_Oadd__def,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [X: numeral_bit0(A),Y: numeral_bit0(A)] : aa(numeral_bit0(A),numeral_bit0(A),aa(numeral_bit0(A),fun(numeral_bit0(A),numeral_bit0(A)),plus_plus(numeral_bit0(A)),X),Y) = aa(int,numeral_bit0(A),numeral_Abs_bit02(A),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(numeral_bit0(A),int,numeral_Rep_bit0(A),X)),aa(numeral_bit0(A),int,numeral_Rep_bit0(A),Y))),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit0(A),top_top(set(numeral_bit0(A))))))) ) ).

% bit0.add_def
tff(fact_7494_bit0_Omult__def,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [X: numeral_bit0(A),Y: numeral_bit0(A)] : aa(numeral_bit0(A),numeral_bit0(A),aa(numeral_bit0(A),fun(numeral_bit0(A),numeral_bit0(A)),times_times(numeral_bit0(A)),X),Y) = aa(int,numeral_bit0(A),numeral_Abs_bit02(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_bit0(A),int,numeral_Rep_bit0(A),X)),aa(numeral_bit0(A),int,numeral_Rep_bit0(A),Y))),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit0(A),top_top(set(numeral_bit0(A))))))) ) ).

% bit0.mult_def
tff(fact_7495_bit0_Odiff__def,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [X: numeral_bit0(A),Y: numeral_bit0(A)] : aa(numeral_bit0(A),numeral_bit0(A),minus_minus(numeral_bit0(A),X),Y) = aa(int,numeral_bit0(A),numeral_Abs_bit02(A),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,minus_minus(int,aa(numeral_bit0(A),int,numeral_Rep_bit0(A),X)),aa(numeral_bit0(A),int,numeral_Rep_bit0(A),Y))),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit0(A),top_top(set(numeral_bit0(A))))))) ) ).

% bit0.diff_def
tff(fact_7496_bit0_Otype,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => type_definition(numeral_bit0(A),int,numeral_Rep_bit0(A),numeral_Abs_bit02(A),set_or7035219750837199246ssThan(int,zero_zero(int),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit0(A),top_top(set(numeral_bit0(A))))))) ) ).

% bit0.type
tff(fact_7497_bit0_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_bit0(A),top_top(set(numeral_bit0(A)))))))
         => ( aa(numeral_bit0(A),int,numeral_Rep_bit0(A),aa(int,numeral_bit0(A),numeral_Abs_bit02(A),M)) = M ) ) ) ).

% bit0.Abs_inverse
tff(fact_7498_Rep__bit0,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [X: numeral_bit0(A)] : member(int,aa(numeral_bit0(A),int,numeral_Rep_bit0(A),X),set_or7035219750837199246ssThan(int,zero_zero(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_bit0
tff(fact_7499_Rep__bit0__cases,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Y: int] :
          ( member(int,Y,set_or7035219750837199246ssThan(int,zero_zero(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)))))))
         => ~ ! [X2: numeral_bit0(A)] : Y != aa(numeral_bit0(A),int,numeral_Rep_bit0(A),X2) ) ) ).

% Rep_bit0_cases
tff(fact_7500_Rep__bit0__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,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)))))))
         => ( ! [X2: numeral_bit0(A)] : aa(int,$o,P,aa(numeral_bit0(A),int,numeral_Rep_bit0(A),X2))
           => aa(int,$o,P,Y) ) ) ) ).

% Rep_bit0_induct
tff(fact_7501_type__definition__bit0,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => type_definition(numeral_bit0(A),int,numeral_Rep_bit0(A),numeral_Abs_bit02(A),set_or7035219750837199246ssThan(int,zero_zero(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_bit0
tff(fact_7502_card__UNIV__char,axiom,
    finite_card(char,top_top(set(char))) = aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2))))))))) ).

% card_UNIV_char
tff(fact_7503_bit0_Omod__ring__axioms,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => numeral_mod_ring(numeral_bit0(A),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit0(A),top_top(set(numeral_bit0(A))))),numeral_Rep_bit0(A),numeral_Abs_bit02(A)) ) ).

% bit0.mod_ring_axioms
tff(fact_7504_mod__ring_ORep__numeral,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Na: int,Rep: fun(A,int),Abs: fun(int,A),W: num] :
          ( numeral_mod_ring(A,Na,Rep,Abs)
         => ( aa(A,int,Rep,aa(num,A,numeral_numeral(A),W)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(num,int,numeral_numeral(int),W)),Na) ) ) ) ).

% mod_ring.Rep_numeral
tff(fact_7505_mod__ring_OCARD__eq,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Na: int,Rep: fun(A,int),Abs: fun(int,A)] :
          ( numeral_mod_ring(A,Na,Rep,Abs)
         => ( finite_card(A,top_top(set(A))) = aa(int,nat,nat2,Na) ) ) ) ).

% mod_ring.CARD_eq
tff(fact_7506_mod__ring_Oof__nat__eq,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Na: int,Rep: fun(A,int),Abs: fun(int,A),K: nat] :
          ( numeral_mod_ring(A,Na,Rep,Abs)
         => ( aa(nat,A,semiring_1_of_nat(A),K) = aa(int,A,Abs,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(nat,int,semiring_1_of_nat(int),K)),Na)) ) ) ) ).

% mod_ring.of_nat_eq
tff(fact_7507_mod__ring_Ocases,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Na: int,Rep: fun(A,int),Abs: fun(int,A),X: A] :
          ( numeral_mod_ring(A,Na,Rep,Abs)
         => ~ ! [Z3: int] :
                ( ( X = aa(int,A,ring_1_of_int(A),Z3) )
               => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z3)
                 => ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z3),Na) ) ) ) ) ).

% mod_ring.cases
tff(fact_7508_mod__ring_Oiszero__numeral,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Na: int,Rep: fun(A,int),Abs: fun(int,A),W: num] :
          ( numeral_mod_ring(A,Na,Rep,Abs)
         => ( ring_1_iszero(A,aa(num,A,numeral_numeral(A),W))
          <=> ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(num,int,numeral_numeral(int),W)),Na) = zero_zero(int) ) ) ) ) ).

% mod_ring.iszero_numeral
tff(fact_7509_UNIV__char__of__nat,axiom,
    top_top(set(char)) = aa(set(nat),set(char),image(nat,char,unique5772411509450598832har_of(nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2))))))))))) ).

% UNIV_char_of_nat
tff(fact_7510_bit1_Omod__ring__axioms,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => numeral_mod_ring(numeral_bit1(A),aa(nat,int,semiring_1_of_nat(int),finite_card(numeral_bit1(A),top_top(set(numeral_bit1(A))))),numeral_Rep_bit1(A),numeral_Abs_bit12(A)) ) ).

% bit1.mod_ring_axioms
tff(fact_7511_char__of__nat,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat] : aa(A,char,unique5772411509450598832har_of(A),aa(nat,A,semiring_1_of_nat(A),Na)) = aa(nat,char,unique5772411509450598832har_of(nat),Na) ) ).

% char_of_nat
tff(fact_7512_char__of__mod__256,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: A] : aa(A,char,unique5772411509450598832har_of(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Na),aa(num,A,numeral_numeral(A),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2))))))))))) = aa(A,char,unique5772411509450598832har_of(A),Na) ) ).

% char_of_mod_256
tff(fact_7513_char__of__quasi__inj,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: A,Na: A] :
          ( ( aa(A,char,unique5772411509450598832har_of(A),M) = aa(A,char,unique5772411509450598832har_of(A),Na) )
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),M),aa(num,A,numeral_numeral(A),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2)))))))))) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Na),aa(num,A,numeral_numeral(A),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2)))))))))) ) ) ) ).

% char_of_quasi_inj
tff(fact_7514_inj__on__char__of__nat,axiom,
    inj_on(nat,char,unique5772411509450598832har_of(nat),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2))))))))))) ).

% inj_on_char_of_nat
tff(fact_7515_char__of__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat,M: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),Na)
         => ( aa(A,char,unique5772411509450598832har_of(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),M)) = aa(A,char,unique5772411509450598832har_of(A),M) ) ) ) ).

% char_of_take_bit_eq
tff(fact_7516_of__char__of,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Aa2: A] : aa(char,A,comm_s6883823935334413003f_char(A),aa(A,char,unique5772411509450598832har_of(A),Aa2)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Aa2),aa(num,A,numeral_numeral(A),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2)))))))))) ) ).

% of_char_of
tff(fact_7517_char__of__def,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: A] : aa(A,char,unique5772411509450598832har_of(A),Na) = aa($o,char,char2(~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Na),one_one(nat)),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Na),aa(num,nat,numeral_numeral(nat),bit1(one2))),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Na),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Na),aa(num,nat,numeral_numeral(nat),bit1(bit0(one2)))),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Na),aa(num,nat,numeral_numeral(nat),bit0(bit1(one2))))),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Na),aa(num,nat,numeral_numeral(nat),bit1(bit1(one2))))) ) ).

% char_of_def
tff(fact_7518_of__char__mod__256,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [C2: char] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(char,A,comm_s6883823935334413003f_char(A),C2)),aa(num,A,numeral_numeral(A),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2)))))))))) = aa(char,A,comm_s6883823935334413003f_char(A),C2) ) ).

% of_char_mod_256
tff(fact_7519_of__char__Char,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] : aa(char,A,comm_s6883823935334413003f_char(A),aa($o,char,char2((B0),(B1),(B22),(B32),(B42),(B52),(B62)),(B72))) = groups4207007520872428315er_sum($o,A,zero_neq_one_of_bool(A),aa(num,A,numeral_numeral(A),bit0(one2)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B0)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B1)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B22)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B32)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B42)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B52)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B62)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B72)),nil($o)))))))))) ) ).

% of_char_Char
tff(fact_7520_char_Osize_I2_J,axiom,
    ! [X12: $o,X23: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] : aa(char,nat,size_size(char),aa($o,char,char2((X12),(X23),(X32),(X42),(X52),(X62),(X72)),(X82))) = zero_zero(nat) ).

% char.size(2)
tff(fact_7521_nat__of__char__less__256,axiom,
    ! [C2: char] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(char,nat,comm_s6883823935334413003f_char(nat),C2)),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2)))))))))) ).

% nat_of_char_less_256
tff(fact_7522_range__nat__of__char,axiom,
    aa(set(char),set(nat),image(char,nat,comm_s6883823935334413003f_char(nat)),top_top(set(char))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2)))))))))) ).

% range_nat_of_char
tff(fact_7523_char__of__eq__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: A,C2: char] :
          ( ( aa(A,char,unique5772411509450598832har_of(A),Na) = C2 )
        <=> ( aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),Na) = aa(char,A,comm_s6883823935334413003f_char(A),C2) ) ) ) ).

% char_of_eq_iff
tff(fact_7524_integer__of__char__code,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] : integer_of_char(aa($o,char,char2((B0),(B1),(B22),(B32),(B42),(B52),(B62)),(B72))) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B72))),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B62)))),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B52)))),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B42)))),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B32)))),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B22)))),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B1)))),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B0))) ).

% integer_of_char_code
tff(fact_7525_less__eq__char__simp,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o,C0: $o,C1: $o,C22: $o,C32: $o,C42: $o,C52: $o,C62: $o,C72: $o] :
      ( aa(char,$o,aa(char,fun(char,$o),ord_less_eq(char),aa($o,char,char2((B0),(B1),(B22),(B32),(B42),(B52),(B62)),(B72))),aa($o,char,char2((C0),(C1),(C22),(C32),(C42),(C52),(C62)),(C72)))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,foldr($o,nat,aTP_Lamp_ajt($o,fun(nat,nat)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B0)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B1)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B22)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B32)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B42)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B52)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B62)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B72)),nil($o)))))))))),zero_zero(nat))),aa(nat,nat,foldr($o,nat,aTP_Lamp_ajt($o,fun(nat,nat)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C0)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C1)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C22)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C32)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C42)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C52)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C62)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C72)),nil($o)))))))))),zero_zero(nat))) ) ).

% less_eq_char_simp
tff(fact_7526_foldr__length,axiom,
    ! [A: $tType,L: list(A)] : aa(nat,nat,foldr(A,nat,aTP_Lamp_aju(A,fun(nat,nat)),L),zero_zero(nat)) = aa(list(A),nat,size_size(list(A)),L) ).

% foldr_length
tff(fact_7527_less__char__simp,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o,C0: $o,C1: $o,C22: $o,C32: $o,C42: $o,C52: $o,C62: $o,C72: $o] :
      ( aa(char,$o,aa(char,fun(char,$o),ord_less(char),aa($o,char,char2((B0),(B1),(B22),(B32),(B42),(B52),(B62)),(B72))),aa($o,char,char2((C0),(C1),(C22),(C32),(C42),(C52),(C62)),(C72)))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,foldr($o,nat,aTP_Lamp_ajt($o,fun(nat,nat)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B0)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B1)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B22)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B32)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B42)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B52)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B62)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B72)),nil($o)))))))))),zero_zero(nat))),aa(nat,nat,foldr($o,nat,aTP_Lamp_ajt($o,fun(nat,nat)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C0)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C1)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C22)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C32)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C42)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C52)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C62)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C72)),nil($o)))))))))),zero_zero(nat))) ) ).

% less_char_simp
tff(fact_7528_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or7035219750837199246ssThan(code_integer,L,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),U),one_one(code_integer))) = set_or1337092689740270186AtMost(code_integer,L,U) ).

% atLeastLessThanPlusOne_atLeastAtMost_integer
tff(fact_7529_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or7035219750837199246ssThan(code_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),L),one_one(code_integer)),U) = set_or5935395276787703475ssThan(code_integer,L,U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_integer
tff(fact_7530_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or1337092689740270186AtMost(code_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),L),one_one(code_integer)),U) = set_or3652927894154168847AtMost(code_integer,L,U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_integer
tff(fact_7531_foldr__cong,axiom,
    ! [B: $tType,A: $tType,Aa2: A,Ba: A,L: list(B),K: list(B),F2: fun(B,fun(A,A)),G: fun(B,fun(A,A))] :
      ( ( Aa2 = Ba )
     => ( ( L = K )
       => ( ! [A3: A,X2: B] :
              ( member(B,X2,aa(list(B),set(B),set2(B),L))
             => ( aa(A,A,aa(B,fun(A,A),F2,X2),A3) = aa(A,A,aa(B,fun(A,A),G,X2),A3) ) )
         => ( aa(A,A,foldr(B,A,F2,L),Aa2) = aa(A,A,foldr(B,A,G,K),Ba) ) ) ) ) ).

% foldr_cong
tff(fact_7532_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_ajv(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_7533_less__eq__char__def,axiom,
    ! [C1: char,C22: char] :
      ( aa(char,$o,aa(char,fun(char,$o),ord_less_eq(char),C1),C22)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(char,nat,comm_s6883823935334413003f_char(nat),C1)),aa(char,nat,comm_s6883823935334413003f_char(nat),C22)) ) ).

% less_eq_char_def
tff(fact_7534_foldr__length__aux,axiom,
    ! [A: $tType,L: list(A),Aa2: nat] : aa(nat,nat,foldr(A,nat,aTP_Lamp_aju(A,fun(nat,nat)),L),Aa2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Aa2),aa(list(A),nat,size_size(list(A)),L)) ).

% foldr_length_aux
tff(fact_7535_horner__sum__foldr,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [F2: fun(B,A),Aa2: A,Xs2: list(B)] : groups4207007520872428315er_sum(B,A,F2,Aa2,Xs2) = aa(A,A,foldr(B,A,aa(A,fun(B,fun(A,A)),aTP_Lamp_ajw(fun(B,A),fun(A,fun(B,fun(A,A))),F2),Aa2),Xs2),zero_zero(A)) ) ).

% horner_sum_foldr
tff(fact_7536_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_ajx(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),Qr) ).

% divmod_step_integer_def
tff(fact_7537_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_7538_finite__atLeastLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : aa(set(code_integer),$o,finite_finite2(code_integer),set_or7035219750837199246ssThan(code_integer,L,U)) ).

% finite_atLeastLessThan_integer
tff(fact_7539_finite__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : aa(set(code_integer),$o,finite_finite2(code_integer),set_or1337092689740270186AtMost(code_integer,L,U)) ).

% finite_atLeastAtMost_integer
tff(fact_7540_finite__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : aa(set(code_integer),$o,finite_finite2(code_integer),set_or3652927894154168847AtMost(code_integer,L,U)) ).

% finite_greaterThanAtMost_integer
tff(fact_7541_finite__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : aa(set(code_integer),$o,finite_finite2(code_integer),set_or5935395276787703475ssThan(code_integer,L,U)) ).

% finite_greaterThanLessThan_integer
tff(fact_7542_VEBT__internal_Oreal__nat__list,axiom,
    ! [A: $tType,F2: fun(A,nat),Xs2: list(A),C2: nat] : aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,foldr(nat,nat,plus_plus(nat),aa(list(A),list(nat),map(A,nat,F2),Xs2)),C2)) = aa(real,real,foldr(real,real,plus_plus(real),aa(list(A),list(real),map(A,real,aTP_Lamp_aed(fun(A,nat),fun(A,real),F2)),Xs2)),aa(nat,real,semiring_1_of_nat(real),C2)) ).

% VEBT_internal.real_nat_list
tff(fact_7543_VEBT__internal_Ofoldr__one,axiom,
    ! [D2: nat,Ysb: list(nat)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),D2),aa(nat,nat,foldr(nat,nat,plus_plus(nat),Ysb),D2)) ).

% VEBT_internal.foldr_one
tff(fact_7544_VEBT__internal_Ofoldr0,axiom,
    ! [Xs2: list(real),C2: real,D2: real] : aa(real,real,foldr(real,real,plus_plus(real),Xs2),aa(real,real,aa(real,fun(real,real),plus_plus(real),C2),D2)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,foldr(real,real,plus_plus(real),Xs2),D2)),C2) ).

% VEBT_internal.foldr0
tff(fact_7545_Misc_Ofoldr__Cons,axiom,
    ! [A: $tType,Xs2: list(A)] : aa(list(A),list(A),foldr(A,list(A),cons(A),Xs2),nil(A)) = Xs2 ).

% Misc.foldr_Cons
tff(fact_7546_finite__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] : aa(set(code_integer),$o,finite_finite2(code_integer),set_or7035219750837199246ssThan(code_integer,zero_zero(code_integer),U)) ).

% finite_atLeastZeroLessThan_integer
tff(fact_7547_one__natural_Orsp,axiom,
    one_one(nat) = one_one(nat) ).

% one_natural.rsp
tff(fact_7548_one__integer_Orsp,axiom,
    one_one(int) = one_one(int) ).

% one_integer.rsp
tff(fact_7549_plus__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),zero_zero(code_integer)),L) = L ).

% plus_integer_code(2)
tff(fact_7550_plus__integer__code_I1_J,axiom,
    ! [K: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),K),zero_zero(code_integer)) = K ).

% plus_integer_code(1)
tff(fact_7551_VEBT__internal_Olistsum__bound,axiom,
    ! [A: $tType,Xs2: list(A),F2: fun(A,real),Y: real] :
      ( ! [X2: A] :
          ( member(A,X2,aa(list(A),set(A),set2(A),Xs2))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(A,real,F2,X2)) )
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,foldr(real,real,plus_plus(real),aa(list(A),list(real),map(A,real,F2),Xs2)),Y)) ) ).

% VEBT_internal.listsum_bound
tff(fact_7552_VEBT__internal_Ofoldr__mono,axiom,
    ! [Xs2: list(nat),Ysb: list(nat),C2: nat,D2: nat] :
      ( ( aa(list(nat),nat,size_size(list(nat)),Xs2) = aa(list(nat),nat,size_size(list(nat)),Ysb) )
     => ( ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(nat),nat,size_size(list(nat)),Xs2))
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,nth(nat,Xs2),I2)),aa(nat,nat,nth(nat,Ysb),I2)) )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),C2),D2)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,foldr(nat,nat,plus_plus(nat),Xs2),C2)),aa(list(nat),nat,size_size(list(nat)),Ysb))),aa(nat,nat,foldr(nat,nat,plus_plus(nat),Ysb),D2)) ) ) ) ).

% VEBT_internal.foldr_mono
tff(fact_7553_VEBT__internal_Ofoldr__same,axiom,
    ! [Xs2: list(real),Y: real] :
      ( ! [X2: real,Y2: real] :
          ( member(real,X2,aa(list(real),set(real),set2(real),Xs2))
         => ( member(real,Y2,aa(list(real),set(real),set2(real),Xs2))
           => ( X2 = Y2 ) ) )
     => ( ! [X2: real] :
            ( member(real,X2,aa(list(real),set(real),set2(real),Xs2))
           => ( X2 = Y ) )
       => ( aa(real,real,foldr(real,real,plus_plus(real),Xs2),zero_zero(real)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(list(real),nat,size_size(list(real)),Xs2))),Y) ) ) ) ).

% VEBT_internal.foldr_same
tff(fact_7554_VEBT__internal_Ofoldr__same__int,axiom,
    ! [Xs2: list(nat),Y: nat] :
      ( ! [X2: nat,Y2: nat] :
          ( member(nat,X2,aa(list(nat),set(nat),set2(nat),Xs2))
         => ( member(nat,Y2,aa(list(nat),set(nat),set2(nat),Xs2))
           => ( X2 = Y2 ) ) )
     => ( ! [X2: nat] :
            ( member(nat,X2,aa(list(nat),set(nat),set2(nat),Xs2))
           => ( X2 = Y ) )
       => ( aa(nat,nat,foldr(nat,nat,plus_plus(nat),Xs2),zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(nat),nat,size_size(list(nat)),Xs2)),Y) ) ) ) ).

% VEBT_internal.foldr_same_int
tff(fact_7555_length__product__lists,axiom,
    ! [A: $tType,Xss2: list(list(A))] : aa(list(list(A)),nat,size_size(list(list(A))),product_lists(A,Xss2)) = aa(nat,nat,foldr(nat,nat,times_times(nat),aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xss2)),one_one(nat)) ).

% length_product_lists
tff(fact_7556_VEBT__internal_Of__g__map__foldr__bound,axiom,
    ! [A: $tType,Xs2: list(A),F2: fun(A,real),C2: real,G: fun(A,real),D2: real] :
      ( ! [X2: A] :
          ( member(A,X2,aa(list(A),set(A),set2(A),Xs2))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(A,real,F2,X2)),aa(real,real,aa(real,fun(real,real),times_times(real),C2),aa(A,real,G,X2))) )
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,foldr(real,real,plus_plus(real),aa(list(A),list(real),map(A,real,F2),Xs2)),D2)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),C2),aa(real,real,foldr(real,real,plus_plus(real),aa(list(A),list(real),map(A,real,G),Xs2)),zero_zero(real)))),D2)) ) ).

% VEBT_internal.f_g_map_foldr_bound
tff(fact_7557_VEBT__internal_Olist__every__elemnt__bound__sum__bound,axiom,
    ! [A: $tType,Xs2: list(A),F2: fun(A,nat),Bound: nat,I: nat] :
      ( ! [X2: A] :
          ( member(A,X2,aa(list(A),set(A),set2(A),Xs2))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,X2)),Bound) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,foldr(nat,nat,plus_plus(nat),aa(list(A),list(nat),map(A,nat,F2),Xs2)),I)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(A),nat,size_size(list(A)),Xs2)),Bound)),I)) ) ).

% VEBT_internal.list_every_elemnt_bound_sum_bound
tff(fact_7558_VEBT__internal_Ofoldr__zero,axiom,
    ! [Xs2: list(nat),D2: nat] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(nat),nat,size_size(list(nat)),Xs2))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,nth(nat,Xs2),I2)) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(nat),nat,size_size(list(nat)),Xs2)),aa(nat,nat,minus_minus(nat,aa(nat,nat,foldr(nat,nat,plus_plus(nat),Xs2),D2)),D2)) ) ).

% VEBT_internal.foldr_zero
tff(fact_7559_VEBT__internal_Olist__every__elemnt__bound__sum__bound__real,axiom,
    ! [A: $tType,Xs2: list(A),F2: fun(A,real),Bound: real,I: real] :
      ( ! [X2: A] :
          ( member(A,X2,aa(list(A),set(A),set2(A),Xs2))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(A,real,F2,X2)),Bound) )
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,foldr(real,real,plus_plus(real),aa(list(A),list(real),map(A,real,F2),Xs2)),I)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(list(A),nat,size_size(list(A)),Xs2))),Bound)),I)) ) ).

% VEBT_internal.list_every_elemnt_bound_sum_bound_real
tff(fact_7560_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_7561_integer__of__int__code,axiom,
    ! [K: int] :
      aa(int,code_integer,code_integer_of_int,K) = $ite(
        aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)),
        aa(code_integer,code_integer,uminus_uminus(code_integer),aa(int,code_integer,code_integer_of_int,aa(int,int,uminus_uminus(int),K))),
        $ite(
          K = zero_zero(int),
          zero_zero(code_integer),
          $let(
            l: code_integer,
            l:= aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2))))),
            $ite(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),aa(num,int,numeral_numeral(int),bit0(one2))) = zero_zero(int),l,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),l),one_one(code_integer))) ) ) ) ).

% integer_of_int_code
tff(fact_7562_String_Ochar__of__ascii__of,axiom,
    ! [C2: char] : aa(char,nat,comm_s6883823935334413003f_char(nat),ascii_of(C2)) = aa(nat,nat,bit_se2584673776208193580ke_bit(nat,aa(num,nat,numeral_numeral(nat),bit1(bit1(one2)))),aa(char,nat,comm_s6883823935334413003f_char(nat),C2)) ).

% String.char_of_ascii_of
tff(fact_7563_less__eq__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),aa(int,code_integer,code_integer_of_int,Xa2)),aa(int,code_integer,code_integer_of_int,X))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xa2),X) ) ).

% less_eq_integer.abs_eq
tff(fact_7564_times__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(int,code_integer,code_integer_of_int,Xa2)),aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),times_times(int),Xa2),X)) ).

% times_integer.abs_eq
tff(fact_7565_plus__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(int,code_integer,code_integer_of_int,Xa2)),aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa2),X)) ).

% plus_integer.abs_eq
tff(fact_7566_one__integer__def,axiom,
    one_one(code_integer) = aa(int,code_integer,code_integer_of_int,one_one(int)) ).

% one_integer_def
tff(fact_7567_Bit__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: $o] : bits_Bit_integer(aa(int,code_integer,code_integer_of_int,Xa2),(X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa($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_7568_dup__1,axiom,
    aa(code_integer,code_integer,code_dup,one_one(code_integer)) = aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)) ).

% dup_1
tff(fact_7569_dup_Oabs__eq,axiom,
    ! [X: int] : aa(code_integer,code_integer,code_dup,aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),X)) ).

% dup.abs_eq
tff(fact_7570_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_7571_Bit__integer__code_I2_J,axiom,
    ! [I: code_integer] : bits_Bit_integer(I,$true) = aa(code_integer,code_integer,aa(code_integer,fun(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_7572_bin__rest__integer_Oabs__eq,axiom,
    ! [X: int] : aa(code_integer,code_integer,bits_b2549910563261871055nteger,aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),divide_divide(int),X),aa(num,int,numeral_numeral(int),bit0(one2)))) ).

% bin_rest_integer.abs_eq
tff(fact_7573_bin__last__integer_Oabs__eq,axiom,
    ! [X: int] :
      ( bits_b8758750999018896077nteger(aa(int,code_integer,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_7574_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_7575_bin__rest__integer__code,axiom,
    ! [I: code_integer] : aa(code_integer,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_7576_int__of__integer__code,axiom,
    ! [K: code_integer] :
      aa(code_integer,int,code_int_of_integer,K) = $ite(
        aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),zero_zero(code_integer)),
        aa(int,int,uminus_uminus(int),aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),K))),
        $ite(K = zero_zero(code_integer),zero_zero(int),aa(product_prod(code_integer,code_integer),int,product_case_prod(code_integer,code_integer,int,aTP_Lamp_ajy(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_7577_integer__of__num_I3_J,axiom,
    ! [Na: num] :
      code_integer_of_num(bit1(Na)) = $let(
        k: code_integer,
        k:= code_integer_of_num(Na),
        aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),k),k)),one_one(code_integer)) ) ).

% integer_of_num(3)
tff(fact_7578_int__of__integer__of__nat,axiom,
    ! [Na: nat] : aa(code_integer,int,code_int_of_integer,aa(nat,code_integer,semiring_1_of_nat(code_integer),Na)) = aa(nat,int,semiring_1_of_nat(int),Na) ).

% int_of_integer_of_nat
tff(fact_7579_int__of__integer__numeral,axiom,
    ! [K: num] : aa(code_integer,int,code_int_of_integer,aa(num,code_integer,numeral_numeral(code_integer),K)) = aa(num,int,numeral_numeral(int),K) ).

% int_of_integer_numeral
tff(fact_7580_one__integer_Orep__eq,axiom,
    aa(code_integer,int,code_int_of_integer,one_one(code_integer)) = one_one(int) ).

% one_integer.rep_eq
tff(fact_7581_plus__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),X),Xa2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(code_integer,int,code_int_of_integer,X)),aa(code_integer,int,code_int_of_integer,Xa2)) ).

% plus_integer.rep_eq
tff(fact_7582_times__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),X),Xa2)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(code_integer,int,code_int_of_integer,X)),aa(code_integer,int,code_int_of_integer,Xa2)) ).

% times_integer.rep_eq
tff(fact_7583_int__of__integer__less__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(code_integer,int,code_int_of_integer,X)),aa(code_integer,int,code_int_of_integer,Y))
    <=> aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),X),Y) ) ).

% int_of_integer_less_iff
tff(fact_7584_integer__less__eq__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),K),L)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(code_integer,int,code_int_of_integer,K)),aa(code_integer,int,code_int_of_integer,L)) ) ).

% integer_less_eq_iff
tff(fact_7585_less__eq__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),X),Xa2)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(code_integer,int,code_int_of_integer,X)),aa(code_integer,int,code_int_of_integer,Xa2)) ) ).

% less_eq_integer.rep_eq
tff(fact_7586_dup_Orep__eq,axiom,
    ! [X: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,code_dup,X)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(code_integer,int,code_int_of_integer,X)),aa(code_integer,int,code_int_of_integer,X)) ).

% dup.rep_eq
tff(fact_7587_integer__of__num__triv_I1_J,axiom,
    code_integer_of_num(one2) = one_one(code_integer) ).

% integer_of_num_triv(1)
tff(fact_7588_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))),aa(code_integer,int,code_int_of_integer,X)) ) ).

% bin_last_integer.rep_eq
tff(fact_7589_bin__rest__integer_Orep__eq,axiom,
    ! [X: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,bits_b2549910563261871055nteger,X)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(code_integer,int,code_int_of_integer,X)),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% bin_rest_integer.rep_eq
tff(fact_7590_integer__of__num_I2_J,axiom,
    ! [Na: num] :
      code_integer_of_num(bit0(Na)) = $let(
        k: code_integer,
        k:= code_integer_of_num(Na),
        aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),k),k) ) ).

% integer_of_num(2)
tff(fact_7591_integer__of__num__triv_I2_J,axiom,
    code_integer_of_num(bit0(one2)) = aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)) ).

% integer_of_num_triv(2)
tff(fact_7592_Bit__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: $o] : aa(code_integer,int,code_int_of_integer,bits_Bit_integer(X,(Xa2))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa($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))),aa(code_integer,int,code_int_of_integer,X))) ).

% Bit_integer.rep_eq
tff(fact_7593_num__of__integer__code,axiom,
    ! [K: code_integer] :
      code_num_of_integer(K) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),K),one_one(code_integer)),one2,aa(product_prod(code_integer,code_integer),num,product_case_prod(code_integer,code_integer,num,aTP_Lamp_ajz(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_7594_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      code_nat_of_integer(K) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),K),zero_zero(code_integer)),zero_zero(nat),aa(product_prod(code_integer,code_integer),nat,product_case_prod(code_integer,code_integer,nat,aTP_Lamp_aka(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_7595_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_7596_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_7597_nat__of__integer__1,axiom,
    code_nat_of_integer(one_one(code_integer)) = one_one(nat) ).

% nat_of_integer_1
tff(fact_7598_nat__of__integer_Orep__eq,axiom,
    ! [X: code_integer] : code_nat_of_integer(X) = aa(int,nat,nat2,aa(code_integer,int,code_int_of_integer,X)) ).

% nat_of_integer.rep_eq
tff(fact_7599_nat__of__integer_Oabs__eq,axiom,
    ! [X: int] : code_nat_of_integer(aa(int,code_integer,code_integer_of_int,X)) = aa(int,nat,nat2,X) ).

% nat_of_integer.abs_eq
tff(fact_7600_nat__of__integer__less__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),zero_zero(code_integer)),X)
     => ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),zero_zero(code_integer)),Y)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),code_nat_of_integer(X)),code_nat_of_integer(Y))
        <=> aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),X),Y) ) ) ) ).

% nat_of_integer_less_iff
tff(fact_7601_image__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),zero_zero(code_integer)),U)
     => ( set_or7035219750837199246ssThan(code_integer,zero_zero(code_integer),U) = aa(set(nat),set(code_integer),image(nat,code_integer,semiring_1_of_nat(code_integer)),set_ord_lessThan(nat,code_nat_of_integer(U))) ) ) ).

% image_atLeastZeroLessThan_integer
tff(fact_7602_exp__dvd__iff__exp__udvd,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W: word(A)] :
          ( aa(word(A),$o,dvd_dvd(word(A),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),W)
        <=> udvd(A,aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na),W) ) ) ).

% exp_dvd_iff_exp_udvd
tff(fact_7603_Code__Target__Int_Opositive__def,axiom,
    code_Target_positive = numeral_numeral(int) ).

% Code_Target_Int.positive_def
tff(fact_7604_udvdI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [W: word(A),V: word(A),U: word(B)] :
          ( ( aa(word(A),nat,semiring_1_unsigned(A,nat),W) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),V)),aa(word(B),nat,semiring_1_unsigned(B,nat),U)) )
         => udvd(A,V,W) ) ) ).

% udvdI
tff(fact_7605_udvd__imp__dvd,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: word(A),W: word(A)] :
          ( udvd(A,V,W)
         => aa(word(A),$o,dvd_dvd(word(A),V),W) ) ) ).

% udvd_imp_dvd
tff(fact_7606_udvdE,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: word(A),W: word(A)] :
          ( udvd(A,V,W)
         => ~ ! [U3: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),W) != aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),V)),aa(word(A),nat,semiring_1_unsigned(A,nat),U3)) ) ) ).

% udvdE
tff(fact_7607_udvd__nat__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] :
          ( udvd(A,Aa2,Ba)
        <=> ? [N2: nat] : aa(word(A),nat,semiring_1_unsigned(A,nat),Ba) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N2),aa(word(A),nat,semiring_1_unsigned(A,nat),Aa2)) ) ) ).

% udvd_nat_alt
tff(fact_7608_udvd__iff__dvd,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A)] :
          ( udvd(A,X,Y)
        <=> aa(nat,$o,dvd_dvd(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),X)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) ) ).

% udvd_iff_dvd
tff(fact_7609_udvd__iff__dvd__int,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: word(A),W: word(A)] :
          ( udvd(A,V,W)
        <=> aa(int,$o,dvd_dvd(int,aa(word(A),int,semiring_1_unsigned(A,int),V)),aa(word(A),int,semiring_1_unsigned(A,int),W)) ) ) ).

% udvd_iff_dvd_int
tff(fact_7610_udvd__imp__mod__eq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: word(A),W: word(A)] :
          ( udvd(A,V,W)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),W),V) = zero_zero(word(A)) ) ) ) ).

% udvd_imp_mod_eq_0
tff(fact_7611_mod__eq__0__imp__udvd,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),V: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),W),V) = zero_zero(word(A)) )
         => udvd(A,V,W) ) ) ).

% mod_eq_0_imp_udvd
tff(fact_7612_udvd__minus__le_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xy: word(A),K: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xy),K)
         => ( udvd(A,Z,Xy)
           => ( udvd(A,Z,K)
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xy),aa(word(A),word(A),minus_minus(word(A),K),Z)) ) ) ) ) ).

% udvd_minus_le'
tff(fact_7613_udvd__unfold__int,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Ba: word(A)] :
          ( udvd(A,Aa2,Ba)
        <=> ? [N2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),N2)
              & ( aa(word(A),int,semiring_1_unsigned(A,int),Ba) = aa(int,int,aa(int,fun(int,int),times_times(int),N2),aa(word(A),int,semiring_1_unsigned(A,int),Aa2)) ) ) ) ) ).

% udvd_unfold_int
tff(fact_7614_udvd_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa2: int,X: int] :
          ( udvd(A,aa(int,word(A),word2(A),Xa2),aa(int,word(A),word2(A),X))
        <=> aa(int,$o,dvd_dvd(int,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),Xa2)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),X)) ) ) ).

% udvd.abs_eq
tff(fact_7615_udvd__incr2__K,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),Aa2: word(A),S: word(A),K3: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),P2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Aa2),S))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Aa2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Aa2),S))
           => ( udvd(A,K3,S)
             => ( udvd(A,K3,aa(word(A),word(A),minus_minus(word(A),P2),Aa2))
               => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Aa2),P2)
                 => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),K3)
                   => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),P2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P2),K3))
                      & aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P2),K3)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Aa2),S)) ) ) ) ) ) ) ) ) ).

% udvd_incr2_K
tff(fact_7616_ord_OLeast__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),P: fun(A,$o)] : aa(fun(A,$o),A,least(A,Less_eq),P) = the(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_akb(fun(A,fun(A,$o)),fun(fun(A,$o),fun(A,$o)),Less_eq),P)) ).

% ord.Least_def
tff(fact_7617_bin__rest__integer__def,axiom,
    bits_b2549910563261871055nteger = map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int,aTP_Lamp_akc(int,int)) ).

% bin_rest_integer_def
tff(fact_7618_ord_OLeast_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o))] : least(A,Less_eq) = least(A,Less_eq) ).

% ord.Least.cong
tff(fact_7619_signed_OLeast__equality,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(word(A),$o),X: word(A)] :
          ( aa(word(A),$o,P,X)
         => ( ! [Y2: word(A)] :
                ( aa(word(A),$o,P,Y2)
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),X),Y2) )
           => ( aa(fun(word(A),$o),word(A),least(word(A),word_sle(A)),P) = X ) ) ) ) ).

% signed.Least_equality
tff(fact_7620_signed_OLeastI2__order,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(word(A),$o),X: word(A),Q: fun(word(A),$o)] :
          ( aa(word(A),$o,P,X)
         => ( ! [Y2: word(A)] :
                ( aa(word(A),$o,P,Y2)
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),X),Y2) )
           => ( ! [X2: word(A)] :
                  ( aa(word(A),$o,P,X2)
                 => ( ! [Y3: word(A)] :
                        ( aa(word(A),$o,P,Y3)
                       => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),X2),Y3) )
                   => aa(word(A),$o,Q,X2) ) )
             => aa(word(A),$o,Q,aa(fun(word(A),$o),word(A),least(word(A),word_sle(A)),P)) ) ) ) ) ).

% signed.LeastI2_order
tff(fact_7621_signed_OLeast1__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(word(A),$o),Z: word(A)] :
          ( ? [X4: word(A)] :
              ( aa(word(A),$o,P,X4)
              & ! [Y2: word(A)] :
                  ( aa(word(A),$o,P,Y2)
                 => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),X4),Y2) )
              & ! [Y2: word(A)] :
                  ( ( aa(word(A),$o,P,Y2)
                    & ! [Ya2: word(A)] :
                        ( aa(word(A),$o,P,Ya2)
                       => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Y2),Ya2) ) )
                 => ( Y2 = X4 ) ) )
         => ( aa(word(A),$o,P,Z)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(fun(word(A),$o),word(A),least(word(A),word_sle(A)),P)),Z) ) ) ) ).

% signed.Least1_le
tff(fact_7622_signed_OLeast1I,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(word(A),$o)] :
          ( ? [X4: word(A)] :
              ( aa(word(A),$o,P,X4)
              & ! [Y2: word(A)] :
                  ( aa(word(A),$o,P,Y2)
                 => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),X4),Y2) )
              & ! [Y2: word(A)] :
                  ( ( aa(word(A),$o,P,Y2)
                    & ! [Ya2: word(A)] :
                        ( aa(word(A),$o,P,Ya2)
                       => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Y2),Ya2) ) )
                 => ( Y2 = X4 ) ) )
         => aa(word(A),$o,P,aa(fun(word(A),$o),word(A),least(word(A),word_sle(A)),P)) ) ) ).

% signed.Least1I
tff(fact_7623_signed_OLeast__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(word(A),$o)] : aa(fun(word(A),$o),word(A),least(word(A),word_sle(A)),P) = the(word(A),aTP_Lamp_akd(fun(word(A),$o),fun(word(A),$o),P)) ) ).

% signed.Least_def
tff(fact_7624_Code__Numeral_Odup__def,axiom,
    code_dup = map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int,aTP_Lamp_ake(int,int)) ).

% Code_Numeral.dup_def
tff(fact_7625_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_7626_part__code_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Pivot: B,X: A,Xs2: list(A)] : linorder_part(A,B,F2,Pivot,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs2)) = aa(product_prod(list(A),product_prod(list(A),list(A))),product_prod(list(A),product_prod(list(A),list(A))),product_case_prod(list(A),product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A))),aa(A,fun(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A))))),aa(B,fun(A,fun(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A)))))),aTP_Lamp_akg(fun(A,B),fun(B,fun(A,fun(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A))))))),F2),Pivot),X)),linorder_part(A,B,F2,Pivot,Xs2)) ) ).

% part_code(2)
tff(fact_7627_part__code_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Pivot: B] : linorder_part(A,B,F2,Pivot,nil(A)) = aa(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A))),aa(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A)))),product_Pair(list(A),product_prod(list(A),list(A))),nil(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A))) ) ).

% part_code(1)
tff(fact_7628_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,$o),aTP_Lamp_akh(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,$o))),K)),code_divmod_abs(K,aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))))) ).

% bit_cut_integer_code
tff(fact_7629_char__of__integer__code,axiom,
    ! [K: code_integer] : char_of_integer(K) = aa(product_prod(code_integer,$o),char,product_case_prod(code_integer,$o,char,aTP_Lamp_akp(code_integer,fun($o,char))),code_bit_cut_integer(K)) ).

% char_of_integer_code
tff(fact_7630_divmod__integer__code,axiom,
    ! [K: code_integer,L: code_integer] :
      code_divmod_integer(K,L) = $ite(
        K = zero_zero(code_integer),
        aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),zero_zero(code_integer)),
        $ite(
          aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),L),
          $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),K),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer),aTP_Lamp_akq(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L))),
          $ite(
            L = zero_zero(code_integer),
            aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),K),
            aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_apsnd(code_integer,code_integer,code_integer),uminus_uminus(code_integer)),
              $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),zero_zero(code_integer)),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer),aTP_Lamp_akr(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_7631_signed_OGreatest__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(word(A),$o)] : aa(fun(word(A),$o),word(A),greatest(word(A),word_sle(A)),P) = the(word(A),aTP_Lamp_aks(fun(word(A),$o),fun(word(A),$o),P)) ) ).

% signed.Greatest_def
tff(fact_7632_order_OGreatest_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o))] : greatest(A,Less_eq) = greatest(A,Less_eq) ).

% order.Greatest.cong
tff(fact_7633_signed_OGreatestI2__order,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(word(A),$o),X: word(A),Q: fun(word(A),$o)] :
          ( aa(word(A),$o,P,X)
         => ( ! [Y2: word(A)] :
                ( aa(word(A),$o,P,Y2)
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Y2),X) )
           => ( ! [X2: word(A)] :
                  ( aa(word(A),$o,P,X2)
                 => ( ! [Y3: word(A)] :
                        ( aa(word(A),$o,P,Y3)
                       => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Y3),X2) )
                   => aa(word(A),$o,Q,X2) ) )
             => aa(word(A),$o,Q,aa(fun(word(A),$o),word(A),greatest(word(A),word_sle(A)),P)) ) ) ) ) ).

% signed.GreatestI2_order
tff(fact_7634_signed_OGreatest__equality,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(word(A),$o),X: word(A)] :
          ( aa(word(A),$o,P,X)
         => ( ! [Y2: word(A)] :
                ( aa(word(A),$o,P,Y2)
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Y2),X) )
           => ( aa(fun(word(A),$o),word(A),greatest(word(A),word_sle(A)),P) = X ) ) ) ) ).

% signed.Greatest_equality
tff(fact_7635_sum__list__map__eq__sum__count2,axiom,
    ! [A: $tType,Xs2: list(A),X5: set(A),F2: fun(A,nat)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs2)),X5)
     => ( aa(set(A),$o,finite_finite2(A),X5)
       => ( groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F2),Xs2)) = aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_akt(list(A),fun(fun(A,nat),fun(A,nat)),Xs2),F2)),X5) ) ) ) ).

% sum_list_map_eq_sum_count2
tff(fact_7636_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [Na: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,minus_minus(nat,J),I))
     => ( aa(nat,nat,nth(nat,linord4507533701916653071of_set(nat,set_or3652927894154168847AtMost(nat,I,J))),Na) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),Na)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_7637_sum__list_ONil,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ( groups8242544230860333062m_list(A,nil(A)) = zero_zero(A) ) ) ).

% sum_list.Nil
tff(fact_7638_sum__list_OCons,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [X: A,Xs2: list(A)] : groups8242544230860333062m_list(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),X),groups8242544230860333062m_list(A,Xs2)) ) ).

% sum_list.Cons
tff(fact_7639_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( linord4507533701916653071of_set(A,bot_bot(set(A))) = nil(A) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_empty
tff(fact_7640_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: set(A)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A2)
         => ( linord4507533701916653071of_set(A,A2) = nil(A) ) ) ) ).

% sorted_list_of_set.fold_insort_key.infinite
tff(fact_7641_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A2)
         => ( aa(list(A),set(A),set2(A),linord4507533701916653071of_set(A,A2)) = A2 ) ) ) ).

% sorted_list_of_set.set_sorted_key_list_of_set
tff(fact_7642_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: set(A)] : aa(list(A),nat,size_size(list(A)),linord4507533701916653071of_set(A,A2)) = finite_card(A,A2) ) ).

% sorted_list_of_set.length_sorted_key_list_of_set
tff(fact_7643_sum__list__0,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Xs2: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aTP_Lamp_aku(B,A)),Xs2)) = zero_zero(A) ) ).

% sum_list_0
tff(fact_7644_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A2)
         => ( ( linord4507533701916653071of_set(A,A2) = nil(A) )
          <=> ( A2 = bot_bot(set(A)) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
tff(fact_7645_sum__list_Oeq__foldr,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [Xs2: list(A)] : groups8242544230860333062m_list(A,Xs2) = aa(A,A,foldr(A,A,plus_plus(A),Xs2),zero_zero(A)) ) ).

% sum_list.eq_foldr
tff(fact_7646_sorted__list__of__set_Odistinct__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: set(A)] : distinct(A,linord4507533701916653071of_set(A,A2)) ) ).

% sorted_list_of_set.distinct_sorted_key_list_of_set
tff(fact_7647_sum__list__const__mult,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [C2: A,F2: fun(B,A),Xs2: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_hq(A,fun(fun(B,A),fun(B,A)),C2),F2)),Xs2)) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,F2),Xs2))) ) ).

% sum_list_const_mult
tff(fact_7648_sum__list__mult__const,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_0(A)
     => ! [F2: fun(B,A),C2: A,Xs2: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aa(A,fun(B,A),aTP_Lamp_hp(fun(B,A),fun(A,fun(B,A)),F2),C2)),Xs2)) = aa(A,A,aa(A,fun(A,A),times_times(A),groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,F2),Xs2))),C2) ) ).

% sum_list_mult_const
tff(fact_7649_sum__list__addf,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(B,A),G: fun(B,A),Xs2: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_hm(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G)),Xs2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,F2),Xs2))),groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,G),Xs2))) ) ).

% sum_list_addf
tff(fact_7650_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: set(A),B2: set(A)] :
          ( ( linord4507533701916653071of_set(A,A2) = linord4507533701916653071of_set(A,B2) )
         => ( aa(set(A),$o,finite_finite2(A),A2)
           => ( aa(set(A),$o,finite_finite2(A),B2)
             => ( A2 = B2 ) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_inject
tff(fact_7651_sum__list__abs,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Xs2: list(A)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),groups8242544230860333062m_list(A,Xs2))),groups8242544230860333062m_list(A,aa(list(A),list(A),map(A,A,abs_abs(A)),Xs2))) ) ).

% sum_list_abs
tff(fact_7652_sum__list__subtractf,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(B,A),G: fun(B,A),Xs2: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_hr(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G)),Xs2)) = aa(A,A,minus_minus(A,groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,F2),Xs2))),groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,G),Xs2))) ) ).

% sum_list_subtractf
tff(fact_7653_member__le__sum__list,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [X: A,Xs2: list(A)] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs2))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),groups8242544230860333062m_list(A,Xs2)) ) ) ).

% member_le_sum_list
tff(fact_7654_Groups__List_Osum__list__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs2: list(A)] :
          ( ! [X2: A] :
              ( member(A,X2,aa(list(A),set(A),set2(A),Xs2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X2) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),groups8242544230860333062m_list(A,Xs2)) ) ) ).

% Groups_List.sum_list_nonneg
tff(fact_7655_sum__list__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs2: list(A)] :
          ( ! [X2: A] :
              ( member(A,X2,aa(list(A),set(A),set2(A),Xs2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X2) )
         => ( ( groups8242544230860333062m_list(A,Xs2) = zero_zero(A) )
          <=> ! [X6: A] :
                ( member(A,X6,aa(list(A),set(A),set2(A),Xs2))
               => ( X6 = zero_zero(A) ) ) ) ) ) ).

% sum_list_nonneg_eq_0_iff
tff(fact_7656_sum__list__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs2: list(A)] :
          ( ! [X2: A] :
              ( member(A,X2,aa(list(A),set(A),set2(A),Xs2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),zero_zero(A)) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),groups8242544230860333062m_list(A,Xs2)),zero_zero(A)) ) ) ).

% sum_list_nonpos
tff(fact_7657_sum__list__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( monoid_add(B)
        & ordere6658533253407199908up_add(B) )
     => ! [Xs2: list(A),F2: fun(A,B),G: fun(A,B)] :
          ( ! [X2: A] :
              ( member(A,X2,aa(list(A),set(A),set2(A),Xs2))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X2)),aa(A,B,G,X2)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,F2),Xs2))),groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,G),Xs2))) ) ) ).

% sum_list_mono
tff(fact_7658_distinct__sum__list__conv__Sum,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Xs2: list(A)] :
          ( distinct(A,Xs2)
         => ( groups8242544230860333062m_list(A,Xs2) = aa(set(A),A,groups7311177749621191930dd_sum(A,A,aTP_Lamp_akv(A,A)),aa(list(A),set(A),set2(A),Xs2)) ) ) ) ).

% distinct_sum_list_conv_Sum
tff(fact_7659_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I)),J)
     => ( linord4507533701916653071of_set(nat,set_or5935395276787703475ssThan(nat,I,J)) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(nat,nat,suc,I)),linord4507533701916653071of_set(nat,set_or5935395276787703475ssThan(nat,aa(nat,nat,suc,I),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_7660_elem__le__sum__list,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [K: nat,Ns: list(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Ns))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Ns),K)),groups8242544230860333062m_list(A,Ns)) ) ) ).

% elem_le_sum_list
tff(fact_7661_sum__list__strict__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( monoid_add(B)
        & strict9044650504122735259up_add(B) )
     => ! [Xs2: list(A),F2: fun(A,B),G: fun(A,B)] :
          ( ( Xs2 != nil(A) )
         => ( ! [X2: A] :
                ( member(A,X2,aa(list(A),set(A),set2(A),Xs2))
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X2)),aa(A,B,G,X2)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,F2),Xs2))),groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,G),Xs2))) ) ) ) ).

% sum_list_strict_mono
tff(fact_7662_sum__list__Suc,axiom,
    ! [A: $tType,F2: fun(A,nat),Xs2: list(A)] : groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,aTP_Lamp_ajj(fun(A,nat),fun(A,nat),F2)),Xs2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F2),Xs2))),aa(list(A),nat,size_size(list(A)),Xs2)) ).

% sum_list_Suc
tff(fact_7663_sum__list__triv,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [R: A,Xs2: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aTP_Lamp_aja(A,fun(B,A),R)),Xs2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(list(B),nat,size_size(list(B)),Xs2))),R) ) ).

% sum_list_triv
tff(fact_7664_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,I)),J)
     => ( linord4507533701916653071of_set(nat,set_or3652927894154168847AtMost(nat,I,J)) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(nat,nat,suc,I)),linord4507533701916653071of_set(nat,set_or3652927894154168847AtMost(nat,aa(nat,nat,suc,I),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_7665_card__length__sum__list__rec,axiom,
    ! [M: nat,N3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),M)
     => ( finite_card(list(nat),collect(list(nat),aa(nat,fun(list(nat),$o),aTP_Lamp_akw(nat,fun(nat,fun(list(nat),$o)),M),N3))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),finite_card(list(nat),collect(list(nat),aa(nat,fun(list(nat),$o),aTP_Lamp_akx(nat,fun(nat,fun(list(nat),$o)),M),N3)))),finite_card(list(nat),collect(list(nat),aa(nat,fun(list(nat),$o),aTP_Lamp_aky(nat,fun(nat,fun(list(nat),$o)),M),N3)))) ) ) ).

% card_length_sum_list_rec
tff(fact_7666_sum__list__sum__nth,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Xs2: list(A)] : groups8242544230860333062m_list(A,Xs2) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,nth(A,Xs2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs2))) ) ).

% sum_list_sum_nth
tff(fact_7667_card__length__sum__list,axiom,
    ! [M: nat,N3: nat] : finite_card(list(nat),collect(list(nat),aa(nat,fun(list(nat),$o),aTP_Lamp_akw(nat,fun(nat,fun(list(nat),$o)),M),N3))) = aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),M)),one_one(nat))),N3) ).

% card_length_sum_list
tff(fact_7668_sum__list__map__eq__sum__count,axiom,
    ! [A: $tType,F2: fun(A,nat),Xs2: list(A)] : groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F2),Xs2)) = aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aa(list(A),fun(A,nat),aTP_Lamp_akz(fun(A,nat),fun(list(A),fun(A,nat)),F2),Xs2)),aa(list(A),set(A),set2(A),Xs2)) ).

% sum_list_map_eq_sum_count
tff(fact_7669_sum__list__update,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [K: nat,Xs2: list(A),X: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Xs2))
         => ( groups8242544230860333062m_list(A,list_update(A,Xs2,K,X)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),groups8242544230860333062m_list(A,Xs2)),X)),aa(nat,A,nth(A,Xs2),K)) ) ) ) ).

% sum_list_update
tff(fact_7670_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [Na: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,minus_minus(nat,J),aa(nat,nat,suc,I)))
     => ( aa(nat,nat,nth(nat,linord4507533701916653071of_set(nat,set_or5935395276787703475ssThan(nat,I,J))),Na) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),Na)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_7671_sorted__wrt__less__sum__mono__lowerbound,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [F2: fun(nat,A),Ns: list(nat)] :
          ( ! [X2: nat,Y2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X2),Y2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,X2)),aa(nat,A,F2,Y2)) )
         => ( sorted_wrt(nat,ord_less(nat),Ns)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(nat),nat,size_size(list(nat)),Ns)))),groups8242544230860333062m_list(A,aa(list(nat),list(A),map(nat,A,F2),Ns))) ) ) ) ).

% sorted_wrt_less_sum_mono_lowerbound
tff(fact_7672_Suc__0__div__numeral,axiom,
    ! [K: num] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),K)) = aa(product_prod(nat,nat),nat,product_fst(nat,nat),unique8689654367752047608divmod(nat,one2,K)) ).

% Suc_0_div_numeral
tff(fact_7673_map__fst__mk__snd,axiom,
    ! [B: $tType,A: $tType,K: B,L: list(A)] : aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_ala(B,fun(A,product_prod(A,B))),K)),L)) = L ).

% map_fst_mk_snd
tff(fact_7674_img__fst,axiom,
    ! [B: $tType,A: $tType,Aa2: A,Ba: B,Sa: set(product_prod(A,B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba),Sa)
     => member(A,Aa2,aa(set(product_prod(A,B)),set(A),image(product_prod(A,B),A,product_fst(A,B)),Sa)) ) ).

% img_fst
tff(fact_7675_sorted__wrt__map__linord,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [L: list(product_prod(A,B))] :
          ( sorted_wrt(product_prod(A,B),aTP_Lamp_alb(product_prod(A,B),fun(product_prod(A,B),$o)),L)
        <=> sorted_wrt(A,ord_less_eq(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),L)) ) ) ).

% sorted_wrt_map_linord
tff(fact_7676_numeral__div__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [K: num,L: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),K)),aa(num,A,numeral_numeral(A),L)) = aa(product_prod(A,A),A,product_fst(A,A),unique8689654367752047608divmod(A,K,L)) ) ).

% numeral_div_numeral
tff(fact_7677_one__div__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Na: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),Na)) = aa(product_prod(A,A),A,product_fst(A,A),unique8689654367752047608divmod(A,one2,Na)) ) ).

% one_div_numeral
tff(fact_7678_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: set(A)] : sorted_wrt(A,ord_less_eq(A),linord4507533701916653071of_set(A,A2)) ) ).

% sorted_list_of_set.sorted_sorted_key_list_of_set
tff(fact_7679_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: set(A)] : sorted_wrt(A,ord_less(A),linord4507533701916653071of_set(A,A2)) ) ).

% sorted_list_of_set.strict_sorted_key_list_of_set
tff(fact_7680_fn__fst__conv,axiom,
    ! [B: $tType,C: $tType,A: $tType,F2: fun(A,C)] : aTP_Lamp_alc(fun(A,C),fun(product_prod(A,B),C),F2) = product_case_prod(A,B,C,aTP_Lamp_ald(fun(A,C),fun(A,fun(B,C)),F2)) ).

% fn_fst_conv
tff(fact_7681_fst__def,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod(A,B)] : aa(product_prod(A,B),A,product_fst(A,B),Prod) = aa(product_prod(A,B),A,product_case_prod(A,B,A,aTP_Lamp_pl(A,fun(B,A))),Prod) ).

% fst_def
tff(fact_7682_sorted__wrt01,axiom,
    ! [A: $tType,Xs2: list(A),P: fun(A,fun(A,$o))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs2)),one_one(nat))
     => sorted_wrt(A,P,Xs2) ) ).

% sorted_wrt01
tff(fact_7683_sorted__map,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs2: list(B)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F2),Xs2))
        <=> sorted_wrt(B,aTP_Lamp_ale(fun(B,A),fun(B,fun(B,$o)),F2),Xs2) ) ) ).

% sorted_map
tff(fact_7684_sorted__distinct__set__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs2: list(A),Ysb: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs2)
         => ( distinct(A,Xs2)
           => ( sorted_wrt(A,ord_less_eq(A),Ysb)
             => ( distinct(A,Ysb)
               => ( ( aa(list(A),set(A),set2(A),Xs2) = aa(list(A),set(A),set2(A),Ysb) )
                 => ( Xs2 = Ysb ) ) ) ) ) ) ) ).

% sorted_distinct_set_unique
tff(fact_7685_sorted__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Ysb: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ysb))
        <=> ( ! [X6: A] :
                ( member(A,X6,aa(list(A),set(A),set2(A),Ysb))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),X6) )
            & sorted_wrt(A,ord_less_eq(A),Ysb) ) ) ) ).

% sorted_simps(2)
tff(fact_7686_sorted__wrt__iff__nth__less,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),Xs2: list(A)] :
      ( sorted_wrt(A,P,Xs2)
    <=> ! [I4: nat,J3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),J3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs2))
           => aa(A,$o,aa(A,fun(A,$o),P,aa(nat,A,nth(A,Xs2),I4)),aa(nat,A,nth(A,Xs2),J3)) ) ) ) ).

% sorted_wrt_iff_nth_less
tff(fact_7687_sorted__wrt__nth__less,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),Xs2: list(A),I: nat,J: nat] :
      ( sorted_wrt(A,P,Xs2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs2))
         => aa(A,$o,aa(A,fun(A,$o),P,aa(nat,A,nth(A,Xs2),I)),aa(nat,A,nth(A,Xs2),J)) ) ) ) ).

% sorted_wrt_nth_less
tff(fact_7688_strict__sorted__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Ysb: list(A)] :
          ( sorted_wrt(A,ord_less(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ysb))
        <=> ( ! [X6: A] :
                ( member(A,X6,aa(list(A),set(A),set2(A),Ysb))
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),X6) )
            & sorted_wrt(A,ord_less(A),Ysb) ) ) ) ).

% strict_sorted_simps(2)
tff(fact_7689_strict__sorted__equal,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs2: list(A),Ysb: list(A)] :
          ( sorted_wrt(A,ord_less(A),Xs2)
         => ( sorted_wrt(A,ord_less(A),Ysb)
           => ( ( aa(list(A),set(A),set2(A),Ysb) = aa(list(A),set(A),set2(A),Xs2) )
             => ( Ysb = Xs2 ) ) ) ) ) ).

% strict_sorted_equal
tff(fact_7690_sorted__wrt__mono__rel,axiom,
    ! [A: $tType,Xs2: list(A),P: fun(A,fun(A,$o)),Q: fun(A,fun(A,$o))] :
      ( ! [X2: A,Y2: A] :
          ( member(A,X2,aa(list(A),set(A),set2(A),Xs2))
         => ( member(A,Y2,aa(list(A),set(A),set2(A),Xs2))
           => ( aa(A,$o,aa(A,fun(A,$o),P,X2),Y2)
             => aa(A,$o,aa(A,fun(A,$o),Q,X2),Y2) ) ) )
     => ( sorted_wrt(A,P,Xs2)
       => sorted_wrt(A,Q,Xs2) ) ) ).

% sorted_wrt_mono_rel
tff(fact_7691_signed_Osorted__map,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(A)
     => ! [F2: fun(B,word(A)),Xs2: list(B)] :
          ( sorted_wrt(word(A),word_sle(A),aa(list(B),list(word(A)),map(B,word(A),F2),Xs2))
        <=> sorted_wrt(B,aTP_Lamp_alf(fun(B,word(A)),fun(B,fun(B,$o)),F2),Xs2) ) ) ).

% signed.sorted_map
tff(fact_7692_sorted__wrt__map,axiom,
    ! [A: $tType,B: $tType,R3: fun(A,fun(A,$o)),F2: fun(B,A),Xs2: list(B)] :
      ( sorted_wrt(A,R3,aa(list(B),list(A),map(B,A,F2),Xs2))
    <=> sorted_wrt(B,aa(fun(B,A),fun(B,fun(B,$o)),aTP_Lamp_alg(fun(A,fun(A,$o)),fun(fun(B,A),fun(B,fun(B,$o))),R3),F2),Xs2) ) ).

% sorted_wrt_map
tff(fact_7693_fst__divmod,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Na: num] : aa(product_prod(A,A),A,product_fst(A,A),unique8689654367752047608divmod(A,M,Na)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Na)) ) ).

% fst_divmod
tff(fact_7694_sorted__wrt_Osimps_I1_J,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o))] : sorted_wrt(A,P,nil(A)) ).

% sorted_wrt.simps(1)
tff(fact_7695_sorted__wrt1,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),X: A] : sorted_wrt(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) ).

% sorted_wrt1
tff(fact_7696_sorted__wrt__true,axiom,
    ! [A: $tType,Xs2: list(A)] : sorted_wrt(A,aTP_Lamp_alh(A,fun(A,$o)),Xs2) ).

% sorted_wrt_true
tff(fact_7697_continuous__fst,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F4: filter(A),F2: fun(A,product_prod(B,C))] :
          ( topolo3448309680560233919inuous(A,product_prod(B,C),F4,F2)
         => topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_ali(fun(A,product_prod(B,C)),fun(A,B),F2)) ) ) ).

% continuous_fst
tff(fact_7698_strict__sorted__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => sorted_wrt(A,ord_less(A),nil(A)) ) ).

% strict_sorted_simps(1)
tff(fact_7699_sorted1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) ) ).

% sorted1
tff(fact_7700_sorted0,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => sorted_wrt(A,ord_less_eq(A),nil(A)) ) ).

% sorted0
tff(fact_7701_strict__sorted__imp__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs2: list(A)] :
          ( sorted_wrt(A,ord_less(A),Xs2)
         => sorted_wrt(A,ord_less_eq(A),Xs2) ) ) ).

% strict_sorted_imp_sorted
tff(fact_7702_sorted2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Zs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y)
            & sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)) ) ) ) ).

% sorted2
tff(fact_7703_strict__sorted__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A)] :
          ( sorted_wrt(A,ord_less(A),L)
        <=> ( sorted_wrt(A,ord_less_eq(A),L)
            & distinct(A,L) ) ) ) ).

% strict_sorted_iff
tff(fact_7704_fst__image__mp,axiom,
    ! [B: $tType,A: $tType,A2: set(product_prod(A,B)),B2: set(A),X: A,Y: B] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(product_prod(A,B)),set(A),image(product_prod(A,B),A,product_fst(A,B)),A2)),B2)
     => ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y),A2)
       => member(A,X,B2) ) ) ).

% fst_image_mp
tff(fact_7705_in__fst__imageE,axiom,
    ! [B: $tType,A: $tType,X: A,Sa: set(product_prod(A,B))] :
      ( member(A,X,aa(set(product_prod(A,B)),set(A),image(product_prod(A,B),A,product_fst(A,B)),Sa))
     => ~ ! [Y2: B] : ~ member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y2),Sa) ) ).

% in_fst_imageE
tff(fact_7706_fstE,axiom,
    ! [B: $tType,A: $tType,X: product_prod(A,B),Aa2: A,Ba: B,P: fun(A,$o)] :
      ( ( X = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba) )
     => ( aa(A,$o,P,aa(product_prod(A,B),A,product_fst(A,B),X))
       => aa(A,$o,P,Aa2) ) ) ).

% fstE
tff(fact_7707_distinct__map__fstD,axiom,
    ! [A: $tType,B: $tType,Xs2: list(product_prod(A,B)),X: A,Y: B,Z: B] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs2))
     => ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs2))
       => ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Z),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs2))
         => ( Y = Z ) ) ) ) ).

% distinct_map_fstD
tff(fact_7708_sorted01,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs2: list(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs2)),one_one(nat))
         => sorted_wrt(A,ord_less_eq(A),Xs2) ) ) ).

% sorted01
tff(fact_7709_sorted__iff__nth__mono__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs2: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs2)
        <=> ! [I4: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),J3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs2))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs2),I4)),aa(nat,A,nth(A,Xs2),J3)) ) ) ) ) ).

% sorted_iff_nth_mono_less
tff(fact_7710_finite__sorted__distinct__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A2)
         => ? [X2: list(A)] :
              ( ( aa(list(A),set(A),set2(A),X2) = A2 )
              & sorted_wrt(A,ord_less_eq(A),X2)
              & distinct(A,X2)
              & ! [Y3: list(A)] :
                  ( ( ( aa(list(A),set(A),set2(A),Y3) = A2 )
                    & sorted_wrt(A,ord_less_eq(A),Y3)
                    & distinct(A,Y3) )
                 => ( Y3 = X2 ) ) ) ) ) ).

% finite_sorted_distinct_unique
tff(fact_7711_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs2: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs2)
         => ( distinct(A,Xs2)
           => ( linord4507533701916653071of_set(A,aa(list(A),set(A),set2(A),Xs2)) = Xs2 ) ) ) ) ).

% sorted_list_of_set.idem_if_sorted_distinct
tff(fact_7712_sorted__wrt__less__idx,axiom,
    ! [Ns: list(nat),I: nat] :
      ( sorted_wrt(nat,ord_less(nat),Ns)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(nat),nat,size_size(list(nat)),Ns))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),aa(nat,nat,nth(nat,Ns),I)) ) ) ).

% sorted_wrt_less_idx
tff(fact_7713_sorted__iff__nth__Suc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs2: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs2)
        <=> ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I4)),aa(list(A),nat,size_size(list(A)),Xs2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs2),I4)),aa(nat,A,nth(A,Xs2),aa(nat,nat,suc,I4))) ) ) ) ).

% sorted_iff_nth_Suc
tff(fact_7714_sorted__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs2: list(A),I: nat,J: nat] :
          ( sorted_wrt(A,ord_less_eq(A),Xs2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs2),I)),aa(nat,A,nth(A,Xs2),J)) ) ) ) ) ).

% sorted_nth_mono
tff(fact_7715_sorted__iff__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs2: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs2)
        <=> ! [I4: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I4),J3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs2))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs2),I4)),aa(nat,A,nth(A,Xs2),J3)) ) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_7716_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A2)
         => ~ ! [L2: list(A)] :
                ( sorted_wrt(A,ord_less(A),L2)
               => ( ( aa(list(A),set(A),set2(A),L2) = A2 )
                 => ( aa(list(A),nat,size_size(list(A)),L2) != finite_card(A,A2) ) ) ) ) ) ).

% sorted_list_of_set.finite_set_strict_sorted
tff(fact_7717_isCont__fst,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [Aa2: A,F2: fun(A,product_prod(B,C))] :
          ( topolo3448309680560233919inuous(A,product_prod(B,C),topolo174197925503356063within(A,Aa2,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Aa2,top_top(set(A))),aTP_Lamp_ali(fun(A,product_prod(B,C)),fun(A,B),F2)) ) ) ).

% isCont_fst
tff(fact_7718_distinct__sorted__strict__mono__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A),I: nat,J: nat] :
          ( distinct(A,L)
         => ( sorted_wrt(A,ord_less_eq(A),L)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),L))
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,nth(A,L),I)),aa(nat,A,nth(A,L),J))
                <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J) ) ) ) ) ) ) ).

% distinct_sorted_strict_mono_iff
tff(fact_7719_distinct__sorted__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A),I: nat,J: nat] :
          ( sorted_wrt(A,ord_less_eq(A),L)
         => ( distinct(A,L)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),L))
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,nth(A,L),I)),aa(nat,A,nth(A,L),J)) ) ) ) ) ) ).

% distinct_sorted_mono
tff(fact_7720_distinct__sorted__mono__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A),I: nat,J: nat] :
          ( distinct(A,L)
         => ( sorted_wrt(A,ord_less_eq(A),L)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),L))
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,L),I)),aa(nat,A,nth(A,L),J))
                <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J) ) ) ) ) ) ) ).

% distinct_sorted_mono_iff
tff(fact_7721_map__sorted__distinct__set__unique,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xs2: list(A),Ysb: list(A)] :
          ( inj_on(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs2)),aa(list(A),set(A),set2(A),Ysb)))
         => ( sorted_wrt(B,ord_less_eq(B),aa(list(A),list(B),map(A,B,F2),Xs2))
           => ( distinct(B,aa(list(A),list(B),map(A,B,F2),Xs2))
             => ( sorted_wrt(B,ord_less_eq(B),aa(list(A),list(B),map(A,B,F2),Ysb))
               => ( distinct(B,aa(list(A),list(B),map(A,B,F2),Ysb))
                 => ( ( aa(list(A),set(A),set2(A),Xs2) = aa(list(A),set(A),set2(A),Ysb) )
                   => ( Xs2 = Ysb ) ) ) ) ) ) ) ) ).

% map_sorted_distinct_set_unique
tff(fact_7722_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: set(A),L: list(A)] :
          ( aa(set(A),$o,finite_finite2(A),A2)
         => ( ( sorted_wrt(A,ord_less(A),L)
              & ( aa(list(A),set(A),set2(A),L) = A2 )
              & ( aa(list(A),nat,size_size(list(A)),L) = finite_card(A,A2) ) )
          <=> ( linord4507533701916653071of_set(A,A2) = L ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_unique
tff(fact_7723_tendsto__fst,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,product_prod(B,C)),Aa2: product_prod(B,C),F4: filter(A)] :
          ( filterlim(A,product_prod(B,C),F2,topolo7230453075368039082e_nhds(product_prod(B,C),Aa2),F4)
         => filterlim(A,B,aTP_Lamp_alj(fun(A,product_prod(B,C)),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(product_prod(B,C),B,product_fst(B,C),Aa2)),F4) ) ) ).

% tendsto_fst
tff(fact_7724_sorted__in__between,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [I: nat,J: nat,L: list(A),X: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),I)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),L))
             => ( sorted_wrt(A,ord_less_eq(A),L)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,L),I)),X)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(nat,A,nth(A,L),J))
                   => ~ ! [K5: nat] :
                          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),K5)
                         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K5),J)
                           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,L),K5)),X)
                             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(nat,A,nth(A,L),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K5),one_one(nat)))) ) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
tff(fact_7725_mergesort__remdups__correct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A)] :
          ( distinct(A,mergesort_remdups(A,L))
          & sorted_wrt(A,ord_less_eq(A),mergesort_remdups(A,L))
          & ( aa(list(A),set(A),set2(A),mergesort_remdups(A,L)) = aa(list(A),set(A),set2(A),L) ) ) ) ).

% mergesort_remdups_correct
tff(fact_7726_folding__insort__key_Ofinite__set__strict__sorted,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Sa: set(B),F2: fun(B,A),A2: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,Sa,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A2),Sa)
       => ( aa(set(B),$o,finite_finite2(B),A2)
         => ~ ! [L2: list(B)] :
                ( sorted_wrt(A,Less,aa(list(B),list(A),map(B,A,F2),L2))
               => ( ( aa(list(B),set(B),set2(B),L2) = A2 )
                 => ( aa(list(B),nat,size_size(list(B)),L2) != finite_card(B,A2) ) ) ) ) ) ) ).

% folding_insort_key.finite_set_strict_sorted
tff(fact_7727_signed_Ostrict__sorted__equal,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs2: list(word(A)),Ysb: list(word(A))] :
          ( sorted_wrt(word(A),word_sless(A),Xs2)
         => ( sorted_wrt(word(A),word_sless(A),Ysb)
           => ( ( aa(list(word(A)),set(word(A)),set2(word(A)),Ysb) = aa(list(word(A)),set(word(A)),set2(word(A)),Xs2) )
             => ( Ysb = Xs2 ) ) ) ) ) ).

% signed.strict_sorted_equal
tff(fact_7728_folding__insort__key_Odistinct__if__distinct__map,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Sa: set(B),F2: fun(B,A),Xs2: list(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,Sa,F2)
     => ( distinct(A,aa(list(B),list(A),map(B,A,F2),Xs2))
       => distinct(B,Xs2) ) ) ).

% folding_insort_key.distinct_if_distinct_map
tff(fact_7729_signed_Osorted__distinct__set__unique,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs2: list(word(A)),Ysb: list(word(A))] :
          ( sorted_wrt(word(A),word_sle(A),Xs2)
         => ( distinct(word(A),Xs2)
           => ( sorted_wrt(word(A),word_sle(A),Ysb)
             => ( distinct(word(A),Ysb)
               => ( ( aa(list(word(A)),set(word(A)),set2(word(A)),Xs2) = aa(list(word(A)),set(word(A)),set2(word(A)),Ysb) )
                 => ( Xs2 = Ysb ) ) ) ) ) ) ) ).

% signed.sorted_distinct_set_unique
tff(fact_7730_signed_Ofinite__sorted__distinct__unique,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: set(word(A))] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A2)
         => ? [X2: list(word(A))] :
              ( ( aa(list(word(A)),set(word(A)),set2(word(A)),X2) = A2 )
              & sorted_wrt(word(A),word_sle(A),X2)
              & distinct(word(A),X2)
              & ! [Y3: list(word(A))] :
                  ( ( ( aa(list(word(A)),set(word(A)),set2(word(A)),Y3) = A2 )
                    & sorted_wrt(word(A),word_sle(A),Y3)
                    & distinct(word(A),Y3) )
                 => ( Y3 = X2 ) ) ) ) ) ).

% signed.finite_sorted_distinct_unique
tff(fact_7731_signed_Ostrict__sorted__simps_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Ysb: list(word(A))] :
          ( sorted_wrt(word(A),word_sless(A),aa(list(word(A)),list(word(A)),aa(word(A),fun(list(word(A)),list(word(A))),cons(word(A)),X),Ysb))
        <=> ( ! [X6: word(A)] :
                ( member(word(A),X6,aa(list(word(A)),set(word(A)),set2(word(A)),Ysb))
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),X),X6) )
            & sorted_wrt(word(A),word_sless(A),Ysb) ) ) ) ).

% signed.strict_sorted_simps(2)
tff(fact_7732_signed_Osorted__simps_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Ysb: list(word(A))] :
          ( sorted_wrt(word(A),word_sle(A),aa(list(word(A)),list(word(A)),aa(word(A),fun(list(word(A)),list(word(A))),cons(word(A)),X),Ysb))
        <=> ( ! [X6: word(A)] :
                ( member(word(A),X6,aa(list(word(A)),set(word(A)),set2(word(A)),Ysb))
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),X),X6) )
            & sorted_wrt(word(A),word_sle(A),Ysb) ) ) ) ).

% signed.sorted_simps(2)
tff(fact_7733_signed_Ostrict__sorted__simps_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => sorted_wrt(word(A),word_sless(A),nil(word(A))) ) ).

% signed.strict_sorted_simps(1)
tff(fact_7734_signed_Osorted0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => sorted_wrt(word(A),word_sle(A),nil(word(A))) ) ).

% signed.sorted0
tff(fact_7735_signed_Osorted2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A),Zs: list(word(A))] :
          ( sorted_wrt(word(A),word_sle(A),aa(list(word(A)),list(word(A)),aa(word(A),fun(list(word(A)),list(word(A))),cons(word(A)),X),aa(list(word(A)),list(word(A)),aa(word(A),fun(list(word(A)),list(word(A))),cons(word(A)),Y),Zs)))
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),X),Y)
            & sorted_wrt(word(A),word_sle(A),aa(list(word(A)),list(word(A)),aa(word(A),fun(list(word(A)),list(word(A))),cons(word(A)),Y),Zs)) ) ) ) ).

% signed.sorted2
tff(fact_7736_signed_Osorted1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A)] : sorted_wrt(word(A),word_sle(A),aa(list(word(A)),list(word(A)),aa(word(A),fun(list(word(A)),list(word(A))),cons(word(A)),X),nil(word(A)))) ) ).

% signed.sorted1
tff(fact_7737_signed_Ostrict__sorted__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [L: list(word(A))] :
          ( sorted_wrt(word(A),word_sless(A),L)
        <=> ( sorted_wrt(word(A),word_sle(A),L)
            & distinct(word(A),L) ) ) ) ).

% signed.strict_sorted_iff
tff(fact_7738_signed_Ostrict__sorted__imp__sorted,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs2: list(word(A))] :
          ( sorted_wrt(word(A),word_sless(A),Xs2)
         => sorted_wrt(word(A),word_sle(A),Xs2) ) ) ).

% signed.strict_sorted_imp_sorted
tff(fact_7739_sorted__wrt__upto,axiom,
    ! [I: int,J: int] : sorted_wrt(int,ord_less(int),upto(I,J)) ).

% sorted_wrt_upto
tff(fact_7740_sorted__upto,axiom,
    ! [M: int,Na: int] : sorted_wrt(int,ord_less_eq(int),upto(M,Na)) ).

% sorted_upto
tff(fact_7741_folding__insort__key_Oinj__on,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Sa: set(B),F2: fun(B,A)] :
      ( folding_insort_key(A,B,Less_eq,Less,Sa,F2)
     => inj_on(B,A,F2,Sa) ) ).

% folding_insort_key.inj_on
tff(fact_7742_signed_Osorted01,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs2: list(word(A))] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(word(A)),nat,size_size(list(word(A))),Xs2)),one_one(nat))
         => sorted_wrt(word(A),word_sle(A),Xs2) ) ) ).

% signed.sorted01
tff(fact_7743_signed_Omap__sorted__distinct__set__unique,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [F2: fun(A,word(B)),Xs2: list(A),Ysb: list(A)] :
          ( inj_on(A,word(B),F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs2)),aa(list(A),set(A),set2(A),Ysb)))
         => ( sorted_wrt(word(B),word_sle(B),aa(list(A),list(word(B)),map(A,word(B),F2),Xs2))
           => ( distinct(word(B),aa(list(A),list(word(B)),map(A,word(B),F2),Xs2))
             => ( sorted_wrt(word(B),word_sle(B),aa(list(A),list(word(B)),map(A,word(B),F2),Ysb))
               => ( distinct(word(B),aa(list(A),list(word(B)),map(A,word(B),F2),Ysb))
                 => ( ( aa(list(A),set(A),set2(A),Xs2) = aa(list(A),set(A),set2(A),Ysb) )
                   => ( Xs2 = Ysb ) ) ) ) ) ) ) ) ).

% signed.map_sorted_distinct_set_unique
tff(fact_7744_signed_Osorted__iff__nth__mono__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs2: list(word(A))] :
          ( sorted_wrt(word(A),word_sle(A),Xs2)
        <=> ! [I4: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),J3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(word(A)),nat,size_size(list(word(A))),Xs2))
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(nat,word(A),nth(word(A),Xs2),I4)),aa(nat,word(A),nth(word(A),Xs2),J3)) ) ) ) ) ).

% signed.sorted_iff_nth_mono_less
tff(fact_7745_rat__sgn__code,axiom,
    ! [P2: rat] : quotient_of(aa(rat,rat,sgn_sgn(rat),P2)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,sgn_sgn(int),aa(product_prod(int,int),int,product_fst(int,int),quotient_of(P2)))),one_one(int)) ).

% rat_sgn_code
tff(fact_7746_signed_Osorted__list__of__set_Ofinite__set__strict__sorted,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: set(word(A))] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A2)
         => ~ ! [L2: list(word(A))] :
                ( sorted_wrt(word(A),word_sless(A),L2)
               => ( ( aa(list(word(A)),set(word(A)),set2(word(A)),L2) = A2 )
                 => ( aa(list(word(A)),nat,size_size(list(word(A))),L2) != finite_card(word(A),A2) ) ) ) ) ) ).

% signed.sorted_list_of_set.finite_set_strict_sorted
tff(fact_7747_sorted__list__of__set_Ofolding__insort__key__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => folding_insort_key(A,A,ord_less_eq(A),ord_less(A),top_top(set(A)),aTP_Lamp_qx(A,A)) ) ).

% sorted_list_of_set.folding_insort_key_axioms
tff(fact_7748_signed_Osorted__list__of__set_Ofolding__insort__key__axioms,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => folding_insort_key(word(A),word(A),word_sle(A),word_sless(A),top_top(set(word(A))),aTP_Lamp_rf(word(A),word(A))) ) ).

% signed.sorted_list_of_set.folding_insort_key_axioms
tff(fact_7749_signed_Osorted__iff__nth__Suc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs2: list(word(A))] :
          ( sorted_wrt(word(A),word_sle(A),Xs2)
        <=> ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I4)),aa(list(word(A)),nat,size_size(list(word(A))),Xs2))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(nat,word(A),nth(word(A),Xs2),I4)),aa(nat,word(A),nth(word(A),Xs2),aa(nat,nat,suc,I4))) ) ) ) ).

% signed.sorted_iff_nth_Suc
tff(fact_7750_signed_Osorted__nth__mono,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs2: list(word(A)),I: nat,J: nat] :
          ( sorted_wrt(word(A),word_sle(A),Xs2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(word(A)),nat,size_size(list(word(A))),Xs2))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(nat,word(A),nth(word(A),Xs2),I)),aa(nat,word(A),nth(word(A),Xs2),J)) ) ) ) ) ).

% signed.sorted_nth_mono
tff(fact_7751_signed_Osorted__iff__nth__mono,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs2: list(word(A))] :
          ( sorted_wrt(word(A),word_sle(A),Xs2)
        <=> ! [I4: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I4),J3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(word(A)),nat,size_size(list(word(A))),Xs2))
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(nat,word(A),nth(word(A),Xs2),I4)),aa(nat,word(A),nth(word(A),Xs2),J3)) ) ) ) ) ).

% signed.sorted_iff_nth_mono
tff(fact_7752_folding__insort__key_Osorted__key__list__of__set__unique,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Sa: set(B),F2: fun(B,A),A2: set(B),L: list(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,Sa,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A2),Sa)
       => ( aa(set(B),$o,finite_finite2(B),A2)
         => ( ( sorted_wrt(A,Less,aa(list(B),list(A),map(B,A,F2),L))
              & ( aa(list(B),set(B),set2(B),L) = A2 )
              & ( aa(list(B),nat,size_size(list(B)),L) = finite_card(B,A2) ) )
          <=> ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),A2) = L ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_unique
tff(fact_7753_signed_Osorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: set(word(A)),L: list(word(A))] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A2)
         => ( ( sorted_wrt(word(A),word_sless(A),L)
              & ( aa(list(word(A)),set(word(A)),set2(word(A)),L) = A2 )
              & ( aa(list(word(A)),nat,size_size(list(word(A))),L) = finite_card(word(A),A2) ) )
          <=> ( aa(set(word(A)),list(word(A)),sorted_list_of_set(word(A),word_sle(A)),A2) = L ) ) ) ) ).

% signed.sorted_list_of_set.sorted_key_list_of_set_unique
tff(fact_7754_signed_Osorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(set(word(A)),list(word(A)),sorted_list_of_set(word(A),word_sle(A)),bot_bot(set(word(A)))) = nil(word(A)) ) ) ).

% signed.sorted_list_of_set.sorted_key_list_of_set_empty
tff(fact_7755_signed_Osorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: set(word(A))] :
          ( ~ aa(set(word(A)),$o,finite_finite2(word(A)),A2)
         => ( aa(set(word(A)),list(word(A)),sorted_list_of_set(word(A),word_sle(A)),A2) = nil(word(A)) ) ) ) ).

% signed.sorted_list_of_set.fold_insort_key.infinite
tff(fact_7756_signed_Osorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: set(word(A))] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A2)
         => ( aa(list(word(A)),set(word(A)),set2(word(A)),aa(set(word(A)),list(word(A)),sorted_list_of_set(word(A),word_sle(A)),A2)) = A2 ) ) ) ).

% signed.sorted_list_of_set.set_sorted_key_list_of_set
tff(fact_7757_signed_Osorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: set(word(A))] : aa(list(word(A)),nat,size_size(list(word(A))),aa(set(word(A)),list(word(A)),sorted_list_of_set(word(A),word_sle(A)),A2)) = finite_card(word(A),A2) ) ).

% signed.sorted_list_of_set.length_sorted_key_list_of_set
tff(fact_7758_signed_Osorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: set(word(A))] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A2)
         => ( ( aa(set(word(A)),list(word(A)),sorted_list_of_set(word(A),word_sle(A)),A2) = nil(word(A)) )
          <=> ( A2 = bot_bot(set(word(A))) ) ) ) ) ).

% signed.sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
tff(fact_7759_signed_Osorted__list__of__set_Oidem__if__sorted__distinct,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs2: list(word(A))] :
          ( sorted_wrt(word(A),word_sle(A),Xs2)
         => ( distinct(word(A),Xs2)
           => ( aa(set(word(A)),list(word(A)),sorted_list_of_set(word(A),word_sle(A)),aa(list(word(A)),set(word(A)),set2(word(A)),Xs2)) = Xs2 ) ) ) ) ).

% signed.sorted_list_of_set.idem_if_sorted_distinct
tff(fact_7760_signed_Osorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: set(word(A))] : sorted_wrt(word(A),word_sle(A),aa(set(word(A)),list(word(A)),sorted_list_of_set(word(A),word_sle(A)),A2)) ) ).

% signed.sorted_list_of_set.sorted_sorted_key_list_of_set
tff(fact_7761_signed_Osorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: set(word(A)),B2: set(word(A))] :
          ( ( aa(set(word(A)),list(word(A)),sorted_list_of_set(word(A),word_sle(A)),A2) = aa(set(word(A)),list(word(A)),sorted_list_of_set(word(A),word_sle(A)),B2) )
         => ( aa(set(word(A)),$o,finite_finite2(word(A)),A2)
           => ( aa(set(word(A)),$o,finite_finite2(word(A)),B2)
             => ( A2 = B2 ) ) ) ) ) ).

% signed.sorted_list_of_set.sorted_key_list_of_set_inject
tff(fact_7762_signed_Osorted__list__of__set_Odistinct__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: set(word(A))] : distinct(word(A),aa(set(word(A)),list(word(A)),sorted_list_of_set(word(A),word_sle(A)),A2)) ) ).

% signed.sorted_list_of_set.distinct_sorted_key_list_of_set
tff(fact_7763_signed_Osorted__list__of__set_Odistinct__if__distinct__map,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs2: list(word(A))] :
          ( distinct(word(A),Xs2)
         => distinct(word(A),Xs2) ) ) ).

% signed.sorted_list_of_set.distinct_if_distinct_map
tff(fact_7764_signed_Osorted__list__of__set__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( sorted_list_of_set(word(A),word_sle(A)) = aa(fun(word(A),word(A)),fun(set(word(A)),list(word(A))),sorted8670434370408473282of_set(word(A),word(A),word_sle(A)),aTP_Lamp_rf(word(A),word(A))) ) ) ).

% signed.sorted_list_of_set_def
tff(fact_7765_linorder_Osorted__key__list__of__set_Ocong,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(B,fun(B,$o))] : sorted8670434370408473282of_set(B,A,Less_eq) = sorted8670434370408473282of_set(B,A,Less_eq) ).

% linorder.sorted_key_list_of_set.cong
tff(fact_7766_linorder_Osorted__list__of__set_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o))] : sorted_list_of_set(A,Less_eq) = sorted_list_of_set(A,Less_eq) ).

% linorder.sorted_list_of_set.cong
tff(fact_7767_signed_Osorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: set(word(A))] : sorted_wrt(word(A),word_sless(A),aa(set(word(A)),list(word(A)),sorted_list_of_set(word(A),word_sle(A)),A2)) ) ).

% signed.sorted_list_of_set.strict_sorted_key_list_of_set
tff(fact_7768_folding__insort__key_Osorted__key__list__of__set__empty,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Sa: set(B),F2: fun(B,A)] :
      ( folding_insort_key(A,B,Less_eq,Less,Sa,F2)
     => ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),bot_bot(set(B))) = nil(B) ) ) ).

% folding_insort_key.sorted_key_list_of_set_empty
tff(fact_7769_folding__insort__key_Osorted__key__list__of__set__inject,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Sa: set(B),F2: fun(B,A),A2: set(B),B2: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,Sa,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A2),Sa)
       => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B2),Sa)
         => ( ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),A2) = aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),B2) )
           => ( aa(set(B),$o,finite_finite2(B),A2)
             => ( aa(set(B),$o,finite_finite2(B),B2)
               => ( A2 = B2 ) ) ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_inject
tff(fact_7770_folding__insort__key_Oset__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Sa: set(B),F2: fun(B,A),A2: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,Sa,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A2),Sa)
       => ( aa(set(B),$o,finite_finite2(B),A2)
         => ( aa(list(B),set(B),set2(B),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),A2)) = A2 ) ) ) ) ).

% folding_insort_key.set_sorted_key_list_of_set
tff(fact_7771_folding__insort__key_Olength__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Sa: set(B),F2: fun(B,A),A2: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,Sa,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A2),Sa)
       => ( aa(list(B),nat,size_size(list(B)),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),A2)) = finite_card(B,A2) ) ) ) ).

% folding_insort_key.length_sorted_key_list_of_set
tff(fact_7772_folding__insort__key_Ostrict__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Sa: set(B),F2: fun(B,A),A2: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,Sa,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A2),Sa)
       => sorted_wrt(A,Less,aa(list(B),list(A),map(B,A,F2),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),A2))) ) ) ).

% folding_insort_key.strict_sorted_key_list_of_set
tff(fact_7773_folding__insort__key_Osorted__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Sa: set(B),F2: fun(B,A),A2: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,Sa,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A2),Sa)
       => sorted_wrt(A,Less_eq,aa(list(B),list(A),map(B,A,F2),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),A2))) ) ) ).

% folding_insort_key.sorted_sorted_key_list_of_set
tff(fact_7774_folding__insort__key_Odistinct__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Sa: set(B),F2: fun(B,A),A2: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,Sa,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A2),Sa)
       => distinct(A,aa(list(B),list(A),map(B,A,F2),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),A2))) ) ) ).

% folding_insort_key.distinct_sorted_key_list_of_set
tff(fact_7775_folding__insort__key_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Sa: set(B),F2: fun(B,A),A2: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,Sa,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A2),Sa)
       => ( aa(set(B),$o,finite_finite2(B),A2)
         => ( ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),A2) = nil(B) )
          <=> ( A2 = bot_bot(set(B)) ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_eq_Nil_iff
tff(fact_7776_folding__insort__key_Oidem__if__sorted__distinct,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Sa: set(B),F2: fun(B,A),Xs2: list(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,Sa,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(list(B),set(B),set2(B),Xs2)),Sa)
       => ( sorted_wrt(A,Less_eq,aa(list(B),list(A),map(B,A,F2),Xs2))
         => ( distinct(B,Xs2)
           => ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),aa(list(B),set(B),set2(B),Xs2)) = Xs2 ) ) ) ) ) ).

% folding_insort_key.idem_if_sorted_distinct
tff(fact_7777_folding__insort__key_Osorted__key__list__of__set__insert__remove,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Sa: set(B),F2: fun(B,A),X: B,A2: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,Sa,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),insert(B,X),A2)),Sa)
       => ( aa(set(B),$o,finite_finite2(B),A2)
         => ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),aa(set(B),set(B),insert(B,X),A2)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F2),X),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),aa(set(B),set(B),minus_minus(set(B),A2),aa(set(B),set(B),insert(B,X),bot_bot(set(B)))))) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_insert_remove
tff(fact_7778_folding__insort__key_Osorted__key__list__of__set__insert,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Sa: set(B),F2: fun(B,A),X: B,A2: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,Sa,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),insert(B,X),A2)),Sa)
       => ( aa(set(B),$o,finite_finite2(B),A2)
         => ( ~ member(B,X,A2)
           => ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),aa(set(B),set(B),insert(B,X),A2)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F2),X),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),A2)) ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_insert
tff(fact_7779_signed_Olength__insort,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [F2: fun(A,word(B)),X: A,Xs2: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),aa(fun(A,word(B)),fun(A,fun(list(A),list(A))),insort_key(word(B),A,word_sle(B)),F2),X),Xs2)) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs2)) ) ).

% signed.length_insort
tff(fact_7780_signed_Osorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: set(word(A)),X: word(A)] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A2)
         => ( ~ member(word(A),X,A2)
           => ( aa(set(word(A)),list(word(A)),sorted_list_of_set(word(A),word_sle(A)),aa(set(word(A)),set(word(A)),insert(word(A),X),A2)) = aa(list(word(A)),list(word(A)),aa(word(A),fun(list(word(A)),list(word(A))),aa(fun(word(A),word(A)),fun(word(A),fun(list(word(A)),list(word(A)))),insort_key(word(A),word(A),word_sle(A)),aTP_Lamp_rf(word(A),word(A))),X),aa(set(word(A)),list(word(A)),sorted_list_of_set(word(A),word_sle(A)),A2)) ) ) ) ) ).

% signed.sorted_list_of_set.sorted_key_list_of_set_insert
tff(fact_7781_signed_Osorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: set(word(A)),X: word(A)] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A2)
         => ( aa(set(word(A)),list(word(A)),sorted_list_of_set(word(A),word_sle(A)),aa(set(word(A)),set(word(A)),insert(word(A),X),A2)) = aa(list(word(A)),list(word(A)),aa(word(A),fun(list(word(A)),list(word(A))),aa(fun(word(A),word(A)),fun(word(A),fun(list(word(A)),list(word(A)))),insort_key(word(A),word(A),word_sle(A)),aTP_Lamp_rf(word(A),word(A))),X),aa(set(word(A)),list(word(A)),sorted_list_of_set(word(A),word_sle(A)),aa(set(word(A)),set(word(A)),minus_minus(set(word(A)),A2),aa(set(word(A)),set(word(A)),insert(word(A),X),bot_bot(set(word(A))))))) ) ) ) ).

% signed.sorted_list_of_set.sorted_key_list_of_set_insert_remove
tff(fact_7782_signed_Osorted__insort,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Xs2: list(word(A))] :
          ( sorted_wrt(word(A),word_sle(A),aa(list(word(A)),list(word(A)),aa(word(A),fun(list(word(A)),list(word(A))),aa(fun(word(A),word(A)),fun(word(A),fun(list(word(A)),list(word(A)))),insort_key(word(A),word(A),word_sle(A)),aTP_Lamp_rf(word(A),word(A))),X),Xs2))
        <=> sorted_wrt(word(A),word_sle(A),Xs2) ) ) ).

% signed.sorted_insort
tff(fact_7783_signed_Oinsort__not__Nil,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [F2: fun(A,word(B)),Aa2: A,Xs2: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aa(fun(A,word(B)),fun(A,fun(list(A),list(A))),insort_key(word(B),A,word_sle(B)),F2),Aa2),Xs2) != nil(A) ) ).

% signed.insort_not_Nil
tff(fact_7784_linorder_Oinsort__key_Ocong,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(B,fun(B,$o))] : insort_key(B,A,Less_eq) = insort_key(B,A,Less_eq) ).

% linorder.insort_key.cong
tff(fact_7785_signed_Oinsort__key__left__comm,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(A)
     => ! [F2: fun(B,word(A)),X: B,Y: B,Xs2: list(B)] :
          ( ( aa(B,word(A),F2,X) != aa(B,word(A),F2,Y) )
         => ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),aa(fun(B,word(A)),fun(B,fun(list(B),list(B))),insort_key(word(A),B,word_sle(A)),F2),Y),aa(list(B),list(B),aa(B,fun(list(B),list(B)),aa(fun(B,word(A)),fun(B,fun(list(B),list(B))),insort_key(word(A),B,word_sle(A)),F2),X),Xs2)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),aa(fun(B,word(A)),fun(B,fun(list(B),list(B))),insort_key(word(A),B,word_sle(A)),F2),X),aa(list(B),list(B),aa(B,fun(list(B),list(B)),aa(fun(B,word(A)),fun(B,fun(list(B),list(B))),insort_key(word(A),B,word_sle(A)),F2),Y),Xs2)) ) ) ) ).

% signed.insort_key_left_comm
tff(fact_7786_signed_Oinsort__left__comm,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Y: word(A),Xs2: list(word(A))] : aa(list(word(A)),list(word(A)),aa(word(A),fun(list(word(A)),list(word(A))),aa(fun(word(A),word(A)),fun(word(A),fun(list(word(A)),list(word(A)))),insort_key(word(A),word(A),word_sle(A)),aTP_Lamp_rf(word(A),word(A))),X),aa(list(word(A)),list(word(A)),aa(word(A),fun(list(word(A)),list(word(A))),aa(fun(word(A),word(A)),fun(word(A),fun(list(word(A)),list(word(A)))),insort_key(word(A),word(A),word_sle(A)),aTP_Lamp_rf(word(A),word(A))),Y),Xs2)) = aa(list(word(A)),list(word(A)),aa(word(A),fun(list(word(A)),list(word(A))),aa(fun(word(A),word(A)),fun(word(A),fun(list(word(A)),list(word(A)))),insort_key(word(A),word(A),word_sle(A)),aTP_Lamp_rf(word(A),word(A))),Y),aa(list(word(A)),list(word(A)),aa(word(A),fun(list(word(A)),list(word(A))),aa(fun(word(A),word(A)),fun(word(A),fun(list(word(A)),list(word(A)))),insort_key(word(A),word(A),word_sle(A)),aTP_Lamp_rf(word(A),word(A))),X),Xs2)) ) ).

% signed.insort_left_comm
tff(fact_7787_signed_Oinsort__key_Osimps_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [F2: fun(A,word(B)),X: A,Y: A,Ysb: list(A)] :
          aa(list(A),list(A),aa(A,fun(list(A),list(A)),aa(fun(A,word(B)),fun(A,fun(list(A),list(A))),insort_key(word(B),A,word_sle(B)),F2),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ysb)) = $ite(aa(word(B),$o,aa(word(B),fun(word(B),$o),word_sle(B),aa(A,word(B),F2,X)),aa(A,word(B),F2,Y)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ysb)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),aa(list(A),list(A),aa(A,fun(list(A),list(A)),aa(fun(A,word(B)),fun(A,fun(list(A),list(A))),insort_key(word(B),A,word_sle(B)),F2),X),Ysb))) ) ).

% signed.insort_key.simps(2)
tff(fact_7788_signed_Oinsort__key_Osimps_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [F2: fun(A,word(B)),X: A] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aa(fun(A,word(B)),fun(A,fun(list(A),list(A))),insort_key(word(B),A,word_sle(B)),F2),X),nil(A)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) ) ).

% signed.insort_key.simps(1)
tff(fact_7789_signed_Oinsort__is__Cons,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [Xs2: list(A),F2: fun(A,word(B)),Aa2: A] :
          ( ! [X2: A] :
              ( member(A,X2,aa(list(A),set(A),set2(A),Xs2))
             => aa(word(B),$o,aa(word(B),fun(word(B),$o),word_sle(B),aa(A,word(B),F2,Aa2)),aa(A,word(B),F2,X2)) )
         => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),aa(fun(A,word(B)),fun(A,fun(list(A),list(A))),insort_key(word(B),A,word_sle(B)),F2),Aa2),Xs2) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Aa2),Xs2) ) ) ) ).

% signed.insort_is_Cons
tff(fact_7790_signed_Oset__insort__key,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [F2: fun(A,word(B)),X: A,Xs2: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),aa(fun(A,word(B)),fun(A,fun(list(A),list(A))),insort_key(word(B),A,word_sle(B)),F2),X),Xs2)) = aa(set(A),set(A),insert(A,X),aa(list(A),set(A),set2(A),Xs2)) ) ).

% signed.set_insort_key
tff(fact_7791_signed_Odistinct__insort,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [F2: fun(A,word(B)),X: A,Xs2: list(A)] :
          ( distinct(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),aa(fun(A,word(B)),fun(A,fun(list(A),list(A))),insort_key(word(B),A,word_sle(B)),F2),X),Xs2))
        <=> ( ~ member(A,X,aa(list(A),set(A),set2(A),Xs2))
            & distinct(A,Xs2) ) ) ) ).

% signed.distinct_insort
tff(fact_7792_signed_Osorted__insort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(A)
     => ! [F2: fun(B,word(A)),X: B,Xs2: list(B)] :
          ( sorted_wrt(word(A),word_sle(A),aa(list(B),list(word(A)),map(B,word(A),F2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),aa(fun(B,word(A)),fun(B,fun(list(B),list(B))),insort_key(word(A),B,word_sle(A)),F2),X),Xs2)))
        <=> sorted_wrt(word(A),word_sle(A),aa(list(B),list(word(A)),map(B,word(A),F2),Xs2)) ) ) ).

% signed.sorted_insort_key
tff(fact_7793_signed_Osorted__list__of__set_Ofold__insort__key_Oremove,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: set(word(A)),X: word(A)] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A2)
         => ( member(word(A),X,A2)
           => ( aa(set(word(A)),list(word(A)),sorted_list_of_set(word(A),word_sle(A)),A2) = aa(list(word(A)),list(word(A)),aa(word(A),fun(list(word(A)),list(word(A))),aa(fun(word(A),word(A)),fun(word(A),fun(list(word(A)),list(word(A)))),insort_key(word(A),word(A),word_sle(A)),aTP_Lamp_rf(word(A),word(A))),X),aa(set(word(A)),list(word(A)),sorted_list_of_set(word(A),word_sle(A)),aa(set(word(A)),set(word(A)),minus_minus(set(word(A)),A2),aa(set(word(A)),set(word(A)),insert(word(A),X),bot_bot(set(word(A))))))) ) ) ) ) ).

% signed.sorted_list_of_set.fold_insort_key.remove
tff(fact_7794_signed_Odistinct__insort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(A)
     => ! [F2: fun(B,word(A)),X: B,Xs2: list(B)] :
          ( distinct(word(A),aa(list(B),list(word(A)),map(B,word(A),F2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),aa(fun(B,word(A)),fun(B,fun(list(B),list(B))),insort_key(word(A),B,word_sle(A)),F2),X),Xs2)))
        <=> ( ~ member(word(A),aa(B,word(A),F2,X),aa(set(B),set(word(A)),image(B,word(A),F2),aa(list(B),set(B),set2(B),Xs2)))
            & distinct(word(A),aa(list(B),list(word(A)),map(B,word(A),F2),Xs2)) ) ) ) ).

% signed.distinct_insort_key
tff(fact_7795_folding__insort__key_Osorted__key__list__of__set__remove,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Sa: set(B),F2: fun(B,A),X: B,A2: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,Sa,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),insert(B,X),A2)),Sa)
       => ( aa(set(B),$o,finite_finite2(B),A2)
         => ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),aa(set(B),set(B),minus_minus(set(B),A2),aa(set(B),set(B),insert(B,X),bot_bot(set(B))))) = remove1(B,X,aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F2),A2)) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_remove
tff(fact_7796_Suc__0__mod__numeral,axiom,
    ! [K: num] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),K)) = aa(product_prod(nat,nat),nat,product_snd(nat,nat),unique8689654367752047608divmod(nat,one2,K)) ).

% Suc_0_mod_numeral
tff(fact_7797_in__set__remove1,axiom,
    ! [A: $tType,Aa2: A,Ba: A,Xs2: list(A)] :
      ( ( Aa2 != Ba )
     => ( member(A,Aa2,aa(list(A),set(A),set2(A),remove1(A,Ba,Xs2)))
      <=> member(A,Aa2,aa(list(A),set(A),set2(A),Xs2)) ) ) ).

% in_set_remove1
tff(fact_7798_img__snd,axiom,
    ! [B: $tType,A: $tType,Aa2: A,Ba: B,Sa: set(product_prod(A,B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba),Sa)
     => member(B,Ba,aa(set(product_prod(A,B)),set(B),image(product_prod(A,B),B,product_snd(A,B)),Sa)) ) ).

% img_snd
tff(fact_7799_signed_Oremove1__insort__key,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [X: A,F2: fun(A,word(B)),Xs2: list(A)] : remove1(A,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),aa(fun(A,word(B)),fun(A,fun(list(A),list(A))),insort_key(word(B),A,word_sle(B)),F2),X),Xs2)) = Xs2 ) ).

% signed.remove1_insort_key
tff(fact_7800_numeral__mod__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [K: num,L: num] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),K)),aa(num,A,numeral_numeral(A),L)) = aa(product_prod(A,A),A,product_snd(A,A),unique8689654367752047608divmod(A,K,L)) ) ).

% numeral_mod_numeral
tff(fact_7801_map__snd__mk__fst,axiom,
    ! [B: $tType,A: $tType,K: B,L: list(A)] : aa(list(product_prod(B,A)),list(A),map(product_prod(B,A),A,product_snd(B,A)),aa(list(A),list(product_prod(B,A)),map(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K)),L)) = L ).

% map_snd_mk_fst
tff(fact_7802_one__mod__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Na: num] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),one_one(A)),aa(num,A,numeral_numeral(A),Na)) = aa(product_prod(A,A),A,product_snd(A,A),unique8689654367752047608divmod(A,one2,Na)) ) ).

% one_mod_numeral
tff(fact_7803_set__remove1__eq,axiom,
    ! [A: $tType,Xs2: list(A),X: A] :
      ( distinct(A,Xs2)
     => ( aa(list(A),set(A),set2(A),remove1(A,X,Xs2)) = aa(set(A),set(A),minus_minus(set(A),aa(list(A),set(A),set2(A),Xs2)),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))) ) ) ).

% set_remove1_eq
tff(fact_7804_fn__snd__conv,axiom,
    ! [A: $tType,C: $tType,B: $tType,F2: fun(B,C)] : aTP_Lamp_alk(fun(B,C),fun(product_prod(A,B),C),F2) = product_case_prod(A,B,C,aTP_Lamp_all(fun(B,C),fun(A,fun(B,C)),F2)) ).

% fn_snd_conv
tff(fact_7805_snd__def,axiom,
    ! [A: $tType,B: $tType,Prod: product_prod(B,A)] : aa(product_prod(B,A),A,product_snd(B,A),Prod) = aa(product_prod(B,A),A,product_case_prod(B,A,A,aTP_Lamp_alm(B,fun(A,A))),Prod) ).

% snd_def
tff(fact_7806_remove1_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Y: A,Xs2: list(A)] :
      remove1(A,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs2)) = $ite(X = Y,Xs2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),remove1(A,X,Xs2))) ).

% remove1.simps(2)
tff(fact_7807_remove1__commute,axiom,
    ! [A: $tType,X: A,Y: A,Zs: list(A)] : remove1(A,X,remove1(A,Y,Zs)) = remove1(A,Y,remove1(A,X,Zs)) ).

% remove1_commute
tff(fact_7808_continuous__snd,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F4: filter(A),F2: fun(A,product_prod(B,C))] :
          ( topolo3448309680560233919inuous(A,product_prod(B,C),F4,F2)
         => topolo3448309680560233919inuous(A,C,F4,aTP_Lamp_aln(fun(A,product_prod(B,C)),fun(A,C),F2)) ) ) ).

% continuous_snd
tff(fact_7809_snd__divmod,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Na: num] : aa(product_prod(A,A),A,product_snd(A,A),unique8689654367752047608divmod(A,M,Na)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Na)) ) ).

% snd_divmod
tff(fact_7810_snd__image__mp,axiom,
    ! [B: $tType,A: $tType,A2: set(product_prod(B,A)),B2: set(A),X: B,Y: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(product_prod(B,A)),set(A),image(product_prod(B,A),A,product_snd(B,A)),A2)),B2)
     => ( member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y),A2)
       => member(A,Y,B2) ) ) ).

% snd_image_mp
tff(fact_7811_in__snd__imageE,axiom,
    ! [A: $tType,B: $tType,Y: A,Sa: set(product_prod(B,A))] :
      ( member(A,Y,aa(set(product_prod(B,A)),set(A),image(product_prod(B,A),A,product_snd(B,A)),Sa))
     => ~ ! [X2: B] : ~ member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X2),Y),Sa) ) ).

% in_snd_imageE
tff(fact_7812_sndE,axiom,
    ! [A: $tType,B: $tType,X: product_prod(A,B),Aa2: A,Ba: B,P: fun(B,$o)] :
      ( ( X = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba) )
     => ( aa(B,$o,P,aa(product_prod(A,B),B,product_snd(A,B),X))
       => aa(B,$o,P,Ba) ) ) ).

% sndE
tff(fact_7813_distinct__remove1,axiom,
    ! [A: $tType,Xs2: list(A),X: A] :
      ( distinct(A,Xs2)
     => distinct(A,remove1(A,X,Xs2)) ) ).

% distinct_remove1
tff(fact_7814_remove1_Osimps_I1_J,axiom,
    ! [A: $tType,X: A] : remove1(A,X,nil(A)) = nil(A) ).

% remove1.simps(1)
tff(fact_7815_set__remove1__subset,axiom,
    ! [A: $tType,X: A,Xs2: list(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),remove1(A,X,Xs2))),aa(list(A),set(A),set2(A),Xs2)) ).

% set_remove1_subset
tff(fact_7816_remove1__idem,axiom,
    ! [A: $tType,X: A,Xs2: list(A)] :
      ( ~ member(A,X,aa(list(A),set(A),set2(A),Xs2))
     => ( remove1(A,X,Xs2) = Xs2 ) ) ).

% remove1_idem
tff(fact_7817_notin__set__remove1,axiom,
    ! [A: $tType,X: A,Xs2: list(A),Y: A] :
      ( ~ member(A,X,aa(list(A),set(A),set2(A),Xs2))
     => ~ member(A,X,aa(list(A),set(A),set2(A),remove1(A,Y,Xs2))) ) ).

% notin_set_remove1
tff(fact_7818_sorted__remove1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs2: list(A),Aa2: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xs2)
         => sorted_wrt(A,ord_less_eq(A),remove1(A,Aa2,Xs2)) ) ) ).

% sorted_remove1
tff(fact_7819_Ex__prod__contract,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o))] :
      ( ? [A7: A,X_1: B] : aa(B,$o,aa(A,fun(B,$o),P,A7),X_1)
    <=> ? [Z5: product_prod(A,B)] : aa(B,$o,aa(A,fun(B,$o),P,aa(product_prod(A,B),A,product_fst(A,B),Z5)),aa(product_prod(A,B),B,product_snd(A,B),Z5)) ) ).

% Ex_prod_contract
tff(fact_7820_All__prod__contract,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o))] :
      ( ! [A7: A,X_1: B] : aa(B,$o,aa(A,fun(B,$o),P,A7),X_1)
    <=> ! [Z5: product_prod(A,B)] : aa(B,$o,aa(A,fun(B,$o),P,aa(product_prod(A,B),A,product_fst(A,B),Z5)),aa(product_prod(A,B),B,product_snd(A,B),Z5)) ) ).

% All_prod_contract
tff(fact_7821_signed_Osorted__remove1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs2: list(word(A)),Aa2: word(A)] :
          ( sorted_wrt(word(A),word_sle(A),Xs2)
         => sorted_wrt(word(A),word_sle(A),remove1(word(A),Aa2,Xs2)) ) ) ).

% signed.sorted_remove1
tff(fact_7822_pair__list__eqI,axiom,
    ! [B: $tType,A: $tType,Xs2: list(product_prod(A,B)),Ysb: list(product_prod(A,B))] :
      ( ( aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs2) = aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Ysb) )
     => ( ( aa(list(product_prod(A,B)),list(B),map(product_prod(A,B),B,product_snd(A,B)),Xs2) = aa(list(product_prod(A,B)),list(B),map(product_prod(A,B),B,product_snd(A,B)),Ysb) )
       => ( Xs2 = Ysb ) ) ) ).

% pair_list_eqI
tff(fact_7823_case__prod__unfold,axiom,
    ! [C: $tType,B: $tType,A: $tType,X4: fun(A,fun(B,C)),Xa: product_prod(A,B)] : aa(product_prod(A,B),C,product_case_prod(A,B,C,X4),Xa) = aa(B,C,aa(A,fun(B,C),X4,aa(product_prod(A,B),A,product_fst(A,B),Xa)),aa(product_prod(A,B),B,product_snd(A,B),Xa)) ).

% case_prod_unfold
tff(fact_7824_case__prod__beta_H,axiom,
    ! [C: $tType,B: $tType,A: $tType,F2: fun(A,fun(B,C)),X4: product_prod(A,B)] : aa(product_prod(A,B),C,product_case_prod(A,B,C,F2),X4) = aa(B,C,aa(A,fun(B,C),F2,aa(product_prod(A,B),A,product_fst(A,B),X4)),aa(product_prod(A,B),B,product_snd(A,B),X4)) ).

% case_prod_beta'
tff(fact_7825_split__comp__eq,axiom,
    ! [B: $tType,C: $tType,D: $tType,A: $tType,F2: fun(D,fun(B,C)),G: fun(A,D)] : aa(fun(A,D),fun(product_prod(A,B),C),aTP_Lamp_alo(fun(D,fun(B,C)),fun(fun(A,D),fun(product_prod(A,B),C)),F2),G) = product_case_prod(A,B,C,aa(fun(A,D),fun(A,fun(B,C)),aTP_Lamp_alp(fun(D,fun(B,C)),fun(fun(A,D),fun(A,fun(B,C))),F2),G)) ).

% split_comp_eq
tff(fact_7826_sorted__map__remove1,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs2: list(B),X: B] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F2),Xs2))
         => sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F2),remove1(B,X,Xs2))) ) ) ).

% sorted_map_remove1
tff(fact_7827_Eps__case__prod,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o))] : fChoice(product_prod(A,B),product_case_prod(A,B,$o,P)) = fChoice(product_prod(A,B),aTP_Lamp_alq(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),P)) ).

% Eps_case_prod
tff(fact_7828_signed_Osorted__map__remove1,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(A)
     => ! [F2: fun(B,word(A)),Xs2: list(B),X: B] :
          ( sorted_wrt(word(A),word_sle(A),aa(list(B),list(word(A)),map(B,word(A),F2),Xs2))
         => sorted_wrt(word(A),word_sle(A),aa(list(B),list(word(A)),map(B,word(A),F2),remove1(B,X,Xs2))) ) ) ).

% signed.sorted_map_remove1
tff(fact_7829_The__case__prod,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o))] : the(product_prod(A,B),product_case_prod(A,B,$o,P)) = the(product_prod(A,B),aTP_Lamp_alq(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),P)) ).

% The_case_prod
tff(fact_7830_signed_Oinsort__remove1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Aa2: word(A),Xs2: list(word(A))] :
          ( member(word(A),Aa2,aa(list(word(A)),set(word(A)),set2(word(A)),Xs2))
         => ( sorted_wrt(word(A),word_sle(A),Xs2)
           => ( aa(list(word(A)),list(word(A)),aa(word(A),fun(list(word(A)),list(word(A))),aa(fun(word(A),word(A)),fun(word(A),fun(list(word(A)),list(word(A)))),insort_key(word(A),word(A),word_sle(A)),aTP_Lamp_rf(word(A),word(A))),Aa2),remove1(word(A),Aa2,Xs2)) = Xs2 ) ) ) ) ).

% signed.insort_remove1
tff(fact_7831_length__remove1,axiom,
    ! [A: $tType,X: A,Xs2: list(A)] :
      aa(list(A),nat,size_size(list(A)),remove1(A,X,Xs2)) = $ite(member(A,X,aa(list(A),set(A),set2(A),Xs2)),aa(nat,nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs2)),one_one(nat)),aa(list(A),nat,size_size(list(A)),Xs2)) ).

% length_remove1
tff(fact_7832_isCont__snd,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [Aa2: A,F2: fun(A,product_prod(B,C))] :
          ( topolo3448309680560233919inuous(A,product_prod(B,C),topolo174197925503356063within(A,Aa2,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,C,topolo174197925503356063within(A,Aa2,top_top(set(A))),aTP_Lamp_aln(fun(A,product_prod(B,C)),fun(A,C),F2)) ) ) ).

% isCont_snd
tff(fact_7833_sum__list__map__remove1,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [X: A,Xs2: list(A),F2: fun(A,B)] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs2))
         => ( groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,F2),Xs2)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F2,X)),groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,F2),remove1(A,X,Xs2)))) ) ) ) ).

% sum_list_map_remove1
tff(fact_7834_tendsto__snd,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,product_prod(B,C)),Aa2: product_prod(B,C),F4: filter(A)] :
          ( filterlim(A,product_prod(B,C),F2,topolo7230453075368039082e_nhds(product_prod(B,C),Aa2),F4)
         => filterlim(A,C,aTP_Lamp_alr(fun(A,product_prod(B,C)),fun(A,C),F2),topolo7230453075368039082e_nhds(C,aa(product_prod(B,C),C,product_snd(B,C),Aa2)),F4) ) ) ).

% tendsto_snd
tff(fact_7835_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: set(A),X: A] :
          ( aa(set(A),$o,finite_finite2(A),A2)
         => ( linord4507533701916653071of_set(A,aa(set(A),set(A),minus_minus(set(A),A2),aa(set(A),set(A),insert(A,X),bot_bot(set(A))))) = remove1(A,X,linord4507533701916653071of_set(A,A2)) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_remove
tff(fact_7836_signed_Osorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: set(word(A)),X: word(A)] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A2)
         => ( aa(set(word(A)),list(word(A)),sorted_list_of_set(word(A),word_sle(A)),aa(set(word(A)),set(word(A)),minus_minus(set(word(A)),A2),aa(set(word(A)),set(word(A)),insert(word(A),X),bot_bot(set(word(A)))))) = remove1(word(A),X,aa(set(word(A)),list(word(A)),sorted_list_of_set(word(A),word_sle(A)),A2)) ) ) ) ).

% signed.sorted_list_of_set.sorted_key_list_of_set_remove
tff(fact_7837_exE__realizer,axiom,
    ! [C: $tType,A: $tType,B: $tType,P: fun(A,fun(B,$o)),P2: product_prod(B,A),Q: fun(C,$o),F2: fun(B,fun(A,C))] :
      ( aa(B,$o,aa(A,fun(B,$o),P,aa(product_prod(B,A),A,product_snd(B,A),P2)),aa(product_prod(B,A),B,product_fst(B,A),P2))
     => ( ! [X2: B,Y2: A] :
            ( aa(B,$o,aa(A,fun(B,$o),P,Y2),X2)
           => aa(C,$o,Q,aa(A,C,aa(B,fun(A,C),F2,X2),Y2)) )
       => aa(C,$o,Q,aa(product_prod(B,A),C,product_case_prod(B,A,C,F2),P2)) ) ) ).

% exE_realizer
tff(fact_7838_in__set__enumerate__eq,axiom,
    ! [A: $tType,P2: product_prod(nat,A),Na: nat,Xs2: list(A)] :
      ( member(product_prod(nat,A),P2,aa(list(product_prod(nat,A)),set(product_prod(nat,A)),set2(product_prod(nat,A)),enumerate(A,Na,Xs2)))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(product_prod(nat,A),nat,product_fst(nat,A),P2))
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(product_prod(nat,A),nat,product_fst(nat,A),P2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs2)),Na))
        & ( aa(nat,A,nth(A,Xs2),aa(nat,nat,minus_minus(nat,aa(product_prod(nat,A),nat,product_fst(nat,A),P2)),Na)) = aa(product_prod(nat,A),A,product_snd(nat,A),P2) ) ) ) ).

% in_set_enumerate_eq
tff(fact_7839_enumerate__simps_I1_J,axiom,
    ! [A: $tType,Na: nat] : enumerate(A,Na,nil(A)) = nil(product_prod(nat,A)) ).

% enumerate_simps(1)
tff(fact_7840_length__enumerate,axiom,
    ! [A: $tType,Na: nat,Xs2: list(A)] : aa(list(product_prod(nat,A)),nat,size_size(list(product_prod(nat,A))),enumerate(A,Na,Xs2)) = aa(list(A),nat,size_size(list(A)),Xs2) ).

% length_enumerate
tff(fact_7841_map__snd__enumerate,axiom,
    ! [A: $tType,Na: nat,Xs2: list(A)] : aa(list(product_prod(nat,A)),list(A),map(product_prod(nat,A),A,product_snd(nat,A)),enumerate(A,Na,Xs2)) = Xs2 ).

% map_snd_enumerate
tff(fact_7842_enumerate__simps_I2_J,axiom,
    ! [A: $tType,Na: nat,X: A,Xs2: list(A)] : enumerate(A,Na,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs2)) = aa(list(product_prod(nat,A)),list(product_prod(nat,A)),aa(product_prod(nat,A),fun(list(product_prod(nat,A)),list(product_prod(nat,A))),cons(product_prod(nat,A)),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Na),X)),enumerate(A,aa(nat,nat,suc,Na),Xs2)) ).

% enumerate_simps(2)
tff(fact_7843_distinct__enumerate,axiom,
    ! [A: $tType,Na: nat,Xs2: list(A)] : distinct(product_prod(nat,A),enumerate(A,Na,Xs2)) ).

% distinct_enumerate
tff(fact_7844_sorted__enumerate,axiom,
    ! [A: $tType,Na: nat,Xs2: list(A)] : sorted_wrt(nat,ord_less_eq(nat),aa(list(product_prod(nat,A)),list(nat),map(product_prod(nat,A),nat,product_fst(nat,A)),enumerate(A,Na,Xs2))) ).

% sorted_enumerate
tff(fact_7845_nth__enumerate__eq,axiom,
    ! [A: $tType,M: nat,Xs2: list(A),Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(list(A),nat,size_size(list(A)),Xs2))
     => ( aa(nat,product_prod(nat,A),nth(product_prod(nat,A),enumerate(A,Na,Xs2)),M) = aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M)),aa(nat,A,nth(A,Xs2),M)) ) ) ).

% nth_enumerate_eq
tff(fact_7846_minus__one__mod__numeral,axiom,
    ! [Na: num] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(num,int,numeral_numeral(int),Na)) = adjust_mod(aa(num,int,numeral_numeral(int),Na),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,one2,Na))) ).

% minus_one_mod_numeral
tff(fact_7847_one__mod__minus__numeral,axiom,
    ! [Na: num] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na))) = aa(int,int,uminus_uminus(int),adjust_mod(aa(num,int,numeral_numeral(int),Na),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,one2,Na)))) ).

% one_mod_minus_numeral
tff(fact_7848_numeral__mod__minus__numeral,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na))) = aa(int,int,uminus_uminus(int),adjust_mod(aa(num,int,numeral_numeral(int),Na),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,M,Na)))) ).

% numeral_mod_minus_numeral
tff(fact_7849_minus__numeral__mod__numeral,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),Na)) = adjust_mod(aa(num,int,numeral_numeral(int),Na),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,M,Na))) ).

% minus_numeral_mod_numeral
tff(fact_7850_bezw_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod(int,int)] :
      ( ( bezw(X,Xa2) = Y )
     => ( Y = $ite(Xa2 = zero_zero(nat),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),Xa2)))),aa(int,int,minus_minus(int,aa(product_prod(int,int),int,product_fst(int,int),bezw(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),Xa2)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),Xa2)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),Xa2)))))) ) ) ).

% bezw.elims
tff(fact_7851_bezw_Osimps,axiom,
    ! [X: nat,Y: nat] :
      bezw(X,Y) = $ite(Y = zero_zero(nat),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),Y)))),aa(int,int,minus_minus(int,aa(product_prod(int,int),int,product_fst(int,int),bezw(Y,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),Y)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),Y)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),Y)))))) ).

% bezw.simps
tff(fact_7852_bezw__0,axiom,
    ! [X: nat] : bezw(X,zero_zero(nat)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)) ).

% bezw_0
tff(fact_7853_bezw__non__0,axiom,
    ! [Y: nat,X: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Y)
     => ( bezw(X,Y) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),Y)))),aa(int,int,minus_minus(int,aa(product_prod(int,int),int,product_fst(int,int),bezw(Y,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),Y)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),Y)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),Y))))) ) ) ).

% bezw_non_0
tff(fact_7854_bezw_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod(int,int)] :
      ( ( bezw(X,Xa2) = Y )
     => ( accp(product_prod(nat,nat),bezw_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa2))
       => ~ ( ( Y = $ite(Xa2 = zero_zero(nat),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),Xa2)))),aa(int,int,minus_minus(int,aa(product_prod(int,int),int,product_fst(int,int),bezw(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),Xa2)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa2,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),Xa2)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),Xa2)))))) )
           => ~ accp(product_prod(nat,nat),bezw_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Xa2)) ) ) ) ).

% bezw.pelims
tff(fact_7855_mergesort__by__rel__split__length,axiom,
    ! [A: $tType,Xs12: list(A),Xs23: list(A),Xs2: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23),Xs2))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs12)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Xs2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(list(A),nat,size_size(list(A)),Xs2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
      & ( aa(list(A),nat,size_size(list(A)),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23),Xs2))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs23)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Xs2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% mergesort_by_rel_split_length
tff(fact_7856_mergesort__by__rel__split_Osimps_I1_J,axiom,
    ! [A: $tType,Xs12: list(A),Xs23: list(A)] : merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23),nil(A)) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23) ).

% mergesort_by_rel_split.simps(1)
tff(fact_7857_mergesort__by__rel__split_Osimps_I3_J,axiom,
    ! [A: $tType,Xs12: list(A),Xs23: list(A),X12: A,X23: A,Xs2: list(A)] : merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),Xs2))) = merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),Xs12)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),Xs23)),Xs2) ).

% mergesort_by_rel_split.simps(3)
tff(fact_7858_mergesort__by__rel__split_Osimps_I2_J,axiom,
    ! [A: $tType,Xs12: list(A),Xs23: list(A),X: A] : merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs12)),Xs23) ).

% mergesort_by_rel_split.simps(2)
tff(fact_7859_mergesort__by__rel__split_Oelims,axiom,
    ! [A: $tType,X: product_prod(list(A),list(A)),Xa2: list(A),Y: product_prod(list(A),list(A))] :
      ( ( merges295452479951948502_split(A,X,Xa2) = Y )
     => ( ! [Xs1: list(A),Xs22: list(A)] :
            ( ( X = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22) )
           => ( ( Xa2 = nil(A) )
             => ( Y != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22) ) ) )
       => ( ! [Xs1: list(A),Xs22: list(A)] :
              ( ( X = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22) )
             => ! [X2: A] :
                  ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)) )
                 => ( Y != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs1)),Xs22) ) ) )
         => ~ ! [Xs1: list(A),Xs22: list(A)] :
                ( ( X = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22) )
               => ! [X1: A,X22: A,Xs: list(A)] :
                    ( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),Xs)) )
                   => ( Y != merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),Xs1)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),Xs22)),Xs) ) ) ) ) ) ) ).

% mergesort_by_rel_split.elims
tff(fact_7860_quickcheck__narrowing__samples_Opartial__term__of__sample__def,axiom,
    ! [A: $tType] :
      ( ( code_term_of(A)
        & quickc6926020345158392990erm_of(A) )
     => ! [A_of_integer: fun(code_integer,product_prod(A,A)),Zero: A,I: code_integer] :
          code_T4081349890594273596sample(A,A_of_integer,Zero,I) = $ite(
            aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),I),zero_zero(code_integer)),
            undefined(A),
            $ite(
              I = zero_zero(code_integer),
              Zero,
              $ite(aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),modulo_modulo(code_integer),I),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))) = zero_zero(code_integer),aa(product_prod(A,A),A,product_snd(A,A),aa(code_integer,product_prod(A,A),A_of_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),I),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))))),aa(product_prod(A,A),A,product_fst(A,A),aa(code_integer,product_prod(A,A),A_of_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),I),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),one_one(code_integer))))) ) ) ) ).

% quickcheck_narrowing_samples.partial_term_of_sample_def
tff(fact_7861_normalize__def,axiom,
    ! [P2: product_prod(int,int)] :
      normalize(P2) = $ite(
        aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),P2)),
        $let(
          a3: int,
          a3:= gcd_gcd(int,aa(product_prod(int,int),int,product_fst(int,int),P2),aa(product_prod(int,int),int,product_snd(int,int),P2)),
          aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_fst(int,int),P2)),a3)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_snd(int,int),P2)),a3)) ),
        $ite(
          aa(product_prod(int,int),int,product_snd(int,int),P2) = zero_zero(int),
          aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),
          $let(
            a3: int,
            a3:= aa(int,int,uminus_uminus(int),gcd_gcd(int,aa(product_prod(int,int),int,product_fst(int,int),P2),aa(product_prod(int,int),int,product_snd(int,int),P2))),
            aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_fst(int,int),P2)),a3)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_snd(int,int),P2)),a3)) ) ) ) ).

% normalize_def
tff(fact_7862_gcd__add2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [M: A,Na: A] : gcd_gcd(A,M,aa(A,A,aa(A,fun(A,A),plus_plus(A),M),Na)) = gcd_gcd(A,M,Na) ) ).

% gcd_add2
tff(fact_7863_gcd__add1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [M: A,Na: A] : gcd_gcd(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),M),Na),Na) = gcd_gcd(A,M,Na) ) ).

% gcd_add1
tff(fact_7864_gcd_Obottom__left__bottom,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Aa2: A] : gcd_gcd(A,one_one(A),Aa2) = one_one(A) ) ).

% gcd.bottom_left_bottom
tff(fact_7865_gcd_Obottom__right__bottom,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Aa2: A] : gcd_gcd(A,Aa2,one_one(A)) = one_one(A) ) ).

% gcd.bottom_right_bottom
tff(fact_7866_gcd__1__int,axiom,
    ! [M: int] : gcd_gcd(int,M,one_one(int)) = one_one(int) ).

% gcd_1_int
tff(fact_7867_gcd__neg__numeral__2,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [Aa2: A,Na: num] : gcd_gcd(A,Aa2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))) = gcd_gcd(A,Aa2,aa(num,A,numeral_numeral(A),Na)) ) ).

% gcd_neg_numeral_2
tff(fact_7868_gcd__neg__numeral__1,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [Na: num,Aa2: A] : gcd_gcd(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na)),Aa2) = gcd_gcd(A,aa(num,A,numeral_numeral(A),Na),Aa2) ) ).

% gcd_neg_numeral_1
tff(fact_7869_is__unit__gcd__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Aa2: A,Ba: A] :
          ( aa(A,$o,dvd_dvd(A,gcd_gcd(A,Aa2,Ba)),one_one(A))
        <=> ( gcd_gcd(A,Aa2,Ba) = one_one(A) ) ) ) ).

% is_unit_gcd_iff
tff(fact_7870_gcd__neg__numeral__2__int,axiom,
    ! [X: int,Na: num] : gcd_gcd(int,X,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na))) = gcd_gcd(int,X,aa(num,int,numeral_numeral(int),Na)) ).

% gcd_neg_numeral_2_int
tff(fact_7871_gcd__neg__numeral__1__int,axiom,
    ! [Na: num,X: int] : gcd_gcd(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na)),X) = gcd_gcd(int,aa(num,int,numeral_numeral(int),Na),X) ).

% gcd_neg_numeral_1_int
tff(fact_7872_gcd__div__unit2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,Aa2),one_one(A))
         => ( gcd_gcd(A,Ba,aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),Aa2)) = gcd_gcd(A,Ba,C2) ) ) ) ).

% gcd_div_unit2
tff(fact_7873_gcd__div__unit1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,Aa2),one_one(A))
         => ( gcd_gcd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Ba),Aa2),C2) = gcd_gcd(A,Ba,C2) ) ) ) ).

% gcd_div_unit1
tff(fact_7874_gcd__dvd__prod,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Aa2: A,Ba: A,K: A] : aa(A,$o,dvd_dvd(A,gcd_gcd(A,Aa2,Ba)),aa(A,A,aa(A,fun(A,A),times_times(A),K),Ba)) ) ).

% gcd_dvd_prod
tff(fact_7875_gcd__unique__int,axiom,
    ! [D2: int,Aa2: int,Ba: int] :
      ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),D2)
        & aa(int,$o,dvd_dvd(int,D2),Aa2)
        & aa(int,$o,dvd_dvd(int,D2),Ba)
        & ! [E3: int] :
            ( ( aa(int,$o,dvd_dvd(int,E3),Aa2)
              & aa(int,$o,dvd_dvd(int,E3),Ba) )
           => aa(int,$o,dvd_dvd(int,E3),D2) ) )
    <=> ( D2 = gcd_gcd(int,Aa2,Ba) ) ) ).

% gcd_unique_int
tff(fact_7876_gcd__mult__unit1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,Aa2),one_one(A))
         => ( gcd_gcd(A,aa(A,A,aa(A,fun(A,A),times_times(A),Ba),Aa2),C2) = gcd_gcd(A,Ba,C2) ) ) ) ).

% gcd_mult_unit1
tff(fact_7877_gcd__mult__unit2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Aa2: A,Ba: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,Aa2),one_one(A))
         => ( gcd_gcd(A,Ba,aa(A,A,aa(A,fun(A,A),times_times(A),C2),Aa2)) = gcd_gcd(A,Ba,C2) ) ) ) ).

% gcd_mult_unit2
tff(fact_7878_gcd__ge__0__int,axiom,
    ! [X: int,Y: int] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),gcd_gcd(int,X,Y)) ).

% gcd_ge_0_int
tff(fact_7879_gcd__le1__int,axiom,
    ! [Aa2: int,Ba: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Aa2)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),gcd_gcd(int,Aa2,Ba)),Aa2) ) ).

% gcd_le1_int
tff(fact_7880_gcd__le2__int,axiom,
    ! [Ba: int,Aa2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Ba)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),gcd_gcd(int,Aa2,Ba)),Ba) ) ).

% gcd_le2_int
tff(fact_7881_gcd__add__mult,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [M: A,K: A,Na: A] : gcd_gcd(A,M,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),K),M)),Na)) = gcd_gcd(A,M,Na) ) ).

% gcd_add_mult
tff(fact_7882_gcd__mult__distrib__int,axiom,
    ! [K: int,M: int,Na: int] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),K)),gcd_gcd(int,M,Na)) = gcd_gcd(int,aa(int,int,aa(int,fun(int,int),times_times(int),K),M),aa(int,int,aa(int,fun(int,int),times_times(int),K),Na)) ).

% gcd_mult_distrib_int
tff(fact_7883_bezout__int,axiom,
    ! [X: int,Y: int] :
    ? [U3: int,V4: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),U3),X)),aa(int,int,aa(int,fun(int,int),times_times(int),V4),Y)) = gcd_gcd(int,X,Y) ).

% bezout_int
tff(fact_7884_gcd__cases__int,axiom,
    ! [X: int,Y: int,P: fun(int,$o)] :
      ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
         => aa(int,$o,P,gcd_gcd(int,X,Y)) ) )
     => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Y),zero_zero(int))
           => aa(int,$o,P,gcd_gcd(int,X,aa(int,int,uminus_uminus(int),Y))) ) )
       => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X),zero_zero(int))
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
             => aa(int,$o,P,gcd_gcd(int,aa(int,int,uminus_uminus(int),X),Y)) ) )
         => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X),zero_zero(int))
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Y),zero_zero(int))
               => aa(int,$o,P,gcd_gcd(int,aa(int,int,uminus_uminus(int),X),aa(int,int,uminus_uminus(int),Y))) ) )
           => aa(int,$o,P,gcd_gcd(int,X,Y)) ) ) ) ) ).

% gcd_cases_int
tff(fact_7885_uint32__shiftl__def,axiom,
    ! [X: uint32,Na: code_integer] :
      uint32_shiftl(X,Na) = $ite(
        ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),Na),zero_zero(code_integer))
        | aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(bit0(bit0(bit0(bit0(one2))))))),Na) ),
        aa(code_integer,uint32,aa(uint32,fun(code_integer,uint32),aa(fun(nat,fun(uint32,uint32)),fun(uint32,fun(code_integer,uint32)),undefined(fun(fun(nat,fun(uint32,uint32)),fun(uint32,fun(code_integer,uint32)))),bit_se4730199178511100633sh_bit(uint32)),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_7886_uint32__shiftr__def,axiom,
    ! [X: uint32,Na: code_integer] :
      uint32_shiftr(X,Na) = $ite(
        ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),Na),zero_zero(code_integer))
        | aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(bit0(bit0(bit0(bit0(one2))))))),Na) ),
        aa(code_integer,uint32,aa(uint32,fun(code_integer,uint32),aa(fun(nat,fun(uint32,uint32)),fun(uint32,fun(code_integer,uint32)),undefined(fun(fun(nat,fun(uint32,uint32)),fun(uint32,fun(code_integer,uint32)))),bit_se4197421643247451524op_bit(uint32)),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_7887_gcd__1__nat,axiom,
    ! [M: nat] : gcd_gcd(nat,M,one_one(nat)) = one_one(nat) ).

% gcd_1_nat
tff(fact_7888_gcd__int__int__eq,axiom,
    ! [M: nat,Na: nat] : gcd_gcd(int,aa(nat,int,semiring_1_of_nat(int),M),aa(nat,int,semiring_1_of_nat(int),Na)) = aa(nat,int,semiring_1_of_nat(int),gcd_gcd(nat,M,Na)) ).

% gcd_int_int_eq
tff(fact_7889_gcd__nat__abs__right__eq,axiom,
    ! [Na: nat,K: int] : gcd_gcd(nat,Na,aa(int,nat,nat2,aa(int,int,abs_abs(int),K))) = aa(int,nat,nat2,gcd_gcd(int,aa(nat,int,semiring_1_of_nat(int),Na),K)) ).

% gcd_nat_abs_right_eq
tff(fact_7890_gcd__nat__abs__left__eq,axiom,
    ! [K: int,Na: nat] : gcd_gcd(nat,aa(int,nat,nat2,aa(int,int,abs_abs(int),K)),Na) = aa(int,nat,nat2,gcd_gcd(int,K,aa(nat,int,semiring_1_of_nat(int),Na))) ).

% gcd_nat_abs_left_eq
tff(fact_7891_bezout__gcd__nat_H,axiom,
    ! [Ba: nat,Aa2: nat] :
    ? [X2: nat,Y2: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ba),Y2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Aa2),X2))
        & ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Aa2),X2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ba),Y2)) = gcd_gcd(nat,Aa2,Ba) ) )
      | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Aa2),Y2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ba),X2))
        & ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ba),X2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Aa2),Y2)) = gcd_gcd(nat,Aa2,Ba) ) ) ) ).

% bezout_gcd_nat'
tff(fact_7892_gcd__diff2__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( gcd_gcd(nat,aa(nat,nat,minus_minus(nat,Na),M),Na) = gcd_gcd(nat,M,Na) ) ) ).

% gcd_diff2_nat
tff(fact_7893_gcd__diff1__nat,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
     => ( gcd_gcd(nat,aa(nat,nat,minus_minus(nat,M),Na),Na) = gcd_gcd(nat,M,Na) ) ) ).

% gcd_diff1_nat
tff(fact_7894_bezout__nat,axiom,
    ! [Aa2: nat,Ba: nat] :
      ( ( Aa2 != zero_zero(nat) )
     => ? [X2: nat,Y2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Aa2),X2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Ba),Y2)),gcd_gcd(nat,Aa2,Ba)) ) ).

% bezout_nat
tff(fact_7895_gcd__le1__nat,axiom,
    ! [Aa2: nat,Ba: nat] :
      ( ( Aa2 != zero_zero(nat) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),gcd_gcd(nat,Aa2,Ba)),Aa2) ) ).

% gcd_le1_nat
tff(fact_7896_gcd__le2__nat,axiom,
    ! [Ba: nat,Aa2: nat] :
      ( ( Ba != zero_zero(nat) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),gcd_gcd(nat,Aa2,Ba)),Ba) ) ).

% gcd_le2_nat
tff(fact_7897_gcd__mult__distrib__nat,axiom,
    ! [K: nat,M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),gcd_gcd(nat,M,Na)) = gcd_gcd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Na)) ).

% gcd_mult_distrib_nat
tff(fact_7898_gcd__int__def,axiom,
    ! [X: int,Y: int] : gcd_gcd(int,X,Y) = aa(nat,int,semiring_1_of_nat(int),gcd_gcd(nat,aa(int,nat,nat2,aa(int,int,abs_abs(int),X)),aa(int,nat,nat2,aa(int,int,abs_abs(int),Y)))) ).

% gcd_int_def
tff(fact_7899_bezw__aux,axiom,
    ! [X: nat,Y: nat] : aa(nat,int,semiring_1_of_nat(int),gcd_gcd(nat,X,Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(X,Y))),aa(nat,int,semiring_1_of_nat(int),X))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(X,Y))),aa(nat,int,semiring_1_of_nat(int),Y))) ).

% bezw_aux
tff(fact_7900_uint32__set__bit__def,axiom,
    ! [X: uint32,Na: code_integer,Ba: $o] :
      uint32_set_bit(X,Na,(Ba)) = $ite(
        ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),Na),zero_zero(code_integer))
        | aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit1(bit1(bit1(bit1(one2)))))),Na) ),
        aa($o,uint32,aa(code_integer,fun($o,uint32),aa(uint32,fun(code_integer,fun($o,uint32)),aa(fun(uint32,fun(nat,fun($o,uint32))),fun(uint32,fun(code_integer,fun($o,uint32))),undefined(fun(fun(uint32,fun(nat,fun($o,uint32))),fun(uint32,fun(code_integer,fun($o,uint32))))),generi7602027413899671122et_bit(uint32)),X),Na),(Ba)),
        aa($o,uint32,aa(nat,fun($o,uint32),aa(uint32,fun(nat,fun($o,uint32)),generi7602027413899671122et_bit(uint32),X),code_nat_of_integer(Na)),(Ba)) ) ).

% uint32_set_bit_def
tff(fact_7901_uint32__test__bit__def,axiom,
    ! [X: uint32,Na: code_integer] :
      ( uint32_test_bit(X,Na)
    <=> $ite(
          ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),Na),zero_zero(code_integer))
          | aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit1(bit1(bit1(bit1(one2)))))),Na) ),
          aa(code_integer,$o,aa(uint32,fun(code_integer,$o),aa(fun(uint32,fun(nat,$o)),fun(uint32,fun(code_integer,$o)),undefined(fun(fun(uint32,fun(nat,$o)),fun(uint32,fun(code_integer,$o)))),bit_se5641148757651400278ts_bit(uint32)),X),Na),
          aa(nat,$o,aa(uint32,fun(nat,$o),bit_se5641148757651400278ts_bit(uint32),X),code_nat_of_integer(Na)) ) ) ).

% uint32_test_bit_def
tff(fact_7902_set__bit__uint32__eq,axiom,
    ! [Aa2: uint32,Na: nat,Ba: $o] :
      aa($o,uint32,aa(nat,fun($o,uint32),aa(uint32,fun(nat,fun($o,uint32)),generi7602027413899671122et_bit(uint32),Aa2),Na),(Ba)) = aa(uint32,uint32,
        aa(nat,fun(uint32,uint32),
          $ite((Ba),bit_se5668285175392031749et_bit(uint32),bit_se2638667681897837118et_bit(uint32)),
          Na),
        Aa2) ).

% set_bit_uint32_eq
tff(fact_7903_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_als(A,B)),bot_bot(set(fun(A,B)))) ).

% Func_empty
tff(fact_7904_uint32__shiftr__code,axiom,
    ! [W: uint32,Na: code_integer] :
      aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,uint32_shiftr(W,Na)) = $ite(
        ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),Na),zero_zero(code_integer))
        | aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(bit0(bit0(bit0(bit0(one2))))))),Na) ),
        aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),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)),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,W)) ) ).

% uint32_shiftr_code
tff(fact_7905_uint32_Oword__of__numeral,axiom,
    ! [Na: num] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,aa(num,uint32,numeral_numeral(uint32),Na)) = 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))))))),Na) ).

% uint32.word_of_numeral
tff(fact_7906_uint32_Oword__of__generic__set__bit,axiom,
    ! [P2: uint32,Na: nat,Ba: $o] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,aa($o,uint32,aa(nat,fun($o,uint32),aa(uint32,fun(nat,fun($o,uint32)),generi7602027413899671122et_bit(uint32),P2),Na),(Ba))) = 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))))))),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,P2)),Na),(Ba)) ).

% uint32.word_of_generic_set_bit
tff(fact_7907_uint32_Obit__eq__word__of,axiom,
    ! [P2: uint32] : aa(uint32,fun(nat,$o),bit_se5641148757651400278ts_bit(uint32),P2) = aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),fun(nat,$o),bit_se5641148757651400278ts_bit(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,P2)) ).

% uint32.bit_eq_word_of
tff(fact_7908_bit__uint32_Orep__eq,axiom,
    ! [X: uint32] : aa(uint32,fun(nat,$o),bit_se5641148757651400278ts_bit(uint32),X) = aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),fun(nat,$o),bit_se5641148757651400278ts_bit(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,X)) ).

% bit_uint32.rep_eq
tff(fact_7909_uint32_Oword__of__nat,axiom,
    ! [Na: nat] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,aa(nat,uint32,semiring_1_of_nat(uint32),Na)) = aa(nat,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),semiring_1_of_nat(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),Na) ).

% uint32.word_of_nat
tff(fact_7910_uint32_Oword__of__minus,axiom,
    ! [P2: uint32] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,aa(uint32,uint32,uminus_uminus(uint32),P2)) = 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)))))),uminus_uminus(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,P2)) ).

% uint32.word_of_minus
tff(fact_7911_uminus__uint32_Orep__eq,axiom,
    ! [X: uint32] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,aa(uint32,uint32,uminus_uminus(uint32),X)) = 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)))))),uminus_uminus(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,X)) ).

% uminus_uint32.rep_eq
tff(fact_7912_divide__uint32_Orep__eq,axiom,
    ! [X: uint32,Xa2: uint32] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,aa(uint32,uint32,aa(uint32,fun(uint32,uint32),divide_divide(uint32),X),Xa2)) = 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(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),divide_divide(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,X)),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,Xa2)) ).

% divide_uint32.rep_eq
tff(fact_7913_uint32_Oword__of__div,axiom,
    ! [P2: uint32,Q3: uint32] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,aa(uint32,uint32,aa(uint32,fun(uint32,uint32),divide_divide(uint32),P2),Q3)) = 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(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),divide_divide(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,P2)),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,Q3)) ).

% uint32.word_of_div
tff(fact_7914_uint32_Oword__of__mult,axiom,
    ! [P2: uint32,Q3: uint32] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,aa(uint32,uint32,aa(uint32,fun(uint32,uint32),times_times(uint32),P2),Q3)) = 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(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),times_times(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,P2)),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,Q3)) ).

% uint32.word_of_mult
tff(fact_7915_times__uint32_Orep__eq,axiom,
    ! [X: uint32,Xa2: uint32] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,aa(uint32,uint32,aa(uint32,fun(uint32,uint32),times_times(uint32),X),Xa2)) = 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(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),times_times(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,X)),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,Xa2)) ).

% times_uint32.rep_eq
tff(fact_7916_plus__uint32_Orep__eq,axiom,
    ! [X: uint32,Xa2: uint32] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,aa(uint32,uint32,aa(uint32,fun(uint32,uint32),plus_plus(uint32),X),Xa2)) = 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(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),plus_plus(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,X)),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,Xa2)) ).

% plus_uint32.rep_eq
tff(fact_7917_uint32_Oword__of__add,axiom,
    ! [P2: uint32,Q3: uint32] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,aa(uint32,uint32,aa(uint32,fun(uint32,uint32),plus_plus(uint32),P2),Q3)) = 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(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),plus_plus(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,P2)),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,Q3)) ).

% uint32.word_of_add
tff(fact_7918_uint32_Oword__of__bool,axiom,
    ! [Na: $o] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,aa($o,uint32,zero_neq_one_of_bool(uint32),(Na))) = aa($o,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),zero_neq_one_of_bool(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),(Na)) ).

% uint32.word_of_bool
tff(fact_7919_or__uint32_Orep__eq,axiom,
    ! [X: uint32,Xa2: uint32] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,aa(uint32,uint32,aa(uint32,fun(uint32,uint32),bit_se1065995026697491101ons_or(uint32),X),Xa2)) = 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(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),bit_se1065995026697491101ons_or(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,X)),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,Xa2)) ).

% or_uint32.rep_eq
tff(fact_7920_and__uint32_Orep__eq,axiom,
    ! [X: uint32,Xa2: uint32] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,aa(uint32,uint32,aa(uint32,fun(uint32,uint32),bit_se5824344872417868541ns_and(uint32),X),Xa2)) = 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(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),bit_se5824344872417868541ns_and(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,X)),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,Xa2)) ).

% and_uint32.rep_eq
tff(fact_7921_uint32_Oword__of__or,axiom,
    ! [P2: uint32,Q3: uint32] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,aa(uint32,uint32,aa(uint32,fun(uint32,uint32),bit_se1065995026697491101ons_or(uint32),P2),Q3)) = 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(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),bit_se1065995026697491101ons_or(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,P2)),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,Q3)) ).

% uint32.word_of_or
tff(fact_7922_xor__uint32_Orep__eq,axiom,
    ! [X: uint32,Xa2: uint32] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,aa(uint32,uint32,aa(uint32,fun(uint32,uint32),bit_se5824344971392196577ns_xor(uint32),X),Xa2)) = 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(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),bit_se5824344971392196577ns_xor(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,X)),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,Xa2)) ).

% xor_uint32.rep_eq
tff(fact_7923_uint32_Oword__of__and,axiom,
    ! [P2: uint32,Q3: uint32] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,aa(uint32,uint32,aa(uint32,fun(uint32,uint32),bit_se5824344872417868541ns_and(uint32),P2),Q3)) = 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(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),bit_se5824344872417868541ns_and(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,P2)),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,Q3)) ).

% uint32.word_of_and
tff(fact_7924_uint32_Oword__of__xor,axiom,
    ! [P2: uint32,Q3: uint32] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,aa(uint32,uint32,aa(uint32,fun(uint32,uint32),bit_se5824344971392196577ns_xor(uint32),P2),Q3)) = 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(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),bit_se5824344971392196577ns_xor(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,P2)),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,Q3)) ).

% uint32.word_of_xor
tff(fact_7925_Rep__uint32__inject,axiom,
    ! [X: uint32,Y: uint32] :
      ( ( aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,X) = aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,Y) )
    <=> ( X = Y ) ) ).

% Rep_uint32_inject
tff(fact_7926_uint32_Oeq__iff__word__of,axiom,
    ! [P2: uint32,Q3: uint32] :
      ( ( P2 = Q3 )
    <=> ( aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,P2) = aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,Q3) ) ) ).

% uint32.eq_iff_word_of
tff(fact_7927_uint32_Oword__of__eqI,axiom,
    ! [P2: uint32,Q3: uint32] :
      ( ( aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,P2) = aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,Q3) )
     => ( P2 = Q3 ) ) ).

% uint32.word_of_eqI
tff(fact_7928_not__uint32_Orep__eq,axiom,
    ! [X: uint32] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,aa(uint32,uint32,bit_ri4277139882892585799ns_not(uint32),X)) = 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)))))),bit_ri4277139882892585799ns_not(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,X)) ).

% not_uint32.rep_eq
tff(fact_7929_uint32_Oword__of__not,axiom,
    ! [P2: uint32] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,aa(uint32,uint32,bit_ri4277139882892585799ns_not(uint32),P2)) = 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)))))),bit_ri4277139882892585799ns_not(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,P2)) ).

% uint32.word_of_not
tff(fact_7930_mask__uint32_Orep__eq,axiom,
    ! [X: nat] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,bit_se2239418461657761734s_mask(uint32,X)) = bit_se2239418461657761734s_mask(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),X) ).

% mask_uint32.rep_eq
tff(fact_7931_uint32_Oword__of__mask,axiom,
    ! [Na: nat] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,bit_se2239418461657761734s_mask(uint32,Na)) = bit_se2239418461657761734s_mask(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),Na) ).

% uint32.word_of_mask
tff(fact_7932_uint32_Oword__of__take__bit,axiom,
    ! [Na: nat,P2: uint32] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,aa(uint32,uint32,bit_se2584673776208193580ke_bit(uint32,Na),P2)) = 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)))))),bit_se2584673776208193580ke_bit(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),Na),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,P2)) ).

% uint32.word_of_take_bit
tff(fact_7933_take__bit__uint32_Orep__eq,axiom,
    ! [X: nat,Xa2: uint32] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,aa(uint32,uint32,bit_se2584673776208193580ke_bit(uint32,X),Xa2)) = 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)))))),bit_se2584673776208193580ke_bit(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),X),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,Xa2)) ).

% take_bit_uint32.rep_eq
tff(fact_7934_lsb__uint32_Orep__eq,axiom,
    ! [X: uint32] :
      ( aa(uint32,$o,least_8051144512741203767sb_lsb(uint32),X)
    <=> aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o,least_8051144512741203767sb_lsb(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,X)) ) ).

% lsb_uint32.rep_eq
tff(fact_7935_uint32_Olsb__iff__word__of,axiom,
    ! [P2: uint32] :
      ( aa(uint32,$o,least_8051144512741203767sb_lsb(uint32),P2)
    <=> aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o,least_8051144512741203767sb_lsb(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,P2)) ) ).

% uint32.lsb_iff_word_of
tff(fact_7936_uint32_Oword__of__flip__bit,axiom,
    ! [Na: nat,P2: uint32] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,bit_se8732182000553998342ip_bit(uint32,Na,P2)) = bit_se8732182000553998342ip_bit(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),Na,aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,P2)) ).

% uint32.word_of_flip_bit
tff(fact_7937_flip__bit__uint32_Orep__eq,axiom,
    ! [X: nat,Xa2: uint32] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,bit_se8732182000553998342ip_bit(uint32,X,Xa2)) = bit_se8732182000553998342ip_bit(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),X,aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,Xa2)) ).

% flip_bit_uint32.rep_eq
tff(fact_7938_uint32_Oword__of__set__bit,axiom,
    ! [Na: nat,P2: uint32] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,aa(uint32,uint32,aa(nat,fun(uint32,uint32),bit_se5668285175392031749et_bit(uint32),Na),P2)) = 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_se5668285175392031749et_bit(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),Na),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,P2)) ).

% uint32.word_of_set_bit
tff(fact_7939_set__bit__uint32_Orep__eq,axiom,
    ! [X: nat,Xa2: uint32] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,aa(uint32,uint32,aa(nat,fun(uint32,uint32),bit_se5668285175392031749et_bit(uint32),X),Xa2)) = 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_se5668285175392031749et_bit(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),X),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,Xa2)) ).

% set_bit_uint32.rep_eq
tff(fact_7940_unset__bit__uint32_Orep__eq,axiom,
    ! [X: nat,Xa2: uint32] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,aa(uint32,uint32,aa(nat,fun(uint32,uint32),bit_se2638667681897837118et_bit(uint32),X),Xa2)) = 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_se2638667681897837118et_bit(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),X),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,Xa2)) ).

% unset_bit_uint32.rep_eq
tff(fact_7941_uint32_Oword__of__unset__bit,axiom,
    ! [Na: nat,P2: uint32] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,aa(uint32,uint32,aa(nat,fun(uint32,uint32),bit_se2638667681897837118et_bit(uint32),Na),P2)) = 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_se2638667681897837118et_bit(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),Na),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,P2)) ).

% uint32.word_of_unset_bit
tff(fact_7942_Rep__uint32,axiom,
    ! [X: uint32] : member(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,X),top_top(set(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))))) ).

% Rep_uint32
tff(fact_7943_Rep__uint32__cases,axiom,
    ! [Y: word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))] :
      ( member(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),Y,top_top(set(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))))))
     => ~ ! [X2: uint32] : Y != aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,X2) ) ).

% Rep_uint32_cases
tff(fact_7944_Rep__uint32__induct,axiom,
    ! [Y: word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),P: fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o)] :
      ( member(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),Y,top_top(set(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))))))
     => ( ! [X2: uint32] : aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o,P,aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,X2))
       => aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o,P,Y) ) ) ).

% Rep_uint32_induct
tff(fact_7945_zero__uint32_Orep__eq,axiom,
    aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),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_7946_less__uint32_Orep__eq,axiom,
    ! [X: uint32,Xa2: uint32] :
      ( aa(uint32,$o,aa(uint32,fun(uint32,$o),ord_less(uint32),X),Xa2)
    <=> aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o,aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o),ord_less(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,X)),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,Xa2)) ) ).

% less_uint32.rep_eq
tff(fact_7947_uint32_Oless__iff__word__of,axiom,
    ! [P2: uint32,Q3: uint32] :
      ( aa(uint32,$o,aa(uint32,fun(uint32,$o),ord_less(uint32),P2),Q3)
    <=> aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o,aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o),ord_less(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,P2)),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,Q3)) ) ).

% uint32.less_iff_word_of
tff(fact_7948_uint32_Oword__of__power,axiom,
    ! [P2: uint32,Na: nat] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,aa(nat,uint32,power_power(uint32,P2),Na)) = aa(nat,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),power_power(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,P2)),Na) ).

% uint32.word_of_power
tff(fact_7949_one__uint32_Orep__eq,axiom,
    aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),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_7950_less__eq__uint32_Orep__eq,axiom,
    ! [X: uint32,Xa2: uint32] :
      ( aa(uint32,$o,aa(uint32,fun(uint32,$o),ord_less_eq(uint32),X),Xa2)
    <=> aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o,aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o),ord_less_eq(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,X)),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,Xa2)) ) ).

% less_eq_uint32.rep_eq
tff(fact_7951_uint32_Oless__eq__iff__word__of,axiom,
    ! [P2: uint32,Q3: uint32] :
      ( aa(uint32,$o,aa(uint32,fun(uint32,$o),ord_less_eq(uint32),P2),Q3)
    <=> aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o,aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o),ord_less_eq(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,P2)),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,Q3)) ) ).

% uint32.less_eq_iff_word_of
tff(fact_7952_modulo__uint32_Orep__eq,axiom,
    ! [X: uint32,Xa2: uint32] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,aa(uint32,uint32,aa(uint32,fun(uint32,uint32),modulo_modulo(uint32),X),Xa2)) = 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(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),modulo_modulo(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,X)),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,Xa2)) ).

% modulo_uint32.rep_eq
tff(fact_7953_uint32_Oword__of__mod,axiom,
    ! [P2: uint32,Q3: uint32] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,aa(uint32,uint32,aa(uint32,fun(uint32,uint32),modulo_modulo(uint32),P2),Q3)) = 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(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),modulo_modulo(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,P2)),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,Q3)) ).

% uint32.word_of_mod
tff(fact_7954_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))),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,P2)) ) ).

% uint32.even_iff_word_of
tff(fact_7955_uint32_Oword__of__diff,axiom,
    ! [P2: uint32,Q3: uint32] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,aa(uint32,uint32,minus_minus(uint32,P2),Q3)) = 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)))))),minus_minus(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,P2)),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,Q3)) ).

% uint32.word_of_diff
tff(fact_7956_minus__uint32_Orep__eq,axiom,
    ! [X: uint32,Xa2: uint32] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,aa(uint32,uint32,minus_minus(uint32,X),Xa2)) = 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)))))),minus_minus(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,X)),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,Xa2)) ).

% minus_uint32.rep_eq
tff(fact_7957_size__uint32_Orep__eq,axiom,
    ! [X: uint32] : aa(uint32,nat,size_size(uint32),X) = aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),nat,size_size(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,X)) ).

% size_uint32.rep_eq
tff(fact_7958_uint32_Osize__eq__word__of,axiom,
    ! [P2: uint32] : aa(uint32,nat,size_size(uint32),P2) = aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),nat,size_size(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,P2)) ).

% uint32.size_eq_word_of
tff(fact_7959_uint32_Oword__of__push__bit,axiom,
    ! [Na: nat,P2: uint32] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,aa(uint32,uint32,aa(nat,fun(uint32,uint32),bit_se4730199178511100633sh_bit(uint32),Na),P2)) = 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))))))),Na),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,P2)) ).

% uint32.word_of_push_bit
tff(fact_7960_push__bit__uint32_Orep__eq,axiom,
    ! [X: nat,Xa2: uint32] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,aa(uint32,uint32,aa(nat,fun(uint32,uint32),bit_se4730199178511100633sh_bit(uint32),X),Xa2)) = 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))))))),X),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,Xa2)) ).

% push_bit_uint32.rep_eq
tff(fact_7961_drop__bit__uint32_Orep__eq,axiom,
    ! [X: nat,Xa2: uint32] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,aa(uint32,uint32,aa(nat,fun(uint32,uint32),bit_se4197421643247451524op_bit(uint32),X),Xa2)) = 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))))))),X),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,Xa2)) ).

% drop_bit_uint32.rep_eq
tff(fact_7962_uint32_Oword__of__drop__bit,axiom,
    ! [Na: nat,P2: uint32] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,aa(uint32,uint32,aa(nat,fun(uint32,uint32),bit_se4197421643247451524op_bit(uint32),Na),P2)) = 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))))))),Na),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,P2)) ).

% uint32.word_of_drop_bit
tff(fact_7963_uint32__test__bit__code,axiom,
    ! [W: uint32,Na: code_integer] :
      ( uint32_test_bit(W,Na)
    <=> $ite(
          ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),Na),zero_zero(code_integer))
          | aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit1(bit1(bit1(bit1(one2)))))),Na) ),
          aa(code_integer,$o,aa(uint32,fun(code_integer,$o),aa(fun(uint32,fun(nat,$o)),fun(uint32,fun(code_integer,$o)),undefined(fun(fun(uint32,fun(nat,$o)),fun(uint32,fun(code_integer,$o)))),bit_se5641148757651400278ts_bit(uint32)),W),Na),
          aa(nat,$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))))))),aa(uint32,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_7964_uint32__set__bit__code,axiom,
    ! [W: uint32,Na: code_integer,Ba: $o] :
      aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,uint32_set_bit(W,Na,(Ba))) = $ite(
        ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),Na),zero_zero(code_integer))
        | aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit1(bit1(bit1(bit1(one2)))))),Na) ),
        aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,aa($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),(Ba))),
        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))))))),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,W)),code_nat_of_integer(Na)),(Ba)) ) ).

% uint32_set_bit_code
tff(fact_7965_uint32__shiftl__code,axiom,
    ! [W: uint32,Na: code_integer] :
      aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,uint32_shiftl(W,Na)) = $ite(
        ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),Na),zero_zero(code_integer))
        | aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(bit0(bit0(bit0(bit0(one2))))))),Na) ),
        aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),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)),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,W)) ) ).

% uint32_shiftl_code
tff(fact_7966_set__bit__uint32__code,axiom,
    ! [X: uint32,Na: nat,Ba: $o] :
      aa($o,uint32,aa(nat,fun($o,uint32),aa(uint32,fun(nat,fun($o,uint32)),generi7602027413899671122et_bit(uint32),X),Na),(Ba)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(one2))))))),uint32_set_bit(X,code_integer_of_nat(Na),(Ba)),X) ).

% set_bit_uint32_code
tff(fact_7967_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,aa(nat,fun(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_7968_integer__of__nat_Orep__eq,axiom,
    ! [X: nat] : aa(code_integer,int,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_7969_int__of__integer__integer__of__nat,axiom,
    ! [Na: nat] : aa(code_integer,int,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_7970_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_7971_integer__of__nat__1,axiom,
    code_integer_of_nat(one_one(nat)) = one_one(code_integer) ).

% integer_of_nat_1
tff(fact_7972_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_7973_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_7974_integer__of__nat_Oabs__eq,axiom,
    ! [X: nat] : code_integer_of_nat(X) = aa(int,code_integer,code_integer_of_int,aa(nat,int,semiring_1_of_nat(int),X)) ).

% integer_of_nat.abs_eq
tff(fact_7975_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,$o,aa(nat,fun(nat,$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_7976_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,aa(nat,fun(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_7977_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_7978_uint32__divmod__code,axiom,
    ! [X: uint32,Y: uint32] :
      uint32_divmod(X,Y) = $ite(
        aa(uint32,$o,aa(uint32,fun(uint32,$o),ord_less_eq(uint32),aa(num,uint32,numeral_numeral(uint32),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2))))))))))))))))))))))))))))))))),Y),
        $ite(aa(uint32,$o,aa(uint32,fun(uint32,$o),ord_less(uint32),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,aa(uint32,fun(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,aa(uint32,fun(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_7979_uint32_Oword__of__set__bits__aux,axiom,
    ! [P: fun(nat,$o),Na: nat,P2: uint32] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,set_bits_aux_uint32(P,Na,P2)) = code_T2661198915054445665ts_aux(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))),P,Na,aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,P2)) ).

% uint32.word_of_set_bits_aux
tff(fact_7980_set__bits__aux__uint32_Orep__eq,axiom,
    ! [X: fun(nat,$o),Xa2: nat,Xb: uint32] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,set_bits_aux_uint32(X,Xa2,Xb)) = code_T2661198915054445665ts_aux(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))),X,Xa2,aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,Xb)) ).

% set_bits_aux_uint32.rep_eq
tff(fact_7981_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_7982_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_7983_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_7984_uint32__sdiv__code,axiom,
    ! [X: uint32,Y: uint32] :
      aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,uint32_sdiv(X,Y)) = $ite(Y = zero_zero(uint32),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),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)))))),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,X),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,Y))) ).

% uint32_sdiv_code
tff(fact_7985_sshiftr__uint32__code,axiom,
    ! [Na: nat,X: uint32] :
      aa(uint32,uint32,aa(nat,fun(uint32,uint32),signed489701013188660438uint32,Na),X) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(one2))))))),
        uint32_sshiftr(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_7986_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_7987_signed__drop__bit__uint32_Orep__eq,axiom,
    ! [X: nat,Xa2: uint32] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,aa(uint32,uint32,aa(nat,fun(uint32,uint32),signed489701013188660438uint32,X),Xa2)) = signed_drop_bit(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))),X,aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,Xa2)) ).

% signed_drop_bit_uint32.rep_eq
tff(fact_7988_uint32_Oword__of__signed__drop__bit,axiom,
    ! [Na: nat,P2: uint32] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,aa(uint32,uint32,aa(nat,fun(uint32,uint32),signed489701013188660438uint32,Na),P2)) = signed_drop_bit(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))),Na,aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,P2)) ).

% uint32.word_of_signed_drop_bit
tff(fact_7989_uint32__div__def,axiom,
    ! [X: uint32,Y: uint32] : uint32_div(X,Y) = aa(product_prod(uint32,uint32),uint32,product_fst(uint32,uint32),uint32_divmod(X,Y)) ).

% uint32_div_def
tff(fact_7990_uint32__sshiftr__def,axiom,
    ! [X: uint32,Na: code_integer] :
      uint32_sshiftr(X,Na) = $ite(
        ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),Na),zero_zero(code_integer))
        | aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(bit0(bit0(bit0(bit0(one2))))))),Na) ),
        aa(uint32,uint32,aa(code_integer,fun(uint32,uint32),aa(fun(nat,fun(uint32,uint32)),fun(code_integer,fun(uint32,uint32)),undefined(fun(fun(nat,fun(uint32,uint32)),fun(code_integer,fun(uint32,uint32)))),signed489701013188660438uint32),Na),X),
        aa(uint32,uint32,aa(nat,fun(uint32,uint32),signed489701013188660438uint32,code_nat_of_integer(Na)),X) ) ).

% uint32_sshiftr_def
tff(fact_7991_uint32__sshiftr__code,axiom,
    ! [W: uint32,Na: code_integer] :
      aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,uint32_sshiftr(W,Na)) = $ite(
        ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),Na),zero_zero(code_integer))
        | aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(bit0(bit0(bit0(bit0(one2))))))),Na) ),
        aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),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),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,W)) ) ).

% uint32_sshiftr_code
tff(fact_7992_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_7993_signed_Oinsort__insert__key__def,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [F2: fun(A,word(B)),X: A,Xs2: list(A)] :
          aa(list(A),list(A),aa(A,fun(list(A),list(A)),aa(fun(A,word(B)),fun(A,fun(list(A),list(A))),insort_insert_key(word(B),A,word_sle(B)),F2),X),Xs2) = $ite(member(word(B),aa(A,word(B),F2,X),aa(set(A),set(word(B)),image(A,word(B),F2),aa(list(A),set(A),set2(A),Xs2))),Xs2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),aa(fun(A,word(B)),fun(A,fun(list(A),list(A))),insort_key(word(B),A,word_sle(B)),F2),X),Xs2)) ) ).

% signed.insort_insert_key_def
tff(fact_7994_linorder_Oinsort__insert__key_Ocong,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(B,fun(B,$o))] : insort_insert_key(B,A,Less_eq) = insort_insert_key(B,A,Less_eq) ).

% linorder.insort_insert_key.cong
tff(fact_7995_signed_Osorted__insort__insert,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs2: list(word(A)),X: word(A)] :
          ( sorted_wrt(word(A),word_sle(A),Xs2)
         => sorted_wrt(word(A),word_sle(A),aa(list(word(A)),list(word(A)),aa(word(A),fun(list(word(A)),list(word(A))),aa(fun(word(A),word(A)),fun(word(A),fun(list(word(A)),list(word(A)))),insort_insert_key(word(A),word(A),word_sle(A)),aTP_Lamp_rf(word(A),word(A))),X),Xs2)) ) ) ).

% signed.sorted_insort_insert
tff(fact_7996_signed_Odistinct__insort__insert,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [Xs2: list(A),F2: fun(A,word(B)),X: A] :
          ( distinct(A,Xs2)
         => distinct(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),aa(fun(A,word(B)),fun(A,fun(list(A),list(A))),insort_insert_key(word(B),A,word_sle(B)),F2),X),Xs2)) ) ) ).

% signed.distinct_insort_insert
tff(fact_7997_signed_Oinsort__insert__triv,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Xs2: list(word(A))] :
          ( member(word(A),X,aa(list(word(A)),set(word(A)),set2(word(A)),Xs2))
         => ( aa(list(word(A)),list(word(A)),aa(word(A),fun(list(word(A)),list(word(A))),aa(fun(word(A),word(A)),fun(word(A),fun(list(word(A)),list(word(A)))),insort_insert_key(word(A),word(A),word_sle(A)),aTP_Lamp_rf(word(A),word(A))),X),Xs2) = Xs2 ) ) ) ).

% signed.insort_insert_triv
tff(fact_7998_signed_Oset__insort__insert,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Xs2: list(word(A))] : aa(list(word(A)),set(word(A)),set2(word(A)),aa(list(word(A)),list(word(A)),aa(word(A),fun(list(word(A)),list(word(A))),aa(fun(word(A),word(A)),fun(word(A),fun(list(word(A)),list(word(A)))),insort_insert_key(word(A),word(A),word_sle(A)),aTP_Lamp_rf(word(A),word(A))),X),Xs2)) = aa(set(word(A)),set(word(A)),insert(word(A),X),aa(list(word(A)),set(word(A)),set2(word(A)),Xs2)) ) ).

% signed.set_insort_insert
tff(fact_7999_signed_Oinsort__insert__insort,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X: word(A),Xs2: list(word(A))] :
          ( ~ member(word(A),X,aa(list(word(A)),set(word(A)),set2(word(A)),Xs2))
         => ( aa(list(word(A)),list(word(A)),aa(word(A),fun(list(word(A)),list(word(A))),aa(fun(word(A),word(A)),fun(word(A),fun(list(word(A)),list(word(A)))),insort_insert_key(word(A),word(A),word_sle(A)),aTP_Lamp_rf(word(A),word(A))),X),Xs2) = aa(list(word(A)),list(word(A)),aa(word(A),fun(list(word(A)),list(word(A))),aa(fun(word(A),word(A)),fun(word(A),fun(list(word(A)),list(word(A)))),insort_key(word(A),word(A),word_sle(A)),aTP_Lamp_rf(word(A),word(A))),X),Xs2) ) ) ) ).

% signed.insort_insert_insort
tff(fact_8000_signed_Oinsort__insert__key__triv,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(A)
     => ! [F2: fun(B,word(A)),X: B,Xs2: list(B)] :
          ( member(word(A),aa(B,word(A),F2,X),aa(set(B),set(word(A)),image(B,word(A),F2),aa(list(B),set(B),set2(B),Xs2)))
         => ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),aa(fun(B,word(A)),fun(B,fun(list(B),list(B))),insort_insert_key(word(A),B,word_sle(A)),F2),X),Xs2) = Xs2 ) ) ) ).

% signed.insort_insert_key_triv
tff(fact_8001_signed_Osorted__insort__insert__key,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(A)
     => ! [F2: fun(B,word(A)),Xs2: list(B),X: B] :
          ( sorted_wrt(word(A),word_sle(A),aa(list(B),list(word(A)),map(B,word(A),F2),Xs2))
         => sorted_wrt(word(A),word_sle(A),aa(list(B),list(word(A)),map(B,word(A),F2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),aa(fun(B,word(A)),fun(B,fun(list(B),list(B))),insort_insert_key(word(A),B,word_sle(A)),F2),X),Xs2))) ) ) ).

% signed.sorted_insort_insert_key
tff(fact_8002_uint32__mod__def,axiom,
    ! [X: uint32,Y: uint32] : uint32_mod(X,Y) = aa(product_prod(uint32,uint32),uint32,product_snd(uint32,uint32),uint32_divmod(X,Y)) ).

% uint32_mod_def
tff(fact_8003_signed_Oinsort__insert__insort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(A)
     => ! [F2: fun(B,word(A)),X: B,Xs2: list(B)] :
          ( ~ member(word(A),aa(B,word(A),F2,X),aa(set(B),set(word(A)),image(B,word(A),F2),aa(list(B),set(B),set2(B),Xs2)))
         => ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),aa(fun(B,word(A)),fun(B,fun(list(B),list(B))),insort_insert_key(word(A),B,word_sle(A)),F2),X),Xs2) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),aa(fun(B,word(A)),fun(B,fun(list(B),list(B))),insort_key(word(A),B,word_sle(A)),F2),X),Xs2) ) ) ) ).

% signed.insort_insert_insort_key
tff(fact_8004_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_8005_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_8006_uint32_Ointeger__of__eq__word__of,axiom,
    ! [P2: uint32] : aa(uint32,code_integer,integer_of_uint32,P2) = aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),code_integer,semiring_1_unsigned(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))),code_integer),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,P2)) ).

% uint32.integer_of_eq_word_of
tff(fact_8007_integer__of__uint32_Orep__eq,axiom,
    ! [X: uint32] : aa(code_integer,int,code_int_of_integer,aa(uint32,code_integer,integer_of_uint32,X)) = 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),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,X)) ).

% integer_of_uint32.rep_eq
tff(fact_8008_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_8009_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),aa(int,code_integer,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),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint32,Na)))) ).

% integer_of_uint32_signed_code
tff(fact_8010_set__bits__False__eq,axiom,
    ! [A: $tType] :
      ( bit_bi6583157726757044596ension(A)
     => ( bit_bi4170147762399595738t_bits(A,aTP_Lamp_aew(nat,$o)) = zero_zero(A) ) ) ).

% set_bits_False_eq
tff(fact_8011_int__set__bits__K__False,axiom,
    bit_bi4170147762399595738t_bits(int,aTP_Lamp_aew(nat,$o)) = zero_zero(int) ).

% int_set_bits_K_False
tff(fact_8012_set__bits__K__False,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( bit_bi4170147762399595738t_bits(word(A),aTP_Lamp_aew(nat,$o)) = zero_zero(word(A)) ) ) ).

% set_bits_K_False
tff(fact_8013_int__set__bits__K__True,axiom,
    bit_bi4170147762399595738t_bits(int,aTP_Lamp_alt(nat,$o)) = aa(int,int,uminus_uminus(int),one_one(int)) ).

% int_set_bits_K_True
tff(fact_8014_uint32_Oword__of__set__bits,axiom,
    ! [P: fun(nat,$o)] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,bit_bi4170147762399595738t_bits(uint32,P)) = bit_bi4170147762399595738t_bits(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),P) ).

% uint32.word_of_set_bits
tff(fact_8015_set__bits__uint32_Orep__eq,axiom,
    ! [X: fun(nat,$o)] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,bit_bi4170147762399595738t_bits(uint32,X)) = bit_bi4170147762399595738t_bits(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),X) ).

% set_bits_uint32.rep_eq
tff(fact_8016_Rep__uint32_H__code,axiom,
    ! [X: uint32] : aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),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_8017_Rep__uint32_H__def,axiom,
    rep_uint32 = rep_uint322 ).

% Rep_uint32'_def
tff(fact_8018_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_8019_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,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,F2,Na) ) ) ) ).

% word_test_bit_set_bits
tff(fact_8020_word__type__copy__misc__axioms_Ointro,axiom,
    ! [A: $tType,B: $tType] :
      ( ( bit_ri3973907225187159222ations(B)
        & cl_HOL_Oequal(B)
        & linorder(B)
        & type_len(A) )
     => ! [Size: nat,Word_of: fun(B,word(A)),Set_bits_aux: fun(fun(nat,$o),fun(nat,fun(B,B)))] :
          ( ( Size = type_len0_len_of(A,type2(A)) )
         => ( ! [P4: B] :
                ( most_s684356279273892711sb_msb(B,P4)
              <=> most_s684356279273892711sb_msb(word(A),aa(B,word(A),Word_of,P4)) )
           => ( ! [P4: B] :
                  ( aa(B,$o,least_8051144512741203767sb_lsb(B),P4)
                <=> aa(word(A),$o,least_8051144512741203767sb_lsb(word(A)),aa(B,word(A),Word_of,P4)) )
             => ( ! [P4: B] : aa(B,nat,size_size(B),P4) = aa(word(A),nat,size_size(word(A)),aa(B,word(A),Word_of,P4))
               => ( ! [P4: B,N: nat,B3: $o] : aa(B,word(A),Word_of,aa($o,B,aa(nat,fun($o,B),aa(B,fun(nat,fun($o,B)),generi7602027413899671122et_bit(B),P4),N),(B3))) = aa($o,word(A),aa(nat,fun($o,word(A)),aa(word(A),fun(nat,fun($o,word(A))),generi7602027413899671122et_bit(word(A)),aa(B,word(A),Word_of,P4)),N),(B3))
                 => ( ! [P5: fun(nat,$o)] : aa(B,word(A),Word_of,bit_bi4170147762399595738t_bits(B,P5)) = bit_bi4170147762399595738t_bits(word(A),P5)
                   => ( ! [P5: fun(nat,$o),N: nat,P4: B] : aa(B,word(A),Word_of,aa(B,B,aa(nat,fun(B,B),aa(fun(nat,$o),fun(nat,fun(B,B)),Set_bits_aux,P5),N),P4)) = code_T2661198915054445665ts_aux(A,P5,N,aa(B,word(A),Word_of,P4))
                     => word_T8964210463127689547axioms(B,A,Word_of,Size,Set_bits_aux) ) ) ) ) ) ) ) ) ).

% word_type_copy_misc_axioms.intro
tff(fact_8021_word__type__copy__misc__axioms__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( bit_ri3973907225187159222ations(A)
        & cl_HOL_Oequal(A)
        & linorder(A)
        & type_len(B) )
     => ! [Word_of: fun(A,word(B)),Size: nat,Set_bits_aux: fun(fun(nat,$o),fun(nat,fun(A,A)))] :
          ( word_T8964210463127689547axioms(A,B,Word_of,Size,Set_bits_aux)
        <=> ( ( Size = type_len0_len_of(B,type2(B)) )
            & ! [P7: A] :
                ( most_s684356279273892711sb_msb(A,P7)
              <=> most_s684356279273892711sb_msb(word(B),aa(A,word(B),Word_of,P7)) )
            & ! [P7: A] :
                ( aa(A,$o,least_8051144512741203767sb_lsb(A),P7)
              <=> aa(word(B),$o,least_8051144512741203767sb_lsb(word(B)),aa(A,word(B),Word_of,P7)) )
            & ! [P7: A] : aa(A,nat,size_size(A),P7) = aa(word(B),nat,size_size(word(B)),aa(A,word(B),Word_of,P7))
            & ! [P7: A,N2: nat,B7: $o] : aa(A,word(B),Word_of,aa($o,A,aa(nat,fun($o,A),aa(A,fun(nat,fun($o,A)),generi7602027413899671122et_bit(A),P7),N2),(B7))) = aa($o,word(B),aa(nat,fun($o,word(B)),aa(word(B),fun(nat,fun($o,word(B))),generi7602027413899671122et_bit(word(B)),aa(A,word(B),Word_of,P7)),N2),(B7))
            & ! [P6: fun(nat,$o)] : aa(A,word(B),Word_of,bit_bi4170147762399595738t_bits(A,P6)) = bit_bi4170147762399595738t_bits(word(B),P6)
            & ! [P6: fun(nat,$o),N2: nat,P7: A] : aa(A,word(B),Word_of,aa(A,A,aa(nat,fun(A,A),aa(fun(nat,$o),fun(nat,fun(A,A)),Set_bits_aux,P6),N2),P7)) = code_T2661198915054445665ts_aux(B,P6,N2,aa(A,word(B),Word_of,P7)) ) ) ) ).

% word_type_copy_misc_axioms_def
tff(fact_8022_msb__numeral_I1_J,axiom,
    ! [Na: num] : ~ most_s684356279273892711sb_msb(int,aa(num,int,numeral_numeral(int),Na)) ).

% msb_numeral(1)
tff(fact_8023_msb__1,axiom,
    ~ most_s684356279273892711sb_msb(int,one_one(int)) ).

% msb_1
tff(fact_8024_msb__numeral_I2_J,axiom,
    ! [Na: num] : most_s684356279273892711sb_msb(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na))) ).

% msb_numeral(2)
tff(fact_8025_msb__bin__rest,axiom,
    ! [X: int] :
      ( most_s684356279273892711sb_msb(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),X),aa(num,int,numeral_numeral(int),bit0(one2))))
    <=> most_s684356279273892711sb_msb(int,X) ) ).

% msb_bin_rest
tff(fact_8026_msb__uint32_Orep__eq,axiom,
    ! [X: uint32] :
      ( most_s684356279273892711sb_msb(uint32,X)
    <=> most_s684356279273892711sb_msb(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,X)) ) ).

% msb_uint32.rep_eq
tff(fact_8027_uint32_Omsb__iff__word__of,axiom,
    ! [P2: uint32] :
      ( most_s684356279273892711sb_msb(uint32,P2)
    <=> most_s684356279273892711sb_msb(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),aa(uint32,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322,P2)) ) ).

% uint32.msb_iff_word_of
tff(fact_8028_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_8029_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_8030_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_8031_word__type__copy__misc_Oset__bits__aux__code,axiom,
    ! [A: $tType,B: $tType] :
      ( ( bit_ri3973907225187159222ations(B)
        & cl_HOL_Oequal(B)
        & linorder(B)
        & type_len(A) )
     => ! [Of_word: fun(word(A),B),Word_of: fun(B,word(A)),Signed_drop_bit: fun(nat,fun(B,B)),Of_nat: fun(nat,B),Nat_of: fun(B,nat),Of_int: fun(int,B),Int_of: fun(B,int),Of_integer: fun(code_integer,B),Integer_of: fun(B,code_integer),Size: nat,Set_bits_aux: fun(fun(nat,$o),fun(nat,fun(B,B))),F2: fun(nat,$o),Na: nat,W: B] :
          ( word_T1870391057261780392y_misc(A,B,Of_word,Word_of,Signed_drop_bit,Of_nat,Nat_of,Of_int,Int_of,Of_integer,Integer_of,Size,Set_bits_aux)
         => ( aa(B,B,aa(nat,fun(B,B),aa(fun(nat,$o),fun(nat,fun(B,B)),Set_bits_aux,F2),Na),W) = $ite(
                Na = zero_zero(nat),
                W,
                $let(
                  n2: nat,
                  n2:= aa(nat,nat,minus_minus(nat,Na),one_one(nat)),
                  aa(B,B,aa(nat,fun(B,B),aa(fun(nat,$o),fun(nat,fun(B,B)),Set_bits_aux,F2),n2),
                    aa(B,B,aa(B,fun(B,B),bit_se1065995026697491101ons_or(B),aa(B,B,aa(nat,fun(B,B),bit_se4730199178511100633sh_bit(B),one_one(nat)),W)),
                      $ite(aa(nat,$o,F2,n2),one_one(B),zero_zero(B)))) ) ) ) ) ) ).

% word_type_copy_misc.set_bits_aux_code
tff(fact_8032_wf__set__bits__int__Suc,axiom,
    ! [F2: fun(nat,$o)] :
      ( bit_wf_set_bits_int(aTP_Lamp_aes(fun(nat,$o),fun(nat,$o),F2))
    <=> bit_wf_set_bits_int(F2) ) ).

% wf_set_bits_int_Suc
tff(fact_8033_word__type__copy__misc_Osize__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( bit_ri3973907225187159222ations(B)
        & cl_HOL_Oequal(B)
        & linorder(B)
        & type_len(A) )
     => ! [Of_word: fun(word(A),B),Word_of: fun(B,word(A)),Signed_drop_bit: fun(nat,fun(B,B)),Of_nat: fun(nat,B),Nat_of: fun(B,nat),Of_int: fun(int,B),Int_of: fun(B,int),Of_integer: fun(code_integer,B),Integer_of: fun(B,code_integer),Size: nat,Set_bits_aux: fun(fun(nat,$o),fun(nat,fun(B,B))),P2: B] :
          ( word_T1870391057261780392y_misc(A,B,Of_word,Word_of,Signed_drop_bit,Of_nat,Nat_of,Of_int,Int_of,Of_integer,Integer_of,Size,Set_bits_aux)
         => ( aa(B,nat,size_size(B),P2) = Size ) ) ) ).

% word_type_copy_misc.size_eq
tff(fact_8034_word__type__copy__misc_Osize__eq__word__of,axiom,
    ! [A: $tType,B: $tType] :
      ( ( bit_ri3973907225187159222ations(B)
        & cl_HOL_Oequal(B)
        & linorder(B)
        & type_len(A) )
     => ! [Of_word: fun(word(A),B),Word_of: fun(B,word(A)),Signed_drop_bit: fun(nat,fun(B,B)),Of_nat: fun(nat,B),Nat_of: fun(B,nat),Of_int: fun(int,B),Int_of: fun(B,int),Of_integer: fun(code_integer,B),Integer_of: fun(B,code_integer),Size: nat,Set_bits_aux: fun(fun(nat,$o),fun(nat,fun(B,B))),P2: B] :
          ( word_T1870391057261780392y_misc(A,B,Of_word,Word_of,Signed_drop_bit,Of_nat,Nat_of,Of_int,Int_of,Of_integer,Integer_of,Size,Set_bits_aux)
         => ( aa(B,nat,size_size(B),P2) = aa(word(A),nat,size_size(word(A)),aa(B,word(A),Word_of,P2)) ) ) ) ).

% word_type_copy_misc.size_eq_word_of
tff(fact_8035_wf__set__bits__int__const,axiom,
    ! [Ba: $o] : bit_wf_set_bits_int(aTP_Lamp_alu($o,fun(nat,$o),(Ba))) ).

% wf_set_bits_int_const
tff(fact_8036_wf__set__bits__int__simps,axiom,
    ! [F2: fun(nat,$o)] :
      ( bit_wf_set_bits_int(F2)
    <=> ? [N2: nat] :
          ( ! [N11: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N2),N11)
             => ~ aa(nat,$o,F2,N11) )
          | ! [N11: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N2),N11)
             => aa(nat,$o,F2,N11) ) ) ) ).

% wf_set_bits_int_simps
tff(fact_8037_zeros,axiom,
    ! [Na: nat,F2: fun(nat,$o)] :
      ( ! [N5: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N5)
         => ~ aa(nat,$o,F2,N5) )
     => bit_wf_set_bits_int(F2) ) ).

% zeros
tff(fact_8038_wf__set__bits__int_Osimps,axiom,
    ! [F2: fun(nat,$o)] :
      ( bit_wf_set_bits_int(F2)
    <=> ( ? [N2: nat] :
          ! [N11: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N2),N11)
           => ~ aa(nat,$o,F2,N11) )
        | ? [N2: nat] :
          ! [N11: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N2),N11)
           => aa(nat,$o,F2,N11) ) ) ) ).

% wf_set_bits_int.simps
tff(fact_8039_wf__set__bits__int_Ocases,axiom,
    ! [F2: fun(nat,$o)] :
      ( bit_wf_set_bits_int(F2)
     => ( ! [N: nat] :
            ~ ! [N4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),N4)
               => ~ aa(nat,$o,F2,N4) )
       => ~ ! [N: nat] :
              ~ ! [N4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),N4)
                 => aa(nat,$o,F2,N4) ) ) ) ).

% wf_set_bits_int.cases
tff(fact_8040_ones,axiom,
    ! [Na: nat,F2: fun(nat,$o)] :
      ( ! [N5: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N5)
         => aa(nat,$o,F2,N5) )
     => bit_wf_set_bits_int(F2) ) ).

% ones
tff(fact_8041_int__set__bits__unfold__BIT,axiom,
    ! [F2: fun(nat,$o)] :
      ( bit_wf_set_bits_int(F2)
     => ( bit_bi4170147762399595738t_bits(int,F2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa($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_8042_word__set__bits__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(nat,$o)] : bit_bi4170147762399595738t_bits(word(A),P) = groups4207007520872428315er_sum($o,word(A),zero_neq_one_of_bool(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2)),aa(list(nat),list($o),map(nat,$o,P),upt(zero_zero(nat),type_len0_len_of(A,type2(A))))) ) ).

% word_set_bits_def
tff(fact_8043_map__comp__map,axiom,
    ! [B: $tType,C: $tType,A: $tType,F2: fun(C,B),G: fun(A,C)] : comp(list(C),list(B),list(A),map(C,B,F2),map(A,C,G)) = map(A,B,comp(C,B,A,F2,G)) ).

% map_comp_map
tff(fact_8044_map__map,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(B,A),G: fun(C,B),Xs2: list(C)] : aa(list(B),list(A),map(B,A,F2),aa(list(C),list(B),map(C,B,G),Xs2)) = aa(list(C),list(A),map(C,A,comp(B,A,C,F2,G)),Xs2) ).

% map_map
tff(fact_8045_List_Omap_Ocompositionality,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(B,A),G: fun(C,B),List: list(C)] : aa(list(B),list(A),map(B,A,F2),aa(list(C),list(B),map(C,B,G),List)) = aa(list(C),list(A),map(C,A,comp(B,A,C,F2,G)),List) ).

% List.map.compositionality
tff(fact_8046_list_Omap__comp,axiom,
    ! [B: $tType,A: $tType,C: $tType,G: fun(B,A),F2: fun(C,B),V: list(C)] : aa(list(B),list(A),map(B,A,G),aa(list(C),list(B),map(C,B,F2),V)) = aa(list(C),list(A),map(C,A,comp(B,A,C,G,F2)),V) ).

% list.map_comp
tff(fact_8047_List_Omap_Ocomp,axiom,
    ! [B: $tType,C: $tType,A: $tType,F2: fun(C,B),G: fun(A,C)] : comp(list(C),list(B),list(A),map(C,B,F2),map(A,C,G)) = map(A,B,comp(C,B,A,F2,G)) ).

% List.map.comp
tff(fact_8048_upt__0__eq__Nil__conv,axiom,
    ! [J: nat] :
      ( ( upt(zero_zero(nat),J) = nil(nat) )
    <=> ( J = zero_zero(nat) ) ) ).

% upt_0_eq_Nil_conv
tff(fact_8049_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( upt(I,J) = nil(nat) ) ) ).

% upt_conv_Nil
tff(fact_8050_length__upt,axiom,
    ! [I: nat,J: nat] : aa(list(nat),nat,size_size(list(nat)),upt(I,J)) = aa(nat,nat,minus_minus(nat,J),I) ).

% length_upt
tff(fact_8051_sorted__list__of__set__range,axiom,
    ! [M: nat,Na: nat] : linord4507533701916653071of_set(nat,set_or7035219750837199246ssThan(nat,M,Na)) = upt(M,Na) ).

% sorted_list_of_set_range
tff(fact_8052_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( upt(I,J) = nil(nat) )
    <=> ( ( J = zero_zero(nat) )
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I) ) ) ).

% upt_eq_Nil_conv
tff(fact_8053_nth__upt,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J)
     => ( aa(nat,nat,nth(nat,upt(I,J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K) ) ) ).

% nth_upt
tff(fact_8054_sum__list__upt,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( groups8242544230860333062m_list(nat,upt(M,Na)) = aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_je(nat,nat)),set_or7035219750837199246ssThan(nat,M,Na)) ) ) ).

% sum_list_upt
tff(fact_8055_upt__rec__numeral,axiom,
    ! [M: num,Na: num] :
      upt(aa(num,nat,numeral_numeral(nat),M),aa(num,nat,numeral_numeral(nat),Na)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),Na)),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(num,nat,numeral_numeral(nat),M)),upt(aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),Na))),nil(nat)) ).

% upt_rec_numeral
tff(fact_8056_map__fst__enumerate,axiom,
    ! [A: $tType,Na: nat,Xs2: list(A)] : aa(list(product_prod(nat,A)),list(nat),map(product_prod(nat,A),nat,product_fst(nat,A)),enumerate(A,Na,Xs2)) = upt(Na,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(list(A),nat,size_size(list(A)),Xs2))) ).

% map_fst_enumerate
tff(fact_8057_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_8058_map__Suc__upt,axiom,
    ! [M: nat,Na: nat] : aa(list(nat),list(nat),map(nat,nat,suc),upt(M,Na)) = upt(aa(nat,nat,suc,M),aa(nat,nat,suc,Na)) ).

% map_Suc_upt
tff(fact_8059_foldr_Ofoldr__Cons,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,fun(A,A)),X: B,Xs2: list(B)] : foldr(B,A,F2,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs2)) = comp(A,A,A,aa(B,fun(A,A),F2,X),foldr(B,A,F2,Xs2)) ).

% foldr.foldr_Cons
tff(fact_8060_upt__conv__Cons__Cons,axiom,
    ! [M: nat,Na: nat,Ns: list(nat),Q3: nat] :
      ( ( aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),M),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),Na),Ns)) = upt(M,Q3) )
    <=> ( aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),Na),Ns) = upt(aa(nat,nat,suc,M),Q3) ) ) ).

% upt_conv_Cons_Cons
tff(fact_8061_distinct__upt,axiom,
    ! [I: nat,J: nat] : distinct(nat,upt(I,J)) ).

% distinct_upt
tff(fact_8062_comp__cong__left,axiom,
    ! [B: $tType,A: $tType,C: $tType,X: fun(A,B),Y: fun(A,B),F2: fun(C,A)] :
      ( ( X = Y )
     => ( comp(A,B,C,X,F2) = comp(A,B,C,Y,F2) ) ) ).

% comp_cong_left
tff(fact_8063_comp__cong__right,axiom,
    ! [C: $tType,B: $tType,A: $tType,X: fun(A,B),Y: fun(A,B),F2: fun(B,C)] :
      ( ( X = Y )
     => ( comp(B,C,A,F2,X) = comp(B,C,A,F2,Y) ) ) ).

% comp_cong_right
tff(fact_8064_fun__comp__eq__conv,axiom,
    ! [C: $tType,B: $tType,A: $tType,F2: fun(C,B),G: fun(A,C),Fg: fun(A,B)] :
      ( ( comp(C,B,A,F2,G) = Fg )
    <=> ! [X6: A] : aa(C,B,F2,aa(A,C,G,X6)) = aa(A,B,Fg,X6) ) ).

% fun_comp_eq_conv
tff(fact_8065_atLeastAtMost__upt,axiom,
    ! [Na: nat,M: nat] : set_or1337092689740270186AtMost(nat,Na,M) = aa(list(nat),set(nat),set2(nat),upt(Na,aa(nat,nat,suc,M))) ).

% atLeastAtMost_upt
tff(fact_8066_atLeastLessThan__upt,axiom,
    ! [I: nat,J: nat] : set_or7035219750837199246ssThan(nat,I,J) = aa(list(nat),set(nat),set2(nat),upt(I,J)) ).

% atLeastLessThan_upt
tff(fact_8067_greaterThanAtMost__upt,axiom,
    ! [Na: nat,M: nat] : set_or3652927894154168847AtMost(nat,Na,M) = aa(list(nat),set(nat),set2(nat),upt(aa(nat,nat,suc,Na),aa(nat,nat,suc,M))) ).

% greaterThanAtMost_upt
tff(fact_8068_map__bit__range__eq__if__take__bit__eq,axiom,
    ! [Na: nat,K: int,L: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K) = aa(int,int,bit_se2584673776208193580ke_bit(int,Na),L) )
     => ( aa(list(nat),list($o),map(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K)),upt(zero_zero(nat),Na)) = aa(list(nat),list($o),map(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),L)),upt(zero_zero(nat),Na)) ) ) ).

% map_bit_range_eq_if_take_bit_eq
tff(fact_8069_bit__shiftr__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A,Na: nat] : aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),bit_Sh4282982442137083166shiftr(A,Aa2,Na)) = comp(nat,$o,nat,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),aa(nat,fun(nat,nat),plus_plus(nat),Na)) ) ).

% bit_shiftr_eq
tff(fact_8070_atLeast__upt,axiom,
    ! [Na: nat] : set_ord_lessThan(nat,Na) = aa(list(nat),set(nat),set2(nat),upt(zero_zero(nat),Na)) ).

% atLeast_upt
tff(fact_8071_prod_OatLeastLessThan__shift__bounds,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,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),K))) = groups7121269368397514597t_prod(nat,A,comp(nat,A,nat,G,aa(nat,fun(nat,nat),plus_plus(nat),K)),set_or7035219750837199246ssThan(nat,M,Na)) ) ).

% prod.atLeastLessThan_shift_bounds
tff(fact_8072_prod_OatLeastAtMost__shift__bounds,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,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),K))) = groups7121269368397514597t_prod(nat,A,comp(nat,A,nat,G,aa(nat,fun(nat,nat),plus_plus(nat),K)),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% prod.atLeastAtMost_shift_bounds
tff(fact_8073_sum_OatLeastLessThan__shift__bounds,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,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),K))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,comp(nat,A,nat,G,aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or7035219750837199246ssThan(nat,M,Na)) ) ).

% sum.atLeastLessThan_shift_bounds
tff(fact_8074_sum_OatLeastAtMost__shift__bounds,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,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),K))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,comp(nat,A,nat,G,aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% sum.atLeastAtMost_shift_bounds
tff(fact_8075_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) )
         => ( ! [X2: B,Y2: B] : aa(B,A,H,aa(B,B,aa(B,fun(B,B),plus_plus(B),X2),Y2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,H,X2)),aa(B,A,H,Y2))
           => ( 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_8076_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_alv(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ).

% summable_inverse_divide
tff(fact_8077_sorted__upt,axiom,
    ! [M: nat,Na: nat] : sorted_wrt(nat,ord_less_eq(nat),upt(M,Na)) ).

% sorted_upt
tff(fact_8078_sorted__wrt__upt,axiom,
    ! [M: nat,Na: nat] : sorted_wrt(nat,ord_less(nat),upt(M,Na)) ).

% sorted_wrt_upt
tff(fact_8079_upt__0,axiom,
    ! [I: nat] : upt(I,zero_zero(nat)) = nil(nat) ).

% upt_0
tff(fact_8080_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( upt(I,J) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),I),upt(aa(nat,nat,suc,I),J)) ) ) ).

% upt_conv_Cons
tff(fact_8081_map__add__upt_H,axiom,
    ! [Ofs: nat,Aa2: nat,Ba: nat] : aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_alw(nat,fun(nat,nat),Ofs)),upt(Aa2,Ba)) = upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Aa2),Ofs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ba),Ofs)) ).

% map_add_upt'
tff(fact_8082_bit__drop__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,Aa2: A] : aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),Aa2)) = comp(nat,$o,nat,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2),aa(nat,fun(nat,nat),plus_plus(nat),Na)) ) ).

% bit_drop_bit_eq
tff(fact_8083_greaterThanLessThan__upt,axiom,
    ! [Na: nat,M: nat] : set_or5935395276787703475ssThan(nat,Na,M) = aa(list(nat),set(nat),set2(nat),upt(aa(nat,nat,suc,Na),M)) ).

% greaterThanLessThan_upt
tff(fact_8084_enumerate__map__upt,axiom,
    ! [A: $tType,Na: nat,F2: fun(nat,A),M: nat] : enumerate(A,Na,aa(list(nat),list(A),map(nat,A,F2),upt(Na,M))) = aa(list(nat),list(product_prod(nat,A)),map(nat,product_prod(nat,A),aTP_Lamp_alx(fun(nat,A),fun(nat,product_prod(nat,A)),F2)),upt(Na,M)) ).

% enumerate_map_upt
tff(fact_8085_map__add__upt,axiom,
    ! [Na: nat,M: nat] : aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_alw(nat,fun(nat,nat),Na)),upt(zero_zero(nat),M)) = upt(Na,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)) ).

% map_add_upt
tff(fact_8086_prod_Oreindex__nontrivial,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [A2: set(A),H: fun(A,B),G: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),A2)
         => ( ! [X2: A,Y2: A] :
                ( member(A,X2,A2)
               => ( member(A,Y2,A2)
                 => ( ( X2 != Y2 )
                   => ( ( aa(A,B,H,X2) = aa(A,B,H,Y2) )
                     => ( aa(B,C,G,aa(A,B,H,X2)) = 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_8087_DERIV__at__within__shift__lemma,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Y: A,Z: A,X: A,Sa: set(A)] :
          ( has_field_derivative(A,F2,Y,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),X),aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),Z)),Sa)))
         => has_field_derivative(A,comp(A,A,A,F2,aa(A,fun(A,A),plus_plus(A),Z)),Y,topolo174197925503356063within(A,X,Sa)) ) ) ).

% DERIV_at_within_shift_lemma
tff(fact_8088_DERIV__image__chain,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Da: A,G: fun(A,A),X: A,S: set(A),Db: A] :
          ( has_field_derivative(A,F2,Da,topolo174197925503356063within(A,aa(A,A,G,X),aa(set(A),set(A),image(A,A,G),S)))
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,X,S))
           => has_field_derivative(A,comp(A,A,A,F2,G),aa(A,A,aa(A,fun(A,A),times_times(A),Da),Db),topolo174197925503356063within(A,X,S)) ) ) ) ).

% DERIV_image_chain
tff(fact_8089_DERIV__chain,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Da: A,G: fun(A,A),X: A,Db: A,S: 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,S))
           => has_field_derivative(A,comp(A,A,A,F2,G),aa(A,A,aa(A,fun(A,A),times_times(A),Da),Db),topolo174197925503356063within(A,X,S)) ) ) ) ).

% DERIV_chain
tff(fact_8090_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,X: nat,Xs2: list(nat)] :
      ( ( upt(I,J) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X),Xs2) )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
        & ( I = X )
        & ( upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),one_one(nat)),J) = Xs2 ) ) ) ).

% upt_eq_Cons_conv
tff(fact_8091_atMost__upto,axiom,
    ! [Na: nat] : set_ord_atMost(nat,Na) = aa(list(nat),set(nat),set2(nat),upt(zero_zero(nat),aa(nat,nat,suc,Na))) ).

% atMost_upto
tff(fact_8092_upt__rec,axiom,
    ! [I: nat,J: nat] :
      upt(I,J) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),I),upt(aa(nat,nat,suc,I),J)),nil(nat)) ).

% upt_rec
tff(fact_8093_map__upt__Suc,axiom,
    ! [A: $tType,F2: fun(nat,A),Na: nat] : aa(list(nat),list(A),map(nat,A,F2),upt(zero_zero(nat),aa(nat,nat,suc,Na))) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,F2,zero_zero(nat))),aa(list(nat),list(A),map(nat,A,aTP_Lamp_acm(fun(nat,A),fun(nat,A),F2)),upt(zero_zero(nat),Na))) ).

% map_upt_Suc
tff(fact_8094_map__nth,axiom,
    ! [A: $tType,Xs2: list(A)] : aa(list(nat),list(A),map(nat,A,nth(A,Xs2)),upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs2))) = Xs2 ).

% map_nth
tff(fact_8095_map__decr__upt,axiom,
    ! [M: nat,Na: nat] : aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_aly(nat,nat)),upt(aa(nat,nat,suc,M),aa(nat,nat,suc,Na))) = upt(M,Na) ).

% map_decr_upt
tff(fact_8096_sum__image__le,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(A),G: fun(C,B),F2: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( ! [I2: A] :
                ( member(A,I2,I5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(C,B,G,aa(A,C,F2,I2))) )
           => aa(B,$o,aa(B,fun(B,$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_8097_filterlim__shift__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(A,B),D2: A,F4: filter(B),Aa2: A] :
          ( filterlim(A,B,comp(A,B,A,F2,aa(A,fun(A,A),plus_plus(A),D2)),F4,topolo174197925503356063within(A,aa(A,A,minus_minus(A,Aa2),D2),top_top(set(A))))
        <=> filterlim(A,B,F2,F4,topolo174197925503356063within(A,Aa2,top_top(set(A)))) ) ) ).

% filterlim_shift_iff
tff(fact_8098_filterlim__shift,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(A,B),F4: filter(B),Aa2: A,D2: A] :
          ( filterlim(A,B,F2,F4,topolo174197925503356063within(A,Aa2,top_top(set(A))))
         => filterlim(A,B,comp(A,B,A,F2,aa(A,fun(A,A),plus_plus(A),D2)),F4,topolo174197925503356063within(A,aa(A,A,minus_minus(A,Aa2),D2),top_top(set(A)))) ) ) ).

% filterlim_shift
tff(fact_8099_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,aa(A,fun(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_8100_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,aa(A,fun(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_8101_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_8102_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_8103_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,aa(nat,fun(nat,nat),plus_plus(nat),M))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,minus_minus(nat,Na),M))) ) ).

% sum.atLeastLessThan_shift_0
tff(fact_8104_nth__map__upt,axiom,
    ! [A: $tType,I: nat,Na: nat,M: nat,F2: fun(nat,A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,minus_minus(nat,Na),M))
     => ( aa(nat,A,nth(A,aa(list(nat),list(A),map(nat,A,F2),upt(M,Na))),I) = aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I)) ) ) ).

% nth_map_upt
tff(fact_8105_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,aa(nat,fun(nat,nat),plus_plus(nat),M)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,minus_minus(nat,Na),M))) ) ).

% prod.atLeastLessThan_shift_0
tff(fact_8106_tendsto__compose__at,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),Y: B,F4: filter(A),G: fun(B,C),Z: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Y),F4)
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,Z),topolo174197925503356063within(B,Y,top_top(set(B))))
           => ( eventually(A,aa(C,fun(A,$o),aa(fun(B,C),fun(C,fun(A,$o)),aa(B,fun(fun(B,C),fun(C,fun(A,$o))),aTP_Lamp_alz(fun(A,B),fun(B,fun(fun(B,C),fun(C,fun(A,$o)))),F2),Y),G),Z),F4)
             => filterlim(A,C,comp(B,C,A,G,F2),topolo7230453075368039082e_nhds(C,Z),F4) ) ) ) ) ).

% tendsto_compose_at
tff(fact_8107_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_aly(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_8108_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_aly(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_8109_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_aly(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_8110_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_aly(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_8111_map__upt__eqI,axiom,
    ! [A: $tType,Xs2: list(A),Na: nat,M: nat,F2: fun(nat,A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(nat,nat,minus_minus(nat,Na),M) )
     => ( ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs2))
           => ( aa(nat,A,nth(A,Xs2),I2) = aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I2)) ) )
       => ( aa(list(nat),list(A),map(nat,A,F2),upt(M,Na)) = Xs2 ) ) ) ).

% map_upt_eqI
tff(fact_8112_sum_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(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,aa(nat,fun(nat,nat),plus_plus(nat),M))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,minus_minus(nat,Na),M))) ) ) ) ).

% sum.atLeastAtMost_shift_0
tff(fact_8113_prod_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(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,aa(nat,fun(nat,nat),plus_plus(nat),M)),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,minus_minus(nat,Na),M))) ) ) ) ).

% prod.atLeastAtMost_shift_0
tff(fact_8114_horner__sum__bit__eq__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Aa2: A,Na: nat] : groups4207007520872428315er_sum($o,A,zero_neq_one_of_bool(A),aa(num,A,numeral_numeral(A),bit0(one2)),aa(list(nat),list($o),map(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Aa2)),upt(zero_zero(nat),Na))) = aa(A,A,bit_se2584673776208193580ke_bit(A,Na),Aa2) ) ).

% horner_sum_bit_eq_take_bit
tff(fact_8115_transpose__rectangle,axiom,
    ! [A: $tType,Xs2: list(list(A)),Na: nat] :
      ( ( ( Xs2 = nil(list(A)) )
       => ( Na = zero_zero(nat) ) )
     => ( ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(list(A)),nat,size_size(list(list(A))),Xs2))
           => ( aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs2),I2)) = Na ) )
       => ( transpose(A,Xs2) = aa(list(nat),list(list(A)),map(nat,list(A),aTP_Lamp_amb(list(list(A)),fun(nat,list(A)),Xs2)),upt(zero_zero(nat),Na)) ) ) ) ).

% transpose_rectangle
tff(fact_8116_map__filter__on__comp,axiom,
    ! [A: $tType,C: $tType,B: $tType,G: fun(B,A),Y5: set(B),X5: set(A),F4: filter(B),F2: fun(A,C)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,G),Y5)),X5)
     => ( eventually(B,aTP_Lamp_amc(set(B),fun(B,$o),Y5),F4)
       => ( map_filter_on(A,C,X5,F2,map_filter_on(B,A,Y5,G,F4)) = map_filter_on(B,C,Y5,comp(A,C,B,F2,G),F4) ) ) ) ).

% map_filter_on_comp
tff(fact_8117_fst__diag__fst,axiom,
    ! [B: $tType,A: $tType] : comp(product_prod(A,A),A,product_prod(A,B),product_fst(A,A),comp(A,product_prod(A,A),product_prod(A,B),aTP_Lamp_amd(A,product_prod(A,A)),product_fst(A,B))) = product_fst(A,B) ).

% fst_diag_fst
tff(fact_8118_set__oo__map__alt,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),X4: list(A)] : aa(list(A),set(B),comp(list(B),set(B),list(A),set2(B),map(A,B,F2)),X4) = aa(set(A),set(B),image(A,B,F2),aa(list(A),set(A),set2(A),X4)) ).

% set_oo_map_alt
tff(fact_8119_foldr__map,axiom,
    ! [C: $tType,B: $tType,A: $tType,G: fun(B,fun(A,A)),F2: fun(C,B),Xs2: list(C),Aa2: A] : aa(A,A,foldr(B,A,G,aa(list(C),list(B),map(C,B,F2),Xs2)),Aa2) = aa(A,A,foldr(C,A,comp(B,fun(A,A),C,G,F2),Xs2),Aa2) ).

% foldr_map
tff(fact_8120_transpose__map__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs2: list(list(B))] : transpose(A,aa(list(list(B)),list(list(A)),map(list(B),list(A),map(B,A,F2)),Xs2)) = aa(list(list(B)),list(list(A)),map(list(B),list(A),map(B,A,F2)),transpose(B,Xs2)) ).

% transpose_map_map
tff(fact_8121_folding__insort__key_Oinsort__key__commute,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Sa: set(B),F2: fun(B,A),X: B,Y: B] :
      ( folding_insort_key(A,B,Less_eq,Less,Sa,F2)
     => ( member(B,X,Sa)
       => ( member(B,Y,Sa)
         => ( comp(list(B),list(B),list(B),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F2),Y),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F2),X)) = comp(list(B),list(B),list(B),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F2),X),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F2),Y)) ) ) ) ) ).

% folding_insort_key.insort_key_commute
tff(fact_8122_signed_Osorted__list__of__set_Ofold__insort__key_Ocomp__fun__commute__on,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),X: word(A)] : comp(list(word(A)),list(word(A)),list(word(A)),aa(word(A),fun(list(word(A)),list(word(A))),aa(fun(word(A),word(A)),fun(word(A),fun(list(word(A)),list(word(A)))),insort_key(word(A),word(A),word_sle(A)),aTP_Lamp_rf(word(A),word(A))),Y),aa(word(A),fun(list(word(A)),list(word(A))),aa(fun(word(A),word(A)),fun(word(A),fun(list(word(A)),list(word(A)))),insort_key(word(A),word(A),word_sle(A)),aTP_Lamp_rf(word(A),word(A))),X)) = comp(list(word(A)),list(word(A)),list(word(A)),aa(word(A),fun(list(word(A)),list(word(A))),aa(fun(word(A),word(A)),fun(word(A),fun(list(word(A)),list(word(A)))),insort_key(word(A),word(A),word_sle(A)),aTP_Lamp_rf(word(A),word(A))),X),aa(word(A),fun(list(word(A)),list(word(A))),aa(fun(word(A),word(A)),fun(word(A),fun(list(word(A)),list(word(A)))),insort_key(word(A),word(A),word_sle(A)),aTP_Lamp_rf(word(A),word(A))),Y)) ) ).

% signed.sorted_list_of_set.fold_insort_key.comp_fun_commute_on
tff(fact_8123_lift__def,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] : heap_Time_lift(A,B,F2) = comp(B,heap_Time_Heap(B),A,heap_Time_return(B),F2) ).

% lift_def
tff(fact_8124_snd__diag__snd,axiom,
    ! [B: $tType,A: $tType] : comp(product_prod(B,B),B,product_prod(A,B),product_snd(B,B),comp(B,product_prod(B,B),product_prod(A,B),aTP_Lamp_ame(B,product_prod(B,B)),product_snd(A,B))) = product_snd(A,B) ).

% snd_diag_snd
tff(fact_8125_fst__diag__snd,axiom,
    ! [B: $tType,A: $tType] : comp(product_prod(B,B),B,product_prod(A,B),product_fst(B,B),comp(B,product_prod(B,B),product_prod(A,B),aTP_Lamp_ame(B,product_prod(B,B)),product_snd(A,B))) = product_snd(A,B) ).

% fst_diag_snd
tff(fact_8126_snd__diag__fst,axiom,
    ! [B: $tType,A: $tType] : comp(product_prod(A,A),A,product_prod(A,B),product_snd(A,A),comp(A,product_prod(A,A),product_prod(A,B),aTP_Lamp_amd(A,product_prod(A,A)),product_fst(A,B))) = product_fst(A,B) ).

% snd_diag_fst
tff(fact_8127_case__prod__comp,axiom,
    ! [D: $tType,A: $tType,C: $tType,B: $tType,F2: fun(D,fun(C,A)),G: fun(B,D),X: product_prod(B,C)] : aa(product_prod(B,C),A,product_case_prod(B,C,A,comp(D,fun(C,A),B,F2,G)),X) = aa(C,A,aa(D,fun(C,A),F2,aa(B,D,G,aa(product_prod(B,C),B,product_fst(B,C),X))),aa(product_prod(B,C),C,product_snd(B,C),X)) ).

% case_prod_comp
tff(fact_8128_eventually__map__filter__on,axiom,
    ! [B: $tType,A: $tType,X5: set(A),F4: filter(A),P: fun(B,$o),F2: fun(A,B)] :
      ( eventually(A,aTP_Lamp_ab(set(A),fun(A,$o),X5),F4)
     => ( eventually(B,P,map_filter_on(A,B,X5,F2,F4))
      <=> eventually(A,aa(fun(A,B),fun(A,$o),aa(fun(B,$o),fun(fun(A,B),fun(A,$o)),aTP_Lamp_amf(set(A),fun(fun(B,$o),fun(fun(A,B),fun(A,$o))),X5),P),F2),F4) ) ) ).

% eventually_map_filter_on
tff(fact_8129_transpose_Osimps_I2_J,axiom,
    ! [A: $tType,Xss2: list(list(A))] : transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2)) = transpose(A,Xss2) ).

% transpose.simps(2)
tff(fact_8130_transpose_Osimps_I1_J,axiom,
    ! [A: $tType] : transpose(A,nil(list(A))) = nil(list(A)) ).

% transpose.simps(1)
tff(fact_8131_sum_Oeq__fold,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),A2: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A2) = finite_fold(B,A,comp(A,fun(A,A),B,plus_plus(A),G),zero_zero(A),A2) ) ).

% sum.eq_fold
tff(fact_8132_prod_Oeq__fold,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),A2: set(B)] : groups7121269368397514597t_prod(B,A,G,A2) = finite_fold(B,A,comp(A,fun(A,A),B,times_times(A),G),one_one(A),A2) ) ).

% prod.eq_fold
tff(fact_8133_transpose__empty,axiom,
    ! [A: $tType,Xs2: list(list(A))] :
      ( ( transpose(A,Xs2) = nil(list(A)) )
    <=> ! [X6: list(A)] :
          ( member(list(A),X6,aa(list(list(A)),set(list(A)),set2(list(A)),Xs2))
         => ( X6 = nil(A) ) ) ) ).

% transpose_empty
tff(fact_8134_snd__fst__flip,axiom,
    ! [A: $tType,B: $tType,Xy: product_prod(B,A)] : aa(product_prod(B,A),A,product_snd(B,A),Xy) = aa(product_prod(B,A),A,comp(product_prod(A,B),A,product_prod(B,A),product_fst(A,B),product_case_prod(B,A,product_prod(A,B),aTP_Lamp_ala(B,fun(A,product_prod(A,B))))),Xy) ).

% snd_fst_flip
tff(fact_8135_fst__snd__flip,axiom,
    ! [B: $tType,A: $tType,Xy: product_prod(A,B)] : aa(product_prod(A,B),A,product_fst(A,B),Xy) = aa(product_prod(A,B),A,comp(product_prod(B,A),A,product_prod(A,B),product_snd(B,A),product_case_prod(A,B,product_prod(B,A),aTP_Lamp_rj(A,fun(B,product_prod(B,A))))),Xy) ).

% fst_snd_flip
tff(fact_8136_infinite__int__iff__infinite__nat__abs,axiom,
    ! [Sa: set(int)] :
      ( ~ aa(set(int),$o,finite_finite2(int),Sa)
    <=> ~ aa(set(nat),$o,finite_finite2(nat),aa(set(int),set(nat),image(int,nat,comp(int,nat,int,nat2,abs_abs(int))),Sa)) ) ).

% infinite_int_iff_infinite_nat_abs
tff(fact_8137_finite__range__prod,axiom,
    ! [A: $tType,C: $tType,B: $tType,F2: fun(B,product_prod(A,C))] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image(B,A,comp(product_prod(A,C),A,B,product_fst(A,C),F2)),top_top(set(B))))
     => ( aa(set(C),$o,finite_finite2(C),aa(set(B),set(C),image(B,C,comp(product_prod(A,C),C,B,product_snd(A,C),F2)),top_top(set(B))))
       => aa(set(product_prod(A,C)),$o,finite_finite2(product_prod(A,C)),aa(set(B),set(product_prod(A,C)),image(B,product_prod(A,C),F2),top_top(set(B)))) ) ) ).

% finite_range_prod
tff(fact_8138_summable__reindex,axiom,
    ! [F2: fun(nat,real),G: fun(nat,nat)] :
      ( summable(real,F2)
     => ( inj_on(nat,nat,G,top_top(set(nat)))
       => ( ! [X2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,F2,X2))
         => summable(real,comp(nat,real,nat,F2,G)) ) ) ) ).

% summable_reindex
tff(fact_8139_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_8140_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_8141_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_8142_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_8143_length__transpose,axiom,
    ! [A: $tType,Xs2: list(list(A))] : aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs2)) = aa(nat,nat,foldr(list(A),nat,aTP_Lamp_amg(list(A),fun(nat,nat)),Xs2),zero_zero(nat)) ).

% length_transpose
tff(fact_8144_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_amh(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_8145_suminf__reindex__mono,axiom,
    ! [F2: fun(nat,real),G: fun(nat,nat)] :
      ( summable(real,F2)
     => ( inj_on(nat,nat,G,top_top(set(nat)))
       => ( ! [X2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,F2,X2))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),suminf(real,comp(nat,real,nat,F2,G))),suminf(real,F2)) ) ) ) ).

% suminf_reindex_mono
tff(fact_8146_filterlim__at__top__iff__inverse__0,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( eventually(A,aTP_Lamp_aha(fun(A,real),fun(A,$o),F2),F4)
     => ( filterlim(A,real,F2,at_top(real),F4)
      <=> filterlim(A,real,comp(real,real,A,inverse_inverse(real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% filterlim_at_top_iff_inverse_0
tff(fact_8147_suminf__reindex,axiom,
    ! [F2: fun(nat,real),G: fun(nat,nat)] :
      ( summable(real,F2)
     => ( inj_on(nat,nat,G,top_top(set(nat)))
       => ( ! [X2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,F2,X2))
         => ( ! [X2: nat] :
                ( ~ member(nat,X2,aa(set(nat),set(nat),image(nat,nat,G),top_top(set(nat))))
               => ( aa(nat,real,F2,X2) = zero_zero(real) ) )
           => ( suminf(real,comp(nat,real,nat,F2,G)) = suminf(real,F2) ) ) ) ) ) ).

% suminf_reindex
tff(fact_8148_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_8149_length__transpose__sorted,axiom,
    ! [A: $tType,Xs2: list(list(A))] :
      ( sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs2)))
     => ( aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs2)) = $ite(Xs2 = nil(list(A)),zero_zero(nat),aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs2),zero_zero(nat)))) ) ) ).

% length_transpose_sorted
tff(fact_8150_rev__rev__ident,axiom,
    ! [A: $tType,Xs2: list(A)] : aa(list(A),list(A),rev(A),aa(list(A),list(A),rev(A),Xs2)) = Xs2 ).

% rev_rev_ident
tff(fact_8151_rev__is__rev__conv,axiom,
    ! [A: $tType,Xs2: list(A),Ysb: list(A)] :
      ( ( aa(list(A),list(A),rev(A),Xs2) = aa(list(A),list(A),rev(A),Ysb) )
    <=> ( Xs2 = Ysb ) ) ).

% rev_is_rev_conv
tff(fact_8152_Nil__is__rev__conv,axiom,
    ! [A: $tType,Xs2: list(A)] :
      ( ( nil(A) = aa(list(A),list(A),rev(A),Xs2) )
    <=> ( Xs2 = nil(A) ) ) ).

% Nil_is_rev_conv
tff(fact_8153_rev__is__Nil__conv,axiom,
    ! [A: $tType,Xs2: list(A)] :
      ( ( aa(list(A),list(A),rev(A),Xs2) = nil(A) )
    <=> ( Xs2 = nil(A) ) ) ).

% rev_is_Nil_conv
tff(fact_8154_length__rev,axiom,
    ! [A: $tType,Xs2: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),rev(A),Xs2)) = aa(list(A),nat,size_size(list(A)),Xs2) ).

% length_rev
tff(fact_8155_set__rev,axiom,
    ! [A: $tType,Xs2: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),rev(A),Xs2)) = aa(list(A),set(A),set2(A),Xs2) ).

% set_rev
tff(fact_8156_distinct__rev,axiom,
    ! [A: $tType,Xs2: list(A)] :
      ( distinct(A,aa(list(A),list(A),rev(A),Xs2))
    <=> distinct(A,Xs2) ) ).

% distinct_rev
tff(fact_8157_inj__on__rev,axiom,
    ! [A: $tType,A2: set(list(A))] : inj_on(list(A),list(A),rev(A),A2) ).

% inj_on_rev
tff(fact_8158_singleton__rev__conv,axiom,
    ! [A: $tType,X: A,Xs2: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) = aa(list(A),list(A),rev(A),Xs2) )
    <=> ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) = Xs2 ) ) ).

% singleton_rev_conv
tff(fact_8159_rev__singleton__conv,axiom,
    ! [A: $tType,Xs2: list(A),X: A] :
      ( ( aa(list(A),list(A),rev(A),Xs2) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) )
    <=> ( Xs2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) ) ) ).

% rev_singleton_conv
tff(fact_8160_sorted__wrt__rev__linord,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A)] :
          ( sorted_wrt(A,aTP_Lamp_afv(A,fun(A,$o)),L)
        <=> sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),L)) ) ) ).

% sorted_wrt_rev_linord
tff(fact_8161_sorted__wrt__map__rev__linord,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [L: list(product_prod(A,B))] :
          ( sorted_wrt(product_prod(A,B),aTP_Lamp_ami(product_prod(A,B),fun(product_prod(A,B),$o)),L)
        <=> sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),L))) ) ) ).

% sorted_wrt_map_rev_linord
tff(fact_8162_rev__map,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs2: list(B)] : aa(list(A),list(A),rev(A),aa(list(B),list(A),map(B,A,F2),Xs2)) = aa(list(B),list(A),map(B,A,F2),aa(list(B),list(B),rev(B),Xs2)) ).

% rev_map
tff(fact_8163_rev_Osimps_I1_J,axiom,
    ! [A: $tType] : aa(list(A),list(A),rev(A),nil(A)) = nil(A) ).

% rev.simps(1)
tff(fact_8164_sorted__wrt__rev,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),Xs2: list(A)] :
      ( sorted_wrt(A,P,aa(list(A),list(A),rev(A),Xs2))
    <=> sorted_wrt(A,aTP_Lamp_amj(fun(A,fun(A,$o)),fun(A,fun(A,$o)),P),Xs2) ) ).

% sorted_wrt_rev
tff(fact_8165_rev__swap,axiom,
    ! [A: $tType,Xs2: list(A),Ysb: list(A)] :
      ( ( aa(list(A),list(A),rev(A),Xs2) = Ysb )
    <=> ( Xs2 = aa(list(A),list(A),rev(A),Ysb) ) ) ).

% rev_swap
tff(fact_8166_card_Oeq__fold,axiom,
    ! [A: $tType,A2: set(A)] : finite_card(A,A2) = finite_fold(A,nat,aTP_Lamp_aju(A,fun(nat,nat)),zero_zero(nat),A2) ).

% card.eq_fold
tff(fact_8167_image__fold__insert,axiom,
    ! [B: $tType,A: $tType,A2: set(A),F2: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),A2)
     => ( aa(set(A),set(B),image(A,B,F2),A2) = finite_fold(A,set(B),aTP_Lamp_amk(fun(A,B),fun(A,fun(set(B),set(B))),F2),bot_bot(set(B)),A2) ) ) ).

% image_fold_insert
tff(fact_8168_rev__nth,axiom,
    ! [A: $tType,Na: nat,Xs2: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs2))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),rev(A),Xs2)),Na) = aa(nat,A,nth(A,Xs2),aa(nat,nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs2)),aa(nat,nat,suc,Na))) ) ) ).

% rev_nth
tff(fact_8169_rev__update,axiom,
    ! [A: $tType,K: nat,Xs2: list(A),Y: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Xs2))
     => ( aa(list(A),list(A),rev(A),list_update(A,Xs2,K,Y)) = list_update(A,aa(list(A),list(A),rev(A),Xs2),aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs2)),K)),one_one(nat)),Y) ) ) ).

% rev_update
tff(fact_8170_signed_Osorted__list__of__set_Ofold__insort__key_Oeq__fold,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: set(word(A))] : aa(set(word(A)),list(word(A)),sorted_list_of_set(word(A),word_sle(A)),A2) = finite_fold(word(A),list(word(A)),aa(fun(word(A),word(A)),fun(word(A),fun(list(word(A)),list(word(A)))),insort_key(word(A),word(A),word_sle(A)),aTP_Lamp_rf(word(A),word(A))),nil(word(A)),A2) ) ).

% signed.sorted_list_of_set.fold_insort_key.eq_fold
tff(fact_8171_sorted__rev__iff__nth__Suc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs2: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),Xs2))
        <=> ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I4)),aa(list(A),nat,size_size(list(A)),Xs2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs2),aa(nat,nat,suc,I4))),aa(nat,A,nth(A,Xs2),I4)) ) ) ) ).

% sorted_rev_iff_nth_Suc
tff(fact_8172_sorted__rev__iff__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs2: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),Xs2))
        <=> ! [I4: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I4),J3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs2))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs2),J3)),aa(nat,A,nth(A,Xs2),I4)) ) ) ) ) ).

% sorted_rev_iff_nth_mono
tff(fact_8173_sorted__rev__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs2: list(A),I: nat,J: nat] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),Xs2))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs2),J)),aa(nat,A,nth(A,Xs2),I)) ) ) ) ) ).

% sorted_rev_nth_mono
tff(fact_8174_sorted__transpose,axiom,
    ! [A: $tType,Xs2: list(list(A))] : sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),transpose(A,Xs2)))) ).

% sorted_transpose
tff(fact_8175_foldr__max__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs2: list(A),Y: A] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),Xs2))
         => ( aa(A,A,foldr(A,A,ord_max(A),Xs2),Y) = $ite(Xs2 = nil(A),Y,aa(A,A,aa(A,fun(A,A),ord_max(A),aa(nat,A,nth(A,Xs2),zero_zero(nat))),Y)) ) ) ) ).

% foldr_max_sorted
tff(fact_8176_nth__nth__transpose__sorted,axiom,
    ! [A: $tType,Xs2: list(list(A)),I: nat,J: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs2)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs2)))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(list(A)),nat,size_size(list(list(A))),filter2(list(A),aTP_Lamp_aml(nat,fun(list(A),$o),I),Xs2)))
         => ( aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),transpose(A,Xs2)),I)),J) = aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),Xs2),J)),I) ) ) ) ) ).

% nth_nth_transpose_sorted
tff(fact_8177_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_amm(C,set(B)),F2) = comp(set(D),set(B),A,image(D,B,G),aTP_Lamp_amn(A,set(D))) ).

% empty_natural
tff(fact_8178_ATP_Olambda__1,axiom,
    ! [Uu2: nat] : aa(nat,heap_Time_Heap(nat),aTP_Lamp_aa(nat,heap_Time_Heap(nat)),Uu2) = vEBT_Example_mIf(nat,vEBT_vebt_memberi(ti,Uu2),aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),Uu2),heap_Time_bind(option(nat),nat,vEBT_vebt_predi(ti,Uu2),aTP_Lamp_a(option(nat),heap_Time_Heap(nat)))) ).

% ATP.lambda_1
tff(fact_8179_ATP_Olambda__2,axiom,
    ! [Uu2: nat] : aa(nat,real,aTP_Lamp_be(nat,real),Uu2) = 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))),Uu2)),one_one(real))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uu2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat)))) ).

% ATP.lambda_2
tff(fact_8180_ATP_Olambda__3,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A] : aa(A,A,aTP_Lamp_zy(A,A),Uu2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,exp(A),Uu2)),one_one(A))),Uu2) ) ).

% ATP.lambda_3
tff(fact_8181_ATP_Olambda__4,axiom,
    ! [Uu2: nat] : aa(nat,real,aTP_Lamp_cn(nat,real),Uu2) = 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,Uu2)) ).

% ATP.lambda_4
tff(fact_8182_ATP_Olambda__5,axiom,
    ! [Uu2: real] :
      ( aa(real,$o,aTP_Lamp_dq(real,$o),Uu2)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Uu2)
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uu2),aa(num,real,numeral_numeral(real),bit0(one2)))
        & ( cos(real,Uu2) = zero_zero(real) ) ) ) ).

% ATP.lambda_5
tff(fact_8183_ATP_Olambda__6,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: nat] : aa(nat,A,aTP_Lamp_abt(nat,A),Uu2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Uu2))),aa(nat,A,semiring_1_of_nat(A),Uu2)) ) ).

% ATP.lambda_6
tff(fact_8184_ATP_Olambda__7,axiom,
    ! [Uu2: nat] : aa(nat,real,aTP_Lamp_hw(nat,real),Uu2) = aa(real,real,aa(real,fun(real,real),times_times(real),cos_coeff(Uu2)),aa(nat,real,power_power(real,zero_zero(real)),Uu2)) ).

% ATP.lambda_7
tff(fact_8185_ATP_Olambda__8,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: nat] : aa(nat,A,aTP_Lamp_abs(nat,A),Uu2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),Uu2)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Uu2))) ) ).

% ATP.lambda_8
tff(fact_8186_ATP_Olambda__9,axiom,
    ! [Uu2: real] : aa(real,real,aTP_Lamp_aac(real,real),Uu2) = aa(real,real,aa(real,fun(real,real),divide_divide(real),cos(real,Uu2)),sin(real,Uu2)) ).

% ATP.lambda_9
tff(fact_8187_ATP_Olambda__10,axiom,
    ! [Uu2: real] : aa(real,real,aTP_Lamp_aej(real,real),Uu2) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),Uu2)),Uu2) ).

% ATP.lambda_10
tff(fact_8188_ATP_Olambda__11,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Uu2: word(B)] :
          ( aa(word(B),$o,aTP_Lamp_qf(word(B),$o),Uu2)
        <=> aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less(word(B)),Uu2),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_11
tff(fact_8189_ATP_Olambda__12,axiom,
    ! [Uu2: int] : aa(int,int,aTP_Lamp_akc(int,int),Uu2) = aa(int,int,aa(int,fun(int,int),divide_divide(int),Uu2),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% ATP.lambda_12
tff(fact_8190_ATP_Olambda__13,axiom,
    ! [Uu2: nat] : aa(nat,real,aTP_Lamp_aba(nat,real),Uu2) = aa(real,real,root(Uu2),aa(nat,real,semiring_1_of_nat(real),Uu2)) ).

% ATP.lambda_13
tff(fact_8191_ATP_Olambda__14,axiom,
    ! [Uu2: nat] : aa(nat,nat,aTP_Lamp_aly(nat,nat),Uu2) = aa(nat,nat,minus_minus(nat,Uu2),aa(nat,nat,suc,zero_zero(nat))) ).

% ATP.lambda_14
tff(fact_8192_ATP_Olambda__15,axiom,
    ! [Uu2: int] : aa(int,int,aTP_Lamp_ake(int,int),Uu2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uu2),Uu2) ).

% ATP.lambda_15
tff(fact_8193_ATP_Olambda__16,axiom,
    ! [B: $tType,Uu2: B] : aa(B,product_prod(B,B),aTP_Lamp_ame(B,product_prod(B,B)),Uu2) = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uu2),Uu2) ).

% ATP.lambda_16
tff(fact_8194_ATP_Olambda__17,axiom,
    ! [A: $tType,Uu2: A] : aa(A,product_prod(A,A),aTP_Lamp_amd(A,product_prod(A,A)),Uu2) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu2),Uu2) ).

% ATP.lambda_17
tff(fact_8195_ATP_Olambda__18,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu2: A] : aa(A,A,aTP_Lamp_as(A,A),Uu2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu2),one_one(A)) ) ).

% ATP.lambda_18
tff(fact_8196_ATP_Olambda__19,axiom,
    ! [A: $tType,Uu2: A] : aa(A,list(A),aTP_Lamp_qs(A,list(A)),Uu2) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu2),nil(A)) ).

% ATP.lambda_19
tff(fact_8197_ATP_Olambda__20,axiom,
    ! [A: $tType,Uu2: A] : aa(A,set(A),aTP_Lamp_rc(A,set(A)),Uu2) = aa(set(A),set(A),insert(A,Uu2),bot_bot(set(A))) ).

% ATP.lambda_20
tff(fact_8198_ATP_Olambda__21,axiom,
    ! [Uu2: nat] : aa(nat,real,aTP_Lamp_abk(nat,real),Uu2) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),Uu2)) ).

% ATP.lambda_21
tff(fact_8199_ATP_Olambda__22,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: nat] : aa(nat,A,aTP_Lamp_abr(nat,A),Uu2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),Uu2)) ) ).

% ATP.lambda_22
tff(fact_8200_ATP_Olambda__23,axiom,
    ! [Uu2: int] :
      ( aa(int,$o,aTP_Lamp_kv(int,$o),Uu2)
    <=> ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Uu2) ) ).

% ATP.lambda_23
tff(fact_8201_ATP_Olambda__24,axiom,
    ! [Uu2: real] : aa(real,real,aTP_Lamp_tb(real,real),Uu2) = suminf(real,aTP_Lamp_bd(real,fun(nat,real),Uu2)) ).

% ATP.lambda_24
tff(fact_8202_ATP_Olambda__25,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,B)] : aa(fun(A,B),set(product_prod(A,B)),aTP_Lamp_re(fun(A,B),set(product_prod(A,B))),Uu2) = collect(product_prod(A,B),product_case_prod(A,B,$o,aTP_Lamp_rd(fun(A,B),fun(A,fun(B,$o)),Uu2))) ).

% ATP.lambda_25
tff(fact_8203_ATP_Olambda__26,axiom,
    ! [Uu2: nat] : aa(nat,real,aTP_Lamp_abl(nat,real),Uu2) = aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uu2))) ).

% ATP.lambda_26
tff(fact_8204_ATP_Olambda__27,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: nat] : aa(nat,A,aTP_Lamp_ed(nat,A),Uu2) = aa(A,A,inverse_inverse(A),semiring_char_0_fact(A,Uu2)) ) ).

% ATP.lambda_27
tff(fact_8205_ATP_Olambda__28,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: nat] : aa(nat,A,aTP_Lamp_abd(nat,A),Uu2) = aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Uu2)) ) ).

% ATP.lambda_28
tff(fact_8206_ATP_Olambda__29,axiom,
    ! [A: $tType,Uu2: list(A)] : aa(list(A),fun(nat,nat),aTP_Lamp_amg(list(A),fun(nat,nat)),Uu2) = aa(nat,fun(nat,nat),ord_max(nat),aa(list(A),nat,size_size(list(A)),Uu2)) ).

% ATP.lambda_29
tff(fact_8207_ATP_Olambda__30,axiom,
    ! [Uu2: option(nat)] : aa(option(nat),heap_Time_Heap(nat),aTP_Lamp_a(option(nat),heap_Time_Heap(nat)),Uu2) = aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),the2(nat,Uu2)) ).

% ATP.lambda_30
tff(fact_8208_ATP_Olambda__31,axiom,
    ! [Uu2: int] : aa(int,fun(int,product_prod(int,int)),aTP_Lamp_lw(int,fun(int,product_prod(int,int))),Uu2) = aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),Uu2)) ).

% ATP.lambda_31
tff(fact_8209_ATP_Olambda__32,axiom,
    ! [Uu2: int] : aa(int,fun(int,product_prod(int,int)),aTP_Lamp_lv(int,fun(int,product_prod(int,int))),Uu2) = aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,abs_abs(int),Uu2)) ).

% ATP.lambda_32
tff(fact_8210_ATP_Olambda__33,axiom,
    ! [Uu2: nat] : aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_lc(nat,fun(nat,product_prod(nat,nat))),Uu2) = aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,suc,Uu2)) ).

% ATP.lambda_33
tff(fact_8211_ATP_Olambda__34,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat] :
          aa(nat,A,aTP_Lamp_es(A,fun(nat,A),Uu2),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,Uu2),Uua)),zero_zero(A)) ) ).

% ATP.lambda_34
tff(fact_8212_ATP_Olambda__35,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu2: nat,Uua: nat] :
          aa(nat,A,aTP_Lamp_gw(nat,fun(nat,A),Uu2),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(Uu2),Uua)),zero_zero(A)) ) ).

% ATP.lambda_35
tff(fact_8213_ATP_Olambda__36,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat] :
          aa(nat,A,aTP_Lamp_en(A,fun(nat,A),Uu2),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,Uu2),Uua))) ) ).

% ATP.lambda_36
tff(fact_8214_ATP_Olambda__37,axiom,
    ! [Uu2: fun(nat,real),Uua: nat] :
      aa(nat,real,aTP_Lamp_co(fun(nat,real),fun(nat,real),Uu2),Uua) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua),zero_zero(real),aa(nat,real,Uu2,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_37
tff(fact_8215_ATP_Olambda__38,axiom,
    ! [Uu2: vEBT_VEBTi,Uua: nat] : aa(nat,heap_Time_Heap(nat),aTP_Lamp_ap(vEBT_VEBTi,fun(nat,heap_Time_Heap(nat)),Uu2),Uua) = vEBT_Example_mIf(nat,vEBT_vebt_memberi(Uu2,Uua),aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),Uua),heap_Time_bind(option(nat),nat,vEBT_vebt_predi(Uu2,Uua),aTP_Lamp_a(option(nat),heap_Time_Heap(nat)))) ).

% ATP.lambda_38
tff(fact_8216_ATP_Olambda__39,axiom,
    ! [Uu2: int,Uua: int] :
      aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_lu(int,fun(int,product_prod(int,int))),Uu2),Uua) = $ite(Uu2 = 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),Uu2)),Uua)),aa(int,int,abs_abs(int),Uu2))) ).

% ATP.lambda_39
tff(fact_8217_ATP_Olambda__40,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu2: nat,Uua: nat] :
          aa(nat,A,aTP_Lamp_gv(nat,fun(nat,A),Uu2),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(Uu2),Uua)),zero_zero(A)) ) ).

% ATP.lambda_40
tff(fact_8218_ATP_Olambda__41,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: fun(nat,A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_adf(fun(nat,A),fun(A,$o),Uu2),Uua)
        <=> filterlim(nat,A,Uu2,topolo7230453075368039082e_nhds(A,Uua),at_top(nat)) ) ) ).

% ATP.lambda_41
tff(fact_8219_ATP_Olambda__42,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,fun(B,$o)),Uua: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aTP_Lamp_alq(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),Uu2),Uua)
    <=> aa(B,$o,aa(A,fun(B,$o),Uu2,aa(product_prod(A,B),A,product_fst(A,B),Uua)),aa(product_prod(A,B),B,product_snd(A,B),Uua)) ) ).

% ATP.lambda_42
tff(fact_8220_ATP_Olambda__43,axiom,
    ! [A: $tType,Uu2: list(list(A)),Uua: nat] : aa(nat,list(A),aTP_Lamp_amb(list(list(A)),fun(nat,list(A)),Uu2),Uua) = aa(list(nat),list(A),map(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ama(list(list(A)),fun(nat,fun(nat,A)),Uu2),Uua)),upt(zero_zero(nat),aa(list(list(A)),nat,size_size(list(list(A))),Uu2))) ).

% ATP.lambda_43
tff(fact_8221_ATP_Olambda__44,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_lj(fun(nat,fun(nat,A)),fun(nat,A),Uu2),Uua) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_li(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu2),Uua),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_44
tff(fact_8222_ATP_Olambda__45,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_lh(fun(nat,fun(nat,A)),fun(nat,A),Uu2),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_lg(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu2),Uua)),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_45
tff(fact_8223_ATP_Olambda__46,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_ay(real,fun(nat,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,power_power(real,aa(real,real,uminus_uminus(real),one_one(real))),Uua)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),one_one(nat)))))),aa(nat,real,power_power(real,aa(real,real,minus_minus(real,Uu2),one_one(real))),aa(nat,nat,suc,Uua))) ).

% ATP.lambda_46
tff(fact_8224_ATP_Olambda__47,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_cs(real,fun(nat,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),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),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,Uu2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),one_one(nat)))) ).

% ATP.lambda_47
tff(fact_8225_ATP_Olambda__48,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_cy(real,fun(nat,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),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,Uu2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))) ).

% ATP.lambda_48
tff(fact_8226_ATP_Olambda__49,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu2: nat,Uua: nat] : aa(nat,A,aTP_Lamp_gp(nat,fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,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(Uu2),Uua))) ) ).

% ATP.lambda_49
tff(fact_8227_ATP_Olambda__50,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: nat,Uua: nat] : aa(nat,A,aTP_Lamp_gq(nat,fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu2),Uua))),Uua)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Uua)) ) ).

% ATP.lambda_50
tff(fact_8228_ATP_Olambda__51,axiom,
    ! [Uu2: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_dr(real,fun(real,$o),Uu2),Uua)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Uua)
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uua),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
        & ( sin(real,Uua) = Uu2 ) ) ) ).

% ATP.lambda_51
tff(fact_8229_ATP_Olambda__52,axiom,
    ! [Uu2: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_dj(real,fun(real,$o),Uu2),Uua)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Uua)
        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Uua),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
        & ( aa(real,real,tan(real),Uua) = Uu2 ) ) ) ).

% ATP.lambda_52
tff(fact_8230_ATP_Olambda__53,axiom,
    ! [Uu2: code_integer,Uua: code_integer] :
      aa(code_integer,int,aa(code_integer,fun(code_integer,int),aTP_Lamp_ajy(code_integer,fun(code_integer,int)),Uu2),Uua) = $let(
        l2: int,
        l2:= aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(code_integer,int,code_int_of_integer,Uu2)),
        $ite(Uua = zero_zero(code_integer),l2,aa(int,int,aa(int,fun(int,int),plus_plus(int),l2),one_one(int))) ) ).

% ATP.lambda_53
tff(fact_8231_ATP_Olambda__54,axiom,
    ! [Uu2: nat,Uua: nat] :
      aa(nat,a,aa(nat,fun(nat,a),aTP_Lamp_lo(nat,fun(nat,a)),Uu2),Uua) = $let(
        m2: a,
        m2:= aa(a,a,aa(a,fun(a,a),times_times(a),aa(num,a,numeral_numeral(a),bit0(one2))),aa(nat,a,semiring_1_of_nat(a),Uu2)),
        $ite(Uua = zero_zero(nat),m2,aa(a,a,aa(a,fun(a,a),plus_plus(a),m2),one_one(a))) ) ).

% ATP.lambda_54
tff(fact_8232_ATP_Olambda__55,axiom,
    ! [Uu2: complex,Uua: real] :
      ( aa(real,$o,aTP_Lamp_ex(complex,fun(real,$o),Uu2),Uua)
    <=> ( ( aa(complex,complex,sgn_sgn(complex),Uu2) = cis(Uua) )
        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),Uua)
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uua),pi) ) ) ).

% ATP.lambda_55
tff(fact_8233_ATP_Olambda__56,axiom,
    ! [Uu2: real,Uua: int] :
      ( aa(int,$o,aTP_Lamp_dv(real,fun(int,$o),Uu2),Uua)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(int,real,ring_1_of_int(real),Uua)),Uu2)
        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Uu2),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),one_one(int)))) ) ) ).

% ATP.lambda_56
tff(fact_8234_ATP_Olambda__57,axiom,
    ! [Uu2: rat,Uua: int] :
      ( aa(int,$o,aTP_Lamp_dz(rat,fun(int,$o),Uu2),Uua)
    <=> ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),aa(int,rat,ring_1_of_int(rat),Uua)),Uu2)
        & aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),Uu2),aa(int,rat,ring_1_of_int(rat),aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),one_one(int)))) ) ) ).

% ATP.lambda_57
tff(fact_8235_ATP_Olambda__58,axiom,
    ! [A: $tType,Uu2: set(A),Uua: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_ajr(set(A),fun(list(A),$o),Uu2),Uua)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uua)),Uu2)
        & distinct(A,Uua) ) ) ).

% ATP.lambda_58
tff(fact_8236_ATP_Olambda__59,axiom,
    ! [A: $tType,Uu2: set(A),Uua: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_ajq(set(A),fun(list(A),$o),Uu2),Uua)
    <=> ( ( aa(list(A),set(A),set2(A),Uua) = Uu2 )
        & distinct(A,Uua) ) ) ).

% ATP.lambda_59
tff(fact_8237_ATP_Olambda__60,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_bd(real,fun(nat,real),Uu2),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,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat))))),aa(nat,real,power_power(real,Uu2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat))))) ).

% ATP.lambda_60
tff(fact_8238_ATP_Olambda__61,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_tc(real,fun(nat,real),Uu2),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,Uu2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% ATP.lambda_61
tff(fact_8239_ATP_Olambda__62,axiom,
    ! [Uu2: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_aai(fun(nat,real),fun(nat,real),Uu2),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,Uu2,Uua)) ).

% ATP.lambda_62
tff(fact_8240_ATP_Olambda__63,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu2: nat,Uua: nat] : aa(nat,A,aTP_Lamp_gt(nat,fun(nat,A),Uu2),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(Uu2),Uua))) ) ).

% ATP.lambda_63
tff(fact_8241_ATP_Olambda__64,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_fp(A,fun(nat,A),Uu2),Uua) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu2),aa(nat,A,semiring_1_of_nat(A),Uua))),Uua) ) ).

% ATP.lambda_64
tff(fact_8242_ATP_Olambda__65,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_gu(A,fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,Uu2),Uua)),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Uu2),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_65
tff(fact_8243_ATP_Olambda__66,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_gb(A,fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,Uu2),Uua)),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),Uua)) ) ).

% ATP.lambda_66
tff(fact_8244_ATP_Olambda__67,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_gg(nat,fun(nat,nat),Uu2),Uua) = aa(nat,nat,power_power(nat,aa(nat,nat,binomial(Uu2),Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% ATP.lambda_67
tff(fact_8245_ATP_Olambda__68,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_im(real,fun(nat,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,power_power(real,Uu2),Uua)),semiring_char_0_fact(real,Uua)) ).

% ATP.lambda_68
tff(fact_8246_ATP_Olambda__69,axiom,
    ! [Uu2: nat,Uua: real] : aa(real,real,aTP_Lamp_ael(nat,fun(real,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,power_power(real,Uua),Uu2)),aa(real,real,exp(real),Uua)) ).

% ATP.lambda_69
tff(fact_8247_ATP_Olambda__70,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_fk(nat,fun(nat,nat),Uu2),Uua) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu2),Uua)),Uu2) ).

% ATP.lambda_70
tff(fact_8248_ATP_Olambda__71,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_fl(nat,fun(nat,nat),Uu2),Uua) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu2),Uua)),Uua) ).

% ATP.lambda_71
tff(fact_8249_ATP_Olambda__72,axiom,
    ! [Uu2: nat,Uua: complex] :
      ( aa(complex,$o,aTP_Lamp_hl(nat,fun(complex,$o),Uu2),Uua)
    <=> ( aa(nat,complex,power_power(complex,Uua),Uu2) = one_one(complex) ) ) ).

% ATP.lambda_72
tff(fact_8250_ATP_Olambda__73,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu2: nat,Uua: A] :
          ( aa(A,$o,aTP_Lamp_nv(nat,fun(A,$o),Uu2),Uua)
        <=> ( aa(nat,A,power_power(A,Uua),Uu2) = one_one(A) ) ) ) ).

% ATP.lambda_73
tff(fact_8251_ATP_Olambda__74,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu2: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_nt(A,fun(A,$o),Uu2),Uua)
        <=> ( member(A,Uua,ring_1_Ints(A))
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Uua)),Uu2) ) ) ) ).

% ATP.lambda_74
tff(fact_8252_ATP_Olambda__75,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_acb(real,fun(nat,real),Uu2),Uua) = aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),Uu2),aa(nat,real,semiring_1_of_nat(real),Uua)))),Uua) ).

% ATP.lambda_75
tff(fact_8253_ATP_Olambda__76,axiom,
    ! [Uu2: real,Uua: real] : aa(real,real,aTP_Lamp_adp(real,fun(real,real),Uu2),Uua) = powr(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),Uu2),Uua)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),Uua)) ).

% ATP.lambda_76
tff(fact_8254_ATP_Olambda__77,axiom,
    ! [Uu2: real,Uua: real] : aa(real,real,aTP_Lamp_aen(real,fun(real,real),Uu2),Uua) = powr(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),Uu2),Uua)),Uua) ).

% ATP.lambda_77
tff(fact_8255_ATP_Olambda__78,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_ax(real,fun(nat,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat))))),aa(nat,real,power_power(real,Uu2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat)))) ).

% ATP.lambda_78
tff(fact_8256_ATP_Olambda__79,axiom,
    ! [Uu2: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_dp(real,fun(real,$o),Uu2),Uua)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Uua)
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uua),pi)
        & ( cos(real,Uua) = Uu2 ) ) ) ).

% ATP.lambda_79
tff(fact_8257_ATP_Olambda__80,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_km(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))),aa(nat,A,Uu2,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)))) ) ).

% ATP.lambda_80
tff(fact_8258_ATP_Olambda__81,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fx(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))),aa(nat,A,Uu2,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)))) ) ).

% ATP.lambda_81
tff(fact_8259_ATP_Olambda__82,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_abh(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,minus_minus(A,aa(nat,A,Uu2,aa(nat,nat,suc,Uua))),aa(nat,A,Uu2,Uua)) ) ).

% ATP.lambda_82
tff(fact_8260_ATP_Olambda__83,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ia(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,minus_minus(A,aa(nat,A,Uu2,aa(nat,nat,suc,Uua))),aa(nat,A,Uu2,Uua)) ) ).

% ATP.lambda_83
tff(fact_8261_ATP_Olambda__84,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_oq(fun(nat,fun(nat,A)),fun(nat,A),Uu2),Uua) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),Uu2,Uua),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_84
tff(fact_8262_ATP_Olambda__85,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_oo(fun(nat,fun(nat,A)),fun(nat,A),Uu2),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),Uu2,Uua)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_85
tff(fact_8263_ATP_Olambda__86,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bw(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uua)),aa(nat,A,power_power(A,zero_zero(A)),Uua)) ) ).

% ATP.lambda_86
tff(fact_8264_ATP_Olambda__87,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ba(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uua)),aa(nat,A,power_power(A,zero_zero(A)),Uua)) ) ).

% ATP.lambda_87
tff(fact_8265_ATP_Olambda__88,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_mq(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uua)),aa(nat,A,power_power(A,zero_zero(A)),Uua)) ) ).

% ATP.lambda_88
tff(fact_8266_ATP_Olambda__89,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bv(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uua)),aa(nat,A,power_power(A,zero_zero(A)),Uua)) ) ).

% ATP.lambda_89
tff(fact_8267_ATP_Olambda__90,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_jc(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,minus_minus(A,aa(nat,A,Uu2,Uua)),aa(nat,A,Uu2,aa(nat,nat,minus_minus(nat,Uua),one_one(nat)))) ) ).

% ATP.lambda_90
tff(fact_8268_ATP_Olambda__91,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_jb(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,minus_minus(A,aa(nat,A,Uu2,Uua)),aa(nat,A,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),one_one(nat)))) ) ).

% ATP.lambda_91
tff(fact_8269_ATP_Olambda__92,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_abg(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,minus_minus(A,aa(nat,A,Uu2,Uua)),aa(nat,A,Uu2,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_92
tff(fact_8270_ATP_Olambda__93,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fb(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,minus_minus(A,aa(nat,A,Uu2,Uua)),aa(nat,A,Uu2,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_93
tff(fact_8271_ATP_Olambda__94,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu2: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_aif(fun(A,$o),fun(A,$o),Uu2),Uua)
        <=> ( aa(A,$o,Uu2,Uua)
            & ! [Y4: A] :
                ( aa(A,$o,Uu2,Y4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y4),Uua) ) ) ) ) ).

% ATP.lambda_94
tff(fact_8272_ATP_Olambda__95,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu2: fun(word(A),$o),Uua: word(A)] :
          ( aa(word(A),$o,aTP_Lamp_akd(fun(word(A),$o),fun(word(A),$o),Uu2),Uua)
        <=> ( aa(word(A),$o,Uu2,Uua)
            & ! [Y4: word(A)] :
                ( aa(word(A),$o,Uu2,Y4)
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Uua),Y4) ) ) ) ) ).

% ATP.lambda_95
tff(fact_8273_ATP_Olambda__96,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu2: fun(word(A),$o),Uua: word(A)] :
          ( aa(word(A),$o,aTP_Lamp_aks(fun(word(A),$o),fun(word(A),$o),Uu2),Uua)
        <=> ( aa(word(A),$o,Uu2,Uua)
            & ! [Y4: word(A)] :
                ( aa(word(A),$o,Uu2,Y4)
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Y4),Uua) ) ) ) ) ).

% ATP.lambda_96
tff(fact_8274_ATP_Olambda__97,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_kf(fun(nat,fun(nat,A)),fun(nat,A),Uu2),Uua) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),Uu2,Uua),set_ord_lessThan(nat,Uua)) ) ).

% ATP.lambda_97
tff(fact_8275_ATP_Olambda__98,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_ix(fun(nat,fun(nat,A)),fun(nat,A),Uu2),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),Uu2,Uua)),set_ord_lessThan(nat,Uua)) ) ).

% ATP.lambda_98
tff(fact_8276_ATP_Olambda__99,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,B),Uua: A] : aa(A,product_prod(B,A),aTP_Lamp_rh(fun(A,B),fun(A,product_prod(B,A)),Uu2),Uua) = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),aa(A,B,Uu2,Uua)),Uua) ).

% ATP.lambda_99
tff(fact_8277_ATP_Olambda__100,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ahb(fun(A,real),fun(A,$o),Uu2),Uua)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(A,real,Uu2,Uua)),zero_zero(real)) ) ).

% ATP.lambda_100
tff(fact_8278_ATP_Olambda__101,axiom,
    ! [B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu2: fun(B,real),Uua: B] : aa(B,real,aTP_Lamp_vm(fun(B,real),fun(B,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(B,real,Uu2,Uua)),zero_zero(real)) ) ).

% ATP.lambda_101
tff(fact_8279_ATP_Olambda__102,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: A] : aa(A,complex,aTP_Lamp_hd(fun(A,real),fun(A,complex),Uu2),Uua) = complex2(aa(A,real,Uu2,Uua),zero_zero(real)) ).

% ATP.lambda_102
tff(fact_8280_ATP_Olambda__103,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu2: fun(A,B),Uua: A] :
          ( aa(A,$o,aTP_Lamp_nq(fun(A,B),fun(A,$o),Uu2),Uua)
        <=> ( aa(A,B,Uu2,Uua) = zero_zero(B) ) ) ) ).

% ATP.lambda_103
tff(fact_8281_ATP_Olambda__104,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: fun(A,B),Uua: A] :
          ( aa(A,$o,aTP_Lamp_ns(fun(A,B),fun(A,$o),Uu2),Uua)
        <=> ( aa(A,B,Uu2,Uua) = one_one(B) ) ) ) ).

% ATP.lambda_104
tff(fact_8282_ATP_Olambda__105,axiom,
    ! [Uu2: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_acl(fun(nat,real),fun(nat,real),Uu2),Uua) = aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_aai(fun(nat,real),fun(nat,real),Uu2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),one_one(nat)))) ).

% ATP.lambda_105
tff(fact_8283_ATP_Olambda__106,axiom,
    ! [Uu2: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_ack(fun(nat,real),fun(nat,real),Uu2),Uua) = aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_aai(fun(nat,real),fun(nat,real),Uu2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))) ).

% ATP.lambda_106
tff(fact_8284_ATP_Olambda__107,axiom,
    ! [Uu2: code_integer,Uua: $o] : aa($o,char,aa(code_integer,fun($o,char),aTP_Lamp_akp(code_integer,fun($o,char)),Uu2),(Uua)) = aa(product_prod(code_integer,$o),char,product_case_prod(code_integer,$o,char,aTP_Lamp_ako($o,fun(code_integer,fun($o,char)),(Uua))),code_bit_cut_integer(Uu2)) ).

% ATP.lambda_107
tff(fact_8285_ATP_Olambda__108,axiom,
    ! [Uu2: list(nat),Uua: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap(list(nat)),aTP_Lamp_aq(list(nat),fun(vEBT_VEBTi,heap_Time_Heap(list(nat))),Uu2),Uua) = vEBT_Example_mmap(nat,nat,aTP_Lamp_ap(vEBT_VEBTi,fun(nat,heap_Time_Heap(nat)),Uua),Uu2) ).

% ATP.lambda_108
tff(fact_8286_ATP_Olambda__109,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_acy(fun(A,B),fun(A,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(B,aa(A,B,Uu2,Uua))),real_V7770717601297561774m_norm(A,Uua)) ) ).

% ATP.lambda_109
tff(fact_8287_ATP_Olambda__110,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_ef(A,fun(nat,A),Uu2),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,A,power_power(A,Uu2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% ATP.lambda_110
tff(fact_8288_ATP_Olambda__111,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_er(A,fun(nat,A),Uu2),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,aa(nat,nat,suc,Uua)))),aa(nat,A,power_power(A,Uu2),aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_111
tff(fact_8289_ATP_Olambda__112,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_ei(A,fun(nat,A),Uu2),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,Uu2),Uua)) ) ).

% ATP.lambda_112
tff(fact_8290_ATP_Olambda__113,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_ee(A,fun(nat,A),Uu2),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,Uu2),Uua)) ) ).

% ATP.lambda_113
tff(fact_8291_ATP_Olambda__114,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_ce(A,fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Uua))),aa(nat,A,power_power(A,Uu2),Uua)) ) ).

% ATP.lambda_114
tff(fact_8292_ATP_Olambda__115,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_eq(A,fun(nat,A),Uu2),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,cos_coeff(Uua)),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),Uu2)),Uua)) ) ).

% ATP.lambda_115
tff(fact_8293_ATP_Olambda__116,axiom,
    ! [Uu2: nat,Uua: real] : aa(real,real,aTP_Lamp_qi(nat,fun(real,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),Uua)),aa(nat,real,power_power(real,aa(real,real,abs_abs(real),Uua)),Uu2)) ).

% ATP.lambda_116
tff(fact_8294_ATP_Olambda__117,axiom,
    ! [Uu2: $o,Uua: nat] : aa(nat,nat,aa($o,fun(nat,nat),aTP_Lamp_ajt($o,fun(nat,nat)),(Uu2)),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa($o,nat,zero_neq_one_of_bool(nat),(Uu2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% ATP.lambda_117
tff(fact_8295_ATP_Olambda__118,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_ej(A,fun(nat,A),Uu2),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,sin_coeff(Uua)),aa(nat,A,power_power(A,Uu2),Uua)) ) ).

% ATP.lambda_118
tff(fact_8296_ATP_Olambda__119,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_ek(A,fun(nat,A),Uu2),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,cos_coeff(Uua)),aa(nat,A,power_power(A,Uu2),Uua)) ) ).

% ATP.lambda_119
tff(fact_8297_ATP_Olambda__120,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_acg(A,fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),Uua)),aa(nat,A,power_power(A,Uu2),Uua)) ) ).

% ATP.lambda_120
tff(fact_8298_ATP_Olambda__121,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_acf(A,fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uua)),aa(nat,A,power_power(A,Uu2),Uua)) ) ).

% ATP.lambda_121
tff(fact_8299_ATP_Olambda__122,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_hv(real,fun(nat,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),sin_coeff(Uua)),aa(nat,real,power_power(real,Uu2),Uua)) ).

% ATP.lambda_122
tff(fact_8300_ATP_Olambda__123,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_hu(real,fun(nat,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),cos_coeff(Uua)),aa(nat,real,power_power(real,Uu2),Uua)) ).

% ATP.lambda_123
tff(fact_8301_ATP_Olambda__124,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [Uu2: product_prod(A,B),Uua: product_prod(A,B)] :
          ( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),aTP_Lamp_alb(product_prod(A,B),fun(product_prod(A,B),$o)),Uu2),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(product_prod(A,B),A,product_fst(A,B),Uu2)),aa(product_prod(A,B),A,product_fst(A,B),Uua)) ) ) ).

% ATP.lambda_124
tff(fact_8302_ATP_Olambda__125,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [Uu2: product_prod(A,B),Uua: product_prod(A,B)] :
          ( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),aTP_Lamp_ami(product_prod(A,B),fun(product_prod(A,B),$o)),Uu2),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(product_prod(A,B),A,product_fst(A,B),Uua)),aa(product_prod(A,B),A,product_fst(A,B),Uu2)) ) ) ).

% ATP.lambda_125
tff(fact_8303_ATP_Olambda__126,axiom,
    ! [Uu2: code_integer,Uua: code_integer] :
      aa(code_integer,num,aa(code_integer,fun(code_integer,num),aTP_Lamp_ajz(code_integer,fun(code_integer,num)),Uu2),Uua) = $let(
        l2: num,
        l2:= code_num_of_integer(Uu2),
        $let(
          l3: num,
          l3:= aa(num,num,aa(num,fun(num,num),plus_plus(num),l2),l2),
          $ite(Uua = zero_zero(code_integer),l3,aa(num,num,aa(num,fun(num,num),plus_plus(num),l3),one2)) ) ) ).

% ATP.lambda_126
tff(fact_8304_ATP_Olambda__127,axiom,
    ! [Uu2: code_integer,Uua: code_integer] :
      aa(code_integer,nat,aa(code_integer,fun(code_integer,nat),aTP_Lamp_aka(code_integer,fun(code_integer,nat)),Uu2),Uua) = $let(
        l2: nat,
        l2:= code_nat_of_integer(Uu2),
        $let(
          l3: nat,
          l3:= aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),l2),l2),
          $ite(Uua = zero_zero(code_integer),l3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),l3),one_one(nat))) ) ) ).

% ATP.lambda_127
tff(fact_8305_ATP_Olambda__128,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: nat,Uua: nat] : aa(nat,A,aTP_Lamp_jd(nat,fun(nat,A),Uu2),Uua) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Uua)),Uu2) ) ).

% ATP.lambda_128
tff(fact_8306_ATP_Olambda__129,axiom,
    ! [Uu2: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_dn(real,fun(real,$o),Uu2),Uua)
    <=> ( aa(real,real,exp(real),Uua) = Uu2 ) ) ).

% ATP.lambda_129
tff(fact_8307_ATP_Olambda__130,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Uu2: A,Uua: A] : aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_kw(A,fun(A,product_prod(A,A))),Uu2),Uua) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)),one_one(A))) ) ).

% ATP.lambda_130
tff(fact_8308_ATP_Olambda__131,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Uu2: A,Uua: A] : aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_ky(A,fun(A,product_prod(A,A))),Uu2),Uua) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)) ) ).

% ATP.lambda_131
tff(fact_8309_ATP_Olambda__132,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_jk(A,fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu2),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),Uua)),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% ATP.lambda_132
tff(fact_8310_ATP_Olambda__133,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_gs(nat,fun(nat,nat),Uu2),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(nat,nat,binomial(Uu2),Uua)) ).

% ATP.lambda_133
tff(fact_8311_ATP_Olambda__134,axiom,
    ! [A: $tType,Uu2: set(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ks(set(A),fun(A,$o),Uu2),Uua)
    <=> ( Uu2 = aa(set(A),set(A),insert(A,Uua),bot_bot(set(A))) ) ) ).

% ATP.lambda_134
tff(fact_8312_ATP_Olambda__135,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_acd(real,fun(nat,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),Uu2),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uua)))))) ).

% ATP.lambda_135
tff(fact_8313_ATP_Olambda__136,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_abw(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu2),set_or1337092689740270186AtMost(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_136
tff(fact_8314_ATP_Olambda__137,axiom,
    ! [A: $tType,Uu2: fun(nat,A),Uua: nat] : aa(nat,product_prod(nat,A),aTP_Lamp_alx(fun(nat,A),fun(nat,product_prod(nat,A)),Uu2),Uua) = aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Uua),aa(nat,A,Uu2,Uua)) ).

% ATP.lambda_137
tff(fact_8315_ATP_Olambda__138,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,B),Uua: A] : aa(A,product_prod(A,B),aTP_Lamp_ri(fun(A,B),fun(A,product_prod(A,B)),Uu2),Uua) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),aa(A,B,Uu2,Uua)) ).

% ATP.lambda_138
tff(fact_8316_ATP_Olambda__139,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_ou(A,fun(nat,A),Uu2),Uua) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Uua),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Uu2),Uua))) ) ).

% ATP.lambda_139
tff(fact_8317_ATP_Olambda__140,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_aby(fun(nat,A),fun(nat,real),Uu2),Uua) = aa(real,real,root(Uua),real_V7770717601297561774m_norm(A,aa(nat,A,Uu2,Uua))) ) ).

% ATP.lambda_140
tff(fact_8318_ATP_Olambda__141,axiom,
    ! [Uu2: int,Uua: int] : aa(int,int,aa(int,fun(int,int),aTP_Lamp_lm(int,fun(int,int)),Uu2),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uu2),aa($o,int,zero_neq_one_of_bool(int),Uua != zero_zero(int))) ).

% ATP.lambda_141
tff(fact_8319_ATP_Olambda__142,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_abx(real,fun(nat,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),plus_plus(real),Uu2),aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uua))))) ).

% ATP.lambda_142
tff(fact_8320_ATP_Olambda__143,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_abn(real,fun(nat,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),plus_plus(real),Uu2),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uua)))) ).

% ATP.lambda_143
tff(fact_8321_ATP_Olambda__144,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_abo(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu2),set_ord_lessThan(nat,Uua)) ) ).

% ATP.lambda_144
tff(fact_8322_ATP_Olambda__145,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_abz(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu2),set_ord_lessThan(nat,Uua)) ) ).

% ATP.lambda_145
tff(fact_8323_ATP_Olambda__146,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_abp(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu2),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_146
tff(fact_8324_ATP_Olambda__147,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_aca(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu2),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_147
tff(fact_8325_ATP_Olambda__148,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_abb(A,fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uu2),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_148
tff(fact_8326_ATP_Olambda__149,axiom,
    ! [A: $tType,Uu2: nat,Uua: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_aml(nat,fun(list(A),$o),Uu2),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uu2),aa(list(A),nat,size_size(list(A)),Uua)) ) ).

% ATP.lambda_149
tff(fact_8327_ATP_Olambda__150,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_kp(A,fun(nat,A),Uu2),Uua) = aa(A,A,minus_minus(A,Uu2),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_150
tff(fact_8328_ATP_Olambda__151,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_ox(A,fun(nat,A),Uu2),Uua) = aa(A,A,minus_minus(A,Uu2),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_151
tff(fact_8329_ATP_Olambda__152,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_kk(A,fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu2),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_152
tff(fact_8330_ATP_Olambda__153,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_afp(fun(A,$o),fun(A,$o),Uu2),Uua)
        <=> eventually(A,Uu2,topolo7230453075368039082e_nhds(A,Uua)) ) ) ).

% ATP.lambda_153
tff(fact_8331_ATP_Olambda__154,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: nat,Uua: A] : aa(A,A,aTP_Lamp_zw(nat,fun(A,A),Uu2),Uua) = comm_s3205402744901411588hammer(A,Uua,Uu2) ) ).

% ATP.lambda_154
tff(fact_8332_ATP_Olambda__155,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: A,Uua: real] : aa(real,A,aTP_Lamp_acr(A,fun(real,A),Uu2),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,Uua),Uu2) ) ).

% ATP.lambda_155
tff(fact_8333_ATP_Olambda__156,axiom,
    ! [Uu2: nat,Uua: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aTP_Lamp_an(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),Uu2),Uua) = vEBT_vebt_inserti(Uua,Uu2) ).

% ATP.lambda_156
tff(fact_8334_ATP_Olambda__157,axiom,
    ! [A: $tType,Uu2: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_od(set(A),fun(set(A),$o),Uu2),Uua)
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uua),Uu2) ) ).

% ATP.lambda_157
tff(fact_8335_ATP_Olambda__158,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_oc(nat,fun(nat,$o),Uu2),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),Uu2) ) ).

% ATP.lambda_158
tff(fact_8336_ATP_Olambda__159,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu2: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_afv(A,fun(A,$o)),Uu2),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu2) ) ) ).

% ATP.lambda_159
tff(fact_8337_ATP_Olambda__160,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu2: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_fm(A,fun(A,$o),Uu2),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu2) ) ) ).

% ATP.lambda_160
tff(fact_8338_ATP_Olambda__161,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_qh(nat,fun(nat,nat),Uu2),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Uua),Uu2) ).

% ATP.lambda_161
tff(fact_8339_ATP_Olambda__162,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_act(A,fun(A,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu2) ) ).

% ATP.lambda_162
tff(fact_8340_ATP_Olambda__163,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_pf(A,fun(A,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu2) ) ).

% ATP.lambda_163
tff(fact_8341_ATP_Olambda__164,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_qt(A,fun(A,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu2) ) ).

% ATP.lambda_164
tff(fact_8342_ATP_Olambda__165,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_gx(nat,fun(nat,$o),Uu2),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uua),Uu2) ) ).

% ATP.lambda_165
tff(fact_8343_ATP_Olambda__166,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [Uu2: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_afu(A,fun(A,$o),Uu2),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu2) ) ) ).

% ATP.lambda_166
tff(fact_8344_ATP_Olambda__167,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu2: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_hx(A,fun(A,$o),Uu2),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu2) ) ) ).

% ATP.lambda_167
tff(fact_8345_ATP_Olambda__168,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_aax(nat,fun(nat,nat),Uu2),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),Uu2) ).

% ATP.lambda_168
tff(fact_8346_ATP_Olambda__169,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_acs(A,fun(A,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu2) ) ).

% ATP.lambda_169
tff(fact_8347_ATP_Olambda__170,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_av(A,fun(A,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu2) ) ).

% ATP.lambda_170
tff(fact_8348_ATP_Olambda__171,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_ps(A,fun(A,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu2) ) ).

% ATP.lambda_171
tff(fact_8349_ATP_Olambda__172,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_pz(nat,fun(nat,nat),Uu2),Uua) = aa(nat,nat,minus_minus(nat,Uua),Uu2) ).

% ATP.lambda_172
tff(fact_8350_ATP_Olambda__173,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_pe(A,fun(A,A),Uu2),Uua) = aa(A,A,minus_minus(A,Uua),Uu2) ) ).

% ATP.lambda_173
tff(fact_8351_ATP_Olambda__174,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_qa(A,fun(A,A),Uu2),Uua) = aa(A,A,minus_minus(A,Uua),Uu2) ) ).

% ATP.lambda_174
tff(fact_8352_ATP_Olambda__175,axiom,
    ! [Uu2: nat,Uua: real] : aa(real,real,aTP_Lamp_sm(nat,fun(real,real),Uu2),Uua) = aa(nat,real,power_power(real,Uua),Uu2) ).

% ATP.lambda_175
tff(fact_8353_ATP_Olambda__176,axiom,
    ! [Uu2: code_integer,Uua: code_integer] : aa(code_integer,code_integer,aTP_Lamp_ajv(code_integer,fun(code_integer,code_integer),Uu2),Uua) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),Uua),Uu2) ).

% ATP.lambda_176
tff(fact_8354_ATP_Olambda__177,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_alw(nat,fun(nat,nat),Uu2),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uu2) ).

% ATP.lambda_177
tff(fact_8355_ATP_Olambda__178,axiom,
    ! [Uu2: int,Uua: int] : aa(int,int,aTP_Lamp_py(int,fun(int,int),Uu2),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),Uu2) ).

% ATP.lambda_178
tff(fact_8356_ATP_Olambda__179,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_qz(A,fun(A,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu2) ) ).

% ATP.lambda_179
tff(fact_8357_ATP_Olambda__180,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_pd(A,fun(A,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu2) ) ).

% ATP.lambda_180
tff(fact_8358_ATP_Olambda__181,axiom,
    ! [Uu2: real,Uua: real] : aa(real,real,aTP_Lamp_sp(real,fun(real,real),Uu2),Uua) = powr(real,Uua,Uu2) ).

% ATP.lambda_181
tff(fact_8359_ATP_Olambda__182,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_nr(nat,fun(nat,$o),Uu2),Uua)
    <=> aa(nat,$o,dvd_dvd(nat,Uua),Uu2) ) ).

% ATP.lambda_182
tff(fact_8360_ATP_Olambda__183,axiom,
    ! [Uu2: int,Uua: int] :
      ( aa(int,$o,aTP_Lamp_oj(int,fun(int,$o),Uu2),Uua)
    <=> aa(int,$o,dvd_dvd(int,Uua),Uu2) ) ).

% ATP.lambda_183
tff(fact_8361_ATP_Olambda__184,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_au(A,fun(A,$o),Uu2),Uua)
        <=> aa(A,$o,dvd_dvd(A,Uua),Uu2) ) ) ).

% ATP.lambda_184
tff(fact_8362_ATP_Olambda__185,axiom,
    ! [A: $tType,B: $tType,Uu2: B,Uua: A] : aa(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_ala(B,fun(A,product_prod(A,B))),Uu2),Uua) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uu2) ).

% ATP.lambda_185
tff(fact_8363_ATP_Olambda__186,axiom,
    ! [B: $tType,A: $tType,Uu2: A,Uua: B] : aa(B,product_prod(B,A),aa(A,fun(B,product_prod(B,A)),aTP_Lamp_rj(A,fun(B,product_prod(B,A))),Uu2),Uua) = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uua),Uu2) ).

% ATP.lambda_186
tff(fact_8364_ATP_Olambda__187,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ff(nat,fun(nat,nat),Uu2),Uua) = aa(nat,nat,binomial(Uua),Uu2) ).

% ATP.lambda_187
tff(fact_8365_ATP_Olambda__188,axiom,
    ! [A: $tType,Uu2: list(A),Uua: A] : aa(A,list(A),aa(list(A),fun(A,list(A)),aTP_Lamp_rg(list(A),fun(A,list(A))),Uu2),Uua) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uu2) ).

% ATP.lambda_188
tff(fact_8366_ATP_Olambda__189,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_abm(real,fun(nat,real),Uu2),Uua) = aa(real,real,root(Uua),Uu2) ).

% ATP.lambda_189
tff(fact_8367_ATP_Olambda__190,axiom,
    ! [B: $tType,Uu2: set(B),Uua: B] :
      ( aa(B,$o,aTP_Lamp_amc(set(B),fun(B,$o),Uu2),Uua)
    <=> member(B,Uua,Uu2) ) ).

% ATP.lambda_190
tff(fact_8368_ATP_Olambda__191,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_afs(set(A),fun(A,$o),Uu2),Uua)
        <=> member(A,Uua,Uu2) ) ) ).

% ATP.lambda_191
tff(fact_8369_ATP_Olambda__192,axiom,
    ! [A: $tType,Uu2: set(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ab(set(A),fun(A,$o),Uu2),Uua)
    <=> member(A,Uua,Uu2) ) ).

% ATP.lambda_192
tff(fact_8370_ATP_Olambda__193,axiom,
    ! [A: $tType,Uu2: A,Uua: A] :
      ( aa(A,$o,aTP_Lamp_dc(A,fun(A,$o),Uu2),Uua)
    <=> ( Uua = Uu2 ) ) ).

% ATP.lambda_193
tff(fact_8371_ATP_Olambda__194,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ahr(fun(A,real),fun(A,$o),Uu2),Uua)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_194
tff(fact_8372_ATP_Olambda__195,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: fun(A,real),Uua: A] :
          ( aa(A,$o,aTP_Lamp_ahg(fun(A,real),fun(A,$o),Uu2),Uua)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(A,real,Uu2,Uua)) ) ) ).

% ATP.lambda_195
tff(fact_8373_ATP_Olambda__196,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ahl(fun(A,real),fun(A,$o),Uu2),Uua)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_196
tff(fact_8374_ATP_Olambda__197,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_aha(fun(A,real),fun(A,$o),Uu2),Uua)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_197
tff(fact_8375_ATP_Olambda__198,axiom,
    ! [A: $tType,Uu2: fun(A,nat),Uua: A] :
      ( aa(A,$o,aTP_Lamp_nj(fun(A,nat),fun(A,$o),Uu2),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,Uu2,Uua)) ) ).

% ATP.lambda_198
tff(fact_8376_ATP_Olambda__199,axiom,
    ! [Uu2: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_il(fun(nat,real),fun(nat,real),Uu2),Uua) = aa(nat,real,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),one_one(nat))) ).

% ATP.lambda_199
tff(fact_8377_ATP_Olambda__200,axiom,
    ! [Uu2: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_ik(fun(nat,real),fun(nat,real),Uu2),Uua) = aa(nat,real,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)) ).

% ATP.lambda_200
tff(fact_8378_ATP_Olambda__201,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_adx(fun(A,A),fun(A,A),Uu2),Uua) = aa(A,A,Uu2,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),Uua)) ) ).

% ATP.lambda_201
tff(fact_8379_ATP_Olambda__202,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: fun(set(nat),A),Uua: nat] : aa(nat,A,aTP_Lamp_aay(fun(set(nat),A),fun(nat,A),Uu2),Uua) = aa(set(nat),A,Uu2,set_ord_lessThan(nat,Uua)) ) ).

% ATP.lambda_202
tff(fact_8380_ATP_Olambda__203,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: fun(set(nat),A),Uua: nat] : aa(nat,A,aTP_Lamp_aaz(fun(set(nat),A),fun(nat,A),Uu2),Uua) = aa(set(nat),A,Uu2,set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_203
tff(fact_8381_ATP_Olambda__204,axiom,
    ! [Uu2: fun(real,$o),Uua: real] :
      ( aa(real,$o,aTP_Lamp_ahk(fun(real,$o),fun(real,$o),Uu2),Uua)
    <=> aa(real,$o,Uu2,aa(real,real,inverse_inverse(real),Uua)) ) ).

% ATP.lambda_204
tff(fact_8382_ATP_Olambda__205,axiom,
    ! [A: $tType,Uu2: fun(real,A),Uua: real] : aa(real,A,aTP_Lamp_aek(fun(real,A),fun(real,A),Uu2),Uua) = aa(real,A,Uu2,aa(real,real,inverse_inverse(real),Uua)) ).

% ATP.lambda_205
tff(fact_8383_ATP_Olambda__206,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: fun(real,A),Uua: nat] : aa(nat,A,aTP_Lamp_aem(fun(real,A),fun(nat,A),Uu2),Uua) = aa(real,A,Uu2,aa(nat,real,semiring_1_of_nat(real),Uua)) ) ).

% ATP.lambda_206
tff(fact_8384_ATP_Olambda__207,axiom,
    ! [A: $tType,Uu2: fun(int,A),Uua: nat] : aa(nat,A,aTP_Lamp_aig(fun(int,A),fun(nat,A),Uu2),Uua) = aa(int,A,Uu2,aa(nat,int,semiring_1_of_nat(int),Uua)) ).

% ATP.lambda_207
tff(fact_8385_ATP_Olambda__208,axiom,
    ! [Uu2: fun(real,real),Uua: real] : aa(real,real,aTP_Lamp_ro(fun(real,real),fun(real,real),Uu2),Uua) = aa(real,real,Uu2,aa(real,real,uminus_uminus(real),Uua)) ).

% ATP.lambda_208
tff(fact_8386_ATP_Olambda__209,axiom,
    ! [Uu2: fun(real,$o),Uua: real] :
      ( aa(real,$o,aTP_Lamp_ahf(fun(real,$o),fun(real,$o),Uu2),Uua)
    <=> aa(real,$o,Uu2,aa(real,real,uminus_uminus(real),Uua)) ) ).

% ATP.lambda_209
tff(fact_8387_ATP_Olambda__210,axiom,
    ! [A: $tType,Uu2: fun(real,A),Uua: real] : aa(real,A,aTP_Lamp_adq(fun(real,A),fun(real,A),Uu2),Uua) = aa(real,A,Uu2,aa(real,real,uminus_uminus(real),Uua)) ).

% ATP.lambda_210
tff(fact_8388_ATP_Olambda__211,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu2: fun(B,C),Uua: product_prod(A,B)] : aa(product_prod(A,B),C,aTP_Lamp_alk(fun(B,C),fun(product_prod(A,B),C),Uu2),Uua) = aa(B,C,Uu2,aa(product_prod(A,B),B,product_snd(A,B),Uua)) ).

% ATP.lambda_211
tff(fact_8389_ATP_Olambda__212,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu2: fun(A,C),Uua: product_prod(A,B)] : aa(product_prod(A,B),C,aTP_Lamp_alc(fun(A,C),fun(product_prod(A,B),C),Uu2),Uua) = aa(A,C,Uu2,aa(product_prod(A,B),A,product_fst(A,B),Uua)) ).

% ATP.lambda_212
tff(fact_8390_ATP_Olambda__213,axiom,
    ! [Uu2: fun(nat,$o),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_aes(fun(nat,$o),fun(nat,$o),Uu2),Uua)
    <=> aa(nat,$o,Uu2,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_213
tff(fact_8391_ATP_Olambda__214,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bl(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(nat,A,Uu2,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_214
tff(fact_8392_ATP_Olambda__215,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_aar(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(nat,A,Uu2,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_215
tff(fact_8393_ATP_Olambda__216,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ck(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(nat,A,Uu2,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_216
tff(fact_8394_ATP_Olambda__217,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ka(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(nat,A,Uu2,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_217
tff(fact_8395_ATP_Olambda__218,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fe(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(nat,A,Uu2,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_218
tff(fact_8396_ATP_Olambda__219,axiom,
    ! [A: $tType,Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_acm(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(nat,A,Uu2,aa(nat,nat,suc,Uua)) ).

% ATP.lambda_219
tff(fact_8397_ATP_Olambda__220,axiom,
    ! [Uu2: list(nat),Uua: nat] : aa(nat,nat,aTP_Lamp_ql(list(nat),fun(nat,nat),Uu2),Uua) = order_Greatest(nat,aa(nat,fun(nat,$o),aTP_Lamp_qk(list(nat),fun(nat,fun(nat,$o)),Uu2),Uua)) ).

% ATP.lambda_220
tff(fact_8398_ATP_Olambda__221,axiom,
    ! [Uu2: fun(nat,real),Uua: real] : aa(real,real,aTP_Lamp_tf(fun(nat,real),fun(real,real),Uu2),Uua) = suminf(real,aa(real,fun(nat,real),aTP_Lamp_te(fun(nat,real),fun(real,fun(nat,real)),Uu2),Uua)) ).

% ATP.lambda_221
tff(fact_8399_ATP_Olambda__222,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(nat,A),Uua: A] : aa(A,A,aTP_Lamp_sn(fun(nat,A),fun(A,A),Uu2),Uua) = suminf(A,aa(A,fun(nat,A),aTP_Lamp_cv(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua)) ) ).

% ATP.lambda_222
tff(fact_8400_ATP_Olambda__223,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,complex,aTP_Lamp_it(nat,fun(nat,complex),Uu2),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),Uu2))) ).

% ATP.lambda_223
tff(fact_8401_ATP_Olambda__224,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_uk(A,fun(A,A),Uu2),Uua) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),Uu2)),Uua)),aa(A,A,inverse_inverse(A),Uu2))) ) ).

% ATP.lambda_224
tff(fact_8402_ATP_Olambda__225,axiom,
    ! [Uu2: fun(real,real),Uua: real] :
      ( aa(real,$o,aTP_Lamp_aep(fun(real,real),fun(real,$o),Uu2),Uua)
    <=> ( aa(real,real,Uu2,Uua) != zero_zero(real) ) ) ).

% ATP.lambda_225
tff(fact_8403_ATP_Olambda__226,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,real,aTP_Lamp_eo(A,fun(nat,real),Uu2),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,Uu2),Uua))) ) ).

% ATP.lambda_226
tff(fact_8404_ATP_Olambda__227,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_db(nat,fun(nat,$o),Uu2),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),Uu2),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))) ) ).

% ATP.lambda_227
tff(fact_8405_ATP_Olambda__228,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_ep(A,fun(nat,A),Uu2),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),Uu2)),Uua))) ) ).

% ATP.lambda_228
tff(fact_8406_ATP_Olambda__229,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,real,aTP_Lamp_el(A,fun(nat,real),Uu2),Uua) = real_V7770717601297561774m_norm(A,aa(A,A,real_V8093663219630862766scaleR(A,sin_coeff(Uua)),aa(nat,A,power_power(A,Uu2),Uua))) ) ).

% ATP.lambda_229
tff(fact_8407_ATP_Olambda__230,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,real,aTP_Lamp_em(A,fun(nat,real),Uu2),Uua) = real_V7770717601297561774m_norm(A,aa(A,A,real_V8093663219630862766scaleR(A,cos_coeff(Uua)),aa(nat,A,power_power(A,Uu2),Uua))) ) ).

% ATP.lambda_230
tff(fact_8408_ATP_Olambda__231,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_da(A,fun(nat,fun(A,A)),Uu2),Uua) = aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Uu2),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_231
tff(fact_8409_ATP_Olambda__232,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_aw(A,fun(nat,fun(A,A)),Uu2),Uua) = aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu2),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_232
tff(fact_8410_ATP_Olambda__233,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_abv(real,fun(nat,real),Uu2),Uua) = aa(real,real,inverse_inverse(real),aa(nat,real,power_power(real,Uu2),Uua)) ).

% ATP.lambda_233
tff(fact_8411_ATP_Olambda__234,axiom,
    ! [A: $tType,Uu2: A,Uua: list(A)] : aa(list(A),heap_Time_Heap(list(A)),aTP_Lamp_ad(A,fun(list(A),heap_Time_Heap(list(A))),Uu2),Uua) = aa(list(A),heap_Time_Heap(list(A)),heap_Time_return(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu2),Uua)) ).

% ATP.lambda_234
tff(fact_8412_ATP_Olambda__235,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_rm(A,fun(A,A),Uu2),Uua) = cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu2)) ) ).

% ATP.lambda_235
tff(fact_8413_ATP_Olambda__236,axiom,
    ! [A: $tType,Uu2: set(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_dt(set(A),fun(A,$o),Uu2),Uua)
    <=> ~ member(A,Uua,Uu2) ) ).

% ATP.lambda_236
tff(fact_8414_ATP_Olambda__237,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_aev(A,fun(A,$o),Uu2),Uua)
        <=> ( Uua != Uu2 ) ) ) ).

% ATP.lambda_237
tff(fact_8415_ATP_Olambda__238,axiom,
    ! [A: $tType] :
      ( topological_t1_space(A)
     => ! [Uu2: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_afq(A,fun(A,$o),Uu2),Uua)
        <=> ( Uua != Uu2 ) ) ) ).

% ATP.lambda_238
tff(fact_8416_ATP_Olambda__239,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [Uu2: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_aex(A,fun(A,$o),Uu2),Uua)
        <=> ( Uua != Uu2 ) ) ) ).

% ATP.lambda_239
tff(fact_8417_ATP_Olambda__240,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [Uu2: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_aft(A,fun(A,$o),Uu2),Uua)
        <=> ( Uua != Uu2 ) ) ) ).

% ATP.lambda_240
tff(fact_8418_ATP_Olambda__241,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_cb(fun(nat,A),fun(nat,real),Uu2),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu2,Uua)) ) ).

% ATP.lambda_241
tff(fact_8419_ATP_Olambda__242,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_fu(fun(nat,A),fun(nat,real),Uu2),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu2,Uua)) ) ).

% ATP.lambda_242
tff(fact_8420_ATP_Olambda__243,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_bs(fun(nat,A),fun(nat,real),Uu2),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu2,Uua)) ) ).

% ATP.lambda_243
tff(fact_8421_ATP_Olambda__244,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(B,A),Uua: B] : aa(B,real,aTP_Lamp_fj(fun(B,A),fun(B,real),Uu2),Uua) = real_V7770717601297561774m_norm(A,aa(B,A,Uu2,Uua)) ) ).

% ATP.lambda_244
tff(fact_8422_ATP_Olambda__245,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: fun(B,A),Uua: B] : aa(B,real,aTP_Lamp_jr(fun(B,A),fun(B,real),Uu2),Uua) = real_V7770717601297561774m_norm(A,aa(B,A,Uu2,Uua)) ) ).

% ATP.lambda_245
tff(fact_8423_ATP_Olambda__246,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_wj(fun(A,B),fun(A,real),Uu2),Uua) = real_V7770717601297561774m_norm(B,aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_246
tff(fact_8424_ATP_Olambda__247,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & comm_semiring_1(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_js(fun(A,B),fun(A,real),Uu2),Uua) = real_V7770717601297561774m_norm(B,aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_247
tff(fact_8425_ATP_Olambda__248,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_xw(fun(A,B),fun(A,real),Uu2),Uua) = real_V7770717601297561774m_norm(B,aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_248
tff(fact_8426_ATP_Olambda__249,axiom,
    ! [Uu2: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_abj(fun(nat,real),fun(nat,real),Uu2),Uua) = aa(real,real,inverse_inverse(real),aa(nat,real,Uu2,Uua)) ).

% ATP.lambda_249
tff(fact_8427_ATP_Olambda__250,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu2: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_ui(fun(B,A),fun(B,A),Uu2),Uua) = aa(A,A,inverse_inverse(A),aa(B,A,Uu2,Uua)) ) ).

% ATP.lambda_250
tff(fact_8428_ATP_Olambda__251,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_aio(fun(A,B),fun(A,B),Uu2),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_251
tff(fact_8429_ATP_Olambda__252,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_sg(fun(A,A),fun(A,A),Uu2),Uua) = aa(A,A,inverse_inverse(A),aa(A,A,Uu2,Uua)) ) ).

% ATP.lambda_252
tff(fact_8430_ATP_Olambda__253,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_zu(fun(A,B),fun(A,B),Uu2),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_253
tff(fact_8431_ATP_Olambda__254,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_aec(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,inverse_inverse(real),aa(A,real,Uu2,Uua)) ).

% ATP.lambda_254
tff(fact_8432_ATP_Olambda__255,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_yr(fun(A,B),fun(A,B),Uu2),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_255
tff(fact_8433_ATP_Olambda__256,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu2: fun(B,nat),Uua: B] : aa(B,A,aTP_Lamp_jn(fun(B,nat),fun(B,A),Uu2),Uua) = aa(nat,A,semiring_1_of_nat(A),aa(B,nat,Uu2,Uua)) ) ).

% ATP.lambda_256
tff(fact_8434_ATP_Olambda__257,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Uu2: fun(B,nat),Uua: B] : aa(B,A,aTP_Lamp_ey(fun(B,nat),fun(B,A),Uu2),Uua) = aa(nat,A,semiring_1_of_nat(A),aa(B,nat,Uu2,Uua)) ) ).

% ATP.lambda_257
tff(fact_8435_ATP_Olambda__258,axiom,
    ! [A: $tType,Uu2: fun(A,nat),Uua: A] : aa(A,real,aTP_Lamp_aed(fun(A,nat),fun(A,real),Uu2),Uua) = aa(nat,real,semiring_1_of_nat(real),aa(A,nat,Uu2,Uua)) ).

% ATP.lambda_258
tff(fact_8436_ATP_Olambda__259,axiom,
    ! [A: $tType,Uu2: fun(A,nat),Uua: A] : aa(A,int,aTP_Lamp_he(fun(A,nat),fun(A,int),Uu2),Uua) = aa(nat,int,semiring_1_of_nat(int),aa(A,nat,Uu2,Uua)) ).

% ATP.lambda_259
tff(fact_8437_ATP_Olambda__260,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ur(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_260
tff(fact_8438_ATP_Olambda__261,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_aaw(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_261
tff(fact_8439_ATP_Olambda__262,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_oa(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu2,Uua)) ).

% ATP.lambda_262
tff(fact_8440_ATP_Olambda__263,axiom,
    ! [Uu2: fun(nat,real),Uua: nat] : aa(nat,complex,aTP_Lamp_cf(fun(nat,real),fun(nat,complex),Uu2),Uua) = aa(real,complex,real_Vector_of_real(complex),aa(nat,real,Uu2,Uua)) ).

% ATP.lambda_263
tff(fact_8441_ATP_Olambda__264,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu2: fun(nat,real),Uua: nat] : aa(nat,A,aTP_Lamp_cg(fun(nat,real),fun(nat,A),Uu2),Uua) = aa(real,A,real_Vector_of_real(A),aa(nat,real,Uu2,Uua)) ) ).

% ATP.lambda_264
tff(fact_8442_ATP_Olambda__265,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu2: fun(nat,real),Uua: nat] : aa(nat,A,aTP_Lamp_ch(fun(nat,real),fun(nat,A),Uu2),Uua) = aa(real,A,real_Vector_of_real(A),aa(nat,real,Uu2,Uua)) ) ).

% ATP.lambda_265
tff(fact_8443_ATP_Olambda__266,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Uu2: fun(nat,real),Uua: nat] : aa(nat,A,aTP_Lamp_ct(fun(nat,real),fun(nat,A),Uu2),Uua) = aa(real,A,real_Vector_of_real(A),aa(nat,real,Uu2,Uua)) ) ).

% ATP.lambda_266
tff(fact_8444_ATP_Olambda__267,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Uu2: fun(B,real),Uua: B] : aa(B,A,aTP_Lamp_fa(fun(B,real),fun(B,A),Uu2),Uua) = aa(real,A,real_Vector_of_real(A),aa(B,real,Uu2,Uua)) ) ).

% ATP.lambda_267
tff(fact_8445_ATP_Olambda__268,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2191834092415804123ebra_1(A) )
     => ! [Uu2: fun(B,real),Uua: B] : aa(B,A,aTP_Lamp_jp(fun(B,real),fun(B,A),Uu2),Uua) = aa(real,A,real_Vector_of_real(A),aa(B,real,Uu2,Uua)) ) ).

% ATP.lambda_268
tff(fact_8446_ATP_Olambda__269,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,B,aTP_Lamp_tv(fun(A,real),fun(A,B),Uu2),Uua) = aa(real,B,real_Vector_of_real(B),aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_269
tff(fact_8447_ATP_Olambda__270,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,B,aTP_Lamp_wi(fun(A,real),fun(A,B),Uu2),Uua) = aa(real,B,real_Vector_of_real(B),aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_270
tff(fact_8448_ATP_Olambda__271,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,B,aTP_Lamp_xz(fun(A,real),fun(A,B),Uu2),Uua) = aa(real,B,real_Vector_of_real(B),aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_271
tff(fact_8449_ATP_Olambda__272,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,B,aTP_Lamp_xy(fun(A,real),fun(A,B),Uu2),Uua) = aa(real,B,real_Vector_of_real(B),aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_272
tff(fact_8450_ATP_Olambda__273,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bo(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,uminus_uminus(A),aa(nat,A,Uu2,Uua)) ) ).

% ATP.lambda_273
tff(fact_8451_ATP_Olambda__274,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cz(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,uminus_uminus(A),aa(nat,A,Uu2,Uua)) ) ).

% ATP.lambda_274
tff(fact_8452_ATP_Olambda__275,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cu(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,uminus_uminus(A),aa(nat,A,Uu2,Uua)) ) ).

% ATP.lambda_275
tff(fact_8453_ATP_Olambda__276,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [Uu2: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_ht(fun(B,A),fun(B,A),Uu2),Uua) = aa(A,A,uminus_uminus(A),aa(B,A,Uu2,Uua)) ) ).

% ATP.lambda_276
tff(fact_8454_ATP_Olambda__277,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_tn(fun(A,B),fun(A,B),Uu2),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_277
tff(fact_8455_ATP_Olambda__278,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_sh(fun(A,A),fun(A,A),Uu2),Uua) = aa(A,A,uminus_uminus(A),aa(A,A,Uu2,Uua)) ) ).

% ATP.lambda_278
tff(fact_8456_ATP_Olambda__279,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_zt(fun(A,B),fun(A,B),Uu2),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_279
tff(fact_8457_ATP_Olambda__280,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_zf(fun(A,B),fun(A,B),Uu2),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_280
tff(fact_8458_ATP_Olambda__281,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_adz(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,uminus_uminus(real),aa(A,real,Uu2,Uua)) ).

% ATP.lambda_281
tff(fact_8459_ATP_Olambda__282,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_zg(fun(A,B),fun(A,B),Uu2),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_282
tff(fact_8460_ATP_Olambda__283,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_ki(fun(nat,A),fun(nat,fun(A,A)),Uu2),Uua) = aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uua)) ) ).

% ATP.lambda_283
tff(fact_8461_ATP_Olambda__284,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_ring_1(A)
     => ! [Uu2: fun(B,int),Uua: B] : aa(B,A,aTP_Lamp_jo(fun(B,int),fun(B,A),Uu2),Uua) = aa(int,A,ring_1_of_int(A),aa(B,int,Uu2,Uua)) ) ).

% ATP.lambda_284
tff(fact_8462_ATP_Olambda__285,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(A)
     => ! [Uu2: fun(B,int),Uua: B] : aa(B,A,aTP_Lamp_ez(fun(B,int),fun(B,A),Uu2),Uua) = aa(int,A,ring_1_of_int(A),aa(B,int,Uu2,Uua)) ) ).

% ATP.lambda_285
tff(fact_8463_ATP_Olambda__286,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,A),Uua: B] : aa(B,heap_Time_Heap(A),aTP_Lamp_ah(fun(B,A),fun(B,heap_Time_Heap(A)),Uu2),Uua) = aa(A,heap_Time_Heap(A),heap_Time_return(A),aa(B,A,Uu2,Uua)) ).

% ATP.lambda_286
tff(fact_8464_ATP_Olambda__287,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_ja(fun(nat,A),fun(nat,fun(A,A)),Uu2),Uua) = aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu2,Uua)) ) ).

% ATP.lambda_287
tff(fact_8465_ATP_Olambda__288,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_adi(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,artanh(real),aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_288
tff(fact_8466_ATP_Olambda__289,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_wn(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,artanh(real),aa(A,real,Uu2,Uua)) ).

% ATP.lambda_289
tff(fact_8467_ATP_Olambda__290,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_aau(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,arsinh(real),aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_290
tff(fact_8468_ATP_Olambda__291,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_xt(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,arsinh(real),aa(A,real,Uu2,Uua)) ).

% ATP.lambda_291
tff(fact_8469_ATP_Olambda__292,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_vd(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,arctan,aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_292
tff(fact_8470_ATP_Olambda__293,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_aap(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,arctan,aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_293
tff(fact_8471_ATP_Olambda__294,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_xv(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,arctan,aa(A,real,Uu2,Uua)) ).

% ATP.lambda_294
tff(fact_8472_ATP_Olambda__295,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_tg(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,arcsin,aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_295
tff(fact_8473_ATP_Olambda__296,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_adh(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,arcosh(real),aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_296
tff(fact_8474_ATP_Olambda__297,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_wt(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,arcosh(real),aa(A,real,Uu2,Uua)) ).

% ATP.lambda_297
tff(fact_8475_ATP_Olambda__298,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_vj(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,arccos,aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_298
tff(fact_8476_ATP_Olambda__299,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: fun(A,product_prod(B,C)),Uua: A] : aa(A,C,aTP_Lamp_aln(fun(A,product_prod(B,C)),fun(A,C),Uu2),Uua) = aa(product_prod(B,C),C,product_snd(B,C),aa(A,product_prod(B,C),Uu2,Uua)) ) ).

% ATP.lambda_299
tff(fact_8477_ATP_Olambda__300,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: fun(A,product_prod(B,C)),Uua: A] : aa(A,C,aTP_Lamp_alr(fun(A,product_prod(B,C)),fun(A,C),Uu2),Uua) = aa(product_prod(B,C),C,product_snd(B,C),aa(A,product_prod(B,C),Uu2,Uua)) ) ).

% ATP.lambda_300
tff(fact_8478_ATP_Olambda__301,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: fun(A,product_prod(B,C)),Uua: A] : aa(A,B,aTP_Lamp_ali(fun(A,product_prod(B,C)),fun(A,B),Uu2),Uua) = aa(product_prod(B,C),B,product_fst(B,C),aa(A,product_prod(B,C),Uu2,Uua)) ) ).

% ATP.lambda_301
tff(fact_8479_ATP_Olambda__302,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: fun(A,product_prod(B,C)),Uua: A] : aa(A,B,aTP_Lamp_alj(fun(A,product_prod(B,C)),fun(A,B),Uu2),Uua) = aa(product_prod(B,C),B,product_fst(B,C),aa(A,product_prod(B,C),Uu2,Uua)) ) ).

% ATP.lambda_302
tff(fact_8480_ATP_Olambda__303,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_zv(fun(A,B),fun(A,B),Uu2),Uua) = aa(B,B,sgn_sgn(B),aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_303
tff(fact_8481_ATP_Olambda__304,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_yh(fun(A,B),fun(A,B),Uu2),Uua) = aa(B,B,sgn_sgn(B),aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_304
tff(fact_8482_ATP_Olambda__305,axiom,
    ! [Uu2: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_bk(fun(nat,real),fun(nat,real),Uu2),Uua) = aa(real,real,abs_abs(real),aa(nat,real,Uu2,Uua)) ).

% ATP.lambda_305
tff(fact_8483_ATP_Olambda__306,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Uu2: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_ha(fun(B,A),fun(B,A),Uu2),Uua) = aa(A,A,abs_abs(A),aa(B,A,Uu2,Uua)) ) ).

% ATP.lambda_306
tff(fact_8484_ATP_Olambda__307,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_field(A)
     => ! [Uu2: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_ju(fun(B,A),fun(B,A),Uu2),Uua) = aa(A,A,abs_abs(A),aa(B,A,Uu2,Uua)) ) ).

% ATP.lambda_307
tff(fact_8485_ATP_Olambda__308,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_aam(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,abs_abs(real),aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_308
tff(fact_8486_ATP_Olambda__309,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ws(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,abs_abs(real),aa(A,real,Uu2,Uua)) ).

% ATP.lambda_309
tff(fact_8487_ATP_Olambda__310,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_aaa(fun(A,B),fun(A,B),Uu2),Uua) = tanh(B,aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_310
tff(fact_8488_ATP_Olambda__311,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_st(fun(A,A),fun(A,A),Uu2),Uua) = tanh(A,aa(A,A,Uu2,Uua)) ) ).

% ATP.lambda_311
tff(fact_8489_ATP_Olambda__312,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_yp(fun(A,B),fun(A,B),Uu2),Uua) = tanh(B,aa(A,B,Uu2,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) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_xh(fun(A,B),fun(A,B),Uu2),Uua) = sinh(B,aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_313
tff(fact_8491_ATP_Olambda__314,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_sb(fun(A,A),fun(A,A),Uu2),Uua) = sinh(A,aa(A,A,Uu2,Uua)) ) ).

% ATP.lambda_314
tff(fact_8492_ATP_Olambda__315,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_xg(fun(A,B),fun(A,B),Uu2),Uua) = sinh(B,aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_315
tff(fact_8493_ATP_Olambda__316,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_yo(fun(A,B),fun(A,B),Uu2),Uua) = cosh(B,aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_316
tff(fact_8494_ATP_Olambda__317,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_sa(fun(A,A),fun(A,A),Uu2),Uua) = cosh(A,aa(A,A,Uu2,Uua)) ) ).

% ATP.lambda_317
tff(fact_8495_ATP_Olambda__318,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_yn(fun(A,B),fun(A,B),Uu2),Uua) = cosh(B,aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_318
tff(fact_8496_ATP_Olambda__319,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_vf(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,tan(real),aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_319
tff(fact_8497_ATP_Olambda__320,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_xi(fun(A,A),fun(A,A),Uu2),Uua) = aa(A,A,tan(A),aa(A,A,Uu2,Uua)) ) ).

% ATP.lambda_320
tff(fact_8498_ATP_Olambda__321,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ue(fun(A,real),fun(A,real),Uu2),Uua) = sin(real,aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_321
tff(fact_8499_ATP_Olambda__322,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_xs(fun(A,B),fun(A,B),Uu2),Uua) = sin(B,aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_322
tff(fact_8500_ATP_Olambda__323,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_rr(fun(A,A),fun(A,A),Uu2),Uua) = sin(A,aa(A,A,Uu2,Uua)) ) ).

% ATP.lambda_323
tff(fact_8501_ATP_Olambda__324,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_uc(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,exp(real),aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_324
tff(fact_8502_ATP_Olambda__325,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_ye(fun(A,B),fun(A,B),Uu2),Uua) = aa(B,B,exp(B),aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_325
tff(fact_8503_ATP_Olambda__326,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_rs(fun(A,A),fun(A,A),Uu2),Uua) = aa(A,A,exp(A),aa(A,A,Uu2,Uua)) ) ).

% ATP.lambda_326
tff(fact_8504_ATP_Olambda__327,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_yf(fun(A,B),fun(A,B),Uu2),Uua) = aa(B,B,exp(B),aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_327
tff(fact_8505_ATP_Olambda__328,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_Vector_banach(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_nu(fun(A,B),fun(A,B),Uu2),Uua) = aa(B,B,exp(B),aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_328
tff(fact_8506_ATP_Olambda__329,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_wo(fun(A,A),fun(A,A),Uu2),Uua) = aa(A,A,cot(A),aa(A,A,Uu2,Uua)) ) ).

% ATP.lambda_329
tff(fact_8507_ATP_Olambda__330,axiom,
    ! [Uu2: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_acj(fun(nat,real),fun(nat,real),Uu2),Uua) = cos(real,aa(nat,real,Uu2,Uua)) ).

% ATP.lambda_330
tff(fact_8508_ATP_Olambda__331,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_un(fun(A,real),fun(A,real),Uu2),Uua) = cos(real,aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_331
tff(fact_8509_ATP_Olambda__332,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_xr(fun(A,B),fun(A,B),Uu2),Uua) = cos(B,aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_332
tff(fact_8510_ATP_Olambda__333,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_rn(fun(A,A),fun(A,A),Uu2),Uua) = cos(A,aa(A,A,Uu2,Uua)) ) ).

% ATP.lambda_333
tff(fact_8511_ATP_Olambda__334,axiom,
    ! [Uu2: fun(nat,complex),Uua: nat] : aa(nat,real,aTP_Lamp_air(fun(nat,complex),fun(nat,real),Uu2),Uua) = re(aa(nat,complex,Uu2,Uua)) ).

% ATP.lambda_334
tff(fact_8512_ATP_Olambda__335,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,complex),Uua: A] : aa(A,real,aTP_Lamp_aiq(fun(A,complex),fun(A,real),Uu2),Uua) = re(aa(A,complex,Uu2,Uua)) ) ).

% ATP.lambda_335
tff(fact_8513_ATP_Olambda__336,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: fun(A,complex),Uua: A] : aa(A,real,aTP_Lamp_ais(fun(A,complex),fun(A,real),Uu2),Uua) = re(aa(A,complex,Uu2,Uua)) ) ).

% ATP.lambda_336
tff(fact_8514_ATP_Olambda__337,axiom,
    ! [A: $tType,Uu2: fun(A,complex),Uua: A] : aa(A,real,aTP_Lamp_aip(fun(A,complex),fun(A,real),Uu2),Uua) = re(aa(A,complex,Uu2,Uua)) ).

% ATP.lambda_337
tff(fact_8515_ATP_Olambda__338,axiom,
    ! [Uu2: fun(nat,complex),Uua: nat] : aa(nat,real,aTP_Lamp_aiu(fun(nat,complex),fun(nat,real),Uu2),Uua) = im(aa(nat,complex,Uu2,Uua)) ).

% ATP.lambda_338
tff(fact_8516_ATP_Olambda__339,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,complex),Uua: A] : aa(A,real,aTP_Lamp_aiw(fun(A,complex),fun(A,real),Uu2),Uua) = im(aa(A,complex,Uu2,Uua)) ) ).

% ATP.lambda_339
tff(fact_8517_ATP_Olambda__340,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: fun(A,complex),Uua: A] : aa(A,real,aTP_Lamp_aiv(fun(A,complex),fun(A,real),Uu2),Uua) = im(aa(A,complex,Uu2,Uua)) ) ).

% ATP.lambda_340
tff(fact_8518_ATP_Olambda__341,axiom,
    ! [A: $tType,Uu2: fun(A,complex),Uua: A] : aa(A,real,aTP_Lamp_ait(fun(A,complex),fun(A,real),Uu2),Uua) = im(aa(A,complex,Uu2,Uua)) ).

% ATP.lambda_341
tff(fact_8519_ATP_Olambda__342,axiom,
    ! [Uu2: fun(real,fun(nat,real)),Uua: real] : aa(real,real,aTP_Lamp_vi(fun(real,fun(nat,real)),fun(real,real),Uu2),Uua) = suminf(real,aa(real,fun(nat,real),Uu2,Uua)) ).

% ATP.lambda_342
tff(fact_8520_ATP_Olambda__343,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_Vector_banach(B) )
     => ! [Uu2: fun(A,fun(nat,B)),Uua: A] : aa(A,B,aTP_Lamp_aab(fun(A,fun(nat,B)),fun(A,B),Uu2),Uua) = suminf(B,aa(A,fun(nat,B),Uu2,Uua)) ) ).

% ATP.lambda_343
tff(fact_8521_ATP_Olambda__344,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [Uu2: fun(A,fun(nat,B)),Uua: A] : aa(A,B,aTP_Lamp_fs(fun(A,fun(nat,B)),fun(A,B),Uu2),Uua) = suminf(B,aa(A,fun(nat,B),Uu2,Uua)) ) ).

% ATP.lambda_344
tff(fact_8522_ATP_Olambda__345,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_vb(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,sqrt,aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_345
tff(fact_8523_ATP_Olambda__346,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_aaq(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,sqrt,aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_346
tff(fact_8524_ATP_Olambda__347,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_xx(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,sqrt,aa(A,real,Uu2,Uua)) ).

% ATP.lambda_347
tff(fact_8525_ATP_Olambda__348,axiom,
    ! [Uu2: fun(nat,complex),Uua: nat] : aa(nat,complex,aTP_Lamp_ajf(fun(nat,complex),fun(nat,complex),Uu2),Uua) = cnj(aa(nat,complex,Uu2,Uua)) ).

% ATP.lambda_348
tff(fact_8526_ATP_Olambda__349,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,complex),Uua: A] : aa(A,complex,aTP_Lamp_ajb(fun(A,complex),fun(A,complex),Uu2),Uua) = cnj(aa(A,complex,Uu2,Uua)) ) ).

% ATP.lambda_349
tff(fact_8527_ATP_Olambda__350,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: fun(A,complex),Uua: A] : aa(A,complex,aTP_Lamp_ajg(fun(A,complex),fun(A,complex),Uu2),Uua) = cnj(aa(A,complex,Uu2,Uua)) ) ).

% ATP.lambda_350
tff(fact_8528_ATP_Olambda__351,axiom,
    ! [A: $tType,Uu2: fun(A,complex),Uua: A] : aa(A,complex,aTP_Lamp_aiy(fun(A,complex),fun(A,complex),Uu2),Uua) = cnj(aa(A,complex,Uu2,Uua)) ).

% ATP.lambda_351
tff(fact_8529_ATP_Olambda__352,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,B),Uua: A] : aa(A,fun(set(B),set(B)),aTP_Lamp_amk(fun(A,B),fun(A,fun(set(B),set(B))),Uu2),Uua) = insert(B,aa(A,B,Uu2,Uua)) ).

% ATP.lambda_352
tff(fact_8530_ATP_Olambda__353,axiom,
    ! [A: $tType,Uu2: fun(A,nat),Uua: A] : aa(A,nat,aTP_Lamp_ajj(fun(A,nat),fun(A,nat),Uu2),Uua) = aa(nat,nat,suc,aa(A,nat,Uu2,Uua)) ).

% ATP.lambda_353
tff(fact_8531_ATP_Olambda__354,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_wg(fun(A,$o),fun(A,$o),Uu2),Uua)
        <=> ~ aa(A,$o,Uu2,Uua) ) ) ).

% ATP.lambda_354
tff(fact_8532_ATP_Olambda__355,axiom,
    ! [A: $tType,Uu2: fun(A,$o),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ds(fun(A,$o),fun(A,$o),Uu2),Uua)
    <=> ~ aa(A,$o,Uu2,Uua) ) ).

% ATP.lambda_355
tff(fact_8533_ATP_Olambda__356,axiom,
    ! [B: $tType,A: $tType] :
      ( finite_finite(A)
     => ! [Uu2: fun(B,fun(A,$o)),Uua: B] :
          ( aa(B,$o,aTP_Lamp_ahw(fun(B,fun(A,$o)),fun(B,$o),Uu2),Uua)
        <=> ! [X_1: A] : aa(A,$o,aa(B,fun(A,$o),Uu2,Uua),X_1) ) ) ).

% ATP.lambda_356
tff(fact_8534_ATP_Olambda__357,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A) )
     => ! [Uu2: fun(real,A),Uua: real] : aa(real,real,aTP_Lamp_aae(fun(real,A),fun(real,real),Uu2),Uua) = aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(A,aa(real,A,Uu2,Uua))) ) ).

% ATP.lambda_357
tff(fact_8535_ATP_Olambda__358,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B)
        & ring_1(C)
        & topolo4958980785337419405_space(C) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,C,aTP_Lamp_ym(fun(A,B),fun(A,C),Uu2),Uua) = aa(int,C,ring_1_of_int(C),archim6421214686448440834_floor(B,aa(A,B,Uu2,Uua))) ) ).

% ATP.lambda_358
tff(fact_8536_ATP_Olambda__359,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B)
        & ring_1(C)
        & topolo4958980785337419405_space(C) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,C,aTP_Lamp_xf(fun(A,B),fun(A,C),Uu2),Uua) = aa(int,C,ring_1_of_int(C),archimedean_ceiling(B,aa(A,B,Uu2,Uua))) ) ).

% ATP.lambda_359
tff(fact_8537_ATP_Olambda__360,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu2: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_aid(fun(A,$o),fun(A,$o),Uu2),Uua)
        <=> ! [Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Y4)
             => aa(A,$o,Uu2,Y4) ) ) ) ).

% ATP.lambda_360
tff(fact_8538_ATP_Olambda__361,axiom,
    ! [Uu2: fun(nat,real),Uua: fun(nat,real),Uub: nat] :
      aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_cp(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Uu2),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,Uu2,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_361
tff(fact_8539_ATP_Olambda__362,axiom,
    ! [Uu2: fun(nat,real),Uua: fun(nat,real),Uub: nat] :
      aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_ij(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Uu2),Uua),Uub) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub),aa(nat,real,Uu2,Uub),aa(nat,real,Uua,Uub)) ).

% ATP.lambda_362
tff(fact_8540_ATP_Olambda__363,axiom,
    ! [Uu2: num,Uua: code_integer,Uub: code_integer] :
      aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_ajx(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu2),Uua),Uub) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),aa(num,code_integer,numeral_numeral(code_integer),Uu2)),Uub),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),Uua)),one_one(code_integer))),aa(code_integer,code_integer,minus_minus(code_integer,Uub),aa(num,code_integer,numeral_numeral(code_integer),Uu2))),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),Uua)),Uub)) ).

% ATP.lambda_363
tff(fact_8541_ATP_Olambda__364,axiom,
    ! [Uu2: num,Uua: nat,Uub: nat] :
      aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_la(num,fun(nat,fun(nat,product_prod(nat,nat))),Uu2),Uua),Uub) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),Uu2)),Uub),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),one_one(nat))),aa(nat,nat,minus_minus(nat,Uub),aa(num,nat,numeral_numeral(nat),Uu2))),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),Uub)) ).

% ATP.lambda_364
tff(fact_8542_ATP_Olambda__365,axiom,
    ! [Uu2: num,Uua: int,Uub: int] :
      aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_lb(num,fun(int,fun(int,product_prod(int,int))),Uu2),Uua),Uub) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),Uu2)),Uub),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Uua)),one_one(int))),aa(int,int,minus_minus(int,Uub),aa(num,int,numeral_numeral(int),Uu2))),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Uua)),Uub)) ).

% ATP.lambda_365
tff(fact_8543_ATP_Olambda__366,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Uu2: num,Uua: A,Uub: A] :
          aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_kx(num,fun(A,fun(A,product_prod(A,A))),Uu2),Uua),Uub) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Uu2)),Uub),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)),one_one(A))),aa(A,A,minus_minus(A,Uub),aa(num,A,numeral_numeral(A),Uu2))),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)),Uub)) ) ).

% ATP.lambda_366
tff(fact_8544_ATP_Olambda__367,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu2: set(nat),Uua: fun(nat,A),Uub: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_mp(set(nat),fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = $ite(member(nat,Uub,Uu2),aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_367
tff(fact_8545_ATP_Olambda__368,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu2: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ml(A,fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = $ite(Uu2 = Uub,aa(A,B,Uua,Uub),zero_zero(B)) ) ).

% ATP.lambda_368
tff(fact_8546_ATP_Olambda__369,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_mn(A,fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = $ite(Uu2 = Uub,aa(A,B,Uua,Uub),one_one(B)) ) ).

% ATP.lambda_369
tff(fact_8547_ATP_Olambda__370,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu2: nat,Uua: fun(nat,A),Uub: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bg(nat,fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = $ite(Uub = Uu2,aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_370
tff(fact_8548_ATP_Olambda__371,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu2: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_mk(A,fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = $ite(Uub = Uu2,aa(A,B,Uua,Uub),zero_zero(B)) ) ).

% ATP.lambda_371
tff(fact_8549_ATP_Olambda__372,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_mm(A,fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = $ite(Uub = Uu2,aa(A,B,Uua,Uub),one_one(B)) ) ).

% ATP.lambda_372
tff(fact_8550_ATP_Olambda__373,axiom,
    ! [Uu2: code_integer,Uua: code_integer,Uub: code_integer] :
      aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_akr(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu2),Uua),Uub) = $ite(Uub = zero_zero(code_integer),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,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),Uu2)),Uub))) ).

% ATP.lambda_373
tff(fact_8551_ATP_Olambda__374,axiom,
    ! [Uu2: code_integer,Uua: code_integer,Uub: code_integer] :
      aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_amh(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu2),Uua),Uub) = $ite(Uub = zero_zero(code_integer),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,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),Uu2)),Uub))) ).

% ATP.lambda_374
tff(fact_8552_ATP_Olambda__375,axiom,
    ! [Uu2: code_integer,Uua: code_integer,Uub: code_integer] :
      aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_akq(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu2),Uua),Uub) = $ite(Uub = zero_zero(code_integer),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,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,Uu2),Uub))) ).

% ATP.lambda_375
tff(fact_8553_ATP_Olambda__376,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu2: fun(nat,$o),Uua: fun(nat,A),Uub: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_mo(fun(nat,$o),fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = $ite(aa(nat,$o,Uu2,Uub),aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_376
tff(fact_8554_ATP_Olambda__377,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,$o),Uub: A] :
          aa(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_nl(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu2),Uua),Uub) = $ite(aa(A,$o,Uua,Uub),aa(A,B,Uu2,Uub),zero_zero(B)) ) ).

% ATP.lambda_377
tff(fact_8555_ATP_Olambda__378,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,$o),Uub: A] :
          aa(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_nm(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu2),Uua),Uub) = $ite(aa(A,$o,Uua,Uub),aa(A,B,Uu2,Uub),one_one(B)) ) ).

% ATP.lambda_378
tff(fact_8556_ATP_Olambda__379,axiom,
    ! [Uu2: fun(real,real),Uua: fun(real,real),Uub: real] :
      ( aa(real,$o,aa(fun(real,real),fun(real,$o),aTP_Lamp_aeq(fun(real,real),fun(fun(real,real),fun(real,$o)),Uu2),Uua),Uub)
    <=> has_field_derivative(real,Uu2,aa(real,real,Uua,Uub),topolo174197925503356063within(real,Uub,top_top(set(real)))) ) ).

% ATP.lambda_379
tff(fact_8557_ATP_Olambda__380,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: filter(B),Uua: fun(B,A),Uub: A] :
          ( aa(A,$o,aa(fun(B,A),fun(A,$o),aTP_Lamp_ade(filter(B),fun(fun(B,A),fun(A,$o)),Uu2),Uua),Uub)
        <=> filterlim(B,A,Uua,topolo7230453075368039082e_nhds(A,Uub),Uu2) ) ) ).

% ATP.lambda_380
tff(fact_8558_ATP_Olambda__381,axiom,
    ! [A: $tType,Uu2: heap_Time_Heap(A),Uua: heap_Time_Heap(A),Uub: $o] :
      aa($o,heap_Time_Heap(A),aa(heap_Time_Heap(A),fun($o,heap_Time_Heap(A)),aTP_Lamp_af(heap_Time_Heap(A),fun(heap_Time_Heap(A),fun($o,heap_Time_Heap(A))),Uu2),Uua),(Uub)) = $ite((Uub),Uu2,Uua) ).

% ATP.lambda_381
tff(fact_8559_ATP_Olambda__382,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,Uu2: fun(D,fun(B,C)),Uua: fun(A,D),Uub: product_prod(A,B)] : aa(product_prod(A,B),C,aa(fun(A,D),fun(product_prod(A,B),C),aTP_Lamp_alo(fun(D,fun(B,C)),fun(fun(A,D),fun(product_prod(A,B),C)),Uu2),Uua),Uub) = aa(B,C,aa(D,fun(B,C),Uu2,aa(A,D,Uua,aa(product_prod(A,B),A,product_fst(A,B),Uub))),aa(product_prod(A,B),B,product_snd(A,B),Uub)) ).

% ATP.lambda_382
tff(fact_8560_ATP_Olambda__383,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_li(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu2,Uub),aa(nat,nat,minus_minus(nat,Uua),Uub)) ) ).

% ATP.lambda_383
tff(fact_8561_ATP_Olambda__384,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_lg(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu2,Uub),aa(nat,nat,minus_minus(nat,Uua),Uub)) ) ).

% ATP.lambda_384
tff(fact_8562_ATP_Olambda__385,axiom,
    ! [Uu2: fun(real,fun(nat,real)),Uua: nat,Uub: real] : aa(real,real,aa(nat,fun(real,real),aTP_Lamp_vh(fun(real,fun(nat,real)),fun(nat,fun(real,real)),Uu2),Uua),Uub) = aa(nat,real,aa(real,fun(nat,real),Uu2,Uub),Uua) ).

% ATP.lambda_385
tff(fact_8563_ATP_Olambda__386,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_kg(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu2,Uub),Uua) ) ).

% ATP.lambda_386
tff(fact_8564_ATP_Olambda__387,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_iy(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu2,Uub),Uua) ) ).

% ATP.lambda_387
tff(fact_8565_ATP_Olambda__388,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu2: fun(B,fun(A,B)),Uua: A,Uub: B] : aa(B,B,aa(A,fun(B,B),aTP_Lamp_si(fun(B,fun(A,B)),fun(A,fun(B,B)),Uu2),Uua),Uub) = aa(A,B,aa(B,fun(A,B),Uu2,Uub),Uua) ) ).

% ATP.lambda_388
tff(fact_8566_ATP_Olambda__389,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(B,fun(C,A)),Uua: C,Uub: B] : aa(B,A,aa(C,fun(B,A),aTP_Lamp_jw(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu2),Uua),Uub) = aa(C,A,aa(B,fun(C,A),Uu2,Uub),Uua) ) ).

% ATP.lambda_389
tff(fact_8567_ATP_Olambda__390,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(B,fun(C,A)),Uua: C,Uub: B] : aa(B,A,aa(C,fun(B,A),aTP_Lamp_hj(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu2),Uua),Uub) = aa(C,A,aa(B,fun(C,A),Uu2,Uub),Uua) ) ).

% ATP.lambda_390
tff(fact_8568_ATP_Olambda__391,axiom,
    ! [B: $tType,A: $tType] :
      ( finite_finite(A)
     => ! [Uu2: fun(B,fun(A,$o)),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_ahv(fun(B,fun(A,$o)),fun(A,fun(B,$o)),Uu2),Uua),Uub)
        <=> aa(A,$o,aa(B,fun(A,$o),Uu2,Uub),Uua) ) ) ).

% ATP.lambda_391
tff(fact_8569_ATP_Olambda__392,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [Uu2: fun(B,fun(A,C)),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_wu(fun(B,fun(A,C)),fun(A,fun(B,C)),Uu2),Uua),Uub) = aa(A,C,aa(B,fun(A,C),Uu2,Uub),Uua) ) ).

% ATP.lambda_392
tff(fact_8570_ATP_Olambda__393,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [Uu2: fun(B,fun(A,C)),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_zl(fun(B,fun(A,C)),fun(A,fun(B,C)),Uu2),Uua),Uub) = aa(A,C,aa(B,fun(A,C),Uu2,Uub),Uua) ) ).

% ATP.lambda_393
tff(fact_8571_ATP_Olambda__394,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [Uu2: fun(A,fun(nat,B)),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_fh(fun(A,fun(nat,B)),fun(nat,fun(A,B)),Uu2),Uua),Uub) = aa(nat,B,aa(A,fun(nat,B),Uu2,Uub),Uua) ) ).

% ATP.lambda_394
tff(fact_8572_ATP_Olambda__395,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Uu2: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_tl(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu2),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu2,Uub),Uua) ) ).

% ATP.lambda_395
tff(fact_8573_ATP_Olambda__396,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(C) )
     => ! [Uu2: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_uv(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu2),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu2,Uub),Uua) ) ).

% ATP.lambda_396
tff(fact_8574_ATP_Olambda__397,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & real_V4412858255891104859lgebra(C)
        & comm_ring_1(C) )
     => ! [Uu2: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_xa(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu2),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu2,Uub),Uua) ) ).

% ATP.lambda_397
tff(fact_8575_ATP_Olambda__398,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & topolo4987421752381908075d_mult(C) )
     => ! [Uu2: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_xc(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu2),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu2,Uub),Uua) ) ).

% ATP.lambda_398
tff(fact_8576_ATP_Olambda__399,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & topolo5987344860129210374id_add(C) )
     => ! [Uu2: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_zj(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu2),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu2,Uub),Uua) ) ).

% ATP.lambda_399
tff(fact_8577_ATP_Olambda__400,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(C)
        & comm_ring_1(C) )
     => ! [Uu2: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_ww(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu2),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu2,Uub),Uua) ) ).

% ATP.lambda_400
tff(fact_8578_ATP_Olambda__401,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [Uu2: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_wy(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu2),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu2,Uub),Uua) ) ).

% ATP.lambda_401
tff(fact_8579_ATP_Olambda__402,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [Uu2: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_zh(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu2),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu2,Uub),Uua) ) ).

% ATP.lambda_402
tff(fact_8580_ATP_Olambda__403,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu2: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_nb(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu2),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu2,Uub),Uua) ) ).

% ATP.lambda_403
tff(fact_8581_ATP_Olambda__404,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu2: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_mx(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu2),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu2,Uub),Uua) ) ).

% ATP.lambda_404
tff(fact_8582_ATP_Olambda__405,axiom,
    ! [A: $tType,Uu2: fun(A,fun(A,$o)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_amj(fun(A,fun(A,$o)),fun(A,fun(A,$o)),Uu2),Uua),Uub)
    <=> aa(A,$o,aa(A,fun(A,$o),Uu2,Uub),Uua) ) ).

% ATP.lambda_405
tff(fact_8583_ATP_Olambda__406,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_gz(A,fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_gy(A,fun(A,fun(nat,fun(nat,A))),Uu2),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_406
tff(fact_8584_ATP_Olambda__407,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu2: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_fw(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu2),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_fv(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu2),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_407
tff(fact_8585_ATP_Olambda__408,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ew(A,fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_ev(A,fun(A,fun(nat,fun(nat,A))),Uu2),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_408
tff(fact_8586_ATP_Olambda__409,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_eu(A,fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_et(A,fun(A,fun(nat,fun(nat,A))),Uu2),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_409
tff(fact_8587_ATP_Olambda__410,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu2: nat,Uua: A,Uub: nat] :
          aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_cm(nat,fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              $ite(Uub = Uu2,one_one(A),zero_zero(A))),
            aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_410
tff(fact_8588_ATP_Olambda__411,axiom,
    ! [Uu2: code_integer,Uua: code_integer,Uub: code_integer] :
      aa(code_integer,product_prod(code_integer,$o),aa(code_integer,fun(code_integer,product_prod(code_integer,$o)),aTP_Lamp_akh(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,$o))),Uu2),Uua),Uub) = aa($o,product_prod(code_integer,$o),
        aa(code_integer,fun($o,product_prod(code_integer,$o)),product_Pair(code_integer,$o),
          $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),Uu2),Uua,aa(code_integer,code_integer,minus_minus(code_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),Uub))),
        Uub = one_one(code_integer)) ).

% ATP.lambda_411
tff(fact_8589_ATP_Olambda__412,axiom,
    ! [Uu2: list(nat),Uua: list(nat),Uub: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap(list(nat)),aa(list(nat),fun(vEBT_VEBTi,heap_Time_Heap(list(nat))),aTP_Lamp_ar(list(nat),fun(list(nat),fun(vEBT_VEBTi,heap_Time_Heap(list(nat)))),Uu2),Uua),Uub) = heap_Time_bind(vEBT_VEBTi,list(nat),aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),vEBT_Example_mfold(nat,vEBT_VEBTi,aTP_Lamp_an(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),zero_zero(nat)),Uu2)),Uub),aTP_Lamp_aq(list(nat),fun(vEBT_VEBTi,heap_Time_Heap(list(nat))),Uua)) ).

% ATP.lambda_412
tff(fact_8590_ATP_Olambda__413,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_kh(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_kg(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu2),Uub),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_413
tff(fact_8591_ATP_Olambda__414,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_iz(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_iy(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu2),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_414
tff(fact_8592_ATP_Olambda__415,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: A] : aa(A,A,aa(nat,fun(A,A),aTP_Lamp_aad(fun(nat,A),fun(nat,fun(A,A)),Uu2),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_bt(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uub)),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_415
tff(fact_8593_ATP_Olambda__416,axiom,
    ! [Uu2: $o,Uua: code_integer,Uub: $o] : aa($o,char,aa(code_integer,fun($o,char),aTP_Lamp_ako($o,fun(code_integer,fun($o,char)),(Uu2)),Uua),(Uub)) = aa(product_prod(code_integer,$o),char,product_case_prod(code_integer,$o,char,aa($o,fun(code_integer,fun($o,char)),aTP_Lamp_akn($o,fun($o,fun(code_integer,fun($o,char))),(Uu2)),(Uub))),code_bit_cut_integer(Uua)) ).

% ATP.lambda_416
tff(fact_8594_ATP_Olambda__417,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_mi(rat,fun(int,fun(int,product_prod(int,int))),Uu2),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_mh(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu2)) ).

% ATP.lambda_417
tff(fact_8595_ATP_Olambda__418,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_mg(rat,fun(int,fun(int,product_prod(int,int))),Uu2),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_mf(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu2)) ).

% ATP.lambda_418
tff(fact_8596_ATP_Olambda__419,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_me(rat,fun(int,fun(int,product_prod(int,int))),Uu2),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_md(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu2)) ).

% ATP.lambda_419
tff(fact_8597_ATP_Olambda__420,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_mc(rat,fun(int,fun(int,product_prod(int,int))),Uu2),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_mb(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu2)) ).

% ATP.lambda_420
tff(fact_8598_ATP_Olambda__421,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_ma(rat,fun(int,fun(int,$o)),Uu2),Uua),Uub)
    <=> aa(product_prod(int,int),$o,product_case_prod(int,int,$o,aa(int,fun(int,fun(int,$o)),aTP_Lamp_lz(int,fun(int,fun(int,fun(int,$o))),Uua),Uub)),quotient_of(Uu2)) ) ).

% ATP.lambda_421
tff(fact_8599_ATP_Olambda__422,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_ly(rat,fun(int,fun(int,$o)),Uu2),Uua),Uub)
    <=> aa(product_prod(int,int),$o,product_case_prod(int,int,$o,aa(int,fun(int,fun(int,$o)),aTP_Lamp_lx(int,fun(int,fun(int,fun(int,$o))),Uua),Uub)),quotient_of(Uu2)) ) ).

% ATP.lambda_422
tff(fact_8600_ATP_Olambda__423,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4987421752381908075d_mult(C)
        & topological_t2_space(B) )
     => ! [Uu2: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_xd(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu2),Uua),Uub) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_xc(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub),Uu2) ) ).

% ATP.lambda_423
tff(fact_8601_ATP_Olambda__424,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(C)
        & comm_ring_1(C)
        & topological_t2_space(B) )
     => ! [Uu2: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_xb(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu2),Uua),Uub) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_xa(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub),Uu2) ) ).

% ATP.lambda_424
tff(fact_8602_ATP_Olambda__425,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [Uu2: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_wz(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu2),Uua),Uub) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_wy(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub),Uu2) ) ).

% ATP.lambda_425
tff(fact_8603_ATP_Olambda__426,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(C)
        & comm_ring_1(C) )
     => ! [Uu2: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_wx(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu2),Uua),Uub) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_ww(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub),Uu2) ) ).

% ATP.lambda_426
tff(fact_8604_ATP_Olambda__427,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_uw(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu2),Uua),Uub) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_uv(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub),Uu2) ) ).

% ATP.lambda_427
tff(fact_8605_ATP_Olambda__428,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(B,fun(C,A)),Uua: set(B),Uub: C] : aa(C,A,aa(set(B),fun(C,A),aTP_Lamp_jx(fun(B,fun(C,A)),fun(set(B),fun(C,A)),Uu2),Uua),Uub) = groups7121269368397514597t_prod(B,A,aa(C,fun(B,A),aTP_Lamp_jw(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu2),Uub),Uua) ) ).

% ATP.lambda_428
tff(fact_8606_ATP_Olambda__429,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(C)
        & topological_t2_space(B) )
     => ! [Uu2: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_zk(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu2),Uua),Uub) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(B,fun(A,C),aTP_Lamp_zj(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub)),Uu2) ) ).

% ATP.lambda_429
tff(fact_8607_ATP_Olambda__430,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [Uu2: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_zi(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu2),Uua),Uub) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(B,fun(A,C),aTP_Lamp_zh(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub)),Uu2) ) ).

% ATP.lambda_430
tff(fact_8608_ATP_Olambda__431,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_tm(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu2),Uua),Uub) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(B,fun(A,C),aTP_Lamp_tl(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub)),Uu2) ) ).

% ATP.lambda_431
tff(fact_8609_ATP_Olambda__432,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(B,fun(C,A)),Uua: set(B),Uub: C] : aa(C,A,aa(set(B),fun(C,A),aTP_Lamp_hk(fun(B,fun(C,A)),fun(set(B),fun(C,A)),Uu2),Uua),Uub) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(C,fun(B,A),aTP_Lamp_hj(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu2),Uub)),Uua) ) ).

% ATP.lambda_432
tff(fact_8610_ATP_Olambda__433,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [Uu2: set(A),Uua: fun(A,fun(nat,B)),Uub: nat] : aa(nat,B,aa(fun(A,fun(nat,B)),fun(nat,B),aTP_Lamp_fi(set(A),fun(fun(A,fun(nat,B)),fun(nat,B)),Uu2),Uua),Uub) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(nat,fun(A,B),aTP_Lamp_fh(fun(A,fun(nat,B)),fun(nat,fun(A,B)),Uua),Uub)),Uu2) ) ).

% ATP.lambda_433
tff(fact_8611_ATP_Olambda__434,axiom,
    ! [Uu2: fun(nat,fun(real,real)),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_su(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uu2,Uub),zero_zero(real))),semiring_char_0_fact(real,Uub))),aa(nat,real,power_power(real,Uua),Uub)) ).

% ATP.lambda_434
tff(fact_8612_ATP_Olambda__435,axiom,
    ! [Uu2: real,Uua: fun(nat,fun(real,real)),Uub: nat] : aa(nat,real,aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_sv(real,fun(fun(nat,fun(real,real)),fun(nat,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uua,Uub),zero_zero(real))),semiring_char_0_fact(real,Uub))),aa(nat,real,power_power(real,Uu2),Uub)) ).

% ATP.lambda_435
tff(fact_8613_ATP_Olambda__436,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [Uu2: real,Uua: fun(nat,fun(A,real)),Uub: nat] : aa(nat,real,aa(fun(nat,fun(A,real)),fun(nat,real),aTP_Lamp_ih(real,fun(fun(nat,fun(A,real)),fun(nat,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(A,real,aa(nat,fun(A,real),Uua,Uub),zero_zero(A))),semiring_char_0_fact(real,Uub))),aa(nat,real,power_power(real,Uu2),Uub)) ) ).

% ATP.lambda_436
tff(fact_8614_ATP_Olambda__437,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: A] :
          ( aa(A,$o,aa(nat,fun(A,$o),aTP_Lamp_ob(fun(nat,A),fun(nat,fun(A,$o)),Uu2),Uua),Uub)
        <=> ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_fn(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uub)),set_ord_atMost(nat,Uua)) = zero_zero(A) ) ) ) ).

% ATP.lambda_437
tff(fact_8615_ATP_Olambda__438,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_zx(fun(A,A),fun(A,fun(A,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,Uu2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub))),aa(A,A,Uu2,Uua))),Uub) ) ).

% ATP.lambda_438
tff(fact_8616_ATP_Olambda__439,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu2: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_zp(fun(A,A),fun(A,fun(A,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,Uu2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub))),aa(A,A,Uu2,Uua))),Uub) ) ).

% ATP.lambda_439
tff(fact_8617_ATP_Olambda__440,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu2: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_mr(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uub)),aa(nat,A,power_power(A,zero_zero(A)),Uub))),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_440
tff(fact_8618_ATP_Olambda__441,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_wm(fun(A,A),fun(A,fun(A,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,Uu2,Uub)),aa(A,A,Uu2,Uua))),aa(A,A,minus_minus(A,Uub),Uua)) ) ).

% ATP.lambda_441
tff(fact_8619_ATP_Olambda__442,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu2: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_zq(fun(A,A),fun(A,fun(A,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,Uu2,Uub)),aa(A,A,Uu2,Uua))),aa(A,A,minus_minus(A,Uub),Uua)) ) ).

% ATP.lambda_442
tff(fact_8620_ATP_Olambda__443,axiom,
    ! [Uu2: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_td(fun(nat,real),fun(real,fun(nat,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,Uu2,Uub)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uub)))),aa(nat,real,power_power(real,Uua),Uub)) ).

% ATP.lambda_443
tff(fact_8621_ATP_Olambda__444,axiom,
    ! [Uu2: real,Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_ii(real,fun(fun(nat,real),fun(nat,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,Uua,Uub)),semiring_char_0_fact(real,Uub))),aa(nat,real,power_power(real,Uu2),Uub)) ).

% ATP.lambda_444
tff(fact_8622_ATP_Olambda__445,axiom,
    ! [Uu2: nat,Uua: nat,Uub: list(nat)] :
      ( aa(list(nat),$o,aa(nat,fun(list(nat),$o),aTP_Lamp_akx(nat,fun(nat,fun(list(nat),$o)),Uu2),Uua),Uub)
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = aa(nat,nat,minus_minus(nat,Uu2),one_one(nat)) )
        & ( groups8242544230860333062m_list(nat,Uub) = Uua ) ) ) ).

% ATP.lambda_445
tff(fact_8623_ATP_Olambda__446,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & ring_1(A) )
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_cr(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uub)),aa(nat,A,Uu2,Uub))),aa(nat,A,power_power(A,Uua),aa(nat,nat,minus_minus(nat,Uub),aa(nat,nat,suc,zero_zero(nat))))) ) ).

% ATP.lambda_446
tff(fact_8624_ATP_Olambda__447,axiom,
    ! [Uu2: nat,Uua: nat,Uub: list(nat)] :
      ( aa(list(nat),$o,aa(nat,fun(list(nat),$o),aTP_Lamp_aky(nat,fun(nat,fun(list(nat),$o)),Uu2),Uua),Uub)
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = Uu2 )
        & ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),groups8242544230860333062m_list(nat,Uub)),one_one(nat)) = Uua ) ) ) ).

% ATP.lambda_447
tff(fact_8625_ATP_Olambda__448,axiom,
    ! [A: $tType,Uu2: nat,Uua: set(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(set(A),fun(list(A),$o),aTP_Lamp_ajs(nat,fun(set(A),fun(list(A),$o)),Uu2),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu2 )
        & distinct(A,Uub)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uua) ) ) ).

% ATP.lambda_448
tff(fact_8626_ATP_Olambda__449,axiom,
    ! [A: $tType,Uu2: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_ajp(set(A),fun(nat,fun(list(A),$o)),Uu2),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uua )
        & distinct(A,Uub)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu2) ) ) ).

% ATP.lambda_449
tff(fact_8627_ATP_Olambda__450,axiom,
    ! [A: $tType,Uu2: nat,Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_at(nat,fun(list(A),fun(list(A),$o)),Uu2),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu2 )
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),aa(list(A),set(A),set2(A),Uua)) ) ) ).

% ATP.lambda_450
tff(fact_8628_ATP_Olambda__451,axiom,
    ! [A: $tType,Uu2: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_ne(set(A),fun(nat,fun(list(A),$o)),Uu2),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu2)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Uub)),Uua) ) ) ).

% ATP.lambda_451
tff(fact_8629_ATP_Olambda__452,axiom,
    ! [A: $tType,Uu2: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_nd(set(A),fun(nat,fun(list(A),$o)),Uu2),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu2)
        & ( aa(list(A),nat,size_size(list(A)),Uub) = Uua ) ) ) ).

% ATP.lambda_452
tff(fact_8630_ATP_Olambda__453,axiom,
    ! [Uu2: nat,Uua: nat,Uub: list(nat)] :
      ( aa(list(nat),$o,aa(nat,fun(list(nat),$o),aTP_Lamp_akw(nat,fun(nat,fun(list(nat),$o)),Uu2),Uua),Uub)
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = Uu2 )
        & ( groups8242544230860333062m_list(nat,Uub) = Uua ) ) ) ).

% ATP.lambda_453
tff(fact_8631_ATP_Olambda__454,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_sq(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,diffs(A,Uu2)),Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_454
tff(fact_8632_ATP_Olambda__455,axiom,
    ! [Uu2: set(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ajh(set(nat),fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> ( member(nat,aa(nat,nat,suc,Uub),Uu2)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uua) ) ) ).

% ATP.lambda_455
tff(fact_8633_ATP_Olambda__456,axiom,
    ! [Uu2: list(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_qk(list(nat),fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> ( member(nat,Uub,aa(set(nat),set(nat),insert(nat,zero_zero(nat)),aa(list(nat),set(nat),set2(nat),Uu2)))
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uub),Uua) ) ) ).

% ATP.lambda_456
tff(fact_8634_ATP_Olambda__457,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ow(A,fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,Uu2),aa(nat,A,semiring_1_of_nat(A),Uub))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,Uua),Uub))) ) ).

% ATP.lambda_457
tff(fact_8635_ATP_Olambda__458,axiom,
    ! [A: $tType,Uu2: set(A),Uua: nat,Uub: set(A)] :
      ( aa(set(A),$o,aa(nat,fun(set(A),$o),aTP_Lamp_ajc(set(A),fun(nat,fun(set(A),$o)),Uu2),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uub),Uu2)
        & ( finite_card(A,Uub) = Uua ) ) ) ).

% ATP.lambda_458
tff(fact_8636_ATP_Olambda__459,axiom,
    ! [Uu2: set(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_aji(set(nat),fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> ( member(nat,Uub,Uu2)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),aa(nat,nat,suc,Uua)) ) ) ).

% ATP.lambda_459
tff(fact_8637_ATP_Olambda__460,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & ring_1(A) )
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_cq(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,Uu2),Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_460
tff(fact_8638_ATP_Olambda__461,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_cw(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,Uu2),Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_461
tff(fact_8639_ATP_Olambda__462,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cx(A,fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,Uua),Uub)),aa(nat,A,power_power(A,Uu2),Uub)) ) ).

% ATP.lambda_462
tff(fact_8640_ATP_Olambda__463,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_fq(nat,fun(nat,fun(nat,nat)),Uu2),Uua),Uub) = aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,Uua),Uub)),aa(nat,nat,minus_minus(nat,Uu2),Uub)) ).

% ATP.lambda_463
tff(fact_8641_ATP_Olambda__464,axiom,
    ! [A: $tType,Uu2: set(A),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_dl(set(A),fun(set(A),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( member(A,Uub,Uu2)
        | member(A,Uub,Uua) ) ) ).

% ATP.lambda_464
tff(fact_8642_ATP_Olambda__465,axiom,
    ! [A: $tType,Uu2: A,Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_de(A,fun(set(A),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( ( Uub = Uu2 )
        | member(A,Uub,Uua) ) ) ).

% ATP.lambda_465
tff(fact_8643_ATP_Olambda__466,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_lq(int,fun(int,fun(int,$o)),Uu2),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu2),Uua)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uua),Uub) ) ) ).

% ATP.lambda_466
tff(fact_8644_ATP_Olambda__467,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_of(int,fun(int,fun(int,$o)),Uu2),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu2),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uub),Uua) ) ) ).

% ATP.lambda_467
tff(fact_8645_ATP_Olambda__468,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_lp(int,fun(int,fun(int,$o)),Uu2),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu2),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uua),Uub) ) ) ).

% ATP.lambda_468
tff(fact_8646_ATP_Olambda__469,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_oh(int,fun(int,fun(int,$o)),Uu2),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu2),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uub),Uua) ) ) ).

% ATP.lambda_469
tff(fact_8647_ATP_Olambda__470,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_oi(int,fun(int,fun(int,$o)),Uu2),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uu2),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uub),Uua) ) ) ).

% ATP.lambda_470
tff(fact_8648_ATP_Olambda__471,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_og(int,fun(int,fun(int,$o)),Uu2),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uu2),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uub),Uua) ) ) ).

% ATP.lambda_471
tff(fact_8649_ATP_Olambda__472,axiom,
    ! [A: $tType,Uu2: set(A),Uua: set(A),Uub: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_mt(set(A),fun(set(A),fun(set(A),$o)),Uu2),Uua),Uub)
    <=> ( aa(set(A),set(A),minus_minus(set(A),Uub),Uu2) = aa(set(A),set(A),minus_minus(set(A),Uua),Uu2) ) ) ).

% ATP.lambda_472
tff(fact_8650_ATP_Olambda__473,axiom,
    ! [A: $tType,Uu2: list(A),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_akt(list(A),fun(fun(A,nat),fun(A,nat)),Uu2),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(A,nat,count_list(A,Uu2),Uub)),aa(A,nat,Uua,Uub)) ).

% ATP.lambda_473
tff(fact_8651_ATP_Olambda__474,axiom,
    ! [A: $tType,Uu2: fun(A,nat),Uua: list(A),Uub: A] : aa(A,nat,aa(list(A),fun(A,nat),aTP_Lamp_akz(fun(A,nat),fun(list(A),fun(A,nat)),Uu2),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(A,nat,count_list(A,Uua),Uub)),aa(A,nat,Uu2,Uub)) ).

% ATP.lambda_474
tff(fact_8652_ATP_Olambda__475,axiom,
    ! [A: $tType,Uu2: set(A),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_is(set(A),fun(fun(A,$o),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( member(A,Uub,Uu2)
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_475
tff(fact_8653_ATP_Olambda__476,axiom,
    ! [A: $tType,Uu2: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_dh(A,fun(fun(A,$o),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( ( Uu2 = Uub )
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_476
tff(fact_8654_ATP_Olambda__477,axiom,
    ! [A: $tType,Uu2: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_di(A,fun(fun(A,$o),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( ( Uub = Uu2 )
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_477
tff(fact_8655_ATP_Olambda__478,axiom,
    ! [A: $tType,Uu2: nat,Uua: heap_Time_Heap(A),Uub: A] : aa(A,heap_Time_Heap(list(A)),aa(heap_Time_Heap(A),fun(A,heap_Time_Heap(list(A))),aTP_Lamp_bb(nat,fun(heap_Time_Heap(A),fun(A,heap_Time_Heap(list(A)))),Uu2),Uua),Uub) = heap_Time_bind(list(A),list(A),vEBT_VEBT_replicatei(A,Uu2,Uua),aTP_Lamp_ad(A,fun(list(A),heap_Time_Heap(list(A))),Uub)) ).

% ATP.lambda_478
tff(fact_8656_ATP_Olambda__479,axiom,
    ! [A: $tType,Uu2: heap_Time_Heap(A),Uua: nat,Uub: A] : aa(A,heap_Time_Heap(list(A)),aa(nat,fun(A,heap_Time_Heap(list(A))),aTP_Lamp_az(heap_Time_Heap(A),fun(nat,fun(A,heap_Time_Heap(list(A)))),Uu2),Uua),Uub) = heap_Time_bind(list(A),list(A),vEBT_VEBT_replicatei(A,Uua,Uu2),aTP_Lamp_ad(A,fun(list(A),heap_Time_Heap(list(A))),Uub)) ).

% ATP.lambda_479
tff(fact_8657_ATP_Olambda__480,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(B,heap_Time_Heap(A)),Uua: list(B),Uub: A] : aa(A,heap_Time_Heap(list(A)),aa(list(B),fun(A,heap_Time_Heap(list(A))),aTP_Lamp_ae(fun(B,heap_Time_Heap(A)),fun(list(B),fun(A,heap_Time_Heap(list(A)))),Uu2),Uua),Uub) = heap_Time_bind(list(A),list(A),vEBT_Example_mmap(B,A,Uu2,Uua),aTP_Lamp_ad(A,fun(list(A),heap_Time_Heap(list(A))),Uub)) ).

% ATP.lambda_480
tff(fact_8658_ATP_Olambda__481,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu2: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_nf(set(A),fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> ( member(A,Uub,Uu2)
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_481
tff(fact_8659_ATP_Olambda__482,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(B)
     => ! [Uu2: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_ok(set(A),fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> ( member(A,Uub,Uu2)
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_482
tff(fact_8660_ATP_Olambda__483,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_nh(set(A),fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> ( member(A,Uub,Uu2)
            & ( aa(A,B,Uua,Uub) != one_one(B) ) ) ) ) ).

% ATP.lambda_483
tff(fact_8661_ATP_Olambda__484,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(B,A),Uua: set(B),Uub: B] :
          ( aa(B,$o,aa(set(B),fun(B,$o),aTP_Lamp_ol(fun(B,A),fun(set(B),fun(B,$o)),Uu2),Uua),Uub)
        <=> ( member(B,Uub,Uua)
            & ( aa(B,A,Uu2,Uub) != zero_zero(A) ) ) ) ) ).

% ATP.lambda_484
tff(fact_8662_ATP_Olambda__485,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_parity(B)
     => ! [Uu2: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_aix(set(A),fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> ( member(A,Uub,Uu2)
            & ~ aa(B,$o,dvd_dvd(B,aa(num,B,numeral_numeral(B),bit0(one2))),aa(A,B,Uua,Uub)) ) ) ) ).

% ATP.lambda_485
tff(fact_8663_ATP_Olambda__486,axiom,
    ! [A: $tType,Uu2: set(A),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_dg(set(A),fun(set(A),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( member(A,Uub,Uu2)
        & ~ member(A,Uub,Uua) ) ) ).

% ATP.lambda_486
tff(fact_8664_ATP_Olambda__487,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_lf(nat,fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu2) ) ).

% ATP.lambda_487
tff(fact_8665_ATP_Olambda__488,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu2: A,Uua: real,Uub: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aTP_Lamp_adk(A,fun(real,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Uu2,Uub)),Uua) ) ) ).

% ATP.lambda_488
tff(fact_8666_ATP_Olambda__489,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_lk(nat,fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu2) ) ).

% ATP.lambda_489
tff(fact_8667_ATP_Olambda__490,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu2: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_pw(A,fun(A,fun(A,A)),Uu2),Uua),Uub) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Uub),Uu2)),Uua) ) ).

% ATP.lambda_490
tff(fact_8668_ATP_Olambda__491,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu2: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_pu(A,fun(A,fun(A,A)),Uu2),Uua),Uub) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),Uu2),Uub)),Uua) ) ).

% ATP.lambda_491
tff(fact_8669_ATP_Olambda__492,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu2: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_pv(A,fun(A,fun(A,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Uub),Uu2)),Uua) ) ).

% ATP.lambda_492
tff(fact_8670_ATP_Olambda__493,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu2: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_pt(A,fun(A,fun(A,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Uu2),Uub)),Uua) ) ).

% ATP.lambda_493
tff(fact_8671_ATP_Olambda__494,axiom,
    ! [A: $tType,B: $tType,Uu2: set(product_prod(B,A)),Uua: B,Uub: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_ku(set(product_prod(B,A)),fun(B,fun(A,$o)),Uu2),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),Uu2) ) ).

% ATP.lambda_494
tff(fact_8672_ATP_Olambda__495,axiom,
    ! [B: $tType,A: $tType,Uu2: set(product_prod(A,B)),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_kt(set(product_prod(A,B)),fun(A,fun(B,$o)),Uu2),Uua),Uub)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub),Uu2) ) ).

% ATP.lambda_495
tff(fact_8673_ATP_Olambda__496,axiom,
    ! [A: $tType,Uu2: list(list(A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ama(list(list(A)),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),Uu2),Uub)),Uua) ).

% ATP.lambda_496
tff(fact_8674_ATP_Olambda__497,axiom,
    ! [Uu2: nat,Uua: complex,Uub: complex] :
      ( aa(complex,$o,aa(complex,fun(complex,$o),aTP_Lamp_hh(nat,fun(complex,fun(complex,$o)),Uu2),Uua),Uub)
    <=> ( aa(nat,complex,power_power(complex,Uub),Uu2) = Uua ) ) ).

% ATP.lambda_497
tff(fact_8675_ATP_Olambda__498,axiom,
    ! [Uu2: complex,Uua: nat,Uub: complex] :
      ( aa(complex,$o,aa(nat,fun(complex,$o),aTP_Lamp_kq(complex,fun(nat,fun(complex,$o)),Uu2),Uua),Uub)
    <=> ( aa(nat,complex,power_power(complex,Uub),Uua) = Uu2 ) ) ).

% ATP.lambda_498
tff(fact_8676_ATP_Olambda__499,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu2: A,Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_nn(A,fun(A,fun(A,$o)),Uu2),Uua),Uub)
        <=> ( member(A,Uub,ring_1_Ints(A))
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uu2),Uub)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uub),Uua) ) ) ) ).

% ATP.lambda_499
tff(fact_8677_ATP_Olambda__500,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu2: 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)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,aa(nat,nat,suc,Uub))),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_500
tff(fact_8678_ATP_Olambda__501,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_by(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,aa(nat,nat,suc,Uub))),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_501
tff(fact_8679_ATP_Olambda__502,axiom,
    ! [Uu2: fun(nat,real),Uua: fun(nat,int),Uub: nat] : aa(nat,real,aa(fun(nat,int),fun(nat,real),aTP_Lamp_aci(fun(nat,real),fun(fun(nat,int),fun(nat,real)),Uu2),Uua),Uub) = aa(real,real,minus_minus(real,aa(nat,real,Uu2,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_502
tff(fact_8680_ATP_Olambda__503,axiom,
    ! [Uu2: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_te(fun(nat,real),fun(real,fun(nat,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,Uu2,Uub)),aa(nat,real,power_power(real,Uua),aa(nat,nat,suc,Uub))) ).

% ATP.lambda_503
tff(fact_8681_ATP_Olambda__504,axiom,
    ! [Uu2: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_cc(fun(nat,real),fun(real,fun(nat,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,Uu2,Uub)),aa(nat,real,power_power(real,Uua),Uub)) ).

% ATP.lambda_504
tff(fact_8682_ATP_Olambda__505,axiom,
    ! [Uu2: fun(nat,nat),Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_gc(fun(nat,nat),fun(nat,fun(nat,nat)),Uu2),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,Uu2,Uub)),aa(nat,nat,power_power(nat,Uua),Uub)) ).

% ATP.lambda_505
tff(fact_8683_ATP_Olambda__506,axiom,
    ! [B: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu2: fun(nat,B),Uua: B,Uub: nat] : aa(nat,B,aa(B,fun(nat,B),aTP_Lamp_aaf(fun(nat,B),fun(B,fun(nat,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,Uu2,Uub)),aa(nat,B,power_power(B,Uua),Uub)) ) ).

% ATP.lambda_506
tff(fact_8684_ATP_Olambda__507,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_fn(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_507
tff(fact_8685_ATP_Olambda__508,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_bt(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_508
tff(fact_8686_ATP_Olambda__509,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_bx(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_509
tff(fact_8687_ATP_Olambda__510,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_cv(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_510
tff(fact_8688_ATP_Olambda__511,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_mult(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_fo(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_511
tff(fact_8689_ATP_Olambda__512,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_fy(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_512
tff(fact_8690_ATP_Olambda__513,axiom,
    ! [Uu2: fun(nat,$o),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ms(fun(nat,$o),fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> ( aa(nat,$o,Uu2,Uub)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uua) ) ) ).

% ATP.lambda_513
tff(fact_8691_ATP_Olambda__514,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,real),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tj(fun(A,real),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_514
tff(fact_8692_ATP_Olambda__515,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,real),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vx(fun(A,real),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_515
tff(fact_8693_ATP_Olambda__516,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu2: fun(A,real),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yi(fun(A,real),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_516
tff(fact_8694_ATP_Olambda__517,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,real,aa(fun(A,B),fun(A,real),aTP_Lamp_adl(fun(A,B),fun(fun(A,B),fun(A,real)),Uu2),Uua),Uub) = real_V557655796197034286t_dist(B,aa(A,B,Uu2,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_517
tff(fact_8695_ATP_Olambda__518,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,real,aa(fun(A,B),fun(A,real),aTP_Lamp_adm(fun(A,B),fun(fun(A,B),fun(A,real)),Uu2),Uua),Uub) = real_V557655796197034286t_dist(B,aa(A,B,Uu2,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_518
tff(fact_8696_ATP_Olambda__519,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu2: fun(B,A),Uua: B,Uub: B] :
          ( aa(B,$o,aa(B,fun(B,$o),aTP_Lamp_ale(fun(B,A),fun(B,fun(B,$o)),Uu2),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,Uu2,Uua)),aa(B,A,Uu2,Uub)) ) ) ).

% ATP.lambda_519
tff(fact_8697_ATP_Olambda__520,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: fun(A,real),Uub: A] :
      ( aa(A,$o,aa(fun(A,real),fun(A,$o),aTP_Lamp_agk(fun(A,real),fun(fun(A,real),fun(A,$o)),Uu2),Uua),Uub)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(A,real,Uu2,Uub)),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_520
tff(fact_8698_ATP_Olambda__521,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_afw(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_521
tff(fact_8699_ATP_Olambda__522,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_agb(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_522
tff(fact_8700_ATP_Olambda__523,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_agw(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uua,Uub)),aa(A,B,Uu2,Uub)) ) ) ).

% ATP.lambda_523
tff(fact_8701_ATP_Olambda__524,axiom,
    ! [Uu2: fun(real,real),Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ahs(fun(real,real),fun(fun(real,real),fun(real,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,Uu2,Uub)),aa(real,real,Uua,Uub)) ).

% ATP.lambda_524
tff(fact_8702_ATP_Olambda__525,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [Uu2: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_jz(fun(B,A),fun(fun(B,A),fun(B,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(B,A,Uu2,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_525
tff(fact_8703_ATP_Olambda__526,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ut(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_526
tff(fact_8704_ATP_Olambda__527,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ug(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_527
tff(fact_8705_ATP_Olambda__528,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_ru(fun(A,A),fun(fun(A,A),fun(A,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,Uu2,Uub)),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_528
tff(fact_8706_ATP_Olambda__529,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_zr(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_529
tff(fact_8707_ATP_Olambda__530,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_aeb(fun(A,real),fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(A,real,Uu2,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_530
tff(fact_8708_ATP_Olambda__531,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aei(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_531
tff(fact_8709_ATP_Olambda__532,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ys(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_532
tff(fact_8710_ATP_Olambda__533,axiom,
    ! [Uu2: fun(real,real),Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aer(fun(real,real),fun(fun(real,real),fun(real,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,Uua,Uub)),aa(real,real,Uu2,Uub)) ).

% ATP.lambda_533
tff(fact_8711_ATP_Olambda__534,axiom,
    ! [Uu2: fun(nat,nat),Uua: fun(nat,nat),Uub: nat] : aa(nat,nat,aa(fun(nat,nat),fun(nat,nat),aTP_Lamp_oz(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),Uu2),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,Uu2,Uub)),aa(nat,nat,Uua,Uub)) ).

% ATP.lambda_534
tff(fact_8712_ATP_Olambda__535,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_aic(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_535
tff(fact_8713_ATP_Olambda__536,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu2: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_oy(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_536
tff(fact_8714_ATP_Olambda__537,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_jt(fun(B,A),fun(fun(B,A),fun(B,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu2,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_537
tff(fact_8715_ATP_Olambda__538,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ty(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_538
tff(fact_8716_ATP_Olambda__539,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_sc(fun(A,A),fun(fun(A,A),fun(A,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,Uu2,Uub)),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_539
tff(fact_8717_ATP_Olambda__540,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_zd(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_540
tff(fact_8718_ATP_Olambda__541,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4211221413907600880p_mult(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_zc(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_541
tff(fact_8719_ATP_Olambda__542,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_adt(fun(A,real),fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uu2,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_542
tff(fact_8720_ATP_Olambda__543,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yw(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_543
tff(fact_8721_ATP_Olambda__544,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aeg(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_544
tff(fact_8722_ATP_Olambda__545,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yx(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_545
tff(fact_8723_ATP_Olambda__546,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ze(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_546
tff(fact_8724_ATP_Olambda__547,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_aee(fun(A,real),fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uub)),aa(A,real,Uu2,Uub)) ).

% ATP.lambda_547
tff(fact_8725_ATP_Olambda__548,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_hb(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uub)),aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_548
tff(fact_8726_ATP_Olambda__549,axiom,
    ! [Uu2: fun(nat,real),Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_abi(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Uu2),Uua),Uub) = aa(real,real,minus_minus(real,aa(nat,real,Uu2,Uub)),aa(nat,real,Uua,Uub)) ).

% ATP.lambda_549
tff(fact_8727_ATP_Olambda__550,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bq(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,minus_minus(A,aa(nat,A,Uu2,Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_550
tff(fact_8728_ATP_Olambda__551,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [Uu2: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_hr(fun(B,A),fun(fun(B,A),fun(B,A)),Uu2),Uua),Uub) = aa(A,A,minus_minus(A,aa(B,A,Uu2,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_551
tff(fact_8729_ATP_Olambda__552,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tx(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,minus_minus(B,aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_552
tff(fact_8730_ATP_Olambda__553,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_rw(fun(A,A),fun(fun(A,A),fun(A,A)),Uu2),Uua),Uub) = aa(A,A,minus_minus(A,aa(A,A,Uu2,Uub)),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_553
tff(fact_8731_ATP_Olambda__554,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_zs(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,minus_minus(B,aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_554
tff(fact_8732_ATP_Olambda__555,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_xn(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,minus_minus(B,aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_555
tff(fact_8733_ATP_Olambda__556,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yl(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,minus_minus(B,aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_556
tff(fact_8734_ATP_Olambda__557,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_xo(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,minus_minus(B,aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_557
tff(fact_8735_ATP_Olambda__558,axiom,
    ! [A: $tType] :
      ( ( topolo1287966508704411220up_add(A)
        & topological_t2_space(A) )
     => ! [Uu2: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_nw(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,minus_minus(A,aa(nat,A,Uua,Uub)),aa(nat,A,Uu2,Uub)) ) ).

% ATP.lambda_558
tff(fact_8736_ATP_Olambda__559,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu2: fun(A,nat),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_hz(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Uu2),Uua),Uub) = aa(nat,nat,minus_minus(nat,aa(A,nat,Uua,Uub)),aa(A,nat,Uu2,Uub)) ) ).

% ATP.lambda_559
tff(fact_8737_ATP_Olambda__560,axiom,
    ! [A: $tType,Uu2: fun(A,nat),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_hf(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Uu2),Uua),Uub) = aa(nat,nat,minus_minus(nat,aa(A,nat,Uua,Uub)),aa(A,nat,Uu2,Uub)) ).

% ATP.lambda_560
tff(fact_8738_ATP_Olambda__561,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yk(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,minus_minus(B,aa(A,B,Uua,Uub)),aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_561
tff(fact_8739_ATP_Olambda__562,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1898628316856586783d_mult(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,nat),Uub: A] : aa(A,B,aa(fun(A,nat),fun(A,B),aTP_Lamp_xk(fun(A,B),fun(fun(A,nat),fun(A,B)),Uu2),Uua),Uub) = aa(nat,B,power_power(B,aa(A,B,Uu2,Uub)),aa(A,nat,Uua,Uub)) ) ).

% ATP.lambda_562
tff(fact_8740_ATP_Olambda__563,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,nat),Uub: A] : aa(A,B,aa(fun(A,nat),fun(A,B),aTP_Lamp_xl(fun(A,B),fun(fun(A,nat),fun(A,B)),Uu2),Uua),Uub) = aa(nat,B,power_power(B,aa(A,B,Uu2,Uub)),aa(A,nat,Uua,Uub)) ) ).

% ATP.lambda_563
tff(fact_8741_ATP_Olambda__564,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_wq(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = ord_min(B,aa(A,B,Uu2,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_564
tff(fact_8742_ATP_Olambda__565,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_wr(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = ord_min(B,aa(A,B,Uu2,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_565
tff(fact_8743_ATP_Olambda__566,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_xq(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),ord_max(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_566
tff(fact_8744_ATP_Olambda__567,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_xp(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),ord_max(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_567
tff(fact_8745_ATP_Olambda__568,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,complex,aa(fun(A,real),fun(A,complex),aTP_Lamp_xu(fun(A,real),fun(fun(A,real),fun(A,complex)),Uu2),Uua),Uub) = complex2(aa(A,real,Uu2,Uub),aa(A,real,Uua,Uub)) ).

% ATP.lambda_568
tff(fact_8746_ATP_Olambda__569,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [Uu2: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_br(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu2,Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_569
tff(fact_8747_ATP_Olambda__570,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_hm(fun(B,A),fun(fun(B,A),fun(B,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,Uu2,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_570
tff(fact_8748_ATP_Olambda__571,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tu(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_571
tff(fact_8749_ATP_Olambda__572,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_rx(fun(A,A),fun(fun(A,A),fun(A,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,Uu2,Uub)),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_572
tff(fact_8750_ATP_Olambda__573,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yd(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_573
tff(fact_8751_ATP_Olambda__574,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ady(fun(A,real),fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(A,real,Uu2,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_574
tff(fact_8752_ATP_Olambda__575,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aea(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_575
tff(fact_8753_ATP_Olambda__576,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yc(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_576
tff(fact_8754_ATP_Olambda__577,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_om(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_577
tff(fact_8755_ATP_Olambda__578,axiom,
    ! [Uu2: fun(real,real),Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ss(fun(real,real),fun(fun(real,real),fun(real,real)),Uu2),Uua),Uub) = powr(real,aa(real,real,Uu2,Uub),aa(real,real,Uua,Uub)) ).

% ATP.lambda_578
tff(fact_8756_ATP_Olambda__579,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_uz(fun(A,real),fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = powr(real,aa(A,real,Uu2,Uub),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_579
tff(fact_8757_ATP_Olambda__580,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_aav(fun(A,real),fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = powr(real,aa(A,real,Uu2,Uub),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_580
tff(fact_8758_ATP_Olambda__581,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_yg(fun(A,real),fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = powr(real,aa(A,real,Uu2,Uub),aa(A,real,Uua,Uub)) ).

% ATP.lambda_581
tff(fact_8759_ATP_Olambda__582,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: 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)),Uu2),Uua),Uub) = aa(real,real,log(aa(A,real,Uu2,Uub)),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_582
tff(fact_8760_ATP_Olambda__583,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_zo(fun(A,real),fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = aa(real,real,log(aa(A,real,Uu2,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_583
tff(fact_8761_ATP_Olambda__584,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,C),Uub: A] : aa(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_vv(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),Uu2),Uua),Uub) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu2,Uub)),aa(A,C,Uua,Uub)) ) ).

% ATP.lambda_584
tff(fact_8762_ATP_Olambda__585,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,C),Uub: A] : aa(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_yq(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),Uu2),Uua),Uub) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu2,Uub)),aa(A,C,Uua,Uub)) ) ).

% ATP.lambda_585
tff(fact_8763_ATP_Olambda__586,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(A)
     => ! [Uu2: fun(B,word(A)),Uua: B,Uub: B] :
          ( aa(B,$o,aa(B,fun(B,$o),aTP_Lamp_alf(fun(B,word(A)),fun(B,fun(B,$o)),Uu2),Uua),Uub)
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(B,word(A),Uu2,Uua)),aa(B,word(A),Uu2,Uub)) ) ) ).

% ATP.lambda_586
tff(fact_8764_ATP_Olambda__587,axiom,
    ! [A: $tType,Uu2: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_dk(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( aa(A,$o,Uu2,Uub)
       => aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_587
tff(fact_8765_ATP_Olambda__588,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: fun(A,$o),Uua: fun(A,$o),Uub: A] :
          ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_adc(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu2),Uua),Uub)
        <=> ( aa(A,$o,Uu2,Uub)
            | aa(A,$o,Uua,Uub) ) ) ) ).

% ATP.lambda_588
tff(fact_8766_ATP_Olambda__589,axiom,
    ! [A: $tType,Uu2: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_dm(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( aa(A,$o,Uu2,Uub)
        | aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_589
tff(fact_8767_ATP_Olambda__590,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: fun(A,$o),Uua: fun(A,$o),Uub: A] :
          ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_add(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu2),Uua),Uub)
        <=> ( aa(A,$o,Uu2,Uub)
            & aa(A,$o,Uua,Uub) ) ) ) ).

% ATP.lambda_590
tff(fact_8768_ATP_Olambda__591,axiom,
    ! [A: $tType,Uu2: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_oe(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( aa(A,$o,Uu2,Uub)
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_591
tff(fact_8769_ATP_Olambda__592,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(nat,A),Uua: fun(nat,A),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_aeu(fun(nat,A),fun(fun(nat,A),fun(nat,$o)),Uu2),Uua),Uub)
        <=> ( aa(nat,A,Uu2,Uub) = aa(nat,A,Uua,Uub) ) ) ) ).

% ATP.lambda_592
tff(fact_8770_ATP_Olambda__593,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,A),Uua: fun(B,A),Uub: B] :
      ( aa(B,$o,aa(fun(B,A),fun(B,$o),aTP_Lamp_afe(fun(B,A),fun(fun(B,A),fun(B,$o)),Uu2),Uua),Uub)
    <=> ( aa(B,A,Uu2,Uub) = aa(B,A,Uua,Uub) ) ) ).

% ATP.lambda_593
tff(fact_8771_ATP_Olambda__594,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_afn(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> ( aa(A,B,Uu2,Uub) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_594
tff(fact_8772_ATP_Olambda__595,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: fun(A,A),Uub: A] :
          ( aa(A,$o,aa(fun(A,A),fun(A,$o),aTP_Lamp_afo(fun(A,A),fun(fun(A,A),fun(A,$o)),Uu2),Uua),Uub)
        <=> ( aa(A,A,Uu2,Uub) = aa(A,A,Uua,Uub) ) ) ) ).

% ATP.lambda_595
tff(fact_8773_ATP_Olambda__596,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_agu(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> ( aa(A,B,Uu2,Uub) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_596
tff(fact_8774_ATP_Olambda__597,axiom,
    ! [A: $tType,Uu2: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_afj(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( aa(A,$o,Uu2,Uub)
      <=> aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_597
tff(fact_8775_ATP_Olambda__598,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_afc(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> ( aa(A,B,Uu2,Uub) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_598
tff(fact_8776_ATP_Olambda__599,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,B),Uua: fun(A,B),Uub: A] :
      ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_afh(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( aa(A,B,Uu2,Uub) = aa(A,B,Uua,Uub) ) ) ).

% ATP.lambda_599
tff(fact_8777_ATP_Olambda__600,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_ahp(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu2,Uub)),aa(int,B,ring_1_of_int(B),archimedean_ceiling(B,Uua))) ) ) ).

% ATP.lambda_600
tff(fact_8778_ATP_Olambda__601,axiom,
    ! [A: $tType,Uu2: fun(A,fun(A,$o)),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_akb(fun(A,fun(A,$o)),fun(fun(A,$o),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( aa(A,$o,Uua,Uub)
        & ! [Y4: A] :
            ( aa(A,$o,Uua,Y4)
           => aa(A,$o,aa(A,fun(A,$o),Uu2,Uub),Y4) ) ) ) ).

% ATP.lambda_601
tff(fact_8779_ATP_Olambda__602,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: nat,Uub: A] : aa(A,A,aa(nat,fun(A,A),aTP_Lamp_sk(fun(A,A),fun(nat,fun(A,A)),Uu2),Uua),Uub) = aa(nat,A,power_power(A,aa(A,A,Uu2,Uub)),aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_602
tff(fact_8780_ATP_Olambda__603,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(nat,real),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_eh(fun(nat,real),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,aa(nat,real,Uu2,Uub)),Uua) ) ).

% ATP.lambda_603
tff(fact_8781_ATP_Olambda__604,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu2: fun(B,real),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_fc(fun(B,real),fun(A,fun(B,A)),Uu2),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,aa(B,real,Uu2,Uub)),Uua) ) ).

% ATP.lambda_604
tff(fact_8782_ATP_Olambda__605,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu2: fun(A,real),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_tq(fun(A,real),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uu2,Uub)),Uua) ) ).

% ATP.lambda_605
tff(fact_8783_ATP_Olambda__606,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(B,fun(C,A)),Uua: set(C),Uub: B] : aa(B,A,aa(set(C),fun(B,A),aTP_Lamp_jv(fun(B,fun(C,A)),fun(set(C),fun(B,A)),Uu2),Uua),Uub) = groups7121269368397514597t_prod(C,A,aa(B,fun(C,A),Uu2,Uub),Uua) ) ).

% ATP.lambda_606
tff(fact_8784_ATP_Olambda__607,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [Uu2: set(A),Uua: fun(B,fun(A,C)),Uub: B] : aa(B,C,aa(fun(B,fun(A,C)),fun(B,C),aTP_Lamp_wv(set(A),fun(fun(B,fun(A,C)),fun(B,C)),Uu2),Uua),Uub) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),Uua,Uub),Uu2) ) ).

% ATP.lambda_607
tff(fact_8785_ATP_Olambda__608,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(B,fun(C,A)),Uua: set(C),Uub: B] : aa(B,A,aa(set(C),fun(B,A),aTP_Lamp_hi(fun(B,fun(C,A)),fun(set(C),fun(B,A)),Uu2),Uua),Uub) = aa(set(C),A,groups7311177749621191930dd_sum(C,A,aa(B,fun(C,A),Uu2,Uub)),Uua) ) ).

% ATP.lambda_608
tff(fact_8786_ATP_Olambda__609,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu2: set(A),Uua: fun(B,fun(A,B)),Uub: B] : aa(B,B,aa(fun(B,fun(A,B)),fun(B,B),aTP_Lamp_sj(set(A),fun(fun(B,fun(A,B)),fun(B,B)),Uu2),Uua),Uub) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(B,fun(A,B),Uua,Uub)),Uu2) ) ).

% ATP.lambda_609
tff(fact_8787_ATP_Olambda__610,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [Uu2: set(A),Uua: fun(B,fun(A,C)),Uub: B] : aa(B,C,aa(fun(B,fun(A,C)),fun(B,C),aTP_Lamp_zm(set(A),fun(fun(B,fun(A,C)),fun(B,C)),Uu2),Uua),Uub) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(B,fun(A,C),Uua,Uub)),Uu2) ) ).

% ATP.lambda_610
tff(fact_8788_ATP_Olambda__611,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,real,aa(B,fun(A,real),aTP_Lamp_adn(fun(A,B),fun(B,fun(A,real)),Uu2),Uua),Uub) = real_V557655796197034286t_dist(B,aa(A,B,Uu2,Uub),Uua) ) ).

% ATP.lambda_611
tff(fact_8789_ATP_Olambda__612,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [Uu2: fun(nat,set(A)),Uua: set(A),Uub: nat] :
          ( aa(nat,$o,aa(set(A),fun(nat,$o),aTP_Lamp_agl(fun(nat,set(A)),fun(set(A),fun(nat,$o)),Uu2),Uua),Uub)
        <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(nat,set(A),Uu2,Uub)),Uua) ) ) ).

% ATP.lambda_612
tff(fact_8790_ATP_Olambda__613,axiom,
    ! [Uu2: fun(nat,nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_mu(fun(nat,nat),fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,Uu2,Uub)),Uua) ) ).

% ATP.lambda_613
tff(fact_8791_ATP_Olambda__614,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_agy(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu2,Uub)),Uua) ) ) ).

% ATP.lambda_614
tff(fact_8792_ATP_Olambda__615,axiom,
    ! [A: $tType,B: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_agv(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu2,Uub)),Uua) ) ) ).

% ATP.lambda_615
tff(fact_8793_ATP_Olambda__616,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_agn(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu2,Uub)),Uua) ) ) ).

% ATP.lambda_616
tff(fact_8794_ATP_Olambda__617,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_agi(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu2,Uub)),Uua) ) ) ).

% ATP.lambda_617
tff(fact_8795_ATP_Olambda__618,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Uu2: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_fg(fun(B,A),fun(A,fun(B,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(B,A,Uu2,Uub)),Uua) ) ).

% ATP.lambda_618
tff(fact_8796_ATP_Olambda__619,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_bj(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,Uu2,Uub)),Uua) ) ).

% ATP.lambda_619
tff(fact_8797_ATP_Olambda__620,axiom,
    ! [B: $tType,A: $tType] :
      ( field(A)
     => ! [Uu2: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_hs(fun(B,A),fun(A,fun(B,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(B,A,Uu2,Uub)),Uua) ) ).

% ATP.lambda_620
tff(fact_8798_ATP_Olambda__621,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_sf(fun(A,A),fun(A,fun(A,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,Uu2,Uub)),Uua) ) ).

% ATP.lambda_621
tff(fact_8799_ATP_Olambda__622,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_yt(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_622
tff(fact_8800_ATP_Olambda__623,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu2: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_aal(A,fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,Uua,Uub)),Uu2) ) ).

% ATP.lambda_623
tff(fact_8801_ATP_Olambda__624,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu2: fun(B,A),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_aga(fun(B,A),fun(A,fun(B,$o)),Uu2),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,Uu2,Uub)),Uua) ) ) ).

% ATP.lambda_624
tff(fact_8802_ATP_Olambda__625,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_afx(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu2,Uub)),Uua) ) ) ).

% ATP.lambda_625
tff(fact_8803_ATP_Olambda__626,axiom,
    ! [A: $tType,B: $tType] :
      ( ( dense_linorder(B)
        & no_bot(B) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_agj(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu2,Uub)),Uua) ) ) ).

% ATP.lambda_626
tff(fact_8804_ATP_Olambda__627,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_bm(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uub)),Uua) ) ).

% ATP.lambda_627
tff(fact_8805_ATP_Olambda__628,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_0(A)
     => ! [Uu2: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_hp(fun(B,A),fun(A,fun(B,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu2,Uub)),Uua) ) ).

% ATP.lambda_628
tff(fact_8806_ATP_Olambda__629,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_tp(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_629
tff(fact_8807_ATP_Olambda__630,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_ain(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_630
tff(fact_8808_ATP_Olambda__631,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_sd(fun(A,A),fun(A,fun(A,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,Uu2,Uub)),Uua) ) ).

% ATP.lambda_631
tff(fact_8809_ATP_Olambda__632,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_za(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_632
tff(fact_8810_ATP_Olambda__633,axiom,
    ! [A: $tType,Uu2: fun(A,assn),Uua: assn,Uub: A] : aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_ec(fun(A,assn),fun(assn,fun(A,assn)),Uu2),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(A,assn,Uu2,Uub)),Uua) ).

% ATP.lambda_633
tff(fact_8811_ATP_Olambda__634,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_yv(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_634
tff(fact_8812_ATP_Olambda__635,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_yz(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_635
tff(fact_8813_ATP_Olambda__636,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: real,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_um(real,fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uub)),Uu2) ) ).

% ATP.lambda_636
tff(fact_8814_ATP_Olambda__637,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu2: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ci(A,fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uua,Uub)),Uu2) ) ).

% ATP.lambda_637
tff(fact_8815_ATP_Olambda__638,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu2: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_aak(A,fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uua,Uub)),Uu2) ) ).

% ATP.lambda_638
tff(fact_8816_ATP_Olambda__639,axiom,
    ! [B: $tType,A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu2: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_vr(A,fun(fun(B,A),fun(B,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uua,Uub)),Uu2) ) ).

% ATP.lambda_639
tff(fact_8817_ATP_Olambda__640,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ib(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,minus_minus(A,aa(nat,A,Uu2,Uub)),Uua) ) ).

% ATP.lambda_640
tff(fact_8818_ATP_Olambda__641,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_yj(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,minus_minus(B,aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_641
tff(fact_8819_ATP_Olambda__642,axiom,
    ! [Uu2: fun(real,real),Uua: nat,Uub: real] : aa(real,real,aa(nat,fun(real,real),aTP_Lamp_so(fun(real,real),fun(nat,fun(real,real)),Uu2),Uua),Uub) = aa(nat,real,power_power(real,aa(real,real,Uu2,Uub)),Uua) ).

% ATP.lambda_642
tff(fact_8820_ATP_Olambda__643,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu2: fun(B,A),Uua: nat,Uub: B] : aa(B,A,aa(nat,fun(B,A),aTP_Lamp_jy(fun(B,A),fun(nat,fun(B,A)),Uu2),Uua),Uub) = aa(nat,A,power_power(A,aa(B,A,Uu2,Uub)),Uua) ) ).

% ATP.lambda_643
tff(fact_8821_ATP_Olambda__644,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu2: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_up(fun(A,B),fun(nat,fun(A,B)),Uu2),Uua),Uub) = aa(nat,B,power_power(B,aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_644
tff(fact_8822_ATP_Olambda__645,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: nat,Uub: A] : aa(A,A,aa(nat,fun(A,A),aTP_Lamp_sl(fun(A,A),fun(nat,fun(A,A)),Uu2),Uua),Uub) = aa(nat,A,power_power(A,aa(A,A,Uu2,Uub)),Uua) ) ).

% ATP.lambda_645
tff(fact_8823_ATP_Olambda__646,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [Uu2: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_xm(fun(A,B),fun(nat,fun(A,B)),Uu2),Uua),Uub) = aa(nat,B,power_power(B,aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_646
tff(fact_8824_ATP_Olambda__647,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu2: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_aeh(fun(A,B),fun(nat,fun(A,B)),Uu2),Uua),Uub) = aa(nat,B,power_power(B,aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_647
tff(fact_8825_ATP_Olambda__648,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [Uu2: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_zn(fun(A,B),fun(nat,fun(A,B)),Uu2),Uua),Uub) = aa(nat,B,power_power(B,aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_648
tff(fact_8826_ATP_Olambda__649,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu2: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_xj(fun(A,B),fun(nat,fun(A,B)),Uu2),Uua),Uub) = aa(nat,B,power_power(B,aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_649
tff(fact_8827_ATP_Olambda__650,axiom,
    ! [Uu2: nat,Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_adw(nat,fun(fun(real,real),fun(real,real)),Uu2),Uua),Uub) = aa(nat,real,power_power(real,aa(real,real,Uua,Uub)),Uu2) ).

% ATP.lambda_650
tff(fact_8828_ATP_Olambda__651,axiom,
    ! [A: $tType,Uu2: nat,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_vt(nat,fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = aa(nat,real,power_power(real,aa(A,real,Uua,Uub)),Uu2) ).

% ATP.lambda_651
tff(fact_8829_ATP_Olambda__652,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_aib(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu2,Uub)),Uua) ) ).

% ATP.lambda_652
tff(fact_8830_ATP_Olambda__653,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu2: fun(C,heap_Time_Heap(B)),Uua: fun(B,heap_Time_Heap(A)),Uub: C] : aa(C,heap_Time_Heap(A),aa(fun(B,heap_Time_Heap(A)),fun(C,heap_Time_Heap(A)),aTP_Lamp_ac(fun(C,heap_Time_Heap(B)),fun(fun(B,heap_Time_Heap(A)),fun(C,heap_Time_Heap(A))),Uu2),Uua),Uub) = heap_Time_bind(B,A,aa(C,heap_Time_Heap(B),Uu2,Uub),Uua) ).

% ATP.lambda_653
tff(fact_8831_ATP_Olambda__654,axiom,
    ! [Uu2: fun(real,real),Uua: real,Uub: real] : aa(real,real,aa(real,fun(real,real),aTP_Lamp_sr(fun(real,real),fun(real,fun(real,real)),Uu2),Uua),Uub) = powr(real,aa(real,real,Uu2,Uub),Uua) ).

% ATP.lambda_654
tff(fact_8832_ATP_Olambda__655,axiom,
    ! [A: $tType,Uu2: real,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_aef(real,fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = powr(real,aa(A,real,Uua,Uub),Uu2) ).

% ATP.lambda_655
tff(fact_8833_ATP_Olambda__656,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: fun(B,A),Uua: set(A),Uub: B] :
          ( aa(B,$o,aa(set(A),fun(B,$o),aTP_Lamp_agh(fun(B,A),fun(set(A),fun(B,$o)),Uu2),Uua),Uub)
        <=> member(A,aa(B,A,Uu2,Uub),Uua) ) ) ).

% ATP.lambda_656
tff(fact_8834_ATP_Olambda__657,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu2: fun(A,B),Uua: set(B),Uub: A] :
          ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_agg(fun(A,B),fun(set(B),fun(A,$o)),Uu2),Uua),Uub)
        <=> member(B,aa(A,B,Uu2,Uub),Uua) ) ) ).

% ATP.lambda_657
tff(fact_8835_ATP_Olambda__658,axiom,
    ! [B: $tType,A: $tType,Uu2: set(A),Uua: fun(B,A),Uub: B] :
      ( aa(B,$o,aa(fun(B,A),fun(B,$o),aTP_Lamp_qw(set(A),fun(fun(B,A),fun(B,$o)),Uu2),Uua),Uub)
    <=> member(A,aa(B,A,Uua,Uub),Uu2) ) ).

% ATP.lambda_658
tff(fact_8836_ATP_Olambda__659,axiom,
    ! [A: $tType,Uu2: fun(A,$o),Uua: $o,Uub: A] :
      ( aa(A,$o,aa($o,fun(A,$o),aTP_Lamp_afk(fun(A,$o),fun($o,fun(A,$o)),Uu2),(Uua)),Uub)
    <=> ( aa(A,$o,Uu2,Uub)
        | (Uua) ) ) ).

% ATP.lambda_659
tff(fact_8837_ATP_Olambda__660,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_aez(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> ( aa(A,B,Uu2,Uub) = Uua ) ) ) ).

% ATP.lambda_660
tff(fact_8838_ATP_Olambda__661,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo8865339358273720382pology(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_afb(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> ( aa(A,B,Uu2,Uub) = Uua ) ) ) ).

% ATP.lambda_661
tff(fact_8839_ATP_Olambda__662,axiom,
    ! [Uu2: list(nat),Uua: list(nat),Uub: list(nat)] : aa(list(nat),assn,aa(list(nat),fun(list(nat),assn),aTP_Lamp_qm(list(nat),fun(list(nat),fun(list(nat),assn)),Uu2),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),pure_assn(Uub = aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_ql(list(nat),fun(nat,nat),Uu2)),Uua))),top_top(assn)) ).

% ATP.lambda_662
tff(fact_8840_ATP_Olambda__663,axiom,
    ! [A: $tType,Uu2: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_df(A,fun(fun(A,$o),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( ( Uub != Uu2 )
       => aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_663
tff(fact_8841_ATP_Olambda__664,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: nat,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ov(nat,fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,Uua),Uub))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,Uu2),Uub))) ) ).

% ATP.lambda_664
tff(fact_8842_ATP_Olambda__665,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: real,Uua: fun(nat,A),Uub: nat] : aa(nat,real,aa(fun(nat,A),fun(nat,real),aTP_Lamp_cd(real,fun(fun(nat,A),fun(nat,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,aa(nat,A,Uua,Uub))),aa(nat,real,power_power(real,Uu2),Uub)) ) ).

% ATP.lambda_665
tff(fact_8843_ATP_Olambda__666,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu2: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_ahj(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),Uu2),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,Uu2,Uub))),aa(nat,real,Uua,Uub)) ) ) ).

% ATP.lambda_666
tff(fact_8844_ATP_Olambda__667,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu2: fun(nat,A),Uua: fun(nat,B),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,B),fun(nat,$o),aTP_Lamp_aie(fun(nat,A),fun(fun(nat,B),fun(nat,$o)),Uu2),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,Uu2,Uub))),real_V7770717601297561774m_norm(B,aa(nat,B,Uua,Uub))) ) ) ).

% ATP.lambda_667
tff(fact_8845_ATP_Olambda__668,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_ahc(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> ( archim6421214686448440834_floor(B,aa(A,B,Uu2,Uub)) = archim6421214686448440834_floor(B,Uua) ) ) ) ).

% ATP.lambda_668
tff(fact_8846_ATP_Olambda__669,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_ahd(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> ( archimedean_ceiling(B,aa(A,B,Uu2,Uub)) = archimedean_ceiling(B,Uua) ) ) ) ).

% ATP.lambda_669
tff(fact_8847_ATP_Olambda__670,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu2: fun(A,B),Uua: real,Uub: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aTP_Lamp_aih(fun(A,B),fun(real,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,Uu2,Uub))),Uua) ) ) ).

% ATP.lambda_670
tff(fact_8848_ATP_Olambda__671,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ip(A,fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)))),aa(nat,A,power_power(A,Uu2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua))) ) ).

% ATP.lambda_671
tff(fact_8849_ATP_Olambda__672,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_aho(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(int,B,ring_1_of_int(B),archim6421214686448440834_floor(B,Uua))),aa(A,B,Uu2,Uub)) ) ) ).

% ATP.lambda_672
tff(fact_8850_ATP_Olambda__673,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu2: nat,Uua: nat,Uub: word(A)] :
          ( aa(word(A),$o,aa(nat,fun(word(A),$o),aTP_Lamp_pa(nat,fun(nat,fun(word(A),$o)),Uu2),Uua),Uub)
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Uub),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Uu2)),aa(nat,word(A),power_power(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Uua))) ) ) ).

% ATP.lambda_673
tff(fact_8851_ATP_Olambda__674,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_pb(nat,fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uu2)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))) ) ).

% ATP.lambda_674
tff(fact_8852_ATP_Olambda__675,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_os(fun(nat,A),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = groups7121269368397514597t_prod(nat,A,Uu2,set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uua))) ) ).

% ATP.lambda_675
tff(fact_8853_ATP_Olambda__676,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ot(fun(nat,A),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu2),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uua))) ) ).

% ATP.lambda_676
tff(fact_8854_ATP_Olambda__677,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_or(fun(nat,A),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu2),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uua))) ) ).

% ATP.lambda_677
tff(fact_8855_ATP_Olambda__678,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Uu2: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_jh(A,fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uub)),Uua)) ) ).

% ATP.lambda_678
tff(fact_8856_ATP_Olambda__679,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu2: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_jf(A,fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uub)),Uua)) ) ).

% ATP.lambda_679
tff(fact_8857_ATP_Olambda__680,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu2: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ig(A,fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,power_power(A,Uu2),aa(nat,nat,minus_minus(nat,Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_680
tff(fact_8858_ATP_Olambda__681,axiom,
    ! [Uu2: real,Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_abu(real,fun(real,fun(nat,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),divide_divide(real),Uua),aa(nat,real,power_power(real,Uu2),Uub)) ).

% ATP.lambda_681
tff(fact_8859_ATP_Olambda__682,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu2: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_gi(A,fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,power_power(A,Uu2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)) ) ).

% ATP.lambda_682
tff(fact_8860_ATP_Olambda__683,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_jg(nat,fun(nat,fun(nat,nat)),Uu2),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)) ).

% ATP.lambda_683
tff(fact_8861_ATP_Olambda__684,axiom,
    ! [Uu2: real,Uua: real,Uub: real] :
      ( aa(real,$o,aa(real,fun(real,$o),aTP_Lamp_ahh(real,fun(real,fun(real,$o)),Uu2),Uua),Uub)
    <=> member(real,Uub,set_or5935395276787703475ssThan(real,Uu2,Uua)) ) ).

% ATP.lambda_684
tff(fact_8862_ATP_Olambda__685,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu2: real,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_fd(real,fun(fun(B,A),fun(B,A)),Uu2),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,Uu2),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_685
tff(fact_8863_ATP_Olambda__686,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(nat,A),Uua: real,Uub: nat] : aa(nat,A,aa(real,fun(nat,A),aTP_Lamp_eg(fun(nat,A),fun(real,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,Uua),aa(nat,A,Uu2,Uub)) ) ).

% ATP.lambda_686
tff(fact_8864_ATP_Olambda__687,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: real,Uub: A] : aa(A,B,aa(real,fun(A,B),aTP_Lamp_tr(fun(A,B),fun(real,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,real_V8093663219630862766scaleR(B,Uua),aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_687
tff(fact_8865_ATP_Olambda__688,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: real,Uub: A] : aa(A,A,aa(real,fun(A,A),aTP_Lamp_rz(fun(A,A),fun(real,fun(A,A)),Uu2),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,Uua),aa(A,A,Uu2,Uub)) ) ).

% ATP.lambda_688
tff(fact_8866_ATP_Olambda__689,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu2: B,Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_ago(B,fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uu2),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_689
tff(fact_8867_ATP_Olambda__690,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_agx(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),aa(A,B,Uu2,Uub)) ) ) ).

% ATP.lambda_690
tff(fact_8868_ATP_Olambda__691,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_agp(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),aa(A,B,Uu2,Uub)) ) ) ).

% ATP.lambda_691
tff(fact_8869_ATP_Olambda__692,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_agc(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),aa(A,B,Uu2,Uub)) ) ) ).

% ATP.lambda_692
tff(fact_8870_ATP_Olambda__693,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_alv(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),aa(nat,A,Uu2,Uub)) ) ).

% ATP.lambda_693
tff(fact_8871_ATP_Olambda__694,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu2: fun(B,A),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_afz(fun(B,A),fun(A,fun(B,$o)),Uu2),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),aa(B,A,Uu2,Uub)) ) ) ).

% ATP.lambda_694
tff(fact_8872_ATP_Olambda__695,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_agd(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),Uua),aa(A,B,Uu2,Uub)) ) ) ).

% ATP.lambda_695
tff(fact_8873_ATP_Olambda__696,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_afy(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),Uua),aa(A,B,Uu2,Uub)) ) ) ).

% ATP.lambda_696
tff(fact_8874_ATP_Olambda__697,axiom,
    ! [B: $tType,Uu2: assn,Uua: fun(B,assn),Uub: B] : aa(B,assn,aa(fun(B,assn),fun(B,assn),aTP_Lamp_ea(assn,fun(fun(B,assn),fun(B,assn)),Uu2),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Uu2),aa(B,assn,Uua,Uub)) ).

% ATP.lambda_697
tff(fact_8875_ATP_Olambda__698,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu2: B,Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aim(B,fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uu2),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_698
tff(fact_8876_ATP_Olambda__699,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bi(A,fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu2),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_699
tff(fact_8877_ATP_Olambda__700,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [Uu2: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_hq(A,fun(fun(B,A),fun(B,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu2),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_700
tff(fact_8878_ATP_Olambda__701,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu2: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cj(A,fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu2),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_701
tff(fact_8879_ATP_Olambda__702,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu2: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_aaj(A,fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu2),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_702
tff(fact_8880_ATP_Olambda__703,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu2: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_vs(A,fun(fun(B,A),fun(B,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu2),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_703
tff(fact_8881_ATP_Olambda__704,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_bn(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),aa(nat,A,Uu2,Uub)) ) ).

% ATP.lambda_704
tff(fact_8882_ATP_Olambda__705,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_to(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_705
tff(fact_8883_ATP_Olambda__706,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_se(fun(A,A),fun(A,fun(A,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),aa(A,A,Uu2,Uub)) ) ).

% ATP.lambda_706
tff(fact_8884_ATP_Olambda__707,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_zb(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_707
tff(fact_8885_ATP_Olambda__708,axiom,
    ! [A: $tType,Uu2: fun(A,assn),Uua: assn,Uub: A] : aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_kr(fun(A,assn),fun(assn,fun(A,assn)),Uu2),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Uua),aa(A,assn,Uu2,Uub)) ).

% ATP.lambda_708
tff(fact_8886_ATP_Olambda__709,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_yu(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_709
tff(fact_8887_ATP_Olambda__710,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_yy(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_710
tff(fact_8888_ATP_Olambda__711,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linordered_field(B)
        & topolo1944317154257567458pology(B) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_ads(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_711
tff(fact_8889_ATP_Olambda__712,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: A,Uua: fun(B,nat),Uub: B] : aa(B,A,aa(fun(B,nat),fun(B,A),aTP_Lamp_kb(A,fun(fun(B,nat),fun(B,A)),Uu2),Uua),Uub) = aa(nat,A,power_power(A,Uu2),aa(B,nat,Uua,Uub)) ) ).

% ATP.lambda_712
tff(fact_8890_ATP_Olambda__713,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [Uu2: fun(A,nat),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_acc(fun(A,nat),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(nat,B,power_power(B,Uua),aa(A,nat,Uu2,Uub)) ) ).

% ATP.lambda_713
tff(fact_8891_ATP_Olambda__714,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [Uu2: B,Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yb(B,fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),Uu2),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_714
tff(fact_8892_ATP_Olambda__715,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: fun(A,real),Uua: nat,Uub: A] : aa(A,real,aa(nat,fun(A,real),aTP_Lamp_aan(fun(A,real),fun(nat,fun(A,real)),Uu2),Uua),Uub) = aa(real,real,root(Uua),aa(A,real,Uu2,Uub)) ) ).

% ATP.lambda_715
tff(fact_8893_ATP_Olambda__716,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: nat,Uub: A] : aa(A,real,aa(nat,fun(A,real),aTP_Lamp_xe(fun(A,real),fun(nat,fun(A,real)),Uu2),Uua),Uub) = aa(real,real,root(Uua),aa(A,real,Uu2,Uub)) ).

% ATP.lambda_716
tff(fact_8894_ATP_Olambda__717,axiom,
    ! [A: $tType,Uu2: $o,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_afm($o,fun(fun(A,$o),fun(A,$o)),(Uu2)),Uua),Uub)
    <=> ( (Uu2)
       => aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_717
tff(fact_8895_ATP_Olambda__718,axiom,
    ! [A: $tType,Uu2: $o,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_afl($o,fun(fun(A,$o),fun(A,$o)),(Uu2)),Uua),Uub)
    <=> ( (Uu2)
        | aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_718
tff(fact_8896_ATP_Olambda__719,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,B),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_rd(fun(A,B),fun(A,fun(B,$o)),Uu2),Uua),Uub)
    <=> ( Uub = aa(A,B,Uu2,Uua) ) ) ).

% ATP.lambda_719
tff(fact_8897_ATP_Olambda__720,axiom,
    ! [A: $tType,Uu2: assn,Uua: A,Uub: A] : aa(A,assn,aa(A,fun(A,assn),aTP_Lamp_dy(assn,fun(A,fun(A,assn)),Uu2),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Uu2),pure_assn(Uub = Uua)) ).

% ATP.lambda_720
tff(fact_8898_ATP_Olambda__721,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu2: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_kl(A,fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu2),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,Uua),Uub))) ) ).

% ATP.lambda_721
tff(fact_8899_ATP_Olambda__722,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu2: fun(A,B),Uua: real,Uub: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aTP_Lamp_ahq(fun(A,B),fun(real,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uua),real_V7770717601297561774m_norm(B,aa(A,B,Uu2,Uub))) ) ) ).

% ATP.lambda_722
tff(fact_8900_ATP_Olambda__723,axiom,
    ! [A: $tType,Uu2: fun(A,nat),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_no(fun(A,nat),fun(A,fun(A,$o)),Uu2),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),
          $ite(Uub = Uua,aa(nat,nat,suc,aa(A,nat,Uu2,Uub)),aa(A,nat,Uu2,Uub))) ) ).

% ATP.lambda_723
tff(fact_8901_ATP_Olambda__724,axiom,
    ! [A: $tType,Uu2: fun(A,nat),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_nk(fun(A,nat),fun(fun(A,$o),fun(A,$o)),Uu2),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),
          $ite(aa(A,$o,Uua,Uub),aa(A,nat,Uu2,Uub),zero_zero(nat))) ) ).

% ATP.lambda_724
tff(fact_8902_ATP_Olambda__725,axiom,
    ! [A: $tType,Uu2: fun(A,nat),Uua: fun(A,nat),Uub: A] :
      ( aa(A,$o,aa(fun(A,nat),fun(A,$o),aTP_Lamp_np(fun(A,nat),fun(fun(A,nat),fun(A,$o)),Uu2),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,minus_minus(nat,aa(A,nat,Uu2,Uub)),aa(A,nat,Uua,Uub))) ) ).

% ATP.lambda_725
tff(fact_8903_ATP_Olambda__726,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_kd(fun(nat,A),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,Uu2,aa(nat,nat,minus_minus(nat,Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_726
tff(fact_8904_ATP_Olambda__727,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_hy(fun(nat,A),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,Uu2,aa(nat,nat,minus_minus(nat,Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_727
tff(fact_8905_ATP_Olambda__728,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_abc(fun(nat,A),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)) ) ).

% ATP.lambda_728
tff(fact_8906_ATP_Olambda__729,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aao(fun(nat,A),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,Uu2,aa(nat,nat,minus_minus(nat,Uub),Uua)) ) ).

% ATP.lambda_729
tff(fact_8907_ATP_Olambda__730,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: fun(A,B),Uua: A,Uub: A] : aa(A,B,aa(A,fun(A,B),aTP_Lamp_vz(fun(A,B),fun(A,fun(A,B)),Uu2),Uua),Uub) = aa(A,B,Uu2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub)) ) ).

% ATP.lambda_730
tff(fact_8908_ATP_Olambda__731,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_rl(fun(A,A),fun(A,fun(A,A)),Uu2),Uua),Uub) = aa(A,A,Uu2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub)) ) ).

% ATP.lambda_731
tff(fact_8909_ATP_Olambda__732,axiom,
    ! [Uu2: fun(real,$o),Uua: real,Uub: real] :
      ( aa(real,$o,aa(real,fun(real,$o),aTP_Lamp_ahe(fun(real,$o),fun(real,fun(real,$o)),Uu2),Uua),Uub)
    <=> aa(real,$o,Uu2,aa(real,real,aa(real,fun(real,real),plus_plus(real),Uub),Uua)) ) ).

% ATP.lambda_732
tff(fact_8910_ATP_Olambda__733,axiom,
    ! [A: $tType,Uu2: fun(real,A),Uua: real,Uub: real] : aa(real,A,aa(real,fun(real,A),aTP_Lamp_adr(fun(real,A),fun(real,fun(real,A)),Uu2),Uua),Uub) = aa(real,A,Uu2,aa(real,real,aa(real,fun(real,real),plus_plus(real),Uub),Uua)) ).

% ATP.lambda_733
tff(fact_8911_ATP_Olambda__734,axiom,
    ! [Uu2: fun(nat,$o),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_aet(fun(nat,$o),fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> aa(nat,$o,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_734
tff(fact_8912_ATP_Olambda__735,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_bh(fun(nat,A),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_735
tff(fact_8913_ATP_Olambda__736,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aat(fun(nat,A),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_736
tff(fact_8914_ATP_Olambda__737,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aia(fun(nat,A),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_737
tff(fact_8915_ATP_Olambda__738,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_kc(fun(nat,A),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_738
tff(fact_8916_ATP_Olambda__739,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_iv(fun(nat,A),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_739
tff(fact_8917_ATP_Olambda__740,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,$o),Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_agm(fun(A,$o),fun(A,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(A,$o,Uu2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua)) ) ) ).

% ATP.lambda_740
tff(fact_8918_ATP_Olambda__741,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: fun(A,B),Uua: A,Uub: A] : aa(A,B,aa(A,fun(A,B),aTP_Lamp_wk(fun(A,B),fun(A,fun(A,B)),Uu2),Uua),Uub) = aa(A,B,Uu2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua)) ) ).

% ATP.lambda_741
tff(fact_8919_ATP_Olambda__742,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,B),Uua: A,Uub: A] : aa(A,B,aa(A,fun(A,B),aTP_Lamp_zz(fun(A,B),fun(A,fun(A,B)),Uu2),Uua),Uub) = aa(A,B,Uu2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua)) ) ).

% ATP.lambda_742
tff(fact_8920_ATP_Olambda__743,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_rt(fun(A,A),fun(A,fun(A,A)),Uu2),Uua),Uub) = aa(A,A,Uu2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua)) ) ).

% ATP.lambda_743
tff(fact_8921_ATP_Olambda__744,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(product_prod(A,B),$o),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_le(fun(product_prod(A,B),$o),fun(A,fun(B,$o)),Uu2),Uua),Uub)
    <=> aa(product_prod(A,B),$o,Uu2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub)) ) ).

% ATP.lambda_744
tff(fact_8922_ATP_Olambda__745,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu2: fun(product_prod(A,B),C),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_ln(fun(product_prod(A,B),C),fun(A,fun(B,C)),Uu2),Uua),Uub) = aa(product_prod(A,B),C,Uu2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub)) ).

% ATP.lambda_745
tff(fact_8923_ATP_Olambda__746,axiom,
    ! [C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu2: B,Uua: fun(B,C),Uub: B] : aa(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_adj(B,fun(fun(B,C),fun(B,C)),Uu2),Uua),Uub) = aa(B,C,Uua,aa(B,B,aa(B,fun(B,B),plus_plus(B),Uu2),Uub)) ) ).

% ATP.lambda_746
tff(fact_8924_ATP_Olambda__747,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: A,Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_wa(A,fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(A,B,Uua,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu2),Uub)) ) ).

% ATP.lambda_747
tff(fact_8925_ATP_Olambda__748,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: nat,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cl(nat,fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,Uua,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uu2)) ) ).

% ATP.lambda_748
tff(fact_8926_ATP_Olambda__749,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(real,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ul(fun(real,real),fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = aa(real,real,Uu2,aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_749
tff(fact_8927_ATP_Olambda__750,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [Uu2: fun(real,A),Uua: fun(nat,real),Uub: nat] : aa(nat,A,aa(fun(nat,real),fun(nat,A),aTP_Lamp_adu(fun(real,A),fun(fun(nat,real),fun(nat,A)),Uu2),Uua),Uub) = aa(real,A,Uu2,aa(nat,real,Uua,Uub)) ) ).

% ATP.lambda_750
tff(fact_8928_ATP_Olambda__751,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(nat,A),Uua: fun(nat,nat),Uub: nat] : aa(nat,A,aa(fun(nat,nat),fun(nat,A),aTP_Lamp_ahz(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,Uu2,aa(nat,nat,Uua,Uub)) ) ).

% ATP.lambda_751
tff(fact_8929_ATP_Olambda__752,axiom,
    ! [B: $tType,C: $tType,D: $tType,A: $tType,Uu2: fun(D,fun(B,C)),Uua: fun(A,D),Uub: A] : aa(A,fun(B,C),aa(fun(A,D),fun(A,fun(B,C)),aTP_Lamp_alp(fun(D,fun(B,C)),fun(fun(A,D),fun(A,fun(B,C))),Uu2),Uua),Uub) = aa(D,fun(B,C),Uu2,aa(A,D,Uua,Uub)) ).

% ATP.lambda_752
tff(fact_8930_ATP_Olambda__753,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu2: fun(C,A),Uua: fun(B,C),Uub: B] : aa(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_qd(fun(C,A),fun(fun(B,C),fun(B,A)),Uu2),Uua),Uub) = aa(C,A,Uu2,aa(B,C,Uua,Uub)) ).

% ATP.lambda_753
tff(fact_8931_ATP_Olambda__754,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu2: fun(B,A),Uua: fun(C,B),Uub: C] : aa(C,A,aa(fun(C,B),fun(C,A),aTP_Lamp_ph(fun(B,A),fun(fun(C,B),fun(C,A)),Uu2),Uua),Uub) = aa(B,A,Uu2,aa(C,B,Uua,Uub)) ).

% ATP.lambda_754
tff(fact_8932_ATP_Olambda__755,axiom,
    ! [A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [Uu2: fun(A,$o),Uua: fun(nat,A),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_aij(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),Uu2),Uua),Uub)
        <=> aa(A,$o,Uu2,aa(nat,A,Uua,Uub)) ) ) ).

% ATP.lambda_755
tff(fact_8933_ATP_Olambda__756,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: fun(A,B),Uua: fun(nat,A),Uub: nat] : aa(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_ail(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu2),Uua),Uub) = aa(A,B,Uu2,aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_756
tff(fact_8934_ATP_Olambda__757,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [Uu2: fun(A,$o),Uua: fun(nat,A),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_aey(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),Uu2),Uua),Uub)
        <=> aa(A,$o,Uu2,aa(nat,A,Uua,Uub)) ) ) ).

% ATP.lambda_757
tff(fact_8935_ATP_Olambda__758,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: fun(A,B),Uua: fun(nat,A),Uub: nat] : aa(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_abe(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu2),Uua),Uub) = aa(A,B,Uu2,aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_758
tff(fact_8936_ATP_Olambda__759,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: fun(nat,A),Uub: nat] : aa(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_acq(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu2),Uua),Uub) = aa(A,B,Uu2,aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_759
tff(fact_8937_ATP_Olambda__760,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ua(fun(A,B),fun(fun(C,A),fun(C,B)),Uu2),Uua),Uub) = aa(A,B,Uu2,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_760
tff(fact_8938_ATP_Olambda__761,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_acp(fun(A,B),fun(fun(C,A),fun(C,B)),Uu2),Uua),Uub) = aa(A,B,Uu2,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_761
tff(fact_8939_ATP_Olambda__762,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_aco(fun(A,B),fun(fun(C,A),fun(C,B)),Uu2),Uua),Uub) = aa(A,B,Uu2,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_762
tff(fact_8940_ATP_Olambda__763,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: fun(A,B),Uua: fun(nat,A),Uub: nat] : aa(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_abf(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu2),Uua),Uub) = aa(A,B,Uu2,aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_763
tff(fact_8941_ATP_Olambda__764,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_we(fun(A,B),fun(fun(C,A),fun(C,B)),Uu2),Uua),Uub) = aa(A,B,Uu2,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_764
tff(fact_8942_ATP_Olambda__765,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_rq(fun(A,A),fun(fun(A,A),fun(A,A)),Uu2),Uua),Uub) = aa(A,A,Uu2,aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_765
tff(fact_8943_ATP_Olambda__766,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_wc(fun(A,B),fun(fun(C,A),fun(C,B)),Uu2),Uua),Uub) = aa(A,B,Uu2,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_766
tff(fact_8944_ATP_Olambda__767,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(A,$o),Uua: fun(B,A),Uub: B] :
      ( aa(B,$o,aa(fun(B,A),fun(B,$o),aTP_Lamp_aff(fun(A,$o),fun(fun(B,A),fun(B,$o)),Uu2),Uua),Uub)
    <=> aa(A,$o,Uu2,aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_767
tff(fact_8945_ATP_Olambda__768,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu2: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_acu(fun(A,B),fun(fun(C,A),fun(C,B)),Uu2),Uua),Uub) = aa(A,B,Uu2,aa(C,A,Uua,Uub)) ).

% ATP.lambda_768
tff(fact_8946_ATP_Olambda__769,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & topological_t2_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: fun(C,A),Uua: fun(A,B),Uub: C] : aa(C,B,aa(fun(A,B),fun(C,B),aTP_Lamp_wb(fun(C,A),fun(fun(A,B),fun(C,B)),Uu2),Uua),Uub) = aa(A,B,Uua,aa(C,A,Uu2,Uub)) ) ).

% ATP.lambda_769
tff(fact_8947_ATP_Olambda__770,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu2: fun(B,A),Uua: fun(A,C),Uub: B] : aa(B,C,aa(fun(A,C),fun(B,C),aTP_Lamp_qe(fun(B,A),fun(fun(A,C),fun(B,C)),Uu2),Uua),Uub) = aa(A,C,Uua,aa(B,A,Uu2,Uub)) ).

% ATP.lambda_770
tff(fact_8948_ATP_Olambda__771,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Uu2: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ti(fun(A,B),fun(fun(B,C),fun(A,C)),Uu2),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_771
tff(fact_8949_ATP_Olambda__772,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu2: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_vw(fun(A,B),fun(fun(B,C),fun(A,C)),Uu2),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_772
tff(fact_8950_ATP_Olambda__773,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu2: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ags(fun(A,B),fun(fun(B,C),fun(A,C)),Uu2),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_773
tff(fact_8951_ATP_Olambda__774,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_rp(fun(A,A),fun(fun(A,A),fun(A,A)),Uu2),Uua),Uub) = aa(A,A,Uua,aa(A,A,Uu2,Uub)) ) ).

% ATP.lambda_774
tff(fact_8952_ATP_Olambda__775,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu2: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ado(fun(A,B),fun(fun(B,C),fun(A,C)),Uu2),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_775
tff(fact_8953_ATP_Olambda__776,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topological_t2_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu2: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_vu(fun(A,B),fun(fun(B,C),fun(A,C)),Uu2),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_776
tff(fact_8954_ATP_Olambda__777,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,B),Uua: fun(B,$o),Uub: A] :
      ( aa(A,$o,aa(fun(B,$o),fun(A,$o),aTP_Lamp_afd(fun(A,B),fun(fun(B,$o),fun(A,$o)),Uu2),Uua),Uub)
    <=> aa(B,$o,Uua,aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_777
tff(fact_8955_ATP_Olambda__778,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu2: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_jq(fun(A,B),fun(fun(B,C),fun(A,C)),Uu2),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_778
tff(fact_8956_ATP_Olambda__779,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu2: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_iu(fun(A,B),fun(fun(B,C),fun(A,C)),Uu2),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_779
tff(fact_8957_ATP_Olambda__780,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( semiring_1(C)
     => ! [Uu2: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ajm(fun(A,B),fun(fun(B,C),fun(A,C)),Uu2),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_780
tff(fact_8958_ATP_Olambda__781,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu2: fun(A,C),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_ald(fun(A,C),fun(A,fun(B,C)),Uu2),Uua),Uub) = aa(A,C,Uu2,Uua) ).

% ATP.lambda_781
tff(fact_8959_ATP_Olambda__782,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu2: fun(A,B),Uua: A,Uub: C] : aa(C,B,aa(A,fun(C,B),aTP_Lamp_qy(fun(A,B),fun(A,fun(C,B)),Uu2),Uua),Uub) = aa(A,B,Uu2,Uua) ).

% ATP.lambda_782
tff(fact_8960_ATP_Olambda__783,axiom,
    ! [A: $tType,B: $tType,Uu2: B,Uua: fun(B,heap_Time_Heap(A)),Uub: product_unit] : aa(product_unit,heap_Time_Heap(A),aa(fun(B,heap_Time_Heap(A)),fun(product_unit,heap_Time_Heap(A)),aTP_Lamp_ai(B,fun(fun(B,heap_Time_Heap(A)),fun(product_unit,heap_Time_Heap(A))),Uu2),Uua),Uub) = aa(B,heap_Time_Heap(A),Uua,Uu2) ).

% ATP.lambda_783
tff(fact_8961_ATP_Olambda__784,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu2: fun(A,B),Uua: fun(nat,B),Uub: A] : aa(A,B,aa(fun(nat,B),fun(A,B),aTP_Lamp_aah(fun(A,B),fun(fun(nat,B),fun(A,B)),Uu2),Uua),Uub) = suminf(B,aa(A,fun(nat,B),aa(fun(nat,B),fun(A,fun(nat,B)),aTP_Lamp_aag(fun(A,B),fun(fun(nat,B),fun(A,fun(nat,B))),Uu2),Uua),Uub)) ) ).

% ATP.lambda_784
tff(fact_8962_ATP_Olambda__785,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(nat,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_vo(fun(nat,A),fun(A,fun(A,A)),Uu2),Uua),Uub) = suminf(A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_vn(fun(nat,A),fun(A,fun(A,fun(nat,A))),Uu2),Uua),Uub)) ) ).

% ATP.lambda_785
tff(fact_8963_ATP_Olambda__786,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,real,aa(A,fun(nat,real),aTP_Lamp_bu(fun(nat,A),fun(A,fun(nat,real)),Uu2),Uua),Uub) = real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uub)),aa(nat,A,power_power(A,Uua),Uub))) ) ).

% ATP.lambda_786
tff(fact_8964_ATP_Olambda__787,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,real,aa(fun(A,B),fun(A,real),aTP_Lamp_kj(fun(A,B),fun(fun(A,B),fun(A,real)),Uu2),Uua),Uub) = real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub))) ) ).

% ATP.lambda_787
tff(fact_8965_ATP_Olambda__788,axiom,
    ! [Uu2: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_ach(fun(nat,real),fun(real,fun(nat,real)),Uu2),Uua),Uub) = cos(real,aa(real,real,minus_minus(real,aa(nat,real,Uu2,Uub)),Uua)) ).

% ATP.lambda_788
tff(fact_8966_ATP_Olambda__789,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(A) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_agr(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> ( aa(A,B,Uu2,Uub) != Uua ) ) ) ).

% ATP.lambda_789
tff(fact_8967_ATP_Olambda__790,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_afg(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> ( aa(A,B,Uu2,Uub) != Uua ) ) ) ).

% ATP.lambda_790
tff(fact_8968_ATP_Olambda__791,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_agq(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> ( aa(A,B,Uu2,Uub) != Uua ) ) ) ).

% ATP.lambda_791
tff(fact_8969_ATP_Olambda__792,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t1_space(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_afa(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> ( aa(A,B,Uu2,Uub) != Uua ) ) ) ).

% ATP.lambda_792
tff(fact_8970_ATP_Olambda__793,axiom,
    ! [C: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: A,Uua: fun(C,A),Uub: C] :
          ( aa(C,$o,aa(fun(C,A),fun(C,$o),aTP_Lamp_agt(A,fun(fun(C,A),fun(C,$o)),Uu2),Uua),Uub)
        <=> ( aa(C,A,Uua,Uub) != Uu2 ) ) ) ).

% ATP.lambda_793
tff(fact_8971_ATP_Olambda__794,axiom,
    ! [B: $tType,A: $tType,Uu2: set(B),Uua: fun(A,fun(B,$o)),Uub: A] : aa(A,nat,aa(fun(A,fun(B,$o)),fun(A,nat),aTP_Lamp_ajd(set(B),fun(fun(A,fun(B,$o)),fun(A,nat)),Uu2),Uua),Uub) = finite_card(B,collect(B,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_mv(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu2),Uua),Uub))) ).

% ATP.lambda_794
tff(fact_8972_ATP_Olambda__795,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A) )
     => ! [Uu2: fun(B,real),Uua: fun(real,A),Uub: B] : aa(B,real,aa(fun(real,A),fun(B,real),aTP_Lamp_vl(fun(B,real),fun(fun(real,A),fun(B,real)),Uu2),Uua),Uub) = aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(A,aa(real,A,Uua,aa(B,real,Uu2,Uub)))) ) ).

% ATP.lambda_795
tff(fact_8973_ATP_Olambda__796,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu2: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_aht(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),Uu2),Uua),Uub)
        <=> ! [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uub),N2)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu2),set_or7035219750837199246ssThan(nat,Uub,N2)))),aa(nat,real,Uua,Uub)) ) ) ) ).

% ATP.lambda_796
tff(fact_8974_ATP_Olambda__797,axiom,
    ! [A: $tType] :
      ( ( real_V8037385150606011577_space(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu2: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_aii(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),Uu2),Uua),Uub)
        <=> ! [A7: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uub),A7)
             => ! [B7: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A7),B7)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu2),set_or3652927894154168847AtMost(nat,A7,B7)))),aa(nat,real,Uua,A7)) ) ) ) ) ).

% ATP.lambda_797
tff(fact_8975_ATP_Olambda__798,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,Uu2: fun(B,fun(C,fun(D,A))),Uua: D,Uub: B,Uuc: C] : aa(C,A,aa(B,fun(C,A),aa(D,fun(B,fun(C,A)),aTP_Lamp_kz(fun(B,fun(C,fun(D,A))),fun(D,fun(B,fun(C,A))),Uu2),Uua),Uub),Uuc) = aa(D,A,aa(C,fun(D,A),aa(B,fun(C,fun(D,A)),Uu2,Uub),Uuc),Uua) ).

% ATP.lambda_798
tff(fact_8976_ATP_Olambda__799,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: A,Uub: nat,Uuc: nat] :
          aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_gy(A,fun(A,fun(nat,fun(nat,A))),Uu2),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,Uu2),Uuc))),aa(nat,A,power_power(A,Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))),
            zero_zero(A) ) ) ).

% ATP.lambda_799
tff(fact_8977_ATP_Olambda__800,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: A,Uub: nat,Uuc: nat] :
          aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_et(A,fun(A,fun(nat,fun(nat,A))),Uu2),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,Uu2),Uuc))),aa(nat,A,power_power(A,Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))),
            zero_zero(A) ) ) ).

% ATP.lambda_800
tff(fact_8978_ATP_Olambda__801,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: A,Uub: nat,Uuc: nat] :
          aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_ev(A,fun(A,fun(nat,fun(nat,A))),Uu2),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,Uu2),Uuc))),aa(nat,A,power_power(A,Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))),zero_zero(A)) ) ).

% ATP.lambda_801
tff(fact_8979_ATP_Olambda__802,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,B),Uua: set(A),Uub: A,Uuc: B] :
      aa(B,A,aa(A,fun(B,A),aa(set(A),fun(A,fun(B,A)),aTP_Lamp_rb(fun(A,B),fun(set(A),fun(A,fun(B,A))),Uu2),Uua),Uub),Uuc) = $ite(member(B,Uuc,aa(set(A),set(B),image(A,B,Uu2),Uua)),the_inv_into(A,B,Uua,Uu2,Uuc),Uub) ).

% ATP.lambda_802
tff(fact_8980_ATP_Olambda__803,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_gj(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu2),Uua),Uub),Uuc) = $ite(
            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu2),
            aa(nat,A,Uua,Uuc),
            $ite(Uuc = Uu2,zero_zero(A),aa(nat,A,Uub,aa(nat,nat,minus_minus(nat,Uuc),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).

% ATP.lambda_803
tff(fact_8981_ATP_Olambda__804,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_kn(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu2),Uua),Uub),Uuc) = $ite(
            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu2),
            aa(nat,A,Uua,Uuc),
            $ite(Uuc = Uu2,one_one(A),aa(nat,A,Uub,aa(nat,nat,minus_minus(nat,Uuc),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).

% ATP.lambda_804
tff(fact_8982_ATP_Olambda__805,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: A,Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_adv(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu2),Uua),Uub),Uuc) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uuc),Uu2),aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_805
tff(fact_8983_ATP_Olambda__806,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_ko(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu2),Uua),Uub),Uuc) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu2),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_806
tff(fact_8984_ATP_Olambda__807,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_gk(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu2),Uua),Uub),Uuc) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu2),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_807
tff(fact_8985_ATP_Olambda__808,axiom,
    ! [A: $tType] :
      ( ( topolo1287966508704411220up_add(A)
        & topological_t2_space(A) )
     => ! [Uu2: fun(nat,A),Uua: set(nat),Uub: fun(nat,A),Uuc: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(set(nat),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_nx(fun(nat,A),fun(set(nat),fun(fun(nat,A),fun(nat,A))),Uu2),Uua),Uub),Uuc) = $ite(member(nat,Uuc,Uua),aa(nat,A,Uub,Uuc),aa(nat,A,Uu2,Uuc)) ) ).

% ATP.lambda_808
tff(fact_8986_ATP_Olambda__809,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: A,Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_nz(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_809
tff(fact_8987_ATP_Olambda__810,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu2: A,Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_ny(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_810
tff(fact_8988_ATP_Olambda__811,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: A,Uua: fun(A,B),Uub: B,Uuc: A] :
          aa(A,B,aa(B,fun(A,B),aa(fun(A,B),fun(B,fun(A,B)),aTP_Lamp_ajo(A,fun(fun(A,B),fun(B,fun(A,B))),Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa(A,B,Uua,Uuc),Uub) ) ).

% ATP.lambda_811
tff(fact_8989_ATP_Olambda__812,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: fun(A,B),Uua: fun(A,$o),Uub: fun(A,B),Uuc: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,$o),fun(fun(A,B),fun(A,B)),aTP_Lamp_wh(fun(A,B),fun(fun(A,$o),fun(fun(A,B),fun(A,B))),Uu2),Uua),Uub),Uuc) = $ite(aa(A,$o,Uua,Uuc),aa(A,B,Uu2,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_812
tff(fact_8990_ATP_Olambda__813,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(A,fun(A,$o)),Uua: fun(B,A),Uub: B,Uuc: B] :
      ( aa(B,$o,aa(B,fun(B,$o),aa(fun(B,A),fun(B,fun(B,$o)),aTP_Lamp_alg(fun(A,fun(A,$o)),fun(fun(B,A),fun(B,fun(B,$o))),Uu2),Uua),Uub),Uuc)
    <=> aa(A,$o,aa(A,fun(A,$o),Uu2,aa(B,A,Uua,Uub)),aa(B,A,Uua,Uuc)) ) ).

% ATP.lambda_813
tff(fact_8991_ATP_Olambda__814,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu2: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_ir(A,fun(A,fun(nat,fun(nat,A))),Uu2),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_iq(A,fun(A,fun(nat,fun(nat,A))),Uu2),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_814
tff(fact_8992_ATP_Olambda__815,axiom,
    ! [Uu2: $o,Uua: $o,Uub: code_integer,Uuc: $o] : aa($o,char,aa(code_integer,fun($o,char),aa($o,fun(code_integer,fun($o,char)),aTP_Lamp_akn($o,fun($o,fun(code_integer,fun($o,char))),(Uu2)),(Uua)),Uub),(Uuc)) = aa(product_prod(code_integer,$o),char,product_case_prod(code_integer,$o,char,aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aTP_Lamp_akm($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),(Uu2)),(Uua)),(Uuc))),code_bit_cut_integer(Uub)) ).

% ATP.lambda_815
tff(fact_8993_ATP_Olambda__816,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( semiring_0(A)
     => ! [Uu2: 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_ho(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),Uu2),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_hn(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu2),Uua),Uuc)),Uub) ) ).

% ATP.lambda_816
tff(fact_8994_ATP_Olambda__817,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu2: nat,Uua: A,Uub: A,Uuc: nat] :
          aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_go(nat,fun(A,fun(A,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              $ite(
                Uuc = zero_zero(nat),
                aa(A,A,uminus_uminus(A),Uub),
                $ite(Uuc = Uu2,one_one(A),zero_zero(A)) )),
            aa(nat,A,power_power(A,Uua),Uuc)) ) ).

% ATP.lambda_817
tff(fact_8995_ATP_Olambda__818,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu2: set(A),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$o)),Uuc: B] : aa(B,C,aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_nc(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),Uu2),Uua),Uub),Uuc) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_nb(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uuc),collect(A,aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_my(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Uu2),Uub),Uuc))) ) ).

% ATP.lambda_818
tff(fact_8996_ATP_Olambda__819,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu2: set(A),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$o)),Uuc: B] : aa(B,C,aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_mz(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),Uu2),Uua),Uub),Uuc) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(B,fun(A,C),aTP_Lamp_mx(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uuc)),collect(A,aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_my(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Uu2),Uub),Uuc))) ) ).

% ATP.lambda_819
tff(fact_8997_ATP_Olambda__820,axiom,
    ! [Uu2: fun(nat,nat),Uua: fun(nat,nat),Uub: nat,Uuc: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aa(fun(nat,nat),fun(nat,fun(nat,nat)),aTP_Lamp_ge(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),Uu2),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aa(fun(nat,nat),fun(nat,fun(nat,nat)),aTP_Lamp_gd(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),Uu2),Uua),Uuc)),set_ord_atMost(nat,Uuc))),aa(nat,nat,power_power(nat,Uub),Uuc)) ).

% ATP.lambda_820
tff(fact_8998_ATP_Olambda__821,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu2: fun(nat,A),Uua: fun(nat,A),Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(fun(nat,A),fun(A,fun(nat,A)),aTP_Lamp_ga(fun(nat,A),fun(fun(nat,A),fun(A,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aTP_Lamp_fz(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu2),Uua),Uuc)),set_ord_atMost(nat,Uuc))),aa(nat,A,power_power(A,Uub),Uuc)) ) ).

% ATP.lambda_821
tff(fact_8999_ATP_Olambda__822,axiom,
    ! [Uu2: fun(nat,fun(real,real)),Uua: nat,Uub: real,Uuc: nat] : aa(nat,real,aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_ta(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uu2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,Uua)),Uuc)),zero_zero(real))),semiring_char_0_fact(real,Uuc))),aa(nat,real,power_power(real,Uub),Uuc)) ).

% ATP.lambda_822
tff(fact_9000_ATP_Olambda__823,axiom,
    ! [Uu2: fun(nat,fun(real,real)),Uua: nat,Uub: real,Uuc: nat] : aa(nat,real,aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_sy(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uu2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uuc)),zero_zero(real))),semiring_char_0_fact(real,Uuc))),aa(nat,real,power_power(real,Uub),Uuc)) ).

% ATP.lambda_823
tff(fact_9001_ATP_Olambda__824,axiom,
    ! [Uu2: fun(nat,fun(real,real)),Uua: real,Uub: real,Uuc: nat] : aa(nat,real,aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_sx(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uu2,Uuc),Uua)),semiring_char_0_fact(real,Uuc))),aa(nat,real,power_power(real,aa(real,real,minus_minus(real,Uub),Uua)),Uuc)) ).

% ATP.lambda_824
tff(fact_9002_ATP_Olambda__825,axiom,
    ! [Uu2: fun(nat,fun(real,real)),Uua: real,Uub: real,Uuc: nat] : aa(nat,real,aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_sw(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uu2,Uuc),Uub)),semiring_char_0_fact(real,Uuc))),aa(nat,real,power_power(real,aa(real,real,minus_minus(real,Uua),Uub)),Uuc)) ).

% ATP.lambda_825
tff(fact_9003_ATP_Olambda__826,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu2: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_ic(A,fun(A,fun(nat,fun(nat,A))),Uu2),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,aa(A,fun(A,A),plus_plus(A),Uu2),Uua)),aa(nat,nat,minus_minus(nat,Uub),Uuc))),aa(nat,A,power_power(A,Uu2),Uuc))),aa(nat,A,power_power(A,Uu2),Uub)) ) ).

% ATP.lambda_826
tff(fact_9004_ATP_Olambda__827,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu2: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_iq(A,fun(A,fun(nat,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu2)),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_827
tff(fact_9005_ATP_Olambda__828,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: A,Uub: fun(nat,A),Uuc: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(A,fun(fun(nat,A),fun(nat,A)),aTP_Lamp_ace(fun(A,A),fun(A,fun(fun(nat,A),fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,Uu2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),aa(nat,A,Uub,Uuc)))),aa(A,A,Uu2,Uua))),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_828
tff(fact_9006_ATP_Olambda__829,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(A,option(B)),Uua: fun(product_prod(A,B),$o),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(product_prod(A,B),$o),fun(A,fun(B,$o)),aTP_Lamp_lt(fun(A,option(B)),fun(fun(product_prod(A,B),$o),fun(A,fun(B,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( ( aa(A,option(B),Uu2,Uub) = some(B,Uuc) )
        & aa(product_prod(A,B),$o,Uua,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uub),Uuc)) ) ) ).

% ATP.lambda_829
tff(fact_9007_ATP_Olambda__830,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(B)
     => ! [Uu2: fun(A,B),Uua: set(B),Uub: B,Uuc: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aa(set(B),fun(B,fun(A,$o)),aTP_Lamp_afi(fun(A,B),fun(set(B),fun(B,fun(A,$o))),Uu2),Uua),Uub),Uuc)
        <=> ( member(B,aa(A,B,Uu2,Uuc),Uua)
            & ( aa(A,B,Uu2,Uuc) != Uub ) ) ) ) ).

% ATP.lambda_830
tff(fact_9008_ATP_Olambda__831,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: set(B),Uuc: A] :
          ( aa(A,$o,aa(set(B),fun(A,$o),aa(B,fun(set(B),fun(A,$o)),aTP_Lamp_age(fun(A,B),fun(B,fun(set(B),fun(A,$o))),Uu2),Uua),Uub),Uuc)
        <=> ( member(B,aa(A,B,Uu2,Uuc),Uub)
            & ( aa(A,B,Uu2,Uuc) != Uua ) ) ) ) ).

% ATP.lambda_831
tff(fact_9009_ATP_Olambda__832,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: real,Uuc: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_ahu(fun(A,B),fun(B,fun(real,fun(A,$o))),Uu2),Uua),Uub),Uuc)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(B,aa(A,B,Uu2,Uuc),Uua)),Uub) ) ) ).

% ATP.lambda_832
tff(fact_9010_ATP_Olambda__833,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: real,Uuc: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_ahi(fun(A,B),fun(B,fun(real,fun(A,$o))),Uu2),Uua),Uub),Uuc)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,Uu2,Uuc),Uua)),Uub) ) ) ).

% ATP.lambda_833
tff(fact_9011_ATP_Olambda__834,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu2: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_gh(A,fun(A,fun(nat,fun(nat,A))),Uu2),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,Uu2,Uuc))),comm_s3205402744901411588hammer(A,Uua,aa(nat,nat,minus_minus(nat,Uub),Uuc))) ) ).

% ATP.lambda_834
tff(fact_9012_ATP_Olambda__835,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_ft(nat,fun(nat,fun(nat,fun(nat,nat))),Uu2),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,Uu2),Uuc))),aa(nat,nat,power_power(nat,Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))) ).

% ATP.lambda_835
tff(fact_9013_ATP_Olambda__836,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu2: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_gf(A,fun(A,fun(nat,fun(nat,A))),Uu2),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,Uu2),Uuc))),aa(nat,A,power_power(A,Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))) ) ).

% ATP.lambda_836
tff(fact_9014_ATP_Olambda__837,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_gn(A,fun(A,fun(nat,fun(nat,A))),Uu2),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,Uu2),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_837
tff(fact_9015_ATP_Olambda__838,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu2: A,Uua: nat,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_if(A,fun(nat,fun(A,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,Uub),aa(nat,nat,minus_minus(nat,Uua),aa(nat,nat,suc,Uuc)))),aa(nat,A,power_power(A,Uu2),Uuc)) ) ).

% ATP.lambda_838
tff(fact_9016_ATP_Olambda__839,axiom,
    ! [A: $tType,Uu2: $o,Uua: A,Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(A,fun(A,fun(A,$o)),aTP_Lamp_du($o,fun(A,fun(A,fun(A,$o))),(Uu2)),Uua),Uub),Uuc)
    <=> ( ( (Uu2)
         => ( Uuc = Uua ) )
        & ( ~ (Uu2)
         => ( Uuc = Uub ) ) ) ) ).

% ATP.lambda_839
tff(fact_9017_ATP_Olambda__840,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(B,A),Uua: set(B),Uub: fun(A,$o),Uuc: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(set(B),fun(fun(A,$o),fun(A,$o)),aTP_Lamp_pi(fun(B,A),fun(set(B),fun(fun(A,$o),fun(A,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,aa(set(B),set(A),image(B,A,Uu2),Uua))
        & aa(A,$o,Uub,Uuc) ) ) ).

% ATP.lambda_840
tff(fact_9018_ATP_Olambda__841,axiom,
    ! [A: $tType,B: $tType,Uu2: set(B),Uua: fun(A,fun(B,$o)),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_mv(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( member(B,Uuc,Uu2)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uub),Uuc) ) ) ).

% ATP.lambda_841
tff(fact_9019_ATP_Olambda__842,axiom,
    ! [A: $tType,B: $tType,Uu2: set(A),Uua: fun(A,fun(B,$o)),Uub: B,Uuc: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_my(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,Uu2)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uuc),Uub) ) ) ).

% ATP.lambda_842
tff(fact_9020_ATP_Olambda__843,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu2: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_id(A,fun(A,fun(nat,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,Uu2),Uuc)),aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu2),Uua)),aa(nat,nat,minus_minus(nat,Uub),Uuc))),aa(nat,A,power_power(A,Uu2),aa(nat,nat,minus_minus(nat,Uub),Uuc)))) ) ).

% ATP.lambda_843
tff(fact_9021_ATP_Olambda__844,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: set(A),Uua: fun(A,A),Uub: fun(A,A),Uuc: A] :
          ( aa(A,$o,aa(fun(A,A),fun(A,$o),aa(fun(A,A),fun(fun(A,A),fun(A,$o)),aTP_Lamp_agf(set(A),fun(fun(A,A),fun(fun(A,A),fun(A,$o))),Uu2),Uua),Uub),Uuc)
        <=> ( member(A,Uuc,Uu2)
           => ( aa(A,A,Uua,Uuc) = aa(A,A,Uub,Uuc) ) ) ) ) ).

% ATP.lambda_844
tff(fact_9022_ATP_Olambda__845,axiom,
    ! [B: $tType,A: $tType,Uu2: set(A),Uua: fun(A,B),Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,B),fun(A,fun(A,$o)),aTP_Lamp_pg(set(A),fun(fun(A,B),fun(A,fun(A,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,Uu2)
        & ( aa(A,B,Uua,Uuc) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_845
tff(fact_9023_ATP_Olambda__846,axiom,
    ! [B: $tType,A: $tType,Uu2: set(B),Uua: fun(B,A),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(B,A),fun(A,fun(B,$o)),aTP_Lamp_qj(set(B),fun(fun(B,A),fun(A,fun(B,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( member(B,Uuc,Uu2)
        & ( aa(B,A,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_846
tff(fact_9024_ATP_Olambda__847,axiom,
    ! [A: $tType,C: $tType,Uu2: set(A),Uua: fun(A,C),Uub: C,Uuc: A] :
      ( aa(A,$o,aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aTP_Lamp_pm(set(A),fun(fun(A,C),fun(C,fun(A,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,Uu2)
        & ( aa(A,C,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_847
tff(fact_9025_ATP_Olambda__848,axiom,
    ! [A: $tType,B: $tType,Uu2: set(A),Uua: fun(A,B),Uub: B,Uuc: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_pp(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,Uu2)
        & ( aa(A,B,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_848
tff(fact_9026_ATP_Olambda__849,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(A,B),Uua: set(A),Uub: B,Uuc: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aa(set(A),fun(B,fun(A,$o)),aTP_Lamp_rk(fun(A,B),fun(set(A),fun(B,fun(A,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,Uua)
        & ( aa(A,B,Uu2,Uuc) = Uub ) ) ) ).

% ATP.lambda_849
tff(fact_9027_ATP_Olambda__850,axiom,
    ! [B: $tType,A: $tType,Uu2: set(A),Uua: fun(B,A),Uub: set(B),Uuc: B] :
      ( aa(B,$o,aa(set(B),fun(B,$o),aa(fun(B,A),fun(set(B),fun(B,$o)),aTP_Lamp_ra(set(A),fun(fun(B,A),fun(set(B),fun(B,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( member(B,Uuc,Uub)
        & member(A,aa(B,A,Uua,Uuc),Uu2) ) ) ).

% ATP.lambda_850
tff(fact_9028_ATP_Olambda__851,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu2: A,Uua: nat,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_ie(A,fun(nat,fun(A,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,Uu2),Uuc)),aa(nat,A,power_power(A,Uub),aa(nat,nat,minus_minus(nat,Uua),Uuc))) ) ).

% ATP.lambda_851
tff(fact_9029_ATP_Olambda__852,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_fr(nat,fun(nat,fun(nat,fun(nat,nat))),Uu2),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(Uu2),Uuc)),aa(nat,nat,binomial(Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))) ).

% ATP.lambda_852
tff(fact_9030_ATP_Olambda__853,axiom,
    ! [Uu2: int,Uua: int,Uub: int,Uuc: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_lz(int,fun(int,fun(int,fun(int,$o))),Uu2),Uua),Uub),Uuc)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu2),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub)) ) ).

% ATP.lambda_853
tff(fact_9031_ATP_Olambda__854,axiom,
    ! [Uu2: int,Uua: int,Uub: int,Uuc: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_lx(int,fun(int,fun(int,fun(int,$o))),Uu2),Uua),Uub),Uuc)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu2),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub)) ) ).

% ATP.lambda_854
tff(fact_9032_ATP_Olambda__855,axiom,
    ! [A: $tType,B: $tType,Uu2: A,Uua: B,Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(B,fun(A,fun(B,$o)),aTP_Lamp_ld(A,fun(B,fun(A,fun(B,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( ( Uu2 = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_855
tff(fact_9033_ATP_Olambda__856,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,A),Uua: set(B),Uub: fun(A,$o),Uuc: B] :
      ( aa(B,$o,aa(fun(A,$o),fun(B,$o),aa(set(B),fun(fun(A,$o),fun(B,$o)),aTP_Lamp_pj(fun(B,A),fun(set(B),fun(fun(A,$o),fun(B,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( member(B,Uuc,Uua)
        & aa(A,$o,Uub,aa(B,A,Uu2,Uuc)) ) ) ).

% ATP.lambda_856
tff(fact_9034_ATP_Olambda__857,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: set(A),Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_ni(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),Uu2),Uua),Uub),Uuc)
        <=> ( member(A,Uuc,Uu2)
            & ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uuc)),aa(A,B,Uub,Uuc)) != one_one(B) ) ) ) ) ).

% ATP.lambda_857
tff(fact_9035_ATP_Olambda__858,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu2: set(A),Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_ng(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),Uu2),Uua),Uub),Uuc)
        <=> ( member(A,Uuc,Uu2)
            & ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uua,Uuc)),aa(A,B,Uub,Uuc)) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_858
tff(fact_9036_ATP_Olambda__859,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: A] : aa(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_acn(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),Uu2),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Uuc),Uub)))),aa(B,B,minus_minus(B,aa(A,B,Uu2,Uuc)),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,aa(A,A,minus_minus(A,Uuc),Uub))))) ) ).

% ATP.lambda_859
tff(fact_9037_ATP_Olambda__860,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu2: 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_ca(fun(nat,A),fun(nat,fun(A,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uuc),Uua))),aa(nat,A,power_power(A,Uub),Uuc)) ) ).

% ATP.lambda_860
tff(fact_9038_ATP_Olambda__861,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu2: fun(B,A),Uua: A,Uub: list(B),Uuc: nat] : aa(nat,A,aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_px(fun(B,A),fun(A,fun(list(B),fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu2,aa(nat,B,nth(B,Uub),Uuc))),aa(nat,A,power_power(A,Uua),Uuc)) ) ).

% ATP.lambda_861
tff(fact_9039_ATP_Olambda__862,axiom,
    ! [B: $tType,A: $tType,Uu2: set(A),Uua: fun(B,$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_amf(set(A),fun(fun(B,$o),fun(fun(A,B),fun(A,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( aa(B,$o,Uua,aa(A,B,Uub,Uuc))
        & member(A,Uuc,Uu2) ) ) ).

% ATP.lambda_862
tff(fact_9040_ATP_Olambda__863,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(nat,A),Uua: A,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_vn(fun(nat,A),fun(A,fun(A,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uuc)),aa(A,A,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,aa(A,fun(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_863
tff(fact_9041_ATP_Olambda__864,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B)
        & topological_t2_space(A) )
     => ! [Uu2: 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_aag(fun(A,B),fun(fun(nat,B),fun(A,fun(nat,B))),Uu2),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,Uu2,Uub)),Uuc)) ) ).

% ATP.lambda_864
tff(fact_9042_ATP_Olambda__865,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: A,Uub: fun(A,real),Uuc: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_vc(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,aa(A,real,Uu2,Uua)))),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ).

% ATP.lambda_865
tff(fact_9043_ATP_Olambda__866,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_ji(fun(nat,A),fun(A,fun(nat,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uuc)),aa(nat,A,power_power(A,Uua),aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,Uuc),Uub)),one_one(nat)))) ) ).

% ATP.lambda_866
tff(fact_9044_ATP_Olambda__867,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: set(B),Uuc: A] :
          ( aa(A,$o,aa(set(B),fun(A,$o),aa(B,fun(set(B),fun(A,$o)),aTP_Lamp_ahm(fun(A,B),fun(B,fun(set(B),fun(A,$o))),Uu2),Uua),Uub),Uuc)
        <=> member(B,aa(A,B,Uu2,Uuc),aa(set(B),set(B),minus_minus(set(B),Uub),aa(set(B),set(B),insert(B,Uua),bot_bot(set(B))))) ) ) ).

% ATP.lambda_867
tff(fact_9045_ATP_Olambda__868,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [Uu2: fun(B,A),Uua: A,Uub: B,Uuc: A] : aa(A,A,aa(B,fun(A,A),aa(A,fun(B,fun(A,A)),aTP_Lamp_ajw(fun(B,A),fun(A,fun(B,fun(A,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,Uu2,Uub)),aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uuc)) ) ).

% ATP.lambda_868
tff(fact_9046_ATP_Olambda__869,axiom,
    ! [Uu2: fun(nat,nat),Uua: fun(nat,nat),Uub: nat,Uuc: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aa(fun(nat,nat),fun(nat,fun(nat,nat)),aTP_Lamp_gd(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),Uu2),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,Uu2,Uuc)),aa(nat,nat,Uua,aa(nat,nat,minus_minus(nat,Uub),Uuc))) ).

% ATP.lambda_869
tff(fact_9047_ATP_Olambda__870,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu2: fun(nat,A),Uua: fun(nat,A),Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aTP_Lamp_fv(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uuc)),aa(nat,A,Uua,aa(nat,nat,minus_minus(nat,Uub),Uuc))) ) ).

% ATP.lambda_870
tff(fact_9048_ATP_Olambda__871,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu2: fun(nat,A),Uua: fun(nat,A),Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aTP_Lamp_fz(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uuc)),aa(nat,A,Uua,aa(nat,nat,minus_minus(nat,Uub),Uuc))) ) ).

% ATP.lambda_871
tff(fact_9049_ATP_Olambda__872,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( semiring_0(A)
     => ! [Uu2: 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_hn(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu2,Uub)),aa(C,A,Uua,Uuc)) ) ).

% ATP.lambda_872
tff(fact_9050_ATP_Olambda__873,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu2: set(B),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$o)),Uuc: A] : aa(A,C,aa(fun(A,fun(B,$o)),fun(A,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C)),aTP_Lamp_na(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),Uu2),Uua),Uub),Uuc) = groups7121269368397514597t_prod(B,C,aa(A,fun(B,C),Uua,Uuc),collect(B,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_mv(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu2),Uub),Uuc))) ) ).

% ATP.lambda_873
tff(fact_9051_ATP_Olambda__874,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu2: set(B),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$o)),Uuc: A] : aa(A,C,aa(fun(A,fun(B,$o)),fun(A,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C)),aTP_Lamp_mw(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),Uu2),Uua),Uub),Uuc) = aa(set(B),C,groups7311177749621191930dd_sum(B,C,aa(A,fun(B,C),Uua,Uuc)),collect(B,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_mv(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu2),Uub),Uuc))) ) ).

% ATP.lambda_874
tff(fact_9052_ATP_Olambda__875,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: A,Uub: fun(A,real),Uuc: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_vg(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,inverse_inverse(real),aa(nat,real,power_power(real,cos(real,aa(A,real,Uu2,Uua))),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% ATP.lambda_875
tff(fact_9053_ATP_Olambda__876,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: 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_ve(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu2),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,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,power_power(real,aa(A,real,Uu2,Uub)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ).

% ATP.lambda_876
tff(fact_9054_ATP_Olambda__877,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: 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_ud(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu2),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,Uu2,Uub))) ) ).

% ATP.lambda_877
tff(fact_9055_ATP_Olambda__878,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: 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_uf(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uuc)),cos(real,aa(A,real,Uu2,Uub))) ) ).

% ATP.lambda_878
tff(fact_9056_ATP_Olambda__879,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: A,Uub: fun(A,real),Uuc: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_us(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,inverse_inverse(real),aa(A,real,Uu2,Uua))) ) ).

% ATP.lambda_879
tff(fact_9057_ATP_Olambda__880,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: A,Uub: fun(A,real),Uuc: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_th(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,power_power(real,aa(A,real,Uu2,Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) ).

% ATP.lambda_880
tff(fact_9058_ATP_Olambda__881,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: 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_uo(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu2),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,Uu2,Uub)))) ) ).

% ATP.lambda_881
tff(fact_9059_ATP_Olambda__882,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: A,Uub: fun(A,real),Uuc: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_vk(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,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,Uu2,Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))) ) ).

% ATP.lambda_882
tff(fact_9060_ATP_Olambda__883,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: A] : aa(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_acz(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(B,B,minus_minus(B,aa(A,B,Uu2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uuc))),aa(A,B,Uu2,Uub))),aa(A,B,Uua,Uuc)))),real_V7770717601297561774m_norm(A,Uuc)) ) ).

% ATP.lambda_883
tff(fact_9061_ATP_Olambda__884,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: A] : aa(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_acv(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(B,B,minus_minus(B,aa(A,B,Uu2,Uuc)),aa(A,B,Uu2,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_884
tff(fact_9062_ATP_Olambda__885,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: fun(A,$o),Uua: A,Uub: set(A),Uuc: A] :
          ( aa(A,$o,aa(set(A),fun(A,$o),aa(A,fun(set(A),fun(A,$o)),aTP_Lamp_afr(fun(A,$o),fun(A,fun(set(A),fun(A,$o))),Uu2),Uua),Uub),Uuc)
        <=> ( ( Uuc != Uua )
           => ( member(A,Uuc,Uub)
             => aa(A,$o,Uu2,Uuc) ) ) ) ) ).

% ATP.lambda_885
tff(fact_9063_ATP_Olambda__886,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Uu2: fun(A,B),Uua: fun(A,C),Uub: real,Uuc: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aa(fun(A,C),fun(real,fun(A,$o)),aTP_Lamp_ahn(fun(A,B),fun(fun(A,C),fun(real,fun(A,$o))),Uu2),Uua),Uub),Uuc)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(C,aa(A,C,Uua,Uuc))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(B,aa(A,B,Uu2,Uuc))),Uub)) ) ) ).

% ATP.lambda_886
tff(fact_9064_ATP_Olambda__887,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: filter(A),Uuc: A] : aa(A,B,aa(filter(A),fun(A,B),aa(fun(A,B),fun(filter(A),fun(A,B)),aTP_Lamp_ada(fun(A,B),fun(fun(A,B),fun(filter(A),fun(A,B))),Uu2),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Uuc),topolo3827282254853284352ce_Lim(A,A,Uub,aTP_Lamp_tw(A,A)))))),aa(B,B,minus_minus(B,aa(B,B,minus_minus(B,aa(A,B,Uu2,Uuc)),aa(A,B,Uu2,topolo3827282254853284352ce_Lim(A,A,Uub,aTP_Lamp_tw(A,A))))),aa(A,B,Uua,aa(A,A,minus_minus(A,Uuc),topolo3827282254853284352ce_Lim(A,A,Uub,aTP_Lamp_tw(A,A)))))) ) ).

% ATP.lambda_887
tff(fact_9065_ATP_Olambda__888,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: A,Uub: fun(A,B),Uuc: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(A,fun(fun(A,B),fun(A,B)),aTP_Lamp_acx(fun(A,B),fun(A,fun(fun(A,B),fun(A,B))),Uu2),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,aa(A,A,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,Uu2,aa(A,A,minus_minus(A,Uuc),Uua)))) ) ).

% ATP.lambda_888
tff(fact_9066_ATP_Olambda__889,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: A] : aa(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_acw(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),Uu2),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Uuc),Uub)))),aa(B,B,minus_minus(B,aa(B,B,minus_minus(B,aa(A,B,Uu2,Uuc)),aa(A,B,Uu2,Uub))),aa(A,B,Uua,aa(A,A,minus_minus(A,Uuc),Uub)))) ) ).

% ATP.lambda_889
tff(fact_9067_ATP_Olambda__890,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [Uu2: set(A),Uua: fun(A,B),Uub: fun(B,C),Uuc: B] : aa(B,C,aa(fun(B,C),fun(B,C),aa(fun(A,B),fun(fun(B,C),fun(B,C)),aTP_Lamp_ajn(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),Uu2),Uua),Uub),Uuc) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(nat,C,semiring_1_of_nat(C),finite_card(A,collect(A,aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_pp(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu2),Uua),Uuc))))),aa(B,C,Uub,Uuc)) ) ).

% ATP.lambda_890
tff(fact_9068_ATP_Olambda__891,axiom,
    ! [A: $tType,B: $tType,Uu2: $o,Uua: fun(A,fun(B,$o)),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_lr($o,fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),(Uu2)),Uua),Uub),Uuc)
    <=> ( (Uu2)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uub),Uuc) ) ) ).

% ATP.lambda_891
tff(fact_9069_ATP_Olambda__892,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: C] : aa(C,B,aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_po(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),Uu2),Uua),Uub),Uuc) = groups7121269368397514597t_prod(A,B,Uua,collect(A,aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aTP_Lamp_pm(set(A),fun(fun(A,C),fun(C,fun(A,$o))),Uu2),Uub),Uuc))) ) ).

% ATP.lambda_892
tff(fact_9070_ATP_Olambda__893,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu2: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: B] : aa(B,C,aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_pr(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),Uu2),Uua),Uub),Uuc) = groups7121269368397514597t_prod(A,C,Uub,collect(A,aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_pp(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu2),Uua),Uuc))) ) ).

% ATP.lambda_893
tff(fact_9071_ATP_Olambda__894,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu2: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: C] : aa(C,B,aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_pn(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),Uu2),Uua),Uub),Uuc) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,Uua),collect(A,aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aTP_Lamp_pm(set(A),fun(fun(A,C),fun(C,fun(A,$o))),Uu2),Uub),Uuc))) ) ).

% ATP.lambda_894
tff(fact_9072_ATP_Olambda__895,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu2: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: B] : aa(B,C,aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_pq(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),Uu2),Uua),Uub),Uuc) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,Uub),collect(A,aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_pp(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu2),Uua),Uuc))) ) ).

% ATP.lambda_895
tff(fact_9073_ATP_Olambda__896,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: real,Uuc: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aa(nat,fun(real,fun(A,$o)),aTP_Lamp_aeo(fun(nat,A),fun(nat,fun(real,fun(A,$o))),Uu2),Uua),Uub),Uuc)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uub),real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_bt(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uuc)),set_ord_atMost(nat,Uua)))) ) ) ).

% ATP.lambda_896
tff(fact_9074_ATP_Olambda__897,axiom,
    ! [B: $tType,A: $tType,Uu2: list(A),Uua: assn,Uub: fun(A,B),Uuc: list(B)] : aa(list(B),assn,aa(fun(A,B),fun(list(B),assn),aa(assn,fun(fun(A,B),fun(list(B),assn)),aTP_Lamp_qo(list(A),fun(assn,fun(fun(A,B),fun(list(B),assn))),Uu2),Uua),Uub),Uuc) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Uua),pure_assn(Uuc = aa(list(A),list(B),map(A,B,Uub),Uu2))) ).

% ATP.lambda_897
tff(fact_9075_ATP_Olambda__898,axiom,
    ! [B: $tType,A: $tType,Uu2: assn,Uua: fun(A,B),Uub: A,Uuc: B] : aa(B,assn,aa(A,fun(B,assn),aa(fun(A,B),fun(A,fun(B,assn)),aTP_Lamp_qn(assn,fun(fun(A,B),fun(A,fun(B,assn))),Uu2),Uua),Uub),Uuc) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Uu2),pure_assn(Uuc = aa(A,B,Uua,Uub))) ).

% ATP.lambda_898
tff(fact_9076_ATP_Olambda__899,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType,Uu2: fun(B,A),Uua: fun(C,fun(D,B)),Uub: C,Uuc: D] : aa(D,A,aa(C,fun(D,A),aa(fun(C,fun(D,B)),fun(C,fun(D,A)),aTP_Lamp_ll(fun(B,A),fun(fun(C,fun(D,B)),fun(C,fun(D,A))),Uu2),Uua),Uub),Uuc) = aa(B,A,Uu2,aa(D,B,aa(C,fun(D,B),Uua,Uub),Uuc)) ).

% ATP.lambda_899
tff(fact_9077_ATP_Olambda__900,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_op(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(nat,A,Uu2,aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),aa(nat,nat,suc,Uuc))) ) ).

% ATP.lambda_900
tff(fact_9078_ATP_Olambda__901,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_on(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(nat,A,Uu2,aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),aa(nat,nat,suc,Uuc))) ) ).

% ATP.lambda_901
tff(fact_9079_ATP_Olambda__902,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_ke(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(nat,A,Uu2,aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),Uuc)) ) ).

% ATP.lambda_902
tff(fact_9080_ATP_Olambda__903,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_iw(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(nat,A,Uu2,aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),Uuc)) ) ).

% ATP.lambda_903
tff(fact_9081_ATP_Olambda__904,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A,Uuc: list(A)] : aa(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A)))),aa(A,fun(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A))))),aa(B,fun(A,fun(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A)))))),aTP_Lamp_akg(fun(A,B),fun(B,fun(A,fun(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A))))))),Uu2),Uua),Uub),Uuc) = product_case_prod(list(A),list(A),product_prod(list(A),product_prod(list(A),list(A))),aa(list(A),fun(list(A),fun(list(A),product_prod(list(A),product_prod(list(A),list(A))))),aa(A,fun(list(A),fun(list(A),fun(list(A),product_prod(list(A),product_prod(list(A),list(A)))))),aa(B,fun(A,fun(list(A),fun(list(A),fun(list(A),product_prod(list(A),product_prod(list(A),list(A))))))),aTP_Lamp_akf(fun(A,B),fun(B,fun(A,fun(list(A),fun(list(A),fun(list(A),product_prod(list(A),product_prod(list(A),list(A)))))))),Uu2),Uua),Uub),Uuc)) ) ).

% ATP.lambda_904
tff(fact_9082_ATP_Olambda__905,axiom,
    ! [Uu2: 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_mb(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu2),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),Uu2),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_905
tff(fact_9083_ATP_Olambda__906,axiom,
    ! [Uu2: 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_md(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu2),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu2),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_906
tff(fact_9084_ATP_Olambda__907,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu2: fun(B,A),Uua: B,Uub: fun(B,A),Uuc: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(B,fun(fun(B,A),fun(B,A)),aTP_Lamp_uj(fun(B,A),fun(B,fun(fun(B,A),fun(B,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(B,A,Uu2,Uua))),aa(B,A,Uub,Uuc))),aa(A,A,inverse_inverse(A),aa(B,A,Uu2,Uua)))) ) ).

% ATP.lambda_907
tff(fact_9085_ATP_Olambda__908,axiom,
    ! [Uu2: 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_mh(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu2),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),Uu2),Uub)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uuc))) ).

% ATP.lambda_908
tff(fact_9086_ATP_Olambda__909,axiom,
    ! [Uu2: 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_mf(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu2),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),Uu2),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub))) ).

% ATP.lambda_909
tff(fact_9087_ATP_Olambda__910,axiom,
    ! [A: $tType,B: $tType,Uu2: set(A),Uua: set(B),Uub: B,Uuc: fun(A,B)] :
      ( aa(fun(A,B),$o,aa(B,fun(fun(A,B),$o),aa(set(B),fun(B,fun(fun(A,B),$o)),aTP_Lamp_ahy(set(A),fun(set(B),fun(B,fun(fun(A,B),$o))),Uu2),Uua),Uub),Uuc)
    <=> ! [X6: A] :
          ( ( member(A,X6,Uu2)
           => member(B,aa(A,B,Uuc,X6),Uua) )
          & ( ~ member(A,X6,Uu2)
           => ( aa(A,B,Uuc,X6) = Uub ) ) ) ) ).

% ATP.lambda_910
tff(fact_9088_ATP_Olambda__911,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,fun(A,B)),Uua: fun(C,A),Uub: C,Uuc: fun(C,A),Uud: C] : aa(C,B,aa(fun(C,A),fun(C,B),aa(C,fun(fun(C,A),fun(C,B)),aa(fun(C,A),fun(C,fun(fun(C,A),fun(C,B))),aTP_Lamp_ub(fun(A,fun(A,B)),fun(fun(C,A),fun(C,fun(fun(C,A),fun(C,B)))),Uu2),Uua),Uub),Uuc),Uud) = aa(A,B,aa(A,fun(A,B),Uu2,aa(C,A,Uua,Uub)),aa(C,A,Uuc,Uud)) ) ).

% ATP.lambda_911
tff(fact_9089_ATP_Olambda__912,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: set(A),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,C)),Uuc: B,Uud: B] : aa(B,C,aa(B,fun(B,C),aa(fun(A,fun(B,C)),fun(B,fun(B,C)),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,C))),aTP_Lamp_uy(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,C)))),Uu2),Uua),Uub),Uuc),Uud) = aa(set(A),C,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_ux(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C))))),Uu2),Uua),Uub),Uuc),Uud)),Uu2) ) ).

% ATP.lambda_912
tff(fact_9090_ATP_Olambda__913,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu2: nat,Uua: fun(nat,A),Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(fun(nat,A),fun(A,fun(A,fun(nat,A))),aTP_Lamp_io(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Uu2),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_in(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,Uu2),Uud))) ) ).

% ATP.lambda_913
tff(fact_9091_ATP_Olambda__914,axiom,
    ! [Uu2: $o,Uua: $o,Uub: $o,Uuc: code_integer,Uud: $o] : aa($o,char,aa(code_integer,fun($o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aTP_Lamp_akm($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),(Uu2)),(Uua)),(Uub)),Uuc),(Uud)) = aa(product_prod(code_integer,$o),char,product_case_prod(code_integer,$o,char,aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aTP_Lamp_akl($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),(Uu2)),(Uua)),(Uub)),(Uud))),code_bit_cut_integer(Uuc)) ).

% ATP.lambda_914
tff(fact_9092_ATP_Olambda__915,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu2: fun(A,fun(B,assn)),Uua: list(A),Uub: list(B),Uuc: fun(C,assn),Uud: C] : aa(C,assn,aa(fun(C,assn),fun(C,assn),aa(list(B),fun(fun(C,assn),fun(C,assn)),aa(list(A),fun(list(B),fun(fun(C,assn),fun(C,assn))),aTP_Lamp_qr(fun(A,fun(B,assn)),fun(list(A),fun(list(B),fun(fun(C,assn),fun(C,assn)))),Uu2),Uua),Uub),Uuc),Uud) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),vEBT_List_list_assn(A,B,Uu2,Uua,Uub)),aa(C,assn,Uuc,Uud)) ).

% ATP.lambda_915
tff(fact_9093_ATP_Olambda__916,axiom,
    ! [Uu2: nat,Uua: set(nat),Uub: vEBT_VEBTi,Uuc: nat,Uud: $o] : aa($o,assn,aa(nat,fun($o,assn),aa(vEBT_VEBTi,fun(nat,fun($o,assn)),aa(set(nat),fun(vEBT_VEBTi,fun(nat,fun($o,assn))),aTP_Lamp_ak(nat,fun(set(nat),fun(vEBT_VEBTi,fun(nat,fun($o,assn)))),Uu2),Uua),Uub),Uuc),(Uud)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,vEBT_Intf_vebt_assn(Uu2,Uua),Uub)),pure_assn((Uud) = member(nat,Uuc,Uua))) ).

% ATP.lambda_916
tff(fact_9094_ATP_Olambda__917,axiom,
    ! [Uu2: nat,Uua: fun(nat,fun(real,real)),Uub: real,Uuc: nat,Uud: real] : aa(real,real,aa(nat,fun(real,real),aa(real,fun(nat,fun(real,real)),aa(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real))),aTP_Lamp_sz(nat,fun(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real)))),Uu2),Uua),Uub),Uuc),Uud) = aa(real,real,minus_minus(real,aa(real,real,aa(nat,fun(real,real),Uua,Uuc),Uud)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_sy(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,Uu2),Uuc)))),aa(real,real,aa(real,fun(real,real),times_times(real),Uub),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,power_power(real,Uud),aa(nat,nat,minus_minus(nat,Uu2),Uuc))),semiring_char_0_fact(real,aa(nat,nat,minus_minus(nat,Uu2),Uuc)))))) ).

% ATP.lambda_917
tff(fact_9095_ATP_Olambda__918,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu2: nat,Uua: fun(nat,A),Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(fun(nat,A),fun(A,fun(A,fun(nat,A))),aTP_Lamp_jj(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Uu2),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_ji(fun(nat,A),fun(A,fun(nat,fun(nat,A))),Uua),Uuc),Uud)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uud),Uu2))),aa(nat,A,power_power(A,Uub),Uud)) ) ).

% ATP.lambda_918
tff(fact_9096_ATP_Olambda__919,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: 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_gr(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Uu2),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Uud)),Uua)),one_one(A))),Uud)),aa(nat,A,power_power(A,Uub),Uud))),aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uuc)),aa(nat,nat,minus_minus(nat,Uu2),Uud))) ) ).

% ATP.lambda_919
tff(fact_9097_ATP_Olambda__920,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: 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_gl(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Uu2),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Uu2)),Uua)),Uud)),aa(nat,A,power_power(A,Uub),Uud))),aa(nat,A,power_power(A,Uuc),aa(nat,nat,minus_minus(nat,Uu2),Uud))) ) ).

% ATP.lambda_920
tff(fact_9098_ATP_Olambda__921,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: 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_gm(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Uu2),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(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,aa(A,fun(A,A),plus_plus(A),Uub),Uuc)),aa(nat,nat,minus_minus(nat,Uu2),Uud))) ) ).

% ATP.lambda_921
tff(fact_9099_ATP_Olambda__922,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu2: fun(nat,A),Uua: A,Uub: A,Uuc: nat,Uud: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aa(A,fun(A,fun(nat,fun(nat,A))),aTP_Lamp_in(fun(nat,A),fun(A,fun(A,fun(nat,fun(nat,A)))),Uu2),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uuc),Uud)),one_one(nat)))),aa(nat,A,power_power(A,Uub),Uud))),aa(nat,A,power_power(A,Uua),Uuc)) ) ).

% ATP.lambda_922
tff(fact_9100_ATP_Olambda__923,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( real_V7819770556892013058_space(B)
        & real_V7819770556892013058_space(C) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: fun(A,C),Uuc: C,Uud: A] :
          ( aa(A,$o,aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aa(B,fun(fun(A,C),fun(C,fun(A,$o))),aTP_Lamp_agz(fun(A,B),fun(B,fun(fun(A,C),fun(C,fun(A,$o)))),Uu2),Uua),Uub),Uuc),Uud)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(C,aa(A,C,Uub,Uud),Uuc)),real_V557655796197034286t_dist(B,aa(A,B,Uu2,Uud),Uua)) ) ) ).

% ATP.lambda_923
tff(fact_9101_ATP_Olambda__924,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: fun(B,C),Uuc: C,Uud: A] :
          ( aa(A,$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,$o))),aTP_Lamp_alz(fun(A,B),fun(B,fun(fun(B,C),fun(C,fun(A,$o)))),Uu2),Uua),Uub),Uuc),Uud)
        <=> ( ( aa(A,B,Uu2,Uud) = Uua )
           => ( aa(B,C,Uub,Uua) = Uuc ) ) ) ) ).

% ATP.lambda_924
tff(fact_9102_ATP_Olambda__925,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu2: 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_uq(fun(A,B),fun(fun(A,B),fun(A,fun(nat,fun(A,B)))),Uu2),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,Uu2,Uub)),aa(nat,nat,minus_minus(nat,Uuc),one_one(nat)))) ) ).

% ATP.lambda_925
tff(fact_9103_ATP_Olambda__926,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(A,fun(B,assn)),Uua: list(A),Uub: list(B),Uuc: nat,Uud: assn] : aa(assn,assn,aa(nat,fun(assn,assn),aa(list(B),fun(nat,fun(assn,assn)),aa(list(A),fun(list(B),fun(nat,fun(assn,assn))),aTP_Lamp_qp(fun(A,fun(B,assn)),fun(list(A),fun(list(B),fun(nat,fun(assn,assn)))),Uu2),Uua),Uub),Uuc),Uud) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Uud),aa(B,assn,aa(A,fun(B,assn),Uu2,aa(nat,A,nth(A,Uua),Uuc)),aa(nat,B,nth(B,Uub),Uuc))) ).

% ATP.lambda_926
tff(fact_9104_ATP_Olambda__927,axiom,
    ! [Uu2: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: code_integer,Uue: $o] : aa($o,char,aa(code_integer,fun($o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aTP_Lamp_akl($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),(Uu2)),(Uua)),(Uub)),(Uuc)),Uud),(Uue)) = aa(product_prod(code_integer,$o),char,product_case_prod(code_integer,$o,char,aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aa($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),aTP_Lamp_akk($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))),(Uu2)),(Uua)),(Uub)),(Uuc)),(Uue))),code_bit_cut_integer(Uud)) ).

% ATP.lambda_927
tff(fact_9105_ATP_Olambda__928,axiom,
    ! [Uu2: nat,Uua: set(nat),Uub: vEBT_VEBTi,Uuc: nat,Uud: nat,Uue: option(nat)] :
      aa(option(nat),assn,aa(nat,fun(option(nat),assn),aa(nat,fun(nat,fun(option(nat),assn)),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(option(nat),assn))),aa(set(nat),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(option(nat),assn)))),aTP_Lamp_ao(nat,fun(set(nat),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(option(nat),assn))))),Uu2),Uua),Uub),Uuc),Uud),Uue) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,vEBT_Intf_vebt_assn(Uu2,Uua),Uub)),
        pure_assn(( ( Uue = some(nat,Uud) )
          = vEBT_is_succ_in_set(Uua,Uuc,Uud) ))) ).

% ATP.lambda_928
tff(fact_9106_ATP_Olambda__929,axiom,
    ! [Uu2: nat,Uua: set(nat),Uub: vEBT_VEBTi,Uuc: nat,Uud: nat,Uue: option(nat)] :
      aa(option(nat),assn,aa(nat,fun(option(nat),assn),aa(nat,fun(nat,fun(option(nat),assn)),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(option(nat),assn))),aa(set(nat),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(option(nat),assn)))),aTP_Lamp_al(nat,fun(set(nat),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(option(nat),assn))))),Uu2),Uua),Uub),Uuc),Uud),Uue) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,vEBT_Intf_vebt_assn(Uu2,Uua),Uub)),
        pure_assn(( ( Uue = some(nat,Uud) )
          = vEBT_is_pred_in_set(Uua,Uuc,Uud) ))) ).

% ATP.lambda_929
tff(fact_9107_ATP_Olambda__930,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: set(A),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,C)),Uuc: B,Uud: B,Uue: A] : aa(A,C,aa(B,fun(A,C),aa(B,fun(B,fun(A,C)),aa(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C))),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C)))),aTP_Lamp_ux(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C))))),Uu2),Uua),Uub),Uuc),Uud),Uue) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(B,C,aa(A,fun(B,C),Uub,Uue),Uud)),groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_uv(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uuc),aa(set(A),set(A),minus_minus(set(A),Uu2),aa(set(A),set(A),insert(A,Uue),bot_bot(set(A)))))) ) ).

% ATP.lambda_930
tff(fact_9108_ATP_Olambda__931,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: fun(A,B),Uud: fun(A,B),Uue: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_uh(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu2),Uua),Uub),Uuc),Uud),Uue) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,minus_minus(B,aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uue)),aa(A,B,Uuc,Uub))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),aa(A,B,Uud,Uue)))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uuc,Uub)),aa(A,B,Uuc,Uub))) ) ).

% ATP.lambda_931
tff(fact_9109_ATP_Olambda__932,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: fun(A,real),Uud: fun(A,real),Uue: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(fun(A,real),fun(fun(A,real),fun(A,real)),aa(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))),aa(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real)))),aTP_Lamp_va(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))))),Uu2),Uua),Uub),Uuc),Uud),Uue) = aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,aa(A,real,Uu2,Uub),aa(A,real,Uuc,Uub))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uud,Uue)),aa(real,real,ln_ln(real),aa(A,real,Uu2,Uub)))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uue)),aa(A,real,Uuc,Uub))),aa(A,real,Uu2,Uub)))) ) ).

% ATP.lambda_932
tff(fact_9110_ATP_Olambda__933,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: 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_tk(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu2),Uua),Uub),Uuc),Uud),Uue) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uu2,Uub)),aa(A,B,Uud,Uue))),aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uua,Uue)),aa(A,B,Uuc,Uub))) ) ).

% ATP.lambda_933
tff(fact_9111_ATP_Olambda__934,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: fun(A,B),Uud: fun(A,B),Uue: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_tz(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu2),Uua),Uub),Uuc),Uud),Uue) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,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_934
tff(fact_9112_ATP_Olambda__935,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: fun(A,B),Uud: fun(A,B),Uue: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_uu(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu2),Uua),Uub),Uuc),Uud),Uue) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,uminus_uminus(B),aa(A,B,Uu2,Uub))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,inverse_inverse(B),aa(A,B,Uuc,Uub))),aa(A,B,Uud,Uue))),aa(B,B,inverse_inverse(B),aa(A,B,Uuc,Uub))))),aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uua,Uue)),aa(A,B,Uuc,Uub))) ) ).

% ATP.lambda_935
tff(fact_9113_ATP_Olambda__936,axiom,
    ! [Uu2: fun(a,b),Uua: b,Uub: a,Uuc: list(a),Uud: list(a),Uue: list(a)] :
      aa(list(a),product_prod(list(a),product_prod(list(a),list(a))),aa(list(a),fun(list(a),product_prod(list(a),product_prod(list(a),list(a)))),aa(list(a),fun(list(a),fun(list(a),product_prod(list(a),product_prod(list(a),list(a))))),aa(a,fun(list(a),fun(list(a),fun(list(a),product_prod(list(a),product_prod(list(a),list(a)))))),aa(b,fun(a,fun(list(a),fun(list(a),fun(list(a),product_prod(list(a),product_prod(list(a),list(a))))))),aTP_Lamp_akf(fun(a,b),fun(b,fun(a,fun(list(a),fun(list(a),fun(list(a),product_prod(list(a),product_prod(list(a),list(a)))))))),Uu2),Uua),Uub),Uuc),Uud),Uue) = $let(
        x: b,
        x:= aa(a,b,Uu2,Uub),
        $ite(
          aa(b,$o,aa(b,fun(b,$o),ord_less(b),x),Uua),
          aa(product_prod(list(a),list(a)),product_prod(list(a),product_prod(list(a),list(a))),aa(list(a),fun(product_prod(list(a),list(a)),product_prod(list(a),product_prod(list(a),list(a)))),product_Pair(list(a),product_prod(list(a),list(a))),aa(list(a),list(a),aa(a,fun(list(a),list(a)),cons(a),Uub),Uuc)),aa(list(a),product_prod(list(a),list(a)),aa(list(a),fun(list(a),product_prod(list(a),list(a))),product_Pair(list(a),list(a)),Uud),Uue)),
          $ite(aa(b,$o,aa(b,fun(b,$o),ord_less(b),Uua),x),aa(product_prod(list(a),list(a)),product_prod(list(a),product_prod(list(a),list(a))),aa(list(a),fun(product_prod(list(a),list(a)),product_prod(list(a),product_prod(list(a),list(a)))),product_Pair(list(a),product_prod(list(a),list(a))),Uuc),aa(list(a),product_prod(list(a),list(a)),aa(list(a),fun(list(a),product_prod(list(a),list(a))),product_Pair(list(a),list(a)),Uud),aa(list(a),list(a),aa(a,fun(list(a),list(a)),cons(a),Uub),Uue))),aa(product_prod(list(a),list(a)),product_prod(list(a),product_prod(list(a),list(a))),aa(list(a),fun(product_prod(list(a),list(a)),product_prod(list(a),product_prod(list(a),list(a)))),product_Pair(list(a),product_prod(list(a),list(a))),Uuc),aa(list(a),product_prod(list(a),list(a)),aa(list(a),fun(list(a),product_prod(list(a),list(a))),product_Pair(list(a),list(a)),aa(list(a),list(a),aa(a,fun(list(a),list(a)),cons(a),Uub),Uud)),Uue))) ) ) ).

% ATP.lambda_936
tff(fact_9114_ATP_Olambda__937,axiom,
    ! [Uu2: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: code_integer,Uuf: $o] : aa($o,char,aa(code_integer,fun($o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aa($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),aTP_Lamp_akk($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))),(Uu2)),(Uua)),(Uub)),(Uuc)),(Uud)),Uue),(Uuf)) = aa(product_prod(code_integer,$o),char,product_case_prod(code_integer,$o,char,aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aa($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))),aTP_Lamp_akj($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))))),(Uu2)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uuf))),code_bit_cut_integer(Uue)) ).

% ATP.lambda_937
tff(fact_9115_ATP_Olambda__938,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(A,fun(B,assn)),Uua: A,Uub: list(A),Uuc: B,Uud: list(B),Uue: nat,Uuf: assn] : aa(assn,assn,aa(nat,fun(assn,assn),aa(list(B),fun(nat,fun(assn,assn)),aa(B,fun(list(B),fun(nat,fun(assn,assn))),aa(list(A),fun(B,fun(list(B),fun(nat,fun(assn,assn)))),aa(A,fun(list(A),fun(B,fun(list(B),fun(nat,fun(assn,assn))))),aTP_Lamp_qq(fun(A,fun(B,assn)),fun(A,fun(list(A),fun(B,fun(list(B),fun(nat,fun(assn,assn)))))),Uu2),Uua),Uub),Uuc),Uud),Uue),Uuf) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Uuf),aa(B,assn,aa(A,fun(B,assn),Uu2,aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uub)),Uue)),aa(nat,B,nth(B,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Uuc),Uud)),Uue))) ).

% ATP.lambda_938
tff(fact_9116_ATP_Olambda__939,axiom,
    ! [Uu2: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: $o,Uuf: $o,Uug: code_integer] : aa(code_integer,fun($o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aa($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))))),aTP_Lamp_aki($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))))),(Uu2)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),(Uuf)),Uug) = char2((Uu2),(Uua),(Uub),(Uuc),(Uud),(Uue),(Uuf)) ).

% ATP.lambda_939
tff(fact_9117_ATP_Olambda__940,axiom,
    ! [Uu2: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: $o,Uuf: code_integer,Uug: $o] : aa($o,char,aa(code_integer,fun($o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aa($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))),aTP_Lamp_akj($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))))),(Uu2)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),Uuf),(Uug)) = aa(product_prod(code_integer,$o),char,product_case_prod(code_integer,$o,char,aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aa($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))))),aTP_Lamp_aki($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))))),(Uu2)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),(Uug))),code_bit_cut_integer(Uuf)) ).

% ATP.lambda_940
tff(fact_9118_ATP_Olambda__941,axiom,
    ! [B: $tType,A: $tType,Uu2: $o,Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_qg($o,fun(A,fun(B,$o)),(Uu2)),Uua),Uub)
    <=> (Uu2) ) ).

% ATP.lambda_941
tff(fact_9119_ATP_Olambda__942,axiom,
    ! [A: $tType,Uu2: heap_Time_Heap(A),Uua: product_unit] : aa(product_unit,heap_Time_Heap(A),aTP_Lamp_aj(heap_Time_Heap(A),fun(product_unit,heap_Time_Heap(A)),Uu2),Uua) = Uu2 ).

% ATP.lambda_942
tff(fact_9120_ATP_Olambda__943,axiom,
    ! [B: $tType,Uu2: assn,Uua: list(B)] : aa(list(B),assn,aTP_Lamp_dx(assn,fun(list(B),assn),Uu2),Uua) = Uu2 ).

% ATP.lambda_943
tff(fact_9121_ATP_Olambda__944,axiom,
    ! [B: $tType,Uu2: assn,Uua: B] : aa(B,assn,aTP_Lamp_dw(assn,fun(B,assn),Uu2),Uua) = Uu2 ).

% ATP.lambda_944
tff(fact_9122_ATP_Olambda__945,axiom,
    ! [Uu2: $o,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_alu($o,fun(nat,$o),(Uu2)),Uua)
    <=> (Uu2) ) ).

% ATP.lambda_945
tff(fact_9123_ATP_Olambda__946,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: $o,Uua: A] :
          ( aa(A,$o,aTP_Lamp_adb($o,fun(A,$o),(Uu2)),Uua)
        <=> (Uu2) ) ) ).

% ATP.lambda_946
tff(fact_9124_ATP_Olambda__947,axiom,
    ! [A: $tType,Uu2: $o,Uua: A] :
      ( aa(A,$o,aTP_Lamp_qb($o,fun(A,$o),(Uu2)),Uua)
    <=> (Uu2) ) ).

% ATP.lambda_947
tff(fact_9125_ATP_Olambda__948,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu2: fun(B,C),Uua: A] : aa(A,fun(B,C),aTP_Lamp_all(fun(B,C),fun(A,fun(B,C)),Uu2),Uua) = Uu2 ).

% ATP.lambda_948
tff(fact_9126_ATP_Olambda__949,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu2: B,Uua: A] : aa(A,B,aTP_Lamp_ts(B,fun(A,B),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_949
tff(fact_9127_ATP_Olambda__950,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topological_t2_space(A) )
     => ! [Uu2: B,Uua: A] : aa(A,B,aTP_Lamp_ya(B,fun(A,B),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_950
tff(fact_9128_ATP_Olambda__951,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu2: B,Uua: A] : aa(A,B,aTP_Lamp_vp(B,fun(A,B),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_951
tff(fact_9129_ATP_Olambda__952,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [Uu2: B,Uua: A] : aa(A,B,aTP_Lamp_ahx(B,fun(A,B),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_952
tff(fact_9130_ATP_Olambda__953,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & topolo8386298272705272623_space(A) )
     => ! [Uu2: B,Uua: A] : aa(A,B,aTP_Lamp_wf(B,fun(A,B),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_953
tff(fact_9131_ATP_Olambda__954,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(B)
     => ! [Uu2: B,Uua: A] : aa(A,B,aTP_Lamp_wp(B,fun(A,B),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_954
tff(fact_9132_ATP_Olambda__955,axiom,
    ! [A: $tType,B: $tType,Uu2: B,Uua: A] : aa(A,B,aTP_Lamp_pk(B,fun(A,B),Uu2),Uua) = Uu2 ).

% ATP.lambda_955
tff(fact_9133_ATP_Olambda__956,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_bp(A,fun(nat,A),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_956
tff(fact_9134_ATP_Olambda__957,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_ry(A,fun(A,A),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_957
tff(fact_9135_ATP_Olambda__958,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu2: A,Uua: B] : aa(B,A,aTP_Lamp_ajk(A,fun(B,A),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_958
tff(fact_9136_ATP_Olambda__959,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_aas(A,fun(nat,A),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_959
tff(fact_9137_ATP_Olambda__960,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo8386298272705272623_space(B) )
     => ! [Uu2: A,Uua: B] : aa(B,A,aTP_Lamp_wd(A,fun(B,A),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_960
tff(fact_9138_ATP_Olambda__961,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: A,Uua: B] : aa(B,A,aTP_Lamp_aiz(A,fun(B,A),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_961
tff(fact_9139_ATP_Olambda__962,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Uu2: A,Uua: B] : aa(B,A,aTP_Lamp_aja(A,fun(B,A),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_962
tff(fact_9140_ATP_Olambda__963,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_aik(A,fun(nat,A),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_963
tff(fact_9141_ATP_Olambda__964,axiom,
    ! [B: $tType,A: $tType] :
      ( ( zero(A)
        & topological_t2_space(A)
        & topolo8386298272705272623_space(B) )
     => ! [Uu2: A,Uua: B] : aa(B,A,aTP_Lamp_vy(A,fun(B,A),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_964
tff(fact_9142_ATP_Olambda__965,axiom,
    ! [B: $tType,A: $tType,Uu2: A,Uua: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_pl(A,fun(B,A)),Uu2),Uua) = Uu2 ).

% ATP.lambda_965
tff(fact_9143_ATP_Olambda__966,axiom,
    ! [B: $tType,A: $tType,Uu2: B,Uua: A] : aa(A,A,aa(B,fun(A,A),aTP_Lamp_alm(B,fun(A,A)),Uu2),Uua) = Uua ).

% ATP.lambda_966
tff(fact_9144_ATP_Olambda__967,axiom,
    ! [B: $tType,A: $tType,Uu2: A,Uua: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_ls(A,fun(B,$o)),Uu2),Uua)
    <=> $true ) ).

% ATP.lambda_967
tff(fact_9145_ATP_Olambda__968,axiom,
    ! [A: $tType,Uu2: A,Uua: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_alh(A,fun(A,$o)),Uu2),Uua)
    <=> $true ) ).

% ATP.lambda_968
tff(fact_9146_ATP_Olambda__969,axiom,
    ! [Uu2: complex] : aa(complex,complex,aTP_Lamp_hg(complex,complex),Uu2) = Uu2 ).

% ATP.lambda_969
tff(fact_9147_ATP_Olambda__970,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu2: word(A)] : aa(word(A),word(A),aTP_Lamp_rf(word(A),word(A)),Uu2) = Uu2 ) ).

% ATP.lambda_970
tff(fact_9148_ATP_Olambda__971,axiom,
    ! [Uu2: nat] : aa(nat,nat,aTP_Lamp_je(nat,nat),Uu2) = Uu2 ).

% ATP.lambda_971
tff(fact_9149_ATP_Olambda__972,axiom,
    ! [Uu2: int] : aa(int,int,aTP_Lamp_jl(int,int),Uu2) = Uu2 ).

% ATP.lambda_972
tff(fact_9150_ATP_Olambda__973,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: B] : aa(B,B,aTP_Lamp_qu(B,B),Uu2) = Uu2 ) ).

% ATP.lambda_973
tff(fact_9151_ATP_Olambda__974,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu2: B] : aa(B,B,aTP_Lamp_qv(B,B),Uu2) = Uu2 ) ).

% ATP.lambda_974
tff(fact_9152_ATP_Olambda__975,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: A] : aa(A,A,aTP_Lamp_tw(A,A),Uu2) = Uu2 ) ).

% ATP.lambda_975
tff(fact_9153_ATP_Olambda__976,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: A] : aa(A,A,aTP_Lamp_vq(A,A),Uu2) = Uu2 ) ).

% ATP.lambda_976
tff(fact_9154_ATP_Olambda__977,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: A] : aa(A,A,aTP_Lamp_rv(A,A),Uu2) = Uu2 ) ).

% ATP.lambda_977
tff(fact_9155_ATP_Olambda__978,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: A] : aa(A,A,aTP_Lamp_wl(A,A),Uu2) = Uu2 ) ).

% ATP.lambda_978
tff(fact_9156_ATP_Olambda__979,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A] : aa(A,A,aTP_Lamp_adg(A,A),Uu2) = Uu2 ) ).

% ATP.lambda_979
tff(fact_9157_ATP_Olambda__980,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: A] : aa(A,A,aTP_Lamp_akv(A,A),Uu2) = Uu2 ) ).

% ATP.lambda_980
tff(fact_9158_ATP_Olambda__981,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu2: A] : aa(A,A,aTP_Lamp_qx(A,A),Uu2) = Uu2 ) ).

% ATP.lambda_981
tff(fact_9159_ATP_Olambda__982,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Uu2: A] : aa(A,A,aTP_Lamp_ag(A,A),Uu2) = Uu2 ) ).

% ATP.lambda_982
tff(fact_9160_ATP_Olambda__983,axiom,
    ! [A: $tType,Uu2: A] : aa(A,A,aTP_Lamp_pc(A,A),Uu2) = Uu2 ).

% ATP.lambda_983
tff(fact_9161_ATP_Olambda__984,axiom,
    ! [C: $tType,B: $tType,Uu2: C] : aa(C,set(B),aTP_Lamp_amm(C,set(B)),Uu2) = bot_bot(set(B)) ).

% ATP.lambda_984
tff(fact_9162_ATP_Olambda__985,axiom,
    ! [A: $tType,D: $tType,Uu2: A] : aa(A,set(D),aTP_Lamp_amn(A,set(D)),Uu2) = bot_bot(set(D)) ).

% ATP.lambda_985
tff(fact_9163_ATP_Olambda__986,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu2: nat] : aa(nat,A,aTP_Lamp_bf(nat,A),Uu2) = zero_zero(A) ) ).

% ATP.lambda_986
tff(fact_9164_ATP_Olambda__987,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [Uu2: nat] : aa(nat,A,aTP_Lamp_bc(nat,A),Uu2) = zero_zero(A) ) ).

% ATP.lambda_987
tff(fact_9165_ATP_Olambda__988,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: B] : aa(B,A,aTP_Lamp_hc(B,A),Uu2) = zero_zero(A) ) ).

% ATP.lambda_988
tff(fact_9166_ATP_Olambda__989,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Uu2: B] : aa(B,A,aTP_Lamp_aku(B,A),Uu2) = zero_zero(A) ) ).

% ATP.lambda_989
tff(fact_9167_ATP_Olambda__990,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu2: A] : aa(A,B,aTP_Lamp_tt(A,B),Uu2) = zero_zero(B) ) ).

% ATP.lambda_990
tff(fact_9168_ATP_Olambda__991,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [Uu2: A] : aa(A,A,aTP_Lamp_am(A,A),Uu2) = zero_zero(A) ) ).

% ATP.lambda_991
tff(fact_9169_ATP_Olambda__992,axiom,
    ! [A: $tType,B: $tType] :
      ( zero(B)
     => ! [Uu2: A] : aa(A,B,aTP_Lamp_mj(A,B),Uu2) = zero_zero(B) ) ).

% ATP.lambda_992
tff(fact_9170_ATP_Olambda__993,axiom,
    ! [Uu2: product_unit] : aa(product_unit,assn,aTP_Lamp_eb(product_unit,assn),Uu2) = one_one(assn) ).

% ATP.lambda_993
tff(fact_9171_ATP_Olambda__994,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: B] : aa(B,A,aTP_Lamp_jm(B,A),Uu2) = one_one(A) ) ).

% ATP.lambda_994
tff(fact_9172_ATP_Olambda__995,axiom,
    ! [A: $tType,Uu2: A] : aa(A,real,aTP_Lamp_ajl(A,real),Uu2) = one_one(real) ).

% ATP.lambda_995
tff(fact_9173_ATP_Olambda__996,axiom,
    ! [A: $tType,Uu2: A] : aa(A,nat,aTP_Lamp_aje(A,nat),Uu2) = one_one(nat) ).

% ATP.lambda_996
tff(fact_9174_ATP_Olambda__997,axiom,
    ! [A: $tType,B: $tType,Uu2: A] : aa(A,B,aTP_Lamp_als(A,B),Uu2) = undefined(B) ).

% ATP.lambda_997
tff(fact_9175_ATP_Olambda__998,axiom,
    ! [Uu2: real] :
      ( aa(real,$o,aTP_Lamp_do(real,$o),Uu2)
    <=> $false ) ).

% ATP.lambda_998
tff(fact_9176_ATP_Olambda__999,axiom,
    ! [Uu2: nat] :
      ( aa(nat,$o,aTP_Lamp_aew(nat,$o),Uu2)
    <=> $false ) ).

% ATP.lambda_999
tff(fact_9177_ATP_Olambda__1000,axiom,
    ! [A: $tType,Uu2: A] :
      ( aa(A,$o,aTP_Lamp_dd(A,$o),Uu2)
    <=> $false ) ).

% ATP.lambda_1000
tff(fact_9178_ATP_Olambda__1001,axiom,
    ! [Uu2: nat] :
      ( aa(nat,$o,aTP_Lamp_alt(nat,$o),Uu2)
    <=> $true ) ).

% ATP.lambda_1001
tff(fact_9179_ATP_Olambda__1002,axiom,
    ! [A: $tType,Uu2: A] :
      ( aa(A,$o,aTP_Lamp_qc(A,$o),Uu2)
    <=> $true ) ).

% ATP.lambda_1002
tff(fact_9180_ATP_Olambda__1003,axiom,
    ! [A: $tType,Uu2: A] : aa(A,fun(nat,nat),aTP_Lamp_aju(A,fun(nat,nat)),Uu2) = suc ).

% ATP.lambda_1003

% Type constructors (1393)
tff(tcon_VEBT__BuildupMemImp_OVEBTi___Typerep_Otyperep,axiom,
    typerep(vEBT_VEBTi) ).

tff(tcon_Heap__Time__Monad_OHeap___Typerep_Otyperep_1,axiom,
    ! [A8: $tType] :
      ( typerep(A8)
     => typerep(heap_Time_Heap(A8)) ) ).

tff(tcon_Code__Numeral_Ointeger___Typerep_Otyperep_2,axiom,
    typerep(code_integer) ).

tff(tcon_Heap_Oheap_Oheap__ext___Typerep_Otyperep_3,axiom,
    ! [A8: $tType] :
      ( typerep(A8)
     => typerep(heap_ext(A8)) ) ).

tff(tcon_Product__Type_Ounit___Typerep_Otyperep_4,axiom,
    typerep(product_unit) ).

tff(tcon_Product__Type_Ounit___Enum_Oenum,axiom,
    enum(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder,axiom,
    comple5582772986160207858norder(product_unit) ).

tff(tcon_Product__Type_Oprod___Typerep_Otyperep_5,axiom,
    ! [A8: $tType,A9: $tType] :
      ( ( typerep(A8)
        & typerep(A9) )
     => typerep(product_prod(A8,A9)) ) ).

tff(tcon_Product__Type_Oprod___Enum_Oenum_6,axiom,
    ! [A8: $tType,A9: $tType] :
      ( ( enum(A8)
        & enum(A9) )
     => enum(product_prod(A8,A9)) ) ).

tff(tcon_Numeral__Type_Onum1___Typerep_Otyperep_7,axiom,
    typerep(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Enum_Oenum_8,axiom,
    enum(numeral_num1) ).

tff(tcon_Numeral__Type_Obit1___Typerep_Otyperep_9,axiom,
    ! [A8: $tType] :
      ( typerep(A8)
     => typerep(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Enum_Oenum_10,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => enum(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Typerep_Otyperep_11,axiom,
    ! [A8: $tType] :
      ( typerep(A8)
     => typerep(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Enum_Oenum_12,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => enum(numeral_bit0(A8)) ) ).

tff(tcon_Multiset_Omultiset___Typerep_Otyperep_13,axiom,
    ! [A8: $tType] :
      ( typerep(A8)
     => typerep(multiset(A8)) ) ).

tff(tcon_Extended__Nat_Oenat___Typerep_Otyperep_14,axiom,
    typerep(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__linorder_15,axiom,
    comple5582772986160207858norder(extended_enat) ).

tff(tcon_Complex_Ocomplex___Typerep_Otyperep_16,axiom,
    typerep(complex) ).

tff(tcon_Assertions_Oassn___Typerep_Otyperep_17,axiom,
    typerep(assn) ).

tff(tcon_Uint32_Ouint32___Typerep_Otyperep_18,axiom,
    typerep(uint32) ).

tff(tcon_Option_Ooption___Typerep_Otyperep_19,axiom,
    ! [A8: $tType] :
      ( typerep(A8)
     => typerep(option(A8)) ) ).

tff(tcon_Option_Ooption___Enum_Oenum_20,axiom,
    ! [A8: $tType] :
      ( enum(A8)
     => enum(option(A8)) ) ).

tff(tcon_Option_Ooption___Complete__Lattices_Ocomplete__linorder_21,axiom,
    ! [A8: $tType] :
      ( comple5582772986160207858norder(A8)
     => comple5582772986160207858norder(option(A8)) ) ).

tff(tcon_Filter_Ofilter___Typerep_Otyperep_22,axiom,
    ! [A8: $tType] :
      ( typerep(A8)
     => typerep(filter(A8)) ) ).

tff(tcon_Enum_Ofinite__3___Typerep_Otyperep_23,axiom,
    typerep(finite_3) ).

tff(tcon_Enum_Ofinite__3___Enum_Oenum_24,axiom,
    enum(finite_3) ).

tff(tcon_Enum_Ofinite__3___Complete__Lattices_Ocomplete__linorder_25,axiom,
    comple5582772986160207858norder(finite_3) ).

tff(tcon_Enum_Ofinite__2___Typerep_Otyperep_26,axiom,
    typerep(finite_2) ).

tff(tcon_Enum_Ofinite__2___Enum_Oenum_27,axiom,
    enum(finite_2) ).

tff(tcon_Enum_Ofinite__2___Complete__Lattices_Ocomplete__linorder_28,axiom,
    comple5582772986160207858norder(finite_2) ).

tff(tcon_Enum_Ofinite__1___Typerep_Otyperep_29,axiom,
    typerep(finite_1) ).

tff(tcon_Enum_Ofinite__1___Enum_Oenum_30,axiom,
    enum(finite_1) ).

tff(tcon_Enum_Ofinite__1___Complete__Lattices_Ocomplete__linorder_31,axiom,
    comple5582772986160207858norder(finite_1) ).

tff(tcon_Sum__Type_Osum___Typerep_Otyperep_32,axiom,
    ! [A8: $tType,A9: $tType] :
      ( ( typerep(A8)
        & typerep(A9) )
     => typerep(sum_sum(A8,A9)) ) ).

tff(tcon_Sum__Type_Osum___Enum_Oenum_33,axiom,
    ! [A8: $tType,A9: $tType] :
      ( ( enum(A8)
        & enum(A9) )
     => enum(sum_sum(A8,A9)) ) ).

tff(tcon_String_Ochar___Typerep_Otyperep_34,axiom,
    typerep(char) ).

tff(tcon_String_Ochar___Enum_Oenum_35,axiom,
    enum(char) ).

tff(tcon_Word_Oword___Typerep_Otyperep_36,axiom,
    ! [A8: $tType] :
      ( typerep(A8)
     => typerep(word(A8)) ) ).

tff(tcon_Word_Oword___Enum_Oenum_37,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => enum(word(A8)) ) ).

tff(tcon_Real_Oreal___Typerep_Otyperep_38,axiom,
    typerep(real) ).

tff(tcon_List_Olist___Typerep_Otyperep_39,axiom,
    ! [A8: $tType] :
      ( typerep(A8)
     => typerep(list(A8)) ) ).

tff(tcon_HOL_Obool___Typerep_Otyperep_40,axiom,
    typerep($o) ).

tff(tcon_HOL_Obool___Enum_Oenum_41,axiom,
    enum($o) ).

tff(tcon_Set_Oset___Typerep_Otyperep_42,axiom,
    ! [A8: $tType] :
      ( typerep(A8)
     => typerep(set(A8)) ) ).

tff(tcon_Set_Oset___Enum_Oenum_43,axiom,
    ! [A8: $tType] :
      ( enum(A8)
     => enum(set(A8)) ) ).

tff(tcon_Rat_Orat___Typerep_Otyperep_44,axiom,
    typerep(rat) ).

tff(tcon_Num_Onum___Typerep_Otyperep_45,axiom,
    typerep(num) ).

tff(tcon_Nat_Onat___Typerep_Otyperep_46,axiom,
    typerep(nat) ).

tff(tcon_Int_Oint___Typerep_Otyperep_47,axiom,
    typerep(int) ).

tff(tcon_fun___Typerep_Otyperep_48,axiom,
    ! [A8: $tType,A9: $tType] :
      ( ( typerep(A8)
        & typerep(A9) )
     => typerep(fun(A8,A9)) ) ).

tff(tcon_fun___Enum_Oenum_49,axiom,
    ! [A8: $tType,A9: $tType] :
      ( ( enum(A8)
        & enum(A9) )
     => enum(fun(A8,A9)) ) ).

tff(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
    ! [A8: $tType,A9: $tType] :
      ( boolea8198339166811842893lgebra(A9)
     => boolea8198339166811842893lgebra(fun(A8,A9)) ) ).

tff(tcon_fun___Lattices_Osemilattice__sup,axiom,
    ! [A8: $tType,A9: $tType] :
      ( semilattice_sup(A9)
     => semilattice_sup(fun(A8,A9)) ) ).

tff(tcon_fun___Code__Evaluation_Oterm__of,axiom,
    ! [A8: $tType,A9: $tType] :
      ( ( typerep(A8)
        & typerep(A9) )
     => code_term_of(fun(A8,A9)) ) ).

tff(tcon_fun___Orderings_Oorder__top,axiom,
    ! [A8: $tType,A9: $tType] :
      ( order_top(A9)
     => order_top(fun(A8,A9)) ) ).

tff(tcon_fun___Orderings_Oorder__bot,axiom,
    ! [A8: $tType,A9: $tType] :
      ( order_bot(A9)
     => order_bot(fun(A8,A9)) ) ).

tff(tcon_fun___Orderings_Opreorder,axiom,
    ! [A8: $tType,A9: $tType] :
      ( preorder(A9)
     => preorder(fun(A8,A9)) ) ).

tff(tcon_fun___Finite__Set_Ofinite,axiom,
    ! [A8: $tType,A9: $tType] :
      ( ( finite_finite(A8)
        & finite_finite(A9) )
     => finite_finite(fun(A8,A9)) ) ).

tff(tcon_fun___Lattices_Olattice,axiom,
    ! [A8: $tType,A9: $tType] :
      ( lattice(A9)
     => lattice(fun(A8,A9)) ) ).

tff(tcon_fun___Orderings_Oorder,axiom,
    ! [A8: $tType,A9: $tType] :
      ( order(A9)
     => order(fun(A8,A9)) ) ).

tff(tcon_fun___Orderings_Otop,axiom,
    ! [A8: $tType,A9: $tType] :
      ( top(A9)
     => top(fun(A8,A9)) ) ).

tff(tcon_fun___Orderings_Oord,axiom,
    ! [A8: $tType,A9: $tType] :
      ( ord(A9)
     => ord(fun(A8,A9)) ) ).

tff(tcon_fun___Orderings_Obot,axiom,
    ! [A8: $tType,A9: $tType] :
      ( bot(A9)
     => bot(fun(A8,A9)) ) ).

tff(tcon_fun___Groups_Ouminus,axiom,
    ! [A8: $tType,A9: $tType] :
      ( uminus(A9)
     => uminus(fun(A8,A9)) ) ).

tff(tcon_fun___Groups_Ominus,axiom,
    ! [A8: $tType,A9: $tType] :
      ( minus(A9)
     => minus(fun(A8,A9)) ) ).

tff(tcon_fun___HOL_Oequal,axiom,
    ! [A8: $tType,A9: $tType] :
      ( ( enum(A8)
        & cl_HOL_Oequal(A9) )
     => cl_HOL_Oequal(fun(A8,A9)) ) ).

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_Ounique__euclidean__semiring,axiom,
    euclid3128863361964157862miring(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel,axiom,
    euclid4440199948858584721cancel(int) ).

tff(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
    unique1627219031080169319umeral(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring__cancel,axiom,
    euclid8851590272496341667cancel(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors__cancel,axiom,
    semiri6575147826004484403cancel(int) ).

tff(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
    strict9044650504122735259up_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
    ordere580206878836729694up_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
    ordere2412721322843649153imp_le(int) ).

tff(tcon_Int_Oint___Bit__Operations_Osemiring__bit__operations,axiom,
    bit_se359711467146920520ations(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom,
    linord2810124833399127020strict(int) ).

tff(tcon_Int_Oint___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___Quickcheck__Narrowing_Opartial__term__of,axiom,
    quickc6926020345158392990erm_of(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_50,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___Code__Evaluation_Oterm__of_51,axiom,
    code_term_of(int) ).

tff(tcon_Int_Oint___Rings_Oordered__semiring,axiom,
    ordered_semiring(int) ).

tff(tcon_Int_Oint___Rings_Oordered__ring__abs,axiom,
    ordered_ring_abs(int) ).

tff(tcon_Int_Oint___Parity_Osemiring__parity,axiom,
    semiring_parity(int) ).

tff(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
    comm_monoid_add(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__modulo,axiom,
    semiring_modulo(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__ring,axiom,
    linordered_ring(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
    linordered_idom(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
    comm_semiring_1(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring__0,axiom,
    comm_semiring_0(int) ).

tff(tcon_Int_Oint___Groups_Osemigroup__mult,axiom,
    semigroup_mult(int) ).

tff(tcon_Int_Oint___Rings_Osemidom__modulo,axiom,
    semidom_modulo(int) ).

tff(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
    semidom_divide(int) ).

tff(tcon_Int_Oint___Num_Osemiring__numeral,axiom,
    semiring_numeral(int) ).

tff(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
    semigroup_add(int) ).

tff(tcon_Int_Oint___Rings_Ozero__less__one,axiom,
    zero_less_one(int) ).

tff(tcon_Int_Oint___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_52,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_53,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_54,axiom,
    order(int) ).

tff(tcon_Int_Oint___Num_Oneg__numeral,axiom,
    neg_numeral(int) ).

tff(tcon_Int_Oint___Nat_Oring__char__0,axiom,
    ring_char_0(int) ).

tff(tcon_Int_Oint___Rings_Osemiring,axiom,
    semiring(int) ).

tff(tcon_Int_Oint___Orderings_Oord_55,axiom,
    ord(int) ).

tff(tcon_Int_Oint___Groups_Ouminus_56,axiom,
    uminus(int) ).

tff(tcon_Int_Oint___Rings_Oring__1,axiom,
    ring_1(int) ).

tff(tcon_Int_Oint___Rings_Oabs__if,axiom,
    abs_if(int) ).

tff(tcon_Int_Oint___Groups_Otimes,axiom,
    times(int) ).

tff(tcon_Int_Oint___Groups_Ominus_57,axiom,
    minus(int) ).

tff(tcon_Int_Oint___GCD_Oring__gcd,axiom,
    ring_gcd(int) ).

tff(tcon_Int_Oint___Power_Opower,axiom,
    power(int) ).

tff(tcon_Int_Oint___Num_Onumeral,axiom,
    numeral(int) ).

tff(tcon_Int_Oint___Groups_Ozero,axiom,
    zero(int) ).

tff(tcon_Int_Oint___Groups_Oplus,axiom,
    plus(int) ).

tff(tcon_Int_Oint___Rings_Oring,axiom,
    ring(int) ).

tff(tcon_Int_Oint___Rings_Oidom,axiom,
    idom(int) ).

tff(tcon_Int_Oint___Groups_Oone,axiom,
    one(int) ).

tff(tcon_Int_Oint___Rings_Odvd,axiom,
    dvd(int) ).

tff(tcon_Int_Oint___HOL_Oequal_58,axiom,
    cl_HOL_Oequal(int) ).

tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_59,axiom,
    condit6923001295902523014norder(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_60,axiom,
    bit_un5681908812861735899ations(nat) ).

tff(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_61,axiom,
    semiri1453513574482234551roduct(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_62,axiom,
    euclid5411537665997757685th_nat(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_63,axiom,
    ordere1937475149494474687imp_le(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_64,axiom,
    euclid3128863361964157862miring(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_65,axiom,
    euclid4440199948858584721cancel(nat) ).

tff(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_66,axiom,
    unique1627219031080169319umeral(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_67,axiom,
    semiri6575147826004484403cancel(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_68,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_69,axiom,
    ordere580206878836729694up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_70,axiom,
    ordere2412721322843649153imp_le(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_71,axiom,
    bit_se359711467146920520ations(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_72,axiom,
    linord2810124833399127020strict(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_73,axiom,
    strict7427464778891057005id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_74,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_75,axiom,
    euclid3725896446679973847miring(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Otopological__space_76,axiom,
    topolo4958980785337419405_space(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_77,axiom,
    topolo1944317154257567458pology(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Odiscrete__topology_78,axiom,
    topolo8865339358273720382pology(nat) ).

tff(tcon_Nat_Onat___Quickcheck__Narrowing_Opartial__term__of_79,axiom,
    quickc6926020345158392990erm_of(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__comm__monoid__mult_80,axiom,
    topolo4987421752381908075d_mult(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__comm__monoid__add_81,axiom,
    topolo5987344860129210374id_add(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Oorder__topology_82,axiom,
    topolo2564578578187576103pology(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_83,axiom,
    semiri2026040879449505780visors(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_84,axiom,
    linord181362715937106298miring(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__semigroup__mult_85,axiom,
    topolo4211221413907600880p_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_86,axiom,
    linord8928482502909563296strict(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_87,axiom,
    semiri3467727345109120633visors(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_88,axiom,
    ordere6658533253407199908up_add(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__monoid__mult_89,axiom,
    topolo1898628316856586783d_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_90,axiom,
    ordere6911136660526730532id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_91,axiom,
    cancel2418104881723323429up_add(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__monoid__add_92,axiom,
    topolo6943815403480290642id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_93,axiom,
    cancel1802427076303600483id_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_94,axiom,
    comm_s4317794764714335236cancel(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_95,axiom,
    bit_semiring_bits(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Ot2__space_96,axiom,
    topological_t2_space(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Ot1__space_97,axiom,
    topological_t1_space(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__comm__semiring_98,axiom,
    ordere2520102378445227354miring(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_99,axiom,
    cancel_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring_100,axiom,
    linordered_semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring__0_101,axiom,
    ordered_semiring_0(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semidom_102,axiom,
    linordered_semidom(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__sup_103,axiom,
    semilattice_sup(nat) ).

tff(tcon_Nat_Onat___Groups_Oab__semigroup__mult_104,axiom,
    ab_semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__cancel_105,axiom,
    semiring_1_cancel(nat) ).

tff(tcon_Nat_Onat___Rings_Oalgebraic__semidom_106,axiom,
    algebraic_semidom(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_107,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_108,axiom,
    ab_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Code__Evaluation_Oterm__of_109,axiom,
    code_term_of(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring_110,axiom,
    ordered_semiring(nat) ).

tff(tcon_Nat_Onat___Parity_Osemiring__parity_111,axiom,
    semiring_parity(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__add_112,axiom,
    comm_monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__modulo_113,axiom,
    semiring_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1_114,axiom,
    comm_semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__0_115,axiom,
    comm_semiring_0(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__mult_116,axiom,
    semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__modulo_117,axiom,
    semidom_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__divide_118,axiom,
    semidom_divide(nat) ).

tff(tcon_Nat_Onat___Num_Osemiring__numeral_119,axiom,
    semiring_numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__add_120,axiom,
    semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__less__one_121,axiom,
    zero_less_one(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring_122,axiom,
    comm_semiring(nat) ).

tff(tcon_Nat_Onat___Orderings_Owellorder,axiom,
    wellorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder__bot_123,axiom,
    order_bot(nat) ).

tff(tcon_Nat_Onat___Nat_Osemiring__char__0_124,axiom,
    semiring_char_0(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__neq__one_125,axiom,
    zero_neq_one(nat) ).

tff(tcon_Nat_Onat___Orderings_Opreorder_126,axiom,
    preorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Olinorder_127,axiom,
    linorder(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__mult_128,axiom,
    monoid_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__add_129,axiom,
    monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1_130,axiom,
    semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__0_131,axiom,
    semiring_0(nat) ).

tff(tcon_Nat_Onat___Orderings_Ono__top_132,axiom,
    no_top(nat) ).

tff(tcon_Nat_Onat___Lattices_Olattice_133,axiom,
    lattice(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__gcd_134,axiom,
    semiring_gcd(nat) ).

tff(tcon_Nat_Onat___Rings_Omult__zero_135,axiom,
    mult_zero(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder_136,axiom,
    order(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring_137,axiom,
    semiring(nat) ).

tff(tcon_Nat_Onat___Orderings_Oord_138,axiom,
    ord(nat) ).

tff(tcon_Nat_Onat___Orderings_Obot_139,axiom,
    bot(nat) ).

tff(tcon_Nat_Onat___Groups_Otimes_140,axiom,
    times(nat) ).

tff(tcon_Nat_Onat___Groups_Ominus_141,axiom,
    minus(nat) ).

tff(tcon_Nat_Onat___Power_Opower_142,axiom,
    power(nat) ).

tff(tcon_Nat_Onat___Num_Onumeral_143,axiom,
    numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Ozero_144,axiom,
    zero(nat) ).

tff(tcon_Nat_Onat___Groups_Oplus_145,axiom,
    plus(nat) ).

tff(tcon_Nat_Onat___Groups_Oone_146,axiom,
    one(nat) ).

tff(tcon_Nat_Onat___Rings_Odvd_147,axiom,
    dvd(nat) ).

tff(tcon_Nat_Onat___HOL_Oequal_148,axiom,
    cl_HOL_Oequal(nat) ).

tff(tcon_Nat_Onat___Nat_Osize,axiom,
    size(nat) ).

tff(tcon_Num_Onum___Quickcheck__Narrowing_Opartial__term__of_149,axiom,
    quickc6926020345158392990erm_of(num) ).

tff(tcon_Num_Onum___Code__Evaluation_Oterm__of_150,axiom,
    code_term_of(num) ).

tff(tcon_Num_Onum___Orderings_Opreorder_151,axiom,
    preorder(num) ).

tff(tcon_Num_Onum___Orderings_Olinorder_152,axiom,
    linorder(num) ).

tff(tcon_Num_Onum___Orderings_Oorder_153,axiom,
    order(num) ).

tff(tcon_Num_Onum___Orderings_Oord_154,axiom,
    ord(num) ).

tff(tcon_Num_Onum___Groups_Otimes_155,axiom,
    times(num) ).

tff(tcon_Num_Onum___Groups_Oplus_156,axiom,
    plus(num) ).

tff(tcon_Num_Onum___HOL_Oequal_157,axiom,
    cl_HOL_Oequal(num) ).

tff(tcon_Num_Onum___Nat_Osize_158,axiom,
    size(num) ).

tff(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_159,axiom,
    semiri1453513574482234551roduct(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_160,axiom,
    ordere1937475149494474687imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_161,axiom,
    semiri6575147826004484403cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_162,axiom,
    strict9044650504122735259up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_163,axiom,
    ordere580206878836729694up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_164,axiom,
    ordere2412721322843649153imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_165,axiom,
    linord2810124833399127020strict(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_166,axiom,
    strict7427464778891057005id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_167,axiom,
    ordere8940638589300402666id_add(rat) ).

tff(tcon_Rat_Orat___Quickcheck__Narrowing_Opartial__term__of_168,axiom,
    quickc6926020345158392990erm_of(rat) ).

tff(tcon_Rat_Orat___Archimedean__Field_Oarchimedean__field,axiom,
    archim462609752435547400_field(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1__strict_169,axiom,
    linord715952674999750819strict(rat) ).

tff(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
    unboun7993243217541854897norder(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_170,axiom,
    semiri2026040879449505780visors(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_171,axiom,
    linord181362715937106298miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_172,axiom,
    linord8928482502909563296strict(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_173,axiom,
    semiri3467727345109120633visors(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_174,axiom,
    ordere6658533253407199908up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_175,axiom,
    ordere166539214618696060dd_abs(rat) ).

tff(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
    archim2362893244070406136eiling(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_176,axiom,
    ordere6911136660526730532id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_177,axiom,
    linord5086331880401160121up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_178,axiom,
    cancel2418104881723323429up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_179,axiom,
    ring_15535105094025558882visors(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_180,axiom,
    cancel1802427076303600483id_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring__strict_181,axiom,
    linord4710134922213307826strict(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_182,axiom,
    comm_s4317794764714335236cancel(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__comm__semiring_183,axiom,
    ordere2520102378445227354miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1_184,axiom,
    linord6961819062388156250ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add_185,axiom,
    ordered_ab_group_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_186,axiom,
    cancel_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring_187,axiom,
    linordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring__0_188,axiom,
    ordered_semiring_0(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semidom_189,axiom,
    linordered_semidom(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
    dense_linorder(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__sup_190,axiom,
    semilattice_sup(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__mult_191,axiom,
    ab_semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__cancel_192,axiom,
    semiring_1_cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_193,axiom,
    comm_monoid_mult(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__add_194,axiom,
    ab_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
    linordered_field(rat) ).

tff(tcon_Rat_Orat___Code__Evaluation_Oterm__of_195,axiom,
    code_term_of(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring_196,axiom,
    ordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring__abs_197,axiom,
    ordered_ring_abs(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__add_198,axiom,
    comm_monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring_199,axiom,
    linordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__idom_200,axiom,
    linordered_idom(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1_201,axiom,
    comm_semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__0_202,axiom,
    comm_semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__order,axiom,
    dense_order(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__mult_203,axiom,
    semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom__divide_204,axiom,
    semidom_divide(rat) ).

tff(tcon_Rat_Orat___Num_Osemiring__numeral_205,axiom,
    semiring_numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__add_206,axiom,
    semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
    division_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Ozero__less__one_207,axiom,
    zero_less_one(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring_208,axiom,
    comm_semiring(rat) ).

tff(tcon_Rat_Orat___Nat_Osemiring__char__0_209,axiom,
    semiring_char_0(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__group__add_210,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_211,axiom,
    zero_neq_one(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring_212,axiom,
    ordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__abs__sgn_213,axiom,
    idom_abs_sgn(rat) ).

tff(tcon_Rat_Orat___Orderings_Opreorder_214,axiom,
    preorder(rat) ).

tff(tcon_Rat_Orat___Orderings_Olinorder_215,axiom,
    linorder(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__mult_216,axiom,
    monoid_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__divide_217,axiom,
    idom_divide(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring__1_218,axiom,
    comm_ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__add_219,axiom,
    monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1_220,axiom,
    semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__0_221,axiom,
    semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__top_222,axiom,
    no_top(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__bot_223,axiom,
    no_bot(rat) ).

tff(tcon_Rat_Orat___Lattices_Olattice_224,axiom,
    lattice(rat) ).

tff(tcon_Rat_Orat___Groups_Ogroup__add_225,axiom,
    group_add(rat) ).

tff(tcon_Rat_Orat___Rings_Omult__zero_226,axiom,
    mult_zero(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring_227,axiom,
    comm_ring(rat) ).

tff(tcon_Rat_Orat___Orderings_Oorder_228,axiom,
    order(rat) ).

tff(tcon_Rat_Orat___Num_Oneg__numeral_229,axiom,
    neg_numeral(rat) ).

tff(tcon_Rat_Orat___Nat_Oring__char__0_230,axiom,
    ring_char_0(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring_231,axiom,
    semiring(rat) ).

tff(tcon_Rat_Orat___Fields_Oinverse,axiom,
    inverse(rat) ).

tff(tcon_Rat_Orat___Orderings_Oord_232,axiom,
    ord(rat) ).

tff(tcon_Rat_Orat___Groups_Ouminus_233,axiom,
    uminus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1_234,axiom,
    ring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Oabs__if_235,axiom,
    abs_if(rat) ).

tff(tcon_Rat_Orat___Groups_Otimes_236,axiom,
    times(rat) ).

tff(tcon_Rat_Orat___Groups_Ominus_237,axiom,
    minus(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield,axiom,
    field(rat) ).

tff(tcon_Rat_Orat___Power_Opower_238,axiom,
    power(rat) ).

tff(tcon_Rat_Orat___Num_Onumeral_239,axiom,
    numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Ozero_240,axiom,
    zero(rat) ).

tff(tcon_Rat_Orat___Groups_Oplus_241,axiom,
    plus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring_242,axiom,
    ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom_243,axiom,
    idom(rat) ).

tff(tcon_Rat_Orat___Groups_Oone_244,axiom,
    one(rat) ).

tff(tcon_Rat_Orat___Rings_Odvd_245,axiom,
    dvd(rat) ).

tff(tcon_Rat_Orat___HOL_Oequal_246,axiom,
    cl_HOL_Oequal(rat) ).

tff(tcon_Set_Oset___Quickcheck__Narrowing_Opartial__term__of_247,axiom,
    ! [A8: $tType] :
      ( typerep(A8)
     => quickc6926020345158392990erm_of(set(A8)) ) ).

tff(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_248,axiom,
    ! [A8: $tType] : boolea8198339166811842893lgebra(set(A8)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__sup_249,axiom,
    ! [A8: $tType] : semilattice_sup(set(A8)) ).

tff(tcon_Set_Oset___Code__Evaluation_Oterm__of_250,axiom,
    ! [A8: $tType] :
      ( typerep(A8)
     => code_term_of(set(A8)) ) ).

tff(tcon_Set_Oset___Orderings_Oorder__top_251,axiom,
    ! [A8: $tType] : order_top(set(A8)) ).

tff(tcon_Set_Oset___Orderings_Oorder__bot_252,axiom,
    ! [A8: $tType] : order_bot(set(A8)) ).

tff(tcon_Set_Oset___Orderings_Opreorder_253,axiom,
    ! [A8: $tType] : preorder(set(A8)) ).

tff(tcon_Set_Oset___Finite__Set_Ofinite_254,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => finite_finite(set(A8)) ) ).

tff(tcon_Set_Oset___Lattices_Olattice_255,axiom,
    ! [A8: $tType] : lattice(set(A8)) ).

tff(tcon_Set_Oset___Orderings_Oorder_256,axiom,
    ! [A8: $tType] : order(set(A8)) ).

tff(tcon_Set_Oset___Orderings_Otop_257,axiom,
    ! [A8: $tType] : top(set(A8)) ).

tff(tcon_Set_Oset___Orderings_Oord_258,axiom,
    ! [A8: $tType] : ord(set(A8)) ).

tff(tcon_Set_Oset___Orderings_Obot_259,axiom,
    ! [A8: $tType] : bot(set(A8)) ).

tff(tcon_Set_Oset___Groups_Ouminus_260,axiom,
    ! [A8: $tType] : uminus(set(A8)) ).

tff(tcon_Set_Oset___Groups_Ominus_261,axiom,
    ! [A8: $tType] : minus(set(A8)) ).

tff(tcon_Set_Oset___HOL_Oequal_262,axiom,
    ! [A8: $tType] :
      ( cl_HOL_Oequal(A8)
     => cl_HOL_Oequal(set(A8)) ) ).

tff(tcon_HOL_Obool___Topological__Spaces_Otopological__space_263,axiom,
    topolo4958980785337419405_space($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_264,axiom,
    topolo1944317154257567458pology($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Odiscrete__topology_265,axiom,
    topolo8865339358273720382pology($o) ).

tff(tcon_HOL_Obool___Quickcheck__Narrowing_Opartial__term__of_266,axiom,
    quickc6926020345158392990erm_of($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_267,axiom,
    topolo2564578578187576103pology($o) ).

tff(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_268,axiom,
    boolea8198339166811842893lgebra($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Ot2__space_269,axiom,
    topological_t2_space($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Ot1__space_270,axiom,
    topological_t1_space($o) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__sup_271,axiom,
    semilattice_sup($o) ).

tff(tcon_HOL_Obool___Code__Evaluation_Oterm__of_272,axiom,
    code_term_of($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder__top_273,axiom,
    order_top($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder__bot_274,axiom,
    order_bot($o) ).

tff(tcon_HOL_Obool___Orderings_Opreorder_275,axiom,
    preorder($o) ).

tff(tcon_HOL_Obool___Orderings_Olinorder_276,axiom,
    linorder($o) ).

tff(tcon_HOL_Obool___Finite__Set_Ofinite_277,axiom,
    finite_finite($o) ).

tff(tcon_HOL_Obool___Lattices_Olattice_278,axiom,
    lattice($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder_279,axiom,
    order($o) ).

tff(tcon_HOL_Obool___Orderings_Otop_280,axiom,
    top($o) ).

tff(tcon_HOL_Obool___Orderings_Oord_281,axiom,
    ord($o) ).

tff(tcon_HOL_Obool___Orderings_Obot_282,axiom,
    bot($o) ).

tff(tcon_HOL_Obool___Groups_Ouminus_283,axiom,
    uminus($o) ).

tff(tcon_HOL_Obool___Groups_Ominus_284,axiom,
    minus($o) ).

tff(tcon_HOL_Obool___HOL_Oequal_285,axiom,
    cl_HOL_Oequal($o) ).

tff(tcon_List_Olist___Quickcheck__Narrowing_Opartial__term__of_286,axiom,
    ! [A8: $tType] :
      ( typerep(A8)
     => quickc6926020345158392990erm_of(list(A8)) ) ).

tff(tcon_List_Olist___Code__Evaluation_Oterm__of_287,axiom,
    ! [A8: $tType] :
      ( typerep(A8)
     => code_term_of(list(A8)) ) ).

tff(tcon_List_Olist___HOL_Oequal_288,axiom,
    ! [A8: $tType] : cl_HOL_Oequal(list(A8)) ).

tff(tcon_List_Olist___Nat_Osize_289,axiom,
    ! [A8: $tType] : size(list(A8)) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_290,axiom,
    condit6923001295902523014norder(real) ).

tff(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_291,axiom,
    semiri1453513574482234551roduct(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__monoid__add__imp__le_292,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_293,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_294,axiom,
    strict9044650504122735259up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_295,axiom,
    ordere580206878836729694up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_296,axiom,
    ordere2412721322843649153imp_le(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__comm__semiring__strict_297,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_298,axiom,
    strict7427464778891057005id_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add_299,axiom,
    ordere8940638589300402666id_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Otopological__space_300,axiom,
    topolo4958980785337419405_space(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_301,axiom,
    topolo1944317154257567458pology(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__field,axiom,
    real_V3459762299906320749_field(real) ).

tff(tcon_Real_Oreal___Quickcheck__Narrowing_Opartial__term__of_302,axiom,
    quickc6926020345158392990erm_of(real) ).

tff(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field_303,axiom,
    archim462609752435547400_field(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1__strict_304,axiom,
    linord715952674999750819strict(real) ).

tff(tcon_Real_Oreal___Orderings_Ounbounded__dense__linorder_305,axiom,
    unboun7993243217541854897norder(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__comm__monoid__add_306,axiom,
    topolo5987344860129210374id_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_307,axiom,
    topolo2564578578187576103pology(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_308,axiom,
    semiri2026040879449505780visors(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_309,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_310,axiom,
    topolo4211221413907600880p_mult(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Operfect__space,axiom,
    topolo8386298272705272623_space(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__strict_311,axiom,
    linord8928482502909563296strict(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_312,axiom,
    semiri3467727345109120633visors(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__algebra,axiom,
    real_V6157519004096292374lgebra(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_313,axiom,
    ordere6658533253407199908up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_314,axiom,
    ordere166539214618696060dd_abs(real) ).

tff(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling_315,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_316,axiom,
    ordere6911136660526730532id_add(real) ).

tff(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_317,axiom,
    linord5086331880401160121up_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_318,axiom,
    cancel2418104881723323429up_add(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_319,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_320,axiom,
    topolo6943815403480290642id_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_321,axiom,
    cancel1802427076303600483id_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring__strict_322,axiom,
    linord4710134922213307826strict(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__1__cancel_323,axiom,
    comm_s4317794764714335236cancel(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__group__add,axiom,
    topolo1633459387980952147up_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ot2__space_324,axiom,
    topological_t2_space(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ot1__space_325,axiom,
    topological_t1_space(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__comm__semiring_326,axiom,
    ordere2520102378445227354miring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1_327,axiom,
    linord6961819062388156250ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add_328,axiom,
    ordered_ab_group_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_329,axiom,
    cancel_semigroup_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring_330,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_331,axiom,
    ordered_semiring_0(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semidom_332,axiom,
    linordered_semidom(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__linorder_333,axiom,
    dense_linorder(real) ).

tff(tcon_Real_Oreal___Lattices_Osemilattice__sup_334,axiom,
    semilattice_sup(real) ).

tff(tcon_Real_Oreal___Groups_Oab__semigroup__mult_335,axiom,
    ab_semigroup_mult(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__cancel_336,axiom,
    semiring_1_cancel(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_337,axiom,
    comm_monoid_mult(real) ).

tff(tcon_Real_Oreal___Groups_Oab__semigroup__add_338,axiom,
    ab_semigroup_add(real) ).

tff(tcon_Real_Oreal___Fields_Olinordered__field_339,axiom,
    linordered_field(real) ).

tff(tcon_Real_Oreal___Code__Evaluation_Oterm__of_340,axiom,
    code_term_of(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__semiring_341,axiom,
    ordered_semiring(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring__abs_342,axiom,
    ordered_ring_abs(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__add_343,axiom,
    comm_monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring_344,axiom,
    linordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__idom_345,axiom,
    linordered_idom(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__1_346,axiom,
    comm_semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__0_347,axiom,
    comm_semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__order_348,axiom,
    dense_order(real) ).

tff(tcon_Real_Oreal___Groups_Osemigroup__mult_349,axiom,
    semigroup_mult(real) ).

tff(tcon_Real_Oreal___Rings_Osemidom__divide_350,axiom,
    semidom_divide(real) ).

tff(tcon_Real_Oreal___Num_Osemiring__numeral_351,axiom,
    semiring_numeral(real) ).

tff(tcon_Real_Oreal___Groups_Osemigroup__add_352,axiom,
    semigroup_add(real) ).

tff(tcon_Real_Oreal___Fields_Odivision__ring_353,axiom,
    division_ring(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__less__one_354,axiom,
    zero_less_one(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring_355,axiom,
    comm_semiring(real) ).

tff(tcon_Real_Oreal___Nat_Osemiring__char__0_356,axiom,
    semiring_char_0(real) ).

tff(tcon_Real_Oreal___Groups_Oab__group__add_357,axiom,
    ab_group_add(real) ).

tff(tcon_Real_Oreal___Fields_Ofield__char__0_358,axiom,
    field_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__neq__one_359,axiom,
    zero_neq_one(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring_360,axiom,
    ordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom__abs__sgn_361,axiom,
    idom_abs_sgn(real) ).

tff(tcon_Real_Oreal___Orderings_Opreorder_362,axiom,
    preorder(real) ).

tff(tcon_Real_Oreal___Orderings_Olinorder_363,axiom,
    linorder(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__mult_364,axiom,
    monoid_mult(real) ).

tff(tcon_Real_Oreal___Transcendental_Oln,axiom,
    ln(real) ).

tff(tcon_Real_Oreal___Rings_Oidom__divide_365,axiom,
    idom_divide(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring__1_366,axiom,
    comm_ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__add_367,axiom,
    monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1_368,axiom,
    semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__0_369,axiom,
    semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__top_370,axiom,
    no_top(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__bot_371,axiom,
    no_bot(real) ).

tff(tcon_Real_Oreal___Lattices_Olattice_372,axiom,
    lattice(real) ).

tff(tcon_Real_Oreal___Groups_Ogroup__add_373,axiom,
    group_add(real) ).

tff(tcon_Real_Oreal___Rings_Omult__zero_374,axiom,
    mult_zero(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring_375,axiom,
    comm_ring(real) ).

tff(tcon_Real_Oreal___Orderings_Oorder_376,axiom,
    order(real) ).

tff(tcon_Real_Oreal___Num_Oneg__numeral_377,axiom,
    neg_numeral(real) ).

tff(tcon_Real_Oreal___Nat_Oring__char__0_378,axiom,
    ring_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring_379,axiom,
    semiring(real) ).

tff(tcon_Real_Oreal___Fields_Oinverse_380,axiom,
    inverse(real) ).

tff(tcon_Real_Oreal___Orderings_Oord_381,axiom,
    ord(real) ).

tff(tcon_Real_Oreal___Groups_Ouminus_382,axiom,
    uminus(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1_383,axiom,
    ring_1(real) ).

tff(tcon_Real_Oreal___Rings_Oabs__if_384,axiom,
    abs_if(real) ).

tff(tcon_Real_Oreal___Groups_Otimes_385,axiom,
    times(real) ).

tff(tcon_Real_Oreal___Groups_Ominus_386,axiom,
    minus(real) ).

tff(tcon_Real_Oreal___Fields_Ofield_387,axiom,
    field(real) ).

tff(tcon_Real_Oreal___Power_Opower_388,axiom,
    power(real) ).

tff(tcon_Real_Oreal___Num_Onumeral_389,axiom,
    numeral(real) ).

tff(tcon_Real_Oreal___Groups_Ozero_390,axiom,
    zero(real) ).

tff(tcon_Real_Oreal___Groups_Oplus_391,axiom,
    plus(real) ).

tff(tcon_Real_Oreal___Rings_Oring_392,axiom,
    ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom_393,axiom,
    idom(real) ).

tff(tcon_Real_Oreal___Groups_Oone_394,axiom,
    one(real) ).

tff(tcon_Real_Oreal___Rings_Odvd_395,axiom,
    dvd(real) ).

tff(tcon_Real_Oreal___HOL_Oequal_396,axiom,
    cl_HOL_Oequal(real) ).

tff(tcon_Word_Oword___Bit__Operations_Osemiring__bit__operations_397,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => bit_se359711467146920520ations(word(A8)) ) ).

tff(tcon_Word_Oword___Quickcheck__Narrowing_Opartial__term__of_398,axiom,
    ! [A8: $tType] :
      ( typerep(A8)
     => quickc6926020345158392990erm_of(word(A8)) ) ).

tff(tcon_Word_Oword___Bit__Comprehension_Obit__comprehension_399,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => bit_bi6583157726757044596ension(word(A8)) ) ).

tff(tcon_Word_Oword___Bit__Operations_Oring__bit__operations_400,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => bit_ri3973907225187159222ations(word(A8)) ) ).

tff(tcon_Word_Oword___Groups_Ocancel__ab__semigroup__add_401,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => cancel2418104881723323429up_add(word(A8)) ) ).

tff(tcon_Word_Oword___Groups_Ocancel__comm__monoid__add_402,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => cancel1802427076303600483id_add(word(A8)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__semiring__1__cancel_403,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => comm_s4317794764714335236cancel(word(A8)) ) ).

tff(tcon_Word_Oword___Bit__Operations_Osemiring__bits_404,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => bit_semiring_bits(word(A8)) ) ).

tff(tcon_Word_Oword___Groups_Ocancel__semigroup__add_405,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => cancel_semigroup_add(word(A8)) ) ).

tff(tcon_Word_Oword___Least__significant__bit_Olsb_406,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => least_6119777620449941438nt_lsb(word(A8)) ) ).

tff(tcon_Word_Oword___Groups_Oab__semigroup__mult_407,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => ab_semigroup_mult(word(A8)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring__1__cancel_408,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => semiring_1_cancel(word(A8)) ) ).

tff(tcon_Word_Oword___Groups_Ocomm__monoid__mult_409,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => comm_monoid_mult(word(A8)) ) ).

tff(tcon_Word_Oword___Groups_Oab__semigroup__add_410,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => ab_semigroup_add(word(A8)) ) ).

tff(tcon_Word_Oword___Generic__set__bit_Oset__bit_411,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => generic_set_set_bit(word(A8)) ) ).

tff(tcon_Word_Oword___Code__Evaluation_Oterm__of_412,axiom,
    ! [A8: $tType] :
      ( typerep(A8)
     => code_term_of(word(A8)) ) ).

tff(tcon_Word_Oword___Parity_Osemiring__parity_413,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => semiring_parity(word(A8)) ) ).

tff(tcon_Word_Oword___Groups_Ocomm__monoid__add_414,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => comm_monoid_add(word(A8)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring__modulo_415,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => semiring_modulo(word(A8)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__semiring__1_416,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => comm_semiring_1(word(A8)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__semiring__0_417,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => comm_semiring_0(word(A8)) ) ).

tff(tcon_Word_Oword___Groups_Osemigroup__mult_418,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => semigroup_mult(word(A8)) ) ).

tff(tcon_Word_Oword___Num_Osemiring__numeral_419,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => semiring_numeral(word(A8)) ) ).

tff(tcon_Word_Oword___Groups_Osemigroup__add_420,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => semigroup_add(word(A8)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__semiring_421,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => comm_semiring(word(A8)) ) ).

tff(tcon_Word_Oword___Orderings_Owellorder_422,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => wellorder(word(A8)) ) ).

tff(tcon_Word_Oword___Groups_Oab__group__add_423,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => ab_group_add(word(A8)) ) ).

tff(tcon_Word_Oword___Rings_Ozero__neq__one_424,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => zero_neq_one(word(A8)) ) ).

tff(tcon_Word_Oword___Parity_Oring__parity_425,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => ring_parity(word(A8)) ) ).

tff(tcon_Word_Oword___Orderings_Opreorder_426,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => preorder(word(A8)) ) ).

tff(tcon_Word_Oword___Orderings_Olinorder_427,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => linorder(word(A8)) ) ).

tff(tcon_Word_Oword___Groups_Omonoid__mult_428,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => monoid_mult(word(A8)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__ring__1_429,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => comm_ring_1(word(A8)) ) ).

tff(tcon_Word_Oword___Groups_Omonoid__add_430,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => monoid_add(word(A8)) ) ).

tff(tcon_Word_Oword___Finite__Set_Ofinite_431,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => finite_finite(word(A8)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring__1_432,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => semiring_1(word(A8)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring__0_433,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => semiring_0(word(A8)) ) ).

tff(tcon_Word_Oword___Groups_Ogroup__add_434,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => group_add(word(A8)) ) ).

tff(tcon_Word_Oword___Rings_Omult__zero_435,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => mult_zero(word(A8)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__ring_436,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => comm_ring(word(A8)) ) ).

tff(tcon_Word_Oword___Orderings_Oorder_437,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => order(word(A8)) ) ).

tff(tcon_Word_Oword___Num_Oneg__numeral_438,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => neg_numeral(word(A8)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring_439,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => semiring(word(A8)) ) ).

tff(tcon_Word_Oword___Orderings_Oord_440,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => ord(word(A8)) ) ).

tff(tcon_Word_Oword___Groups_Ouminus_441,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => uminus(word(A8)) ) ).

tff(tcon_Word_Oword___Rings_Oring__1_442,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => ring_1(word(A8)) ) ).

tff(tcon_Word_Oword___Groups_Otimes_443,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => times(word(A8)) ) ).

tff(tcon_Word_Oword___Groups_Ominus_444,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => minus(word(A8)) ) ).

tff(tcon_Word_Oword___Power_Opower_445,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => power(word(A8)) ) ).

tff(tcon_Word_Oword___Num_Onumeral_446,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => numeral(word(A8)) ) ).

tff(tcon_Word_Oword___Groups_Ozero_447,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => zero(word(A8)) ) ).

tff(tcon_Word_Oword___Groups_Oplus_448,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => plus(word(A8)) ) ).

tff(tcon_Word_Oword___Rings_Oring_449,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => ring(word(A8)) ) ).

tff(tcon_Word_Oword___Groups_Oone_450,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => one(word(A8)) ) ).

tff(tcon_Word_Oword___Rings_Odvd_451,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => dvd(word(A8)) ) ).

tff(tcon_Word_Oword___HOL_Oequal_452,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => cl_HOL_Oequal(word(A8)) ) ).

tff(tcon_Word_Oword___Nat_Osize_453,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => size(word(A8)) ) ).

tff(tcon_String_Ochar___Quickcheck__Narrowing_Opartial__term__of_454,axiom,
    quickc6926020345158392990erm_of(char) ).

tff(tcon_String_Ochar___Lattices_Osemilattice__sup_455,axiom,
    semilattice_sup(char) ).

tff(tcon_String_Ochar___Code__Evaluation_Oterm__of_456,axiom,
    code_term_of(char) ).

tff(tcon_String_Ochar___Orderings_Opreorder_457,axiom,
    preorder(char) ).

tff(tcon_String_Ochar___Orderings_Olinorder_458,axiom,
    linorder(char) ).

tff(tcon_String_Ochar___Finite__Set_Ofinite_459,axiom,
    finite_finite(char) ).

tff(tcon_String_Ochar___Lattices_Olattice_460,axiom,
    lattice(char) ).

tff(tcon_String_Ochar___Orderings_Oorder_461,axiom,
    order(char) ).

tff(tcon_String_Ochar___Orderings_Oord_462,axiom,
    ord(char) ).

tff(tcon_String_Ochar___HOL_Oequal_463,axiom,
    cl_HOL_Oequal(char) ).

tff(tcon_String_Ochar___Nat_Osize_464,axiom,
    size(char) ).

tff(tcon_Sum__Type_Osum___Quickcheck__Narrowing_Opartial__term__of_465,axiom,
    ! [A8: $tType,A9: $tType] :
      ( ( typerep(A8)
        & typerep(A9) )
     => quickc6926020345158392990erm_of(sum_sum(A8,A9)) ) ).

tff(tcon_Sum__Type_Osum___Code__Evaluation_Oterm__of_466,axiom,
    ! [A8: $tType,A9: $tType] :
      ( ( typerep(A8)
        & typerep(A9) )
     => code_term_of(sum_sum(A8,A9)) ) ).

tff(tcon_Sum__Type_Osum___Finite__Set_Ofinite_467,axiom,
    ! [A8: $tType,A9: $tType] :
      ( ( finite_finite(A8)
        & finite_finite(A9) )
     => finite_finite(sum_sum(A8,A9)) ) ).

tff(tcon_Sum__Type_Osum___HOL_Oequal_468,axiom,
    ! [A8: $tType,A9: $tType] : cl_HOL_Oequal(sum_sum(A8,A9)) ).

tff(tcon_Sum__Type_Osum___Nat_Osize_469,axiom,
    ! [A8: $tType,A9: $tType] : size(sum_sum(A8,A9)) ).

tff(tcon_Enum_Ofinite__1___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_470,axiom,
    condit6923001295902523014norder(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__ab__semigroup__monoid__add__imp__le_471,axiom,
    ordere1937475149494474687imp_le(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Osemiring__no__zero__divisors__cancel_472,axiom,
    semiri6575147826004484403cancel(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ostrict__ordered__ab__semigroup__add_473,axiom,
    strict9044650504122735259up_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__cancel__comm__monoid__diff_474,axiom,
    ordere1170586879665033532d_diff(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__cancel__ab__semigroup__add_475,axiom,
    ordere580206878836729694up_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__ab__semigroup__add__imp__le_476,axiom,
    ordere2412721322843649153imp_le(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Olinordered__comm__semiring__strict_477,axiom,
    linord2810124833399127020strict(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ostrict__ordered__comm__monoid__add_478,axiom,
    strict7427464778891057005id_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__cancel__comm__monoid__add_479,axiom,
    ordere8940638589300402666id_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ocanonically__ordered__monoid__add_480,axiom,
    canoni5634975068530333245id_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Quickcheck__Narrowing_Opartial__term__of_481,axiom,
    quickc6926020345158392990erm_of(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Olinordered__semiring__strict_482,axiom,
    linord8928482502909563296strict(finite_1) ).

tff(tcon_Enum_Ofinite__1___Boolean__Algebras_Oboolean__algebra_483,axiom,
    boolea8198339166811842893lgebra(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Osemiring__no__zero__divisors_484,axiom,
    semiri3467727345109120633visors(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__ab__semigroup__add_485,axiom,
    ordere6658533253407199908up_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__ab__group__add__abs_486,axiom,
    ordere166539214618696060dd_abs(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__comm__monoid__add_487,axiom,
    ordere6911136660526730532id_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Olinordered__ab__group__add_488,axiom,
    linord5086331880401160121up_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ocancel__ab__semigroup__add_489,axiom,
    cancel2418104881723323429up_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ocancel__comm__monoid__add_490,axiom,
    cancel1802427076303600483id_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Olinordered__ring__strict_491,axiom,
    linord4710134922213307826strict(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oordered__comm__semiring_492,axiom,
    ordere2520102378445227354miring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__ab__group__add_493,axiom,
    ordered_ab_group_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ocancel__semigroup__add_494,axiom,
    cancel_semigroup_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Olinordered__semiring_495,axiom,
    linordered_semiring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oordered__semiring__0_496,axiom,
    ordered_semiring_0(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Odense__linorder_497,axiom,
    dense_linorder(finite_1) ).

tff(tcon_Enum_Ofinite__1___Lattices_Osemilattice__sup_498,axiom,
    semilattice_sup(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oab__semigroup__mult_499,axiom,
    ab_semigroup_mult(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ocomm__monoid__mult_500,axiom,
    comm_monoid_mult(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ocomm__monoid__diff_501,axiom,
    comm_monoid_diff(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oab__semigroup__add_502,axiom,
    ab_semigroup_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Code__Evaluation_Oterm__of_503,axiom,
    code_term_of(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oordered__semiring_504,axiom,
    ordered_semiring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oordered__ring__abs_505,axiom,
    ordered_ring_abs(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ocomm__monoid__add_506,axiom,
    comm_monoid_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Olinordered__ring_507,axiom,
    linordered_ring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Ocomm__semiring__0_508,axiom,
    comm_semiring_0(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Odense__order_509,axiom,
    dense_order(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Osemigroup__mult_510,axiom,
    semigroup_mult(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Osemigroup__add_511,axiom,
    semigroup_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Ocomm__semiring_512,axiom,
    comm_semiring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Owellorder_513,axiom,
    wellorder(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Oorder__top_514,axiom,
    order_top(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Oorder__bot_515,axiom,
    order_bot(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oab__group__add_516,axiom,
    ab_group_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oordered__ring_517,axiom,
    ordered_ring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Opreorder_518,axiom,
    preorder(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Olinorder_519,axiom,
    linorder(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Omonoid__mult_520,axiom,
    monoid_mult(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Omonoid__add_521,axiom,
    monoid_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Finite__Set_Ofinite_522,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_523,axiom,
    semiring_0(finite_1) ).

tff(tcon_Enum_Ofinite__1___Lattices_Olattice_524,axiom,
    lattice(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ogroup__add_525,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_526,axiom,
    mult_zero(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Ocomm__ring_527,axiom,
    comm_ring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Oorder_528,axiom,
    order(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Osemiring_529,axiom,
    semiring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Fields_Oinverse_530,axiom,
    inverse(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Otop_531,axiom,
    top(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Oord_532,axiom,
    ord(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Obot_533,axiom,
    bot(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ouminus_534,axiom,
    uminus(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oabs__if_535,axiom,
    abs_if(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Otimes_536,axiom,
    times(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ominus_537,axiom,
    minus(finite_1) ).

tff(tcon_Enum_Ofinite__1___Power_Opower_538,axiom,
    power(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ozero_539,axiom,
    zero(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oplus_540,axiom,
    plus(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oring_541,axiom,
    ring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oone_542,axiom,
    one(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Odvd_543,axiom,
    dvd(finite_1) ).

tff(tcon_Enum_Ofinite__1___HOL_Oequal_544,axiom,
    cl_HOL_Oequal(finite_1) ).

tff(tcon_Enum_Ofinite__2___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_545,axiom,
    condit6923001295902523014norder(finite_2) ).

tff(tcon_Enum_Ofinite__2___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_546,axiom,
    semiri1453513574482234551roduct(finite_2) ).

tff(tcon_Enum_Ofinite__2___Euclidean__Division_Ounique__euclidean__semiring_547,axiom,
    euclid3128863361964157862miring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Euclidean__Division_Oeuclidean__semiring__cancel_548,axiom,
    euclid4440199948858584721cancel(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__no__zero__divisors__cancel_549,axiom,
    semiri6575147826004484403cancel(finite_2) ).

tff(tcon_Enum_Ofinite__2___Euclidean__Division_Oeuclidean__semiring_550,axiom,
    euclid3725896446679973847miring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Quickcheck__Narrowing_Opartial__term__of_551,axiom,
    quickc6926020345158392990erm_of(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__1__no__zero__divisors_552,axiom,
    semiri2026040879449505780visors(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__no__zero__divisors_553,axiom,
    semiri3467727345109120633visors(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ocancel__ab__semigroup__add_554,axiom,
    cancel2418104881723323429up_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oring__1__no__zero__divisors_555,axiom,
    ring_15535105094025558882visors(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ocancel__comm__monoid__add_556,axiom,
    cancel1802427076303600483id_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring__1__cancel_557,axiom,
    comm_s4317794764714335236cancel(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ocancel__semigroup__add_558,axiom,
    cancel_semigroup_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Lattices_Osemilattice__sup_559,axiom,
    semilattice_sup(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Oab__semigroup__mult_560,axiom,
    ab_semigroup_mult(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__1__cancel_561,axiom,
    semiring_1_cancel(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oalgebraic__semidom_562,axiom,
    algebraic_semidom(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ocomm__monoid__mult_563,axiom,
    comm_monoid_mult(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Oab__semigroup__add_564,axiom,
    ab_semigroup_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Code__Evaluation_Oterm__of_565,axiom,
    code_term_of(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ocomm__monoid__add_566,axiom,
    comm_monoid_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__modulo_567,axiom,
    semiring_modulo(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring__1_568,axiom,
    comm_semiring_1(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring__0_569,axiom,
    comm_semiring_0(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Osemigroup__mult_570,axiom,
    semigroup_mult(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemidom__modulo_571,axiom,
    semidom_modulo(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemidom__divide_572,axiom,
    semidom_divide(finite_2) ).

tff(tcon_Enum_Ofinite__2___Num_Osemiring__numeral_573,axiom,
    semiring_numeral(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Osemigroup__add_574,axiom,
    semigroup_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Fields_Odivision__ring_575,axiom,
    division_ring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring_576,axiom,
    comm_semiring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Owellorder_577,axiom,
    wellorder(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Oorder__top_578,axiom,
    order_top(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Oorder__bot_579,axiom,
    order_bot(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Oab__group__add_580,axiom,
    ab_group_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ozero__neq__one_581,axiom,
    zero_neq_one(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oidom__abs__sgn_582,axiom,
    idom_abs_sgn(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Opreorder_583,axiom,
    preorder(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Olinorder_584,axiom,
    linorder(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Omonoid__mult_585,axiom,
    monoid_mult(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oidom__modulo_586,axiom,
    idom_modulo(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oidom__divide_587,axiom,
    idom_divide(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ocomm__ring__1_588,axiom,
    comm_ring_1(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Omonoid__add_589,axiom,
    monoid_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Finite__Set_Ofinite_590,axiom,
    finite_finite(finite_2) ).

tff(tcon_Enum_Ofinite__2___Type__Length_Olen0_591,axiom,
    type_len0(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__1_592,axiom,
    semiring_1(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__0_593,axiom,
    semiring_0(finite_2) ).

tff(tcon_Enum_Ofinite__2___Lattices_Olattice_594,axiom,
    lattice(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ogroup__add_595,axiom,
    group_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Type__Length_Olen_596,axiom,
    type_len(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Omult__zero_597,axiom,
    mult_zero(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ocomm__ring_598,axiom,
    comm_ring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Oorder_599,axiom,
    order(finite_2) ).

tff(tcon_Enum_Ofinite__2___Num_Oneg__numeral_600,axiom,
    neg_numeral(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring_601,axiom,
    semiring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Fields_Oinverse_602,axiom,
    inverse(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Otop_603,axiom,
    top(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Oord_604,axiom,
    ord(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Obot_605,axiom,
    bot(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ouminus_606,axiom,
    uminus(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oring__1_607,axiom,
    ring_1(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Otimes_608,axiom,
    times(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ominus_609,axiom,
    minus(finite_2) ).

tff(tcon_Enum_Ofinite__2___Fields_Ofield_610,axiom,
    field(finite_2) ).

tff(tcon_Enum_Ofinite__2___Power_Opower_611,axiom,
    power(finite_2) ).

tff(tcon_Enum_Ofinite__2___Num_Onumeral_612,axiom,
    numeral(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ozero_613,axiom,
    zero(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Oplus_614,axiom,
    plus(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oring_615,axiom,
    ring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oidom_616,axiom,
    idom(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Oone_617,axiom,
    one(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Odvd_618,axiom,
    dvd(finite_2) ).

tff(tcon_Enum_Ofinite__2___HOL_Oequal_619,axiom,
    cl_HOL_Oequal(finite_2) ).

tff(tcon_Enum_Ofinite__3___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_620,axiom,
    condit6923001295902523014norder(finite_3) ).

tff(tcon_Enum_Ofinite__3___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_621,axiom,
    semiri1453513574482234551roduct(finite_3) ).

tff(tcon_Enum_Ofinite__3___Euclidean__Division_Ounique__euclidean__semiring_622,axiom,
    euclid3128863361964157862miring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Euclidean__Division_Oeuclidean__semiring__cancel_623,axiom,
    euclid4440199948858584721cancel(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__no__zero__divisors__cancel_624,axiom,
    semiri6575147826004484403cancel(finite_3) ).

tff(tcon_Enum_Ofinite__3___Euclidean__Division_Oeuclidean__semiring_625,axiom,
    euclid3725896446679973847miring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Quickcheck__Narrowing_Opartial__term__of_626,axiom,
    quickc6926020345158392990erm_of(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__1__no__zero__divisors_627,axiom,
    semiri2026040879449505780visors(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__no__zero__divisors_628,axiom,
    semiri3467727345109120633visors(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ocancel__ab__semigroup__add_629,axiom,
    cancel2418104881723323429up_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oring__1__no__zero__divisors_630,axiom,
    ring_15535105094025558882visors(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ocancel__comm__monoid__add_631,axiom,
    cancel1802427076303600483id_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring__1__cancel_632,axiom,
    comm_s4317794764714335236cancel(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ocancel__semigroup__add_633,axiom,
    cancel_semigroup_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Lattices_Osemilattice__sup_634,axiom,
    semilattice_sup(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Oab__semigroup__mult_635,axiom,
    ab_semigroup_mult(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__1__cancel_636,axiom,
    semiring_1_cancel(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oalgebraic__semidom_637,axiom,
    algebraic_semidom(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ocomm__monoid__mult_638,axiom,
    comm_monoid_mult(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Oab__semigroup__add_639,axiom,
    ab_semigroup_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Code__Evaluation_Oterm__of_640,axiom,
    code_term_of(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ocomm__monoid__add_641,axiom,
    comm_monoid_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__modulo_642,axiom,
    semiring_modulo(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring__1_643,axiom,
    comm_semiring_1(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring__0_644,axiom,
    comm_semiring_0(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Osemigroup__mult_645,axiom,
    semigroup_mult(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemidom__modulo_646,axiom,
    semidom_modulo(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemidom__divide_647,axiom,
    semidom_divide(finite_3) ).

tff(tcon_Enum_Ofinite__3___Num_Osemiring__numeral_648,axiom,
    semiring_numeral(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Osemigroup__add_649,axiom,
    semigroup_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Fields_Odivision__ring_650,axiom,
    division_ring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring_651,axiom,
    comm_semiring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Owellorder_652,axiom,
    wellorder(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Oorder__top_653,axiom,
    order_top(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Oorder__bot_654,axiom,
    order_bot(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Oab__group__add_655,axiom,
    ab_group_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ozero__neq__one_656,axiom,
    zero_neq_one(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oidom__abs__sgn_657,axiom,
    idom_abs_sgn(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Opreorder_658,axiom,
    preorder(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Olinorder_659,axiom,
    linorder(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Omonoid__mult_660,axiom,
    monoid_mult(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oidom__modulo_661,axiom,
    idom_modulo(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oidom__divide_662,axiom,
    idom_divide(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ocomm__ring__1_663,axiom,
    comm_ring_1(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Omonoid__add_664,axiom,
    monoid_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Finite__Set_Ofinite_665,axiom,
    finite_finite(finite_3) ).

tff(tcon_Enum_Ofinite__3___Type__Length_Olen0_666,axiom,
    type_len0(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__1_667,axiom,
    semiring_1(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__0_668,axiom,
    semiring_0(finite_3) ).

tff(tcon_Enum_Ofinite__3___Lattices_Olattice_669,axiom,
    lattice(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ogroup__add_670,axiom,
    group_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Type__Length_Olen_671,axiom,
    type_len(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Omult__zero_672,axiom,
    mult_zero(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ocomm__ring_673,axiom,
    comm_ring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Oorder_674,axiom,
    order(finite_3) ).

tff(tcon_Enum_Ofinite__3___Num_Oneg__numeral_675,axiom,
    neg_numeral(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring_676,axiom,
    semiring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Fields_Oinverse_677,axiom,
    inverse(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Otop_678,axiom,
    top(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Oord_679,axiom,
    ord(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Obot_680,axiom,
    bot(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ouminus_681,axiom,
    uminus(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oring__1_682,axiom,
    ring_1(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Otimes_683,axiom,
    times(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ominus_684,axiom,
    minus(finite_3) ).

tff(tcon_Enum_Ofinite__3___Fields_Ofield_685,axiom,
    field(finite_3) ).

tff(tcon_Enum_Ofinite__3___Power_Opower_686,axiom,
    power(finite_3) ).

tff(tcon_Enum_Ofinite__3___Num_Onumeral_687,axiom,
    numeral(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ozero_688,axiom,
    zero(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Oplus_689,axiom,
    plus(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oring_690,axiom,
    ring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oidom_691,axiom,
    idom(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Oone_692,axiom,
    one(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Odvd_693,axiom,
    dvd(finite_3) ).

tff(tcon_Enum_Ofinite__3___HOL_Oequal_694,axiom,
    cl_HOL_Oequal(finite_3) ).

tff(tcon_Filter_Ofilter___Quickcheck__Narrowing_Opartial__term__of_695,axiom,
    ! [A8: $tType] :
      ( typerep(A8)
     => quickc6926020345158392990erm_of(filter(A8)) ) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_696,axiom,
    ! [A8: $tType] : semilattice_sup(filter(A8)) ).

tff(tcon_Filter_Ofilter___Code__Evaluation_Oterm__of_697,axiom,
    ! [A8: $tType] :
      ( typerep(A8)
     => code_term_of(filter(A8)) ) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__top_698,axiom,
    ! [A8: $tType] : order_top(filter(A8)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__bot_699,axiom,
    ! [A8: $tType] : order_bot(filter(A8)) ).

tff(tcon_Filter_Ofilter___Orderings_Opreorder_700,axiom,
    ! [A8: $tType] : preorder(filter(A8)) ).

tff(tcon_Filter_Ofilter___Lattices_Olattice_701,axiom,
    ! [A8: $tType] : lattice(filter(A8)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder_702,axiom,
    ! [A8: $tType] : order(filter(A8)) ).

tff(tcon_Filter_Ofilter___Orderings_Otop_703,axiom,
    ! [A8: $tType] : top(filter(A8)) ).

tff(tcon_Filter_Ofilter___Orderings_Oord_704,axiom,
    ! [A8: $tType] : ord(filter(A8)) ).

tff(tcon_Filter_Ofilter___Orderings_Obot_705,axiom,
    ! [A8: $tType] : bot(filter(A8)) ).

tff(tcon_Filter_Ofilter___HOL_Oequal_706,axiom,
    ! [A8: $tType] :
      ( cl_HOL_Oequal(A8)
     => cl_HOL_Oequal(filter(A8)) ) ).

tff(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_707,axiom,
    ! [A8: $tType] :
      ( comple5582772986160207858norder(A8)
     => condit6923001295902523014norder(option(A8)) ) ).

tff(tcon_Option_Ooption___Quickcheck__Narrowing_Opartial__term__of_708,axiom,
    ! [A8: $tType] :
      ( typerep(A8)
     => quickc6926020345158392990erm_of(option(A8)) ) ).

tff(tcon_Option_Ooption___Lattices_Osemilattice__sup_709,axiom,
    ! [A8: $tType] :
      ( semilattice_sup(A8)
     => semilattice_sup(option(A8)) ) ).

tff(tcon_Option_Ooption___Code__Evaluation_Oterm__of_710,axiom,
    ! [A8: $tType] :
      ( typerep(A8)
     => code_term_of(option(A8)) ) ).

tff(tcon_Option_Ooption___Orderings_Owellorder_711,axiom,
    ! [A8: $tType] :
      ( wellorder(A8)
     => wellorder(option(A8)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder__top_712,axiom,
    ! [A8: $tType] :
      ( order_top(A8)
     => order_top(option(A8)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder__bot_713,axiom,
    ! [A8: $tType] :
      ( order(A8)
     => order_bot(option(A8)) ) ).

tff(tcon_Option_Ooption___Orderings_Opreorder_714,axiom,
    ! [A8: $tType] :
      ( preorder(A8)
     => preorder(option(A8)) ) ).

tff(tcon_Option_Ooption___Orderings_Olinorder_715,axiom,
    ! [A8: $tType] :
      ( linorder(A8)
     => linorder(option(A8)) ) ).

tff(tcon_Option_Ooption___Finite__Set_Ofinite_716,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => finite_finite(option(A8)) ) ).

tff(tcon_Option_Ooption___Lattices_Olattice_717,axiom,
    ! [A8: $tType] :
      ( lattice(A8)
     => lattice(option(A8)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder_718,axiom,
    ! [A8: $tType] :
      ( order(A8)
     => order(option(A8)) ) ).

tff(tcon_Option_Ooption___Orderings_Otop_719,axiom,
    ! [A8: $tType] :
      ( order_top(A8)
     => top(option(A8)) ) ).

tff(tcon_Option_Ooption___Orderings_Oord_720,axiom,
    ! [A8: $tType] :
      ( preorder(A8)
     => ord(option(A8)) ) ).

tff(tcon_Option_Ooption___Orderings_Obot_721,axiom,
    ! [A8: $tType] :
      ( order(A8)
     => bot(option(A8)) ) ).

tff(tcon_Option_Ooption___HOL_Oequal_722,axiom,
    ! [A8: $tType] : cl_HOL_Oequal(option(A8)) ).

tff(tcon_Option_Ooption___Nat_Osize_723,axiom,
    ! [A8: $tType] : size(option(A8)) ).

tff(tcon_Uint32_Ouint32___Bit__Operations_Osemiring__bit__operations_724,axiom,
    bit_se359711467146920520ations(uint32) ).

tff(tcon_Uint32_Ouint32___Quickcheck__Narrowing_Opartial__term__of_725,axiom,
    quickc6926020345158392990erm_of(uint32) ).

tff(tcon_Uint32_Ouint32___Bit__Comprehension_Obit__comprehension_726,axiom,
    bit_bi6583157726757044596ension(uint32) ).

tff(tcon_Uint32_Ouint32___Bit__Operations_Oring__bit__operations_727,axiom,
    bit_ri3973907225187159222ations(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocancel__ab__semigroup__add_728,axiom,
    cancel2418104881723323429up_add(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocancel__comm__monoid__add_729,axiom,
    cancel1802427076303600483id_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__1__cancel_730,axiom,
    comm_s4317794764714335236cancel(uint32) ).

tff(tcon_Uint32_Ouint32___Bit__Operations_Osemiring__bits_731,axiom,
    bit_semiring_bits(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocancel__semigroup__add_732,axiom,
    cancel_semigroup_add(uint32) ).

tff(tcon_Uint32_Ouint32___Least__significant__bit_Olsb_733,axiom,
    least_6119777620449941438nt_lsb(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oab__semigroup__mult_734,axiom,
    ab_semigroup_mult(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring__1__cancel_735,axiom,
    semiring_1_cancel(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocomm__monoid__mult_736,axiom,
    comm_monoid_mult(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oab__semigroup__add_737,axiom,
    ab_semigroup_add(uint32) ).

tff(tcon_Uint32_Ouint32___Generic__set__bit_Oset__bit_738,axiom,
    generic_set_set_bit(uint32) ).

tff(tcon_Uint32_Ouint32___Code__Evaluation_Oterm__of_739,axiom,
    code_term_of(uint32) ).

tff(tcon_Uint32_Ouint32___Parity_Osemiring__parity_740,axiom,
    semiring_parity(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocomm__monoid__add_741,axiom,
    comm_monoid_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring__modulo_742,axiom,
    semiring_modulo(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__1_743,axiom,
    comm_semiring_1(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__0_744,axiom,
    comm_semiring_0(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Osemigroup__mult_745,axiom,
    semigroup_mult(uint32) ).

tff(tcon_Uint32_Ouint32___Num_Osemiring__numeral_746,axiom,
    semiring_numeral(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Osemigroup__add_747,axiom,
    semigroup_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__semiring_748,axiom,
    comm_semiring(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oab__group__add_749,axiom,
    ab_group_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ozero__neq__one_750,axiom,
    zero_neq_one(uint32) ).

tff(tcon_Uint32_Ouint32___Parity_Oring__parity_751,axiom,
    ring_parity(uint32) ).

tff(tcon_Uint32_Ouint32___Orderings_Opreorder_752,axiom,
    preorder(uint32) ).

tff(tcon_Uint32_Ouint32___Orderings_Olinorder_753,axiom,
    linorder(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Omonoid__mult_754,axiom,
    monoid_mult(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__ring__1_755,axiom,
    comm_ring_1(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Omonoid__add_756,axiom,
    monoid_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring__1_757,axiom,
    semiring_1(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring__0_758,axiom,
    semiring_0(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ogroup__add_759,axiom,
    group_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Omult__zero_760,axiom,
    mult_zero(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__ring_761,axiom,
    comm_ring(uint32) ).

tff(tcon_Uint32_Ouint32___Orderings_Oorder_762,axiom,
    order(uint32) ).

tff(tcon_Uint32_Ouint32___Num_Oneg__numeral_763,axiom,
    neg_numeral(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring_764,axiom,
    semiring(uint32) ).

tff(tcon_Uint32_Ouint32___Orderings_Oord_765,axiom,
    ord(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ouminus_766,axiom,
    uminus(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Oring__1_767,axiom,
    ring_1(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Otimes_768,axiom,
    times(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ominus_769,axiom,
    minus(uint32) ).

tff(tcon_Uint32_Ouint32___Power_Opower_770,axiom,
    power(uint32) ).

tff(tcon_Uint32_Ouint32___Num_Onumeral_771,axiom,
    numeral(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ozero_772,axiom,
    zero(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oplus_773,axiom,
    plus(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Oring_774,axiom,
    ring(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oone_775,axiom,
    one(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Odvd_776,axiom,
    dvd(uint32) ).

tff(tcon_Uint32_Ouint32___HOL_Oequal_777,axiom,
    cl_HOL_Oequal(uint32) ).

tff(tcon_Uint32_Ouint32___Nat_Osize_778,axiom,
    size(uint32) ).

tff(tcon_Assertions_Oassn___Boolean__Algebras_Oboolean__algebra_779,axiom,
    boolea8198339166811842893lgebra(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Osemilattice__sup_780,axiom,
    semilattice_sup(assn) ).

tff(tcon_Assertions_Oassn___Groups_Oab__semigroup__mult_781,axiom,
    ab_semigroup_mult(assn) ).

tff(tcon_Assertions_Oassn___Groups_Ocomm__monoid__mult_782,axiom,
    comm_monoid_mult(assn) ).

tff(tcon_Assertions_Oassn___Groups_Osemigroup__mult_783,axiom,
    semigroup_mult(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oorder__top_784,axiom,
    order_top(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oorder__bot_785,axiom,
    order_bot(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Opreorder_786,axiom,
    preorder(assn) ).

tff(tcon_Assertions_Oassn___Groups_Omonoid__mult_787,axiom,
    monoid_mult(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Olattice_788,axiom,
    lattice(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oorder_789,axiom,
    order(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Otop_790,axiom,
    top(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oord_791,axiom,
    ord(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Obot_792,axiom,
    bot(assn) ).

tff(tcon_Assertions_Oassn___Groups_Ouminus_793,axiom,
    uminus(assn) ).

tff(tcon_Assertions_Oassn___Groups_Otimes_794,axiom,
    times(assn) ).

tff(tcon_Assertions_Oassn___Groups_Ominus_795,axiom,
    minus(assn) ).

tff(tcon_Assertions_Oassn___Power_Opower_796,axiom,
    power(assn) ).

tff(tcon_Assertions_Oassn___Groups_Oone_797,axiom,
    one(assn) ).

tff(tcon_Assertions_Oassn___Rings_Odvd_798,axiom,
    dvd(assn) ).

tff(tcon_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_799,axiom,
    semiri1453513574482234551roduct(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ofirst__countable__topology_800,axiom,
    topolo3112930676232923870pology(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__div__algebra_801,axiom,
    real_V8999393235501362500lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra__1_802,axiom,
    real_V2822296259951069270ebra_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors__cancel_803,axiom,
    semiri6575147826004484403cancel(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra_804,axiom,
    real_V4412858255891104859lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_805,axiom,
    real_V822414075346904944vector(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_806,axiom,
    topolo4958980785337419405_space(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_807,axiom,
    real_V3459762299906320749_field(complex) ).

tff(tcon_Complex_Ocomplex___Quickcheck__Narrowing_Opartial__term__of_808,axiom,
    quickc6926020345158392990erm_of(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__comm__monoid__add_809,axiom,
    topolo5987344860129210374id_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1__no__zero__divisors_810,axiom,
    semiri2026040879449505780visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra__1_811,axiom,
    real_V2191834092415804123ebra_1(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ocomplete__space_812,axiom,
    real_V8037385150606011577_space(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__semigroup__mult_813,axiom,
    topolo4211221413907600880p_mult(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Operfect__space_814,axiom,
    topolo8386298272705272623_space(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors_815,axiom,
    semiri3467727345109120633visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra_816,axiom,
    real_V6157519004096292374lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ometric__space_817,axiom,
    real_V7819770556892013058_space(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__ab__group__add_818,axiom,
    topolo1287966508704411220up_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_819,axiom,
    real_V4867850818363320053vector(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add_820,axiom,
    cancel2418104881723323429up_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors_821,axiom,
    ring_15535105094025558882visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__field_822,axiom,
    real_V7773925162809079976_field(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__monoid__add_823,axiom,
    topolo6943815403480290642id_add(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__comm__monoid__add_824,axiom,
    cancel1802427076303600483id_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1__cancel_825,axiom,
    comm_s4317794764714335236cancel(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__group__add_826,axiom,
    topolo1633459387980952147up_add(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ot2__space_827,axiom,
    topological_t2_space(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ot1__space_828,axiom,
    topological_t1_space(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__semigroup__add_829,axiom,
    cancel_semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Obanach_830,axiom,
    real_Vector_banach(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__semigroup__mult_831,axiom,
    ab_semigroup_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1__cancel_832,axiom,
    semiring_1_cancel(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__mult_833,axiom,
    comm_monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__semigroup__add_834,axiom,
    ab_semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Code__Evaluation_Oterm__of_835,axiom,
    code_term_of(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__add_836,axiom,
    comm_monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1_837,axiom,
    comm_semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__0_838,axiom,
    comm_semiring_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Osemigroup__mult_839,axiom,
    semigroup_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemidom__divide_840,axiom,
    semidom_divide(complex) ).

tff(tcon_Complex_Ocomplex___Num_Osemiring__numeral_841,axiom,
    semiring_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Osemigroup__add_842,axiom,
    semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Odivision__ring_843,axiom,
    division_ring(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring_844,axiom,
    comm_semiring(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Osemiring__char__0_845,axiom,
    semiring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__group__add_846,axiom,
    ab_group_add(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield__char__0_847,axiom,
    field_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ozero__neq__one_848,axiom,
    zero_neq_one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom__abs__sgn_849,axiom,
    idom_abs_sgn(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__mult_850,axiom,
    monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom__divide_851,axiom,
    idom_divide(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring__1_852,axiom,
    comm_ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__add_853,axiom,
    monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1_854,axiom,
    semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__0_855,axiom,
    semiring_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ogroup__add_856,axiom,
    group_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Omult__zero_857,axiom,
    mult_zero(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring_858,axiom,
    comm_ring(complex) ).

tff(tcon_Complex_Ocomplex___Num_Oneg__numeral_859,axiom,
    neg_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Oring__char__0_860,axiom,
    ring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring_861,axiom,
    semiring(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Oinverse_862,axiom,
    inverse(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ouminus_863,axiom,
    uminus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1_864,axiom,
    ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Otimes_865,axiom,
    times(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ominus_866,axiom,
    minus(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield_867,axiom,
    field(complex) ).

tff(tcon_Complex_Ocomplex___Power_Opower_868,axiom,
    power(complex) ).

tff(tcon_Complex_Ocomplex___Num_Onumeral_869,axiom,
    numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ozero_870,axiom,
    zero(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oplus_871,axiom,
    plus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring_872,axiom,
    ring(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom_873,axiom,
    idom(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oone_874,axiom,
    one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Odvd_875,axiom,
    dvd(complex) ).

tff(tcon_Complex_Ocomplex___HOL_Oequal_876,axiom,
    cl_HOL_Oequal(complex) ).

tff(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_877,axiom,
    condit6923001295902523014norder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_878,axiom,
    strict9044650504122735259up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_879,axiom,
    strict7427464778891057005id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_880,axiom,
    canoni5634975068530333245id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Quickcheck__Narrowing_Opartial__term__of_881,axiom,
    quickc6926020345158392990erm_of(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_882,axiom,
    linord181362715937106298miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_883,axiom,
    semiri3467727345109120633visors(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_884,axiom,
    ordere6658533253407199908up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_885,axiom,
    ordere6911136660526730532id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__comm__semiring_886,axiom,
    ordere2520102378445227354miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__sup_887,axiom,
    semilattice_sup(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_888,axiom,
    ab_semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__mult_889,axiom,
    comm_monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_890,axiom,
    ab_semigroup_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Code__Evaluation_Oterm__of_891,axiom,
    code_term_of(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_892,axiom,
    ordered_semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_893,axiom,
    comm_monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_894,axiom,
    comm_semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__0_895,axiom,
    comm_semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_896,axiom,
    semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Osemiring__numeral_897,axiom,
    semiring_numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_898,axiom,
    semigroup_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__less__one_899,axiom,
    zero_less_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring_900,axiom,
    comm_semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Owellorder_901,axiom,
    wellorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_902,axiom,
    order_top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__bot_903,axiom,
    order_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_904,axiom,
    semiring_char_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__neq__one_905,axiom,
    zero_neq_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Opreorder_906,axiom,
    preorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Olinorder_907,axiom,
    linorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__mult_908,axiom,
    monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_909,axiom,
    monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_910,axiom,
    semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__0_911,axiom,
    semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Olattice_912,axiom,
    lattice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Omult__zero_913,axiom,
    mult_zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder_914,axiom,
    order(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring_915,axiom,
    semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Otop_916,axiom,
    top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oord_917,axiom,
    ord(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Obot_918,axiom,
    bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Otimes_919,axiom,
    times(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ominus_920,axiom,
    minus(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Power_Opower_921,axiom,
    power(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Onumeral_922,axiom,
    numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ozero_923,axiom,
    zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oplus_924,axiom,
    plus(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oone_925,axiom,
    one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Odvd_926,axiom,
    dvd(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___HOL_Oequal_927,axiom,
    cl_HOL_Oequal(extended_enat) ).

tff(tcon_Multiset_Omultiset___Quickcheck__Narrowing_Opartial__term__of_928,axiom,
    ! [A8: $tType] :
      ( typerep(A8)
     => quickc6926020345158392990erm_of(multiset(A8)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add_929,axiom,
    ! [A8: $tType] :
      ( preorder(A8)
     => ordere6658533253407199908up_add(multiset(A8)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Ocancel__ab__semigroup__add_930,axiom,
    ! [A8: $tType] : cancel2418104881723323429up_add(multiset(A8)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocancel__comm__monoid__add_931,axiom,
    ! [A8: $tType] : cancel1802427076303600483id_add(multiset(A8)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocancel__semigroup__add_932,axiom,
    ! [A8: $tType] : cancel_semigroup_add(multiset(A8)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__diff_933,axiom,
    ! [A8: $tType] : comm_monoid_diff(multiset(A8)) ).

tff(tcon_Multiset_Omultiset___Groups_Oab__semigroup__add_934,axiom,
    ! [A8: $tType] : ab_semigroup_add(multiset(A8)) ).

tff(tcon_Multiset_Omultiset___Code__Evaluation_Oterm__of_935,axiom,
    ! [A8: $tType] :
      ( typerep(A8)
     => code_term_of(multiset(A8)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_936,axiom,
    ! [A8: $tType] : comm_monoid_add(multiset(A8)) ).

tff(tcon_Multiset_Omultiset___Groups_Osemigroup__add_937,axiom,
    ! [A8: $tType] : semigroup_add(multiset(A8)) ).

tff(tcon_Multiset_Omultiset___Orderings_Opreorder_938,axiom,
    ! [A8: $tType] :
      ( preorder(A8)
     => preorder(multiset(A8)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Omonoid__add_939,axiom,
    ! [A8: $tType] : monoid_add(multiset(A8)) ).

tff(tcon_Multiset_Omultiset___Orderings_Oorder_940,axiom,
    ! [A8: $tType] :
      ( preorder(A8)
     => order(multiset(A8)) ) ).

tff(tcon_Multiset_Omultiset___Orderings_Oord_941,axiom,
    ! [A8: $tType] :
      ( preorder(A8)
     => ord(multiset(A8)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Ominus_942,axiom,
    ! [A8: $tType] : minus(multiset(A8)) ).

tff(tcon_Multiset_Omultiset___Groups_Ozero_943,axiom,
    ! [A8: $tType] : zero(multiset(A8)) ).

tff(tcon_Multiset_Omultiset___Groups_Oplus_944,axiom,
    ! [A8: $tType] : plus(multiset(A8)) ).

tff(tcon_Multiset_Omultiset___HOL_Oequal_945,axiom,
    ! [A8: $tType] :
      ( cl_HOL_Oequal(A8)
     => cl_HOL_Oequal(multiset(A8)) ) ).

tff(tcon_Multiset_Omultiset___Nat_Osize_946,axiom,
    ! [A8: $tType] : size(multiset(A8)) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocancel__ab__semigroup__add_947,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => cancel2418104881723323429up_add(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocancel__comm__monoid__add_948,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => cancel1802427076303600483id_add(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__1__cancel_949,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => comm_s4317794764714335236cancel(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocancel__semigroup__add_950,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => cancel_semigroup_add(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oab__semigroup__mult_951,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => ab_semigroup_mult(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Osemiring__1__cancel_952,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => semiring_1_cancel(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocomm__monoid__mult_953,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => comm_monoid_mult(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oab__semigroup__add_954,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => ab_semigroup_add(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Code__Evaluation_Oterm__of_955,axiom,
    ! [A8: $tType] :
      ( typerep(A8)
     => code_term_of(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocomm__monoid__add_956,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => comm_monoid_add(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__1_957,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => comm_semiring_1(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__0_958,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => comm_semiring_0(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Osemigroup__mult_959,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => semigroup_mult(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Num_Osemiring__numeral_960,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => semiring_numeral(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Osemigroup__add_961,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => semigroup_add(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring_962,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => comm_semiring(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Owellorder_963,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => wellorder(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oab__group__add_964,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => ab_group_add(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ozero__neq__one_965,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => zero_neq_one(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Opreorder_966,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => preorder(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Olinorder_967,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => linorder(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Omonoid__mult_968,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => monoid_mult(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__ring__1_969,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => comm_ring_1(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Omonoid__add_970,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => monoid_add(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Finite__Set_Ofinite_971,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => finite_finite(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Cardinality_Ocard2,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => card2(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Type__Length_Olen0_972,axiom,
    ! [A8: $tType] :
      ( type_len0(A8)
     => type_len0(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Osemiring__1_973,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => semiring_1(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Osemiring__0_974,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => semiring_0(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ogroup__add_975,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => group_add(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Type__Length_Olen_976,axiom,
    ! [A8: $tType] :
      ( type_len(A8)
     => type_len(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Omult__zero_977,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => mult_zero(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__ring_978,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => comm_ring(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Oorder_979,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => order(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Num_Oneg__numeral_980,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => neg_numeral(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Osemiring_981,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => semiring(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Oord_982,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => ord(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ouminus_983,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => uminus(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Oring__1_984,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => ring_1(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Otimes_985,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => times(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ominus_986,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => minus(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Power_Opower_987,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => power(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Num_Onumeral_988,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => numeral(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ozero_989,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => zero(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oplus_990,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => plus(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Oring_991,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => ring(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oone_992,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => one(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Odvd_993,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => dvd(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit0___HOL_Oequal_994,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => cl_HOL_Oequal(numeral_bit0(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocancel__ab__semigroup__add_995,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => cancel2418104881723323429up_add(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocancel__comm__monoid__add_996,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => cancel1802427076303600483id_add(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__1__cancel_997,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => comm_s4317794764714335236cancel(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocancel__semigroup__add_998,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => cancel_semigroup_add(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oab__semigroup__mult_999,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => ab_semigroup_mult(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Osemiring__1__cancel_1000,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => semiring_1_cancel(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocomm__monoid__mult_1001,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => comm_monoid_mult(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oab__semigroup__add_1002,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => ab_semigroup_add(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Code__Evaluation_Oterm__of_1003,axiom,
    ! [A8: $tType] :
      ( typerep(A8)
     => code_term_of(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocomm__monoid__add_1004,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => comm_monoid_add(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__1_1005,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => comm_semiring_1(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__0_1006,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => comm_semiring_0(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Osemigroup__mult_1007,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => semigroup_mult(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Num_Osemiring__numeral_1008,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => semiring_numeral(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Osemigroup__add_1009,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => semigroup_add(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring_1010,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => comm_semiring(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Owellorder_1011,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => wellorder(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oab__group__add_1012,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => ab_group_add(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ozero__neq__one_1013,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => zero_neq_one(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Opreorder_1014,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => preorder(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Olinorder_1015,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => linorder(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Omonoid__mult_1016,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => monoid_mult(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__ring__1_1017,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => comm_ring_1(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Omonoid__add_1018,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => monoid_add(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Finite__Set_Ofinite_1019,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => finite_finite(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Cardinality_Ocard2_1020,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => card2(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Type__Length_Olen0_1021,axiom,
    ! [A8: $tType] :
      ( type_len0(A8)
     => type_len0(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Osemiring__1_1022,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => semiring_1(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Osemiring__0_1023,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => semiring_0(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ogroup__add_1024,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => group_add(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Type__Length_Olen_1025,axiom,
    ! [A8: $tType] :
      ( type_len0(A8)
     => type_len(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Omult__zero_1026,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => mult_zero(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__ring_1027,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => comm_ring(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Oorder_1028,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => order(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Num_Oneg__numeral_1029,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => neg_numeral(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Osemiring_1030,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => semiring(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Oord_1031,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => ord(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ouminus_1032,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => uminus(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Oring__1_1033,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => ring_1(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Otimes_1034,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => times(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ominus_1035,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => minus(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Power_Opower_1036,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => power(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Num_Onumeral_1037,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => numeral(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ozero_1038,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => zero(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oplus_1039,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => plus(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Oring_1040,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => ring(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oone_1041,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => one(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Odvd_1042,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => dvd(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Obit1___HOL_Oequal_1043,axiom,
    ! [A8: $tType] :
      ( finite_finite(A8)
     => cl_HOL_Oequal(numeral_bit1(A8)) ) ).

tff(tcon_Numeral__Type_Onum1___Quickcheck__Narrowing_Opartial__term__of_1044,axiom,
    quickc6926020345158392990erm_of(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ocancel__ab__semigroup__add_1045,axiom,
    cancel2418104881723323429up_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ocancel__comm__monoid__add_1046,axiom,
    cancel1802427076303600483id_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ocancel__semigroup__add_1047,axiom,
    cancel_semigroup_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Oab__semigroup__mult_1048,axiom,
    ab_semigroup_mult(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ocomm__monoid__mult_1049,axiom,
    comm_monoid_mult(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Oab__semigroup__add_1050,axiom,
    ab_semigroup_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Code__Evaluation_Oterm__of_1051,axiom,
    code_term_of(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ocomm__monoid__add_1052,axiom,
    comm_monoid_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Ocomm__semiring__0_1053,axiom,
    comm_semiring_0(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Osemigroup__mult_1054,axiom,
    semigroup_mult(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Osemigroup__add_1055,axiom,
    semigroup_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Ocomm__semiring_1056,axiom,
    comm_semiring(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Orderings_Owellorder_1057,axiom,
    wellorder(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Oab__group__add_1058,axiom,
    ab_group_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Orderings_Opreorder_1059,axiom,
    preorder(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Orderings_Olinorder_1060,axiom,
    linorder(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Omonoid__mult_1061,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_1062,axiom,
    monoid_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Finite__Set_Ofinite_1063,axiom,
    finite_finite(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Type__Length_Olen0_1064,axiom,
    type_len0(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Osemiring__0_1065,axiom,
    semiring_0(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ogroup__add_1066,axiom,
    group_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Type__Length_Olen_1067,axiom,
    type_len(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Omult__zero_1068,axiom,
    mult_zero(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Ocomm__ring_1069,axiom,
    comm_ring(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Orderings_Oorder_1070,axiom,
    order(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Osemiring_1071,axiom,
    semiring(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Orderings_Oord_1072,axiom,
    ord(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ouminus_1073,axiom,
    uminus(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Otimes_1074,axiom,
    times(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ominus_1075,axiom,
    minus(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Power_Opower_1076,axiom,
    power(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Num_Onumeral_1077,axiom,
    numeral(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ozero_1078,axiom,
    zero(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Oplus_1079,axiom,
    plus(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Oring_1080,axiom,
    ring(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Oone_1081,axiom,
    one(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Odvd_1082,axiom,
    dvd(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___HOL_Oequal_1083,axiom,
    cl_HOL_Oequal(numeral_num1) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_1084,axiom,
    ! [A8: $tType,A9: $tType] :
      ( ( topolo4958980785337419405_space(A8)
        & topolo4958980785337419405_space(A9) )
     => topolo4958980785337419405_space(product_prod(A8,A9)) ) ).

tff(tcon_Product__Type_Oprod___Quickcheck__Narrowing_Opartial__term__of_1085,axiom,
    ! [A8: $tType,A9: $tType] :
      ( ( typerep(A8)
        & typerep(A9) )
     => quickc6926020345158392990erm_of(product_prod(A8,A9)) ) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Ot2__space_1086,axiom,
    ! [A8: $tType,A9: $tType] :
      ( ( topological_t2_space(A8)
        & topological_t2_space(A9) )
     => topological_t2_space(product_prod(A8,A9)) ) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Ot1__space_1087,axiom,
    ! [A8: $tType,A9: $tType] :
      ( ( topological_t1_space(A8)
        & topological_t1_space(A9) )
     => topological_t1_space(product_prod(A8,A9)) ) ).

tff(tcon_Product__Type_Oprod___Code__Evaluation_Oterm__of_1088,axiom,
    ! [A8: $tType,A9: $tType] :
      ( ( typerep(A8)
        & typerep(A9) )
     => code_term_of(product_prod(A8,A9)) ) ).

tff(tcon_Product__Type_Oprod___Finite__Set_Ofinite_1089,axiom,
    ! [A8: $tType,A9: $tType] :
      ( ( finite_finite(A8)
        & finite_finite(A9) )
     => finite_finite(product_prod(A8,A9)) ) ).

tff(tcon_Product__Type_Oprod___HOL_Oequal_1090,axiom,
    ! [A8: $tType,A9: $tType] : cl_HOL_Oequal(product_prod(A8,A9)) ).

tff(tcon_Product__Type_Oprod___Nat_Osize_1091,axiom,
    ! [A8: $tType,A9: $tType] : size(product_prod(A8,A9)) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_1092,axiom,
    condit6923001295902523014norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Quickcheck__Narrowing_Opartial__term__of_1093,axiom,
    quickc6926020345158392990erm_of(product_unit) ).

tff(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_1094,axiom,
    boolea8198339166811842893lgebra(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_1095,axiom,
    semilattice_sup(product_unit) ).

tff(tcon_Product__Type_Ounit___Code__Evaluation_Oterm__of_1096,axiom,
    code_term_of(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Owellorder_1097,axiom,
    wellorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__top_1098,axiom,
    order_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__bot_1099,axiom,
    order_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Opreorder_1100,axiom,
    preorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Olinorder_1101,axiom,
    linorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Finite__Set_Ofinite_1102,axiom,
    finite_finite(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Olattice_1103,axiom,
    lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder_1104,axiom,
    order(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Otop_1105,axiom,
    top(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oord_1106,axiom,
    ord(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Obot_1107,axiom,
    bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Groups_Ouminus_1108,axiom,
    uminus(product_unit) ).

tff(tcon_Product__Type_Ounit___Groups_Ominus_1109,axiom,
    minus(product_unit) ).

tff(tcon_Product__Type_Ounit___HOL_Oequal_1110,axiom,
    cl_HOL_Oequal(product_unit) ).

tff(tcon_Heap_Oheap_Oheap__ext___Quickcheck__Narrowing_Opartial__term__of_1111,axiom,
    ! [A8: $tType] :
      ( typerep(A8)
     => quickc6926020345158392990erm_of(heap_ext(A8)) ) ).

tff(tcon_Heap_Oheap_Oheap__ext___Code__Evaluation_Oterm__of_1112,axiom,
    ! [A8: $tType] :
      ( typerep(A8)
     => code_term_of(heap_ext(A8)) ) ).

tff(tcon_Heap_Oheap_Oheap__ext___HOL_Oequal_1113,axiom,
    ! [A8: $tType] : cl_HOL_Oequal(heap_ext(A8)) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_1114,axiom,
    bit_un5681908812861735899ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_1115,axiom,
    semiri1453513574482234551roduct(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_1116,axiom,
    euclid5411537665997757685th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_1117,axiom,
    euclid8789492081693882211th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_1118,axiom,
    ordere1937475149494474687imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_1119,axiom,
    euclid3128863361964157862miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_1120,axiom,
    euclid4440199948858584721cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_1121,axiom,
    unique1627219031080169319umeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_1122,axiom,
    euclid8851590272496341667cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_1123,axiom,
    semiri6575147826004484403cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_1124,axiom,
    strict9044650504122735259up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_1125,axiom,
    ordere580206878836729694up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_1126,axiom,
    ordere2412721322843649153imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_1127,axiom,
    bit_se359711467146920520ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_1128,axiom,
    linord2810124833399127020strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_1129,axiom,
    strict7427464778891057005id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_1130,axiom,
    ordere8940638589300402666id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_1131,axiom,
    euclid3725896446679973847miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Quickcheck__Narrowing_Opartial__term__of_1132,axiom,
    quickc6926020345158392990erm_of(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_1133,axiom,
    linord715952674999750819strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_1134,axiom,
    bit_ri3973907225187159222ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_1135,axiom,
    semiri2026040879449505780visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_1136,axiom,
    linord181362715937106298miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_1137,axiom,
    linord8928482502909563296strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_1138,axiom,
    semiri3467727345109120633visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_1139,axiom,
    ordere6658533253407199908up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_1140,axiom,
    ordere166539214618696060dd_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_1141,axiom,
    ordere6911136660526730532id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_1142,axiom,
    linord5086331880401160121up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_1143,axiom,
    cancel2418104881723323429up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_1144,axiom,
    ring_15535105094025558882visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_1145,axiom,
    cancel1802427076303600483id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_1146,axiom,
    linord4710134922213307826strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_1147,axiom,
    comm_s4317794764714335236cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_1148,axiom,
    bit_semiring_bits(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_1149,axiom,
    ordere2520102378445227354miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_1150,axiom,
    linord6961819062388156250ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_1151,axiom,
    ordered_ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_1152,axiom,
    cancel_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_1153,axiom,
    linordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Least__significant__bit_Olsb_1154,axiom,
    least_6119777620449941438nt_lsb(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_1155,axiom,
    ordered_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_1156,axiom,
    linordered_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_1157,axiom,
    ab_semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_1158,axiom,
    semiring_1_cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_1159,axiom,
    algebraic_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_1160,axiom,
    comm_monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_1161,axiom,
    ab_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Generic__set__bit_Oset__bit_1162,axiom,
    generic_set_set_bit(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Code__Evaluation_Oterm__of_1163,axiom,
    code_term_of(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_1164,axiom,
    ordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_1165,axiom,
    ordered_ring_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_1166,axiom,
    semiring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_1167,axiom,
    comm_monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_1168,axiom,
    semiring_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_1169,axiom,
    linordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_1170,axiom,
    linordered_idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_1171,axiom,
    comm_semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_1172,axiom,
    comm_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_1173,axiom,
    semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_1174,axiom,
    semidom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_1175,axiom,
    semidom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_1176,axiom,
    semiring_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_1177,axiom,
    semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_1178,axiom,
    zero_less_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_1179,axiom,
    comm_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_1180,axiom,
    semiring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_1181,axiom,
    ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_1182,axiom,
    zero_neq_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_1183,axiom,
    ordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_1184,axiom,
    idom_abs_sgn(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_1185,axiom,
    ring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_1186,axiom,
    preorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_1187,axiom,
    linorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_1188,axiom,
    monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__modulo_1189,axiom,
    idom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_1190,axiom,
    idom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_1191,axiom,
    comm_ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_1192,axiom,
    monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_1193,axiom,
    semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_1194,axiom,
    semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_1195,axiom,
    group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_1196,axiom,
    mult_zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_1197,axiom,
    comm_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oorder_1198,axiom,
    order(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_1199,axiom,
    neg_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_1200,axiom,
    ring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring_1201,axiom,
    semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oord_1202,axiom,
    ord(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ouminus_1203,axiom,
    uminus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1_1204,axiom,
    ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_1205,axiom,
    abs_if(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Otimes_1206,axiom,
    times(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ominus_1207,axiom,
    minus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Power_Opower_1208,axiom,
    power(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Onumeral_1209,axiom,
    numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ozero_1210,axiom,
    zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oplus_1211,axiom,
    plus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring_1212,axiom,
    ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom_1213,axiom,
    idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oone_1214,axiom,
    one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Odvd_1215,axiom,
    dvd(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___HOL_Oequal_1216,axiom,
    cl_HOL_Oequal(code_integer) ).

tff(tcon_Heap__Time__Monad_OHeap___Quickcheck__Narrowing_Opartial__term__of_1217,axiom,
    ! [A8: $tType] :
      ( typerep(A8)
     => quickc6926020345158392990erm_of(heap_Time_Heap(A8)) ) ).

tff(tcon_Heap__Time__Monad_OHeap___Code__Evaluation_Oterm__of_1218,axiom,
    ! [A8: $tType] :
      ( typerep(A8)
     => code_term_of(heap_Time_Heap(A8)) ) ).

tff(tcon_Heap__Time__Monad_OHeap___HOL_Oequal_1219,axiom,
    ! [A8: $tType] : cl_HOL_Oequal(heap_Time_Heap(A8)) ).

tff(tcon_Heap__Time__Monad_OHeap___Nat_Osize_1220,axiom,
    ! [A8: $tType] : size(heap_Time_Heap(A8)) ).

tff(tcon_VEBT__BuildupMemImp_OVEBTi___Quickcheck__Narrowing_Opartial__term__of_1221,axiom,
    quickc6926020345158392990erm_of(vEBT_VEBTi) ).

tff(tcon_VEBT__BuildupMemImp_OVEBTi___Code__Evaluation_Oterm__of_1222,axiom,
    code_term_of(vEBT_VEBTi) ).

tff(tcon_VEBT__BuildupMemImp_OVEBTi___HOL_Oequal_1223,axiom,
    cl_HOL_Oequal(vEBT_VEBTi) ).

tff(tcon_VEBT__BuildupMemImp_OVEBTi___Nat_Osize_1224,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,fequal(A,X),Y) ) ).

tff(help_fequal_1_1_T,axiom,
    ! [A: $tType,X: A,Y: A] :
      ( ~ aa(A,$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 (2)
tff(tfree_0,hypothesis,
    semiring_1(a) ).

tff(tfree_1,hypothesis,
    linorder(b) ).

% Conjectures (2)
tff(conj_0,hypothesis,
    simple8792101892335181110TBOUND(list(nat),aa(vEBT_VEBTi,assn,vEBT_Intf_vebt_assn(n,s),ti),vEBT_Example_mmap(nat,nat,aTP_Lamp_aa(nat,heap_Time_Heap(nat)),ysa),b2(ysa,n)) ).

tff(conj_1,conjecture,
    aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit1(bit1(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit1(one2))))),aa(int,nat,nat2,archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),n)))))),b2(ysa,n)))),b2(aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),a2),ysa),n)) ).